Applied nonautonomous and random dynamical systems

Tomás Caraballo and Xiaoying Han Applied Nonautonomous and Random Dynamical Systems Applied Dynamical Systems To our families and friends Preface During the past two decades, the theory of nonautonomous dynamical systems and random dynamical systems has made substantial progress in studying the long term dynamics of open systems subject to time-dependent or random forcing. However, most of the existing pertinent literatures are fairly technical and almost impenetrable for a general audience, except the most dedicated specialists. Moreover, the concepts and methods of nonautonomous and random dynamical systems, though well-established, are extremely nontrivial to apply to real-world problems. The aim of this work is to provide an accessible and broad introduction to the theory of nonautonomous and random dynamical systems, with an emphasis on applications of the theory to problems arising in the applied sciences and engineering. The book starts with basic concepts in the theory of autonomous dynamical systems which are easier to understand, and used as the motivation for the study of non-autonomous and random dynamical systems. Then the framework of nonautonomous dynamical systems is set up, including various approaches to analyze the long time behavior of non-autonomous problems. The major emphasis is given to the novel theory of pullback attractors, as it can be regarded as a natural extension of the autonomous theory and allows a larger variety of time-dependence forcing than other alternatives such as skew-product flows or cocycles. In the end the theory of random dynamical systems and random attractors is introduced and shown to fairly informative to the study of long term behavior of stochastic systems with random forcing. Each set of theory is illustrated by being applied to three different models, the chemostat model, the SIR epidemic model, and the Lorenz-84 model, in their autonomous, nonautonomous, and stochastic formulations, respectively. The techniques and methods adopted can be applied to the study of the long term behavior of a wide range of applications arising in applied sciences and engineering. Sevilla, Spain, October 22, 2016 Tom´as Caraballo Auburn, Alabama Xiaoying Han vii Contents 1 Introduction ................................................... 1 2 Autonomous dynamical systems ................................. 11 2.1 Basic Stability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.2 Attractors ................................................. 19 2.3 Applications............................................... 22 2.3.1 Application to ecology: a chemostat model . . . . . . . . . . . . . . . 22 2.3.2 Application to epidemiology: the SIR model . . . . . . . . . . . . . 26 2.3.3 Application to climate change: the Lorenz-84 model . . . . . . . 29 3 Nonautonomous dynamical systems .............................. 35 3.1 Formulations of nonautonomous dynamical systems . . . . . . . . . . . . . 35 3.1.1 Process formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.1.2 Skew product flow formulation . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.2 Nonautonomous Attractors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.2.1 Nonautonomous attractors for processes . . . . . . . . . . . . . . . . . 40 3.2.2 Nonautonomous attractors for skew product flows . . . . . . . . . 48 3.3 Applications............................................... 53 3.3.1 Nonautonomous chemostat model . . . . . . . . . . . . . . . . . . . . . . 53 3.3.2 Nonautonomous SIR model . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.3.3 Nonautonomous Lorenz-84 model . . . . . . . . . . . . . . . . . . . . . . 64 4 Random dynamical systems ..................................... 67 4.1 Noise is present almost everywhere . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 4.2 Formulation of Random Dynamical System and Random Attractor . 68 4.2.1 Some properties of the random attractor . . . . . . . . . . . . . . . . . 74 4.2.2 Generation of random dynamical systems . . . . . . . . . . . . . . . . 76 4.2.3 A brief introduction to stochastic differential equations . . . . . 77 4.2.4 Global asymptotic behavior of SDEs: conjugation of RDS . . 81 4.3 Applications............................................... 86 4.3.1 Random chemostat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 ix x Contents 4.3.2 Random and stochastic SIR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.3.3 Stochastic Lorenz models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.4 Stabilization of dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 References.....................................................104 Chapter 1 Introduction The research on the theories of dynamical systems (autonomous, nonautonomous or stochastic) has been receiving much attention over the last decades, due to the fact that many real world phenomena can be modeled by a system of differential or difference equations which generates one dynamical system. A general theory for dynamical systems allows us to analyze various type of evolutionary problems, but is sometimes over abstract for practitioner research. In particular, while analyzing finite dimensional systems, the underlying theory can be less complicated and more intuitive than the general theory for abstract dynamical systems. In this book, we will set up a unified framework of dynamical systems accessible to researchers with different background, which suffices to analyze finite dimensional systems arising in applied sciences and engineering. Roughly speaking, a dynamical system is a system that evolves in time through the iterated application of an underlying dynamical transition rule. The application of the transition rule can happen either at discrete time, with the time parameter taking integer values, or infinitesimally with continuous time taking real values as in differential equations. The theoretical framework of dynamical systems allows us to develop a unified theory which can be applied directly to real-life evolutionary events to analyze their long term behavior. To motivate the use of the theory of dynamical systems, we start from considering the evolution in time of a certain quantity, e.g., the position of a particle moving on a plane R2(or in the space R3), that can be modeled by the following differential equation in R2(or R3), dx dt=f(x),(1.1) where f:Rd→Rdd=2 (or 3) is a function which ensures existence of solutions to equation(1.1). Usually, additional information is needed to determine properties of solutions to the system (1.1). In fact, knowing the initial position x0∈Rdof the particle is sufficient to obtain its positions in all future instants. This corresponds to the knowledge of the value of xat an initial instant t0. In other words, we can consider an initial 1 2 1 Introduction value problem (IVP) associated with equation (1.1): dx dt=f(x),x(t0)=x0∈Rd.(1.2) Under a set of suitable assumptions on the function f, one can prove in a mathematically rigorous way that the IVP (1.2) possesses a unique solution which is defined for all future instants of time. The solution is a continuously differentiable mapping x(·;t0,x0):R→Rdwith derivative x�(t)=f(x(t)) and satisfies x(t0;t0,x0)=x0. Here and also in the sequel the notation “;t0,x0” is used to specify the explicit dependence of the solution on the initial value (t0,x0). Basic properties of the solution mapping include: (a) The evolution of xdepends on the elapsed time instead of the initial and final times separately. As the vector field fdoes not depend on t, a simple shift on the time variable t→t−t0allows us to prove that the position of the particle only depends on the elapsed time t−t0. Mathematically this is described as x(t;t0,x0)=x(t−t0;0,x0) for t≥t0, which means that the position of the particle at time t(≥t0), starting from the point x0at the initial time t0, is exactly the same as the position of the particle at time t−t0if it starts from point x0at the initial time zero. Thanks to this property we can always choose the initial time for the IVP (1.2) to be the time instant zero, i.e., t0=0. (b) Initial value property. This is a straightforward property meaning that if the particle starts from x0at the initial time t0, then the solution trajectory of xpasses through the point (t0,x0). Mathematically this property reads x(t0;t0,x0)=x0,for all x0∈Rd,t0∈R. (c) Concatenation of solutions. Assume that the particle starts from the point x0at the initial time t0=0, arrives at the point x1=x(t1;0,x0) after t1period of time, and takes offagain from x1 and arrives at the location x2=x(t2;0,x1) after another t2period of time. Then x2will be the same as the location where the particle arrives at if it starts from the point x0at the initial time t0=0 and travels a total of t1+t2period of time. Mathematically this is described as x(t2;0,x(t1;0,x0)) =x(t1+t2;0,x0). (d) Continuity of solutions with respect to initial values. 1 Introduction 9 then be whether or not the maximal solution ϕ(t;·,·) expressed in terms of the initial values is also continuous with respect to the initial value (t0,x0). The answer is YES as stated in the following theorem. Theorem 1.6. (Continuous dependence on the initial values) Assume that Ois a nonempty, open and connected subset of Rdand f :I×O→Rdis continuous with respect to t and locally Lipschitz with respect to x. Let Θbe defined in (1.13). Then the global solution ϕ(·;·,·)of the IVP (1.6) expressed in terms of the initial values is continuous in Θ, i.e. ϕ∈C0(Θ;Rd). Existence and uniqueness of global solutions, along with the continuous dependence of the global solutions on initial conditions, are the essential prerequisites of the study of dynamical systems. In the theories of dynamical systems, we will assume these hold, while in the applications we always need to prove these before using the theories of dynamical systems. The rest of the book is organized as follows. In Chapter 2 we will introduce the basic concepts from the theory of autonomous dynamical systems which are easier to understand, and can serve as the motivation for the study of non-autonomous and random dynamical systems. In Chapter 3 we will set up the framework of nonautonomous dynamical systems, that can be regarded as a natural extension of the autonomous theory, but allows a larger variety of time dependence forces than other alternatives such as skew-product flows or cocycles. In particular we will describe various approaches to analyze the long time behavior of non-autonomous dynamical systems, with an emphasis on the novel theory of pullback attractors. Then in Chapter 4 we will introduce the theory of random dynamical systems and random pullback attractors and show how suitable is to analyze real-world problems with randomness. In each chapter after establishing the main concepts, we will apply them to analyze the long term behavior of several interesting models arising in applied sciences and engineering. In particular, the chemostat model, the SIR epidemic model and the Lorenze-84 model are studied in their autonomous, nonautonomous and stochastic/random formulations. Chapter 2 Autonomous dynamical systems The theory of autonomous dynamical systems is now well established after being studied intensively over the past years. In this chapter we will provide a brief review of autonomous dynamical systems, as the background and motivation to introduce nonautonomous and random dynamical systems which are the major topics of the book. An ordinary differential equation (ODE) is said to be an autonomous differential equation if the right hand side does not depend on time explicitly, i.e., it can be formulated as dx dt=g(x),(2.1) where g:O⊂Rd→Rdis a mapping from an open subset Oof Rdto Rd. This is a particular case of the system in (1.6) where I=Rand the vector field gdoes not depend on the time t. Associating Equation (2.1) with an initial datum x(t0)=x0∈O gives the following IVP dx dt=g(x),x(t0)=x0.(2.2) To apply the general results presented in Chapter 1 to obtain the existence and uniqueness of a maximal solution to equation (2.2), we only need to impose a locally Lipschitz assumption on function g. Note that if a function is locally Lipschitz with respect to all its variables, then it is continuous. But in general, the continuity of gguarantees only the existence of solutions to (2.1) (see e.g., [25]), but not the uniqueness. Remark 2.1. One effective way to check if a function satisfies a Lipschitz condition is to check if it is continuously differentiable. A continuously differentiable function is always locally Lipschitz (see, e.g., [66]), hence every IVP problem (2.2) with g∈ C1(O) possesses a unique maximal solution. Moreover, if the domain Ois convex, then a continuously differentiable function is globally Lipschitz if and only if the partial derivatives ∂gi ∂xj(x),i,j=1,2···,d,are globally bounded. 11 12 2 Autonomous dynamical systems A general theorem on existence and uniqueness of a maximal solution to the IVP (2.2) is stated below. Theorem 2.1. (Existence and uniqueness of a maximal solution) Let Obe an open subset of Rdand assume that gis continuously differentiable on O. Then for any t0∈Rand any x0∈Othe initial value problem (2.2) has a unique maximal solution ϕ(·;t0,x0)defined in its maximal open interval Imax =Imax(t0,x0). Remark 2.2. As our main interest is the long term behavior of solutions to (2.2), we will focus on the cases when Imax contains the interval [t0,+∞). This means that the solution ϕ(·;t0,x0) is globally defined in all future times, i.e., is a global solution. However, note that the existence of a global solution is not free; it requires conditions such as in Theorem 1.1. An important property of autonomous ODEs, which can be easily verified, is that the solution mapping depends only on the elapsed time t−t0but not separately on the initial time t0and current time t(see, e.g., [51]), i.e., ϕ(t−t0;0,x0)=ϕ(t;t0,x0),and Imax(t0,x0)=t0+Imax(0,x0), for all x0∈O,t0∈R, and t∈Imax(t0,x0). Therefore for autonomous ODEs we can always focus on t0=0. With t0=0 the solution can be written as ϕ(t;x0) and the existence interval of maximal solution can be written as Imax(x0). It is straightforward to check that this solution mapping ϕ(·;·):R×O→Rdsatisfies the initial value property (1.3) and the group property (1.4) (when Imax =R), and hence defines a dynamical system, namely an autonomous dynamical system. Next we will provide a survey on the long term behavior for autonomous dynamical systems. In particular, we will start from the stability theory of linear ODE systems, followed by the stability of nonlinear ODE systems by the first approximation method. Then we will introduce basic Lyapunov theory for stability and asymptotic stability. Some comments on globally attracting sets will be provided via the LaSalle theorem and the Poincar´ e–Bendixson theorem in dimension d=2, and serve as a motivation for the analysis of global asymptotic behavior in higher dimensions. In the end we will introduce the general concept of attractors and their main properties for autonomous dynamical systems. 2.1 Basic Stability Theory For ease of understanding, we start from the stability of equilibrium points of system (2.1). Recall that an equilibrium point (or steady state) x∗of system (2.1) is a constant solution to (2.1) satisfying g(x∗)=0. Definition 2.1. An equilibrium x∗is said to be •stable if for any ε>0 there exists δ(ε)>0 such that if |x0−x∗|<δ(ε) then Imax(x0)⊇[0,+∞) and 2.1 Basic Stability Theory 13 |ϕ(t;x0)−x∗|<ε,∀t≥0. (solutions starting close to one equilibrium point remain close to it in the future.) •convergent or attractive if there exists δ>0 such that if |x0−x∗|<δthen Imax(x0)⊇[0,+∞) and lim t→∞ ϕ(t;x0)=x∗. (solutions starting close to one equilibrium point will converge to it when time goes to infinity.) •asymptotically stable if it is both stable and convergent. •exponentially stable if there exist δ>0 and α,λ>0 such that if |x0−x∗|<δthen Imax(x0)⊇[0,+∞) and |ϕ(t;x0)−x∗|<α|x0−x∗|e−λt,∀t≥0. Remark 2.3. Definition 2.1 includes only the stability for an equilibrium point, i.e., x∗is constant, but can be easily generalized to any non-constant particular solution of equation (2.1). More precisely, a particular solution x∗(t) to (2.1) is said to be stable if for any ε>0 there exists δ(ε)>0 such that if |x0−x∗(0)|<δ(ε) then Imax ⊇ [0,+∞) and |ϕ(t;x0)−x∗(t)|<ε,∀t≥0. Following a similar manner all the other concepts in Definition 2.1 can be generalized to any particular solution of (2.1). It is worth mentioning that exponential stability implies asymptotic stability, and asymptotic stability implies stability and convergence. However, stability and convergence are independent properties. Convergence implies stability for linear ODEs (see, e.g., [81]), but does not imply stability in general. For example, the autonomous ODE system dx(t) dt=x2(y−x)+y5 (x2+y2)(1 +(x2+y2)2);dx(t) dt=0 for x=0,y=0,(2.3) dy(t) dt=y2(y−2x) (x2+y2)(1 +(x2+y2)2);dy(t) dt=0 for x=0,y=0.(2.4) has an isolated equilibrium at (0,0) that is convergent but unstable (see Fig. 2.1). Remark 2.4. In the autonomous framework, all the stability concepts in Definition 2.1 are uniform in time, i.e., the choice of δdoes not depend on time. But this does not hold true for nonautonomous ODEs, which requires new definitions to distinguish uniform and non-uniform type of stability (see, e.g., [71]). Notice that a very simple change of variable can reduce the problem of analyzing the stability of any solution to (2.1), to the problem of analyzing the stability of the zero (or trivial) solution ϕ0(t)≡0 (for all t∈R) of a corresponding system of ODEs. 14 2 Autonomous dynamical systems Fig. 2.1: Attractivity and convergence of example (2.3) – (2.4). More precisely, assume that x∗(t) is a non-zero particular solution of system (2.1), and consider the change of variables y(t)=x(t)−x∗(t), then it is easy to check that dy(t) dt=˜g(t,y(t)),where ˜g(t,y)=g(y+x∗(t)) −g(x∗(t)), in which the right hand side satisfies ˜g(t,0) =0 for all t∈R. In the special case that x∗(t)=x∗is a constant for all t∈R,˜g(t,y)=˜g(y)=g(y+x∗)−g(x∗). This observation allows us to focus the stability analysis to the zero solution by assuming that function gin (2.1) satisfies g(0) =0. Stability for linear ODEs One important fact of linear ODE systems is that all solutions possess the same type of stability. Moreover, the type of stability can be characterized by the asymptotic behavior of its fundamental matrix. For the autonomous system (2.1), the fundamental matrix can be determined by the eigenvalues of the matrix on the right hand side. More specifically, consider the following linear system of ODEs dx dt=Ax+b(t),(2.5) where A=(aij)i,j=1,···,dis a d×dmatrix with real coefficients aij ∈R. Clearly if x∗(t) is a solution to (2.5), then y(t)=x(t)−x∗(t) is solution to the ODE system 2.1 Basic Stability Theory 15 dy dt=Ay. (2.6) The same holds true if x∗(t) is a solution to the homogeneous counterpart of equation (2.5), x�=Ax. Consequently, the stability of any solution of the linear ODE system (2.5) is equivalent to the stability of the zero solution to (2.6), regardless whether or not b(t) is zero. The theorem presents the stability of linear ODEs. Theorem 2.2. Let {λj}1≤j≤d⊂Cbe the set of eigenvalues for the matrix A. Then, (i) any solution to (2.6) is exponentially stable if and only if the real parts of all the eigenvalues are negative, i.e., �(λj)<0for all 1≤j≤d. (ii) any solution to (2.6) is (uniformly) stable if and only if �(λj)≤0for all 1≤j≤d and, if for those eigenvalues λj(1≤j≤d) such that �(λj)=0, the dimension of the Jordan boxes associated to them in their canonical forms is 1(in other words, their algebraic and geometric multiplicity coincide). Stability for nonlinear ODEs Theorem 2.2 can characterize completely the stability of linear ODE systems with constant coefficients. It can also be used to analyze the stability of nonlinear differential equations, by the so called first approximation method which can be briefly described as follows. Assume that the function gin (2.1) is continuously differentiable and satisfies g(0) =0. Then according to the Taylor formula, gcan be written as g(x)=Jx+T1(x),J=�∂gi ∂xj (0)�i,j=1,...,d (2.7) where the higher order term T1(·) is sufficiently small for small values of xin the sense that lim x→0 |T1(x)| |x|=0. We now state the following result on stability and instability of solutions to system (2.1). Theorem 2.3. (Stability in first approximation) Assume that g∈C1(O)and g(0) =0. Let J be the Jacobian matrix defined in (2.7). Then, (i) if all the eigenvalues of matrix J have negative real parts, the trivial solution of (2.1) is (locally) exponentially stable. (ii) if one of the eigenvalues of matrix J has positive real part, the trivial solution of (2.1) is (locally) unstable. Stability by the Lyapunov theory Theorem 2.3 is based on a spectrum analysis of the linearization of system (2.1), and provides only local stability. For nonautonomous systems, such linearization 16 2 Autonomous dynamical systems requires additional justification (see, e.g., [55]). We next introduce the Lyapunov theory, that allows us to determine the stability of a system without either explicitly solving the differential equation (2.1) or approximating it by its first approximation linear system. Practically, the theory is a generalization of the idea that if there exists some “measure of energy” in one dynamical system, then we can study the rate of change of the energy to ascertain stability. In fact this “measure of energy” can be characterized by the so-called Lyapunov function. Roughly, if there exists a function V:X→Rsatisfying certain conditions on Vand ˙ V(the derivative along solution trajectories) that proves the Lyapunov stability of a system, we call it a Lyapunov function. Definition 2.2. A continuous function V:O∈Rd→Ris said to be •positive definite around x=0 if V(0) =0 and V(x)>0 for all x∈O\{0}. •positive semi-definite around x=0 if V(0) =0 and V(x)≥0 for all x∈O\{0}. •negative definite or negative semi-definite if −Vis positive definite or positive semi-definite, respectively. The following basic theorems provide sufficient conditions for the stability and instability of the origin of an autonomous dynamical system. First we recall that given a continuously differentiable function V:O→R, the derivative of Valong the trajectories of system (2.1), ˙ V:X→Ris defined by ˙ V(x)= d � i=1 ∂V ∂xi (x)gi(x),x∈O. It is straightforward to check that if x(·) is a solution to (2.1), then d dtV(x(t)) =˙ V(x(t)),∀t∈R. Next we present the Lyapunov theorem on stability and the Tchetaev theorem on instability based on the Laypunov functions. Theorem 2.4. (Lyapunov’s stability theorem) Let V :O→Rbe a continuously differentiable function with derivative ˙ V along the trajectories of the system (2.1). 1. If V is positive definite and ˙ V is negative semi-definite, then the zero solution is stable. 2. If V is positive definite and ˙ V is negative definite, then the zero solution is asymptotically stable. 3. If there exist some positive constants a1,a2,a3and k such that a1|x|k≤V(x)≤a2|x|kand ˙ V(x)≤−a3|x|k,∀x∈O, then the zero solution is exponentially stable. Denote by B(x0;r) the ball centered at x0with radius r. 2.1 Basic Stability Theory 17 Theorem 2.5. (Tchetaev’s instability theorem) Assume that there exists ρ>0and V∈C1(B(0;ρ)) such that B(0;ρ)⊆Oand (i) V(0) =0, (ii) ˙ V is positive definite in B(0;ρ), (iii) for any σ∈(0,ρ)there exists yσ∈B(0;σ)such that V(yσ)>0. Then the zero solution is unstable. Sometimes an equilibrium point can be asymptotically stable even if ˙ Vis not negative definite. In fact, if we can find a Lyapunov function whose derivative along the trajectories of the system is only negative semi-definite, but we can further establish that no trajectory can stay identically at points where ˙ Vvanishes, then the equilibrium must be asymptotically stable. This is the idea of Lasalle’s invariance principle [54]. Before stating the principle we first introduce the definition of ω-limit set and invariant set, which is needed to state the LaSalle theorem. Let ϕ(t;x0) be the autonomous dynamical system generated by the solutions of IVP (2.2). Definition 2.3. A set S⊂Rdis said to be the ω-limit set of ϕ(t;x0) if for every x∈S, there exists a strictly increasing sequence of times tnsuch that ϕ(tn;x0)→xas tn→∞. It is usual to write S=ω(x0).In a similar way, it is defined the omega limit of a set A⊂O, and it is denoted as ω(A), as the set of points x∈Osuch that there exist two sequences {xn}⊂A,tn→+∞such that ϕ(tn;xn)→x,as n→+∞. Definition 2.4. A set M⊂Rdis said to be (positively) invariant if for all x∈Mwe have ϕ(t;x)∈M,∀t≥0. Remark 2.5. The positively invariance means that as long as a solution passes a point inside Mit will remain inside Mforever, although the solution may have been outside of Min some previous instants of time. Theorem 2.6. (LaSalle’s Invariance Principle) Let K ⊂X be a compact and positively invariant set. Let V :K⊂Rd→Rbe continuously differentiable with ˙ V≤0on K. Let M be the largest invariant set in E ={x∈K:˙ V=0}. Then ϕ(t;x0)approaches M as t →∞for every x0∈K. Remark 2.6. LaSalle’s Invariance principle requires Vto be continuously differentiable but not necessarily positive definite. It is applicable to any equilibrium set, rather than just an isolated equilibrium point. But when Mis just a single point, it provides additional information about the type of stability of the equilibrium point. Indeed, when Mis just a single point, and we are able to find a Lyapunov function 18 2 Autonomous dynamical systems which is only negative semi-definite, we can then ensure that this equilibrium is stable (thanks to Theorem 2.4) and also convergent as a consequence of LaSalle’s principle, and hence is asymptotically stable. To illustrate how the LaSalle invariance principle work, consider the following second order differential equation describing the movement of a pendulum with friction d2x dt2+βdx dt+ksin x=0, where kand βare positive constants. First we transform the equation into an equivalent first order system by setting x=y1,dx dt=y2to obtain dy1 dt=y2, dy2 dt=−βy2−ksiny1. Consider the function V(y1,y2)=1 2y2 2+k(1 −cosy1). It is easy to prove that Vis positive definite on B(0;π) and satisfies ˙ V(y)=(ksiny1)y2+y2(−βy2−ksiny1)=−βy2 2≤0,∀(y1,y2)∈B(0;π). Therefore, ˙ Vis negative semi-definite and, consequently, the trivial solution is stable. To prove that the trivial solution is also attractive, we will use LaSalle’s invariance principle. Denote by E:={y∈B(0;π): ˙ V(y)=0}≡{(y1,0) : y1∈(−π,π)}. If we prove that {(0,0)}is the only positively invariant subset of E, then it will be also attractive, and thus the trivial solution will be uniformly asymptotically stable. To this end, pick y0=(y01,0) ∈E, with y01 ∈(−π,π)\{0}, then the solution ϕ(t;0,y0)=(ϕ1(t),ϕ2(t)) satisfies the differential system as well as the initial condition (ϕ1(0),ϕ2(0)) =y0=(y01,0). Notice that ϕ�(0) =ϕ2(0) =0, ϕ� 2(0) =−kϕ2(0) −ksinϕ1(0) =−ksiny01 �0(y01 ∈(−π,π)\{0}). Therefore, the function ϕ2is strictly monotone in a neighborhood of t=0, and since ϕ2(0) =0, there exists ˜ t∈Imax(y0) such that ϕ2(˜ t)�0. Thus, solutions starting from points of E\{(0,0)}leave this set and, therefore, the unique invariant subset is (0,0). Remark 2.7. The LaSalle invariance principle is applicable to autonomous or periodic systems and can be extended to some specific nonautonomous systems (see, e.g., [65]), but not to general nonautonomous systems. 2.3 Applications 25 J(x,y)=                  −D−aλy (λ+x)2−ax λ+x aλy (λ+x)2−D+ax λ+x                   ,J(I,0) =                −D−aI λ+I 0−D+aI λ+I                 . The eigenvalues of J(I,0) are −Dand −D+aI λ+I, which are both negative because a<Dis assumed. The equilibrium (I,0) is then asymptotically exponentially stable. 2. When a>Dand λD a−D<I, system (2.10) – (2.11) has two equilibria, among which the positive equilibrium (x∗,y∗)=�Dλ a−D,I−λD a−D�is globally asymptotically stable. This means that the microorganisms yand the nutrient xco-exist. Sketch of proof: In this case, the axial equilibrium becomes unstable because one of the eigenvalues of J(I,0), −D+aI λ+I, is positive due to the condition λD a−D<I. On the other hand, the positive equilibrium point (x∗,y∗) is now globally asymptotically (exponentially) stable, since the Jacobian evaluated at (x∗,y∗) J(x∗,y∗)=                   −D−aλx∗ (λ+x∗)2−D aλy∗ (λ+x∗)20                    , has two negative eigenvalues −Dand −aλx∗ (λ+x∗)2. Global attractor As each IVP associated to (2.10) – (2.11) corresponding to positive initial values has a positive global solution, this system generates an autonomous dynamical system ϕ(t,x0,y0). Adding (2.10) – (2.11) we obtain immediately that d(x+y) dt=DI −D(x+y), and given x(0) =x0,y(0) =y0we have x(t)+y(t)=I+(x0+y0−I)e−Dt. This implies that Kε:=�(x,y)∈R2 +:x+y≤I+ε� is a bounded absorbing set for the dynamical system ϕgenerated by solutions of (2.10)– (2.11). Hence due to Theorem 2.7 ϕpossesses a global attractor Ainside 26 2 Autonomous dynamical systems the nonnegative quadrant R2 +. Moreover, we obtain the geometric structure of the global attractor as follows. 1. When a<D, the attractor Ahas a single point (I,0). 2. When a>Dand λD a−D<I, the attractor Aconsists of two points, (I,0) and �Dλ a−D,I−Dλ a−D�, and heteroclinic solutions between them (solutions that converge in one time direction to a steady state and in the other direction to different steady state). 2.3.2 Application to epidemiology: the SIR model The modeling of infectious diseases and their spread is crucial in the field of mathematical epidemiology, as it is an important and power tool for gauging the impact of different vaccination programs on the control or eradication of diseases. Here we will only introduce a simple autonomous model, which does not take into account age structure nor environmental fluctuation. More sophisticated models will be discussed in later chapters. The classical work on epidemics is due to Kermack and McKendrick [42, 43, 44]. The Kermack-McKendrick model is essentially a compartmental model based on relatively simple assumptions on the rates of flow between different classes of members of the population. The population is divided into three classes labeled S,Iand R, where S(t) denote the number of individuals who are not yet infected but susceptible to the disease, I(t) denotes the number of infected individuals, assumed infectious and able to spread the disease by contact with susceptible, and R(t) denotes the number of individuals who have been infected and then removed from the possibility of being infected again or of spreading infection. Removal is carried out through isolation from the rest of the population, through immunization against infection, recovery from the disease with full immunity against reinfection, or through death caused by the disease. The terminology “SIR” is used to describe a disease that confers immunity against reinfection, to indicate that the passage of individuals is from the susceptible class Sto the infective class Ito the removed class R. Epidemics are usually diseases of this type. The terminology “SIS” is used to describe a disease with no immunity against re-infection, to indicate that the passage of individuals is from the susceptible class to the infective class and then back to the susceptible class. Usually, diseases caused by a virus are of SIR type, while diseases caused by bacteria are of SIS type. In this book we will use SIR as an example and investigate dynamics of autonomous, nonautonomous and random SIR models. The simplest SIR model assumes that the total population size is held constant, i.e., S(t)+I(t)+R(t)=N, 2.3 Applications 27 by making birth and death rates equal. Denote by νthe birth and death rate of all population, and assume that all new-born are susceptible to disease, we have the following basic SIR model. dS dt=ν(S+I+R)−β NSI−νS,(2.12) dI dt=β NSI−νI−γI,(2.13) dR dt=γI−νR,(2.14) where γis the fraction of infected individuals being removed per unit of time, and βis the contact rate of susceptible and infected individuals (see Fig. 2.3). Fig. 2.3: Transition between SIR compartments The vector field of system (2.12)-(2.14) is continuously differentiable, and hence ensures the maximal solution to the corresponding IVP exists and is unique. Furthermore the solutions are defined globally in time (see Chapter 3 for more details). Notice that assumption of constant total population, S(t)+I(t)+R(t)=N, reduces system (2.12) – (2.14) to the following two dimensional system. dS dt=νN−β NSI−νS,(2.15) dI dt=β NSI−νI−γI.(2.16) Note that such a reduction needs the positiveness of solutions which will be discussed later in Chapter 3. Stability analysis The equilibrium solutions to system (2.15) – (2.16) can be found by solving νN−β NS∗I∗−νS∗=0, β NS∗I∗−νI∗−γI∗=0, 28 2 Autonomous dynamical systems which yields two equilibrium solutions (S∗,I∗)=(N,0),(S∗,I∗)=�N(ν+γ) β,νN�1 ν+γ−1 β��. 1. When ν+γ>β, system (2.15) – (2.16) has only one axial equilibrium, (N,0), in the nonnegative quadrant, which is globally asymptotically stable. This means that all infected and removed individuals are cleared, the system is fully immunized. Sketch of proof: The corresponding Jacobian of system (2.15) – (2.16) and its value at the equilibrium point (N,0) are, respectively, J(S,I)=           −β NI−νβ NS β NIβ NS−ν−γ            ,J(N,0) =          −ν−β 0β−ν−γ           . The eigenvalues of J(N,0) are negative, and hence the axial equilibrium (N,0) is globally asymptotically stable. 2. When ν+γ<β, system (2.15) – (2.16) has two equilibria, among which the positive equilibrium (S∗,I∗)=�N(ν+γ) β,νN�1 ν+γ−1 β��is globally asymptotically stable and the axial equilibrium (N,0) is unstable. This means that the system is at endemic state, where infected, removed and susceptible individuals co-exist. Sktech of proof: The instability of the axial equilibrium is obvious as J(N,0)) now has a positive eigenvalue since ν+γ<β. The Jacobian evaluated at the positive equilibrium (S∗,I∗) is J(S∗,I∗)=            −βν�1 ν+γ−1 β�−ν−(ν+γ) βν�1 ν+γ−1 β�0             , whose eigenvalues are the roots of the quadratic equation r2+�βν�1 ν+γ−1 β�+ν�r+β(ν+γ)�1 ν+γ−1 β�=0, both of which have negative real part. Hence the positive equilibrium (S∗,I∗) is asymptotic exponential stable. Attractors Any IVP of (2.15) – (2.16) associated with positive initial values has positive global solutions (see Chapter 3 for detailed proof). Hence system (2.15) – (2.16) generates an autonomous dynamical system ϕ(t,S0,I0). By adding (2.15) – (2.16) we obtain immediately that 2.3 Applications 29 d(S+I) dt=νN−ν(S+I)−γI≤νN−ν(S+I), and given S(0) =S0,I(0) =I0we have S(t)+I(t)≤N+(S0+I0−N)e−νt. This implies that Kε:=�(S,I)∈R2 +:S+I≤N+ε� is a bounded absorbing set for the dynamical system ϕgenerated by solutions of (2.15)– (2.16). Hence due to Theorem 2.7, ϕpossesses a global attractor Ainside the nonnegative quadrant R2 +. Moreover, at light of the stability properties of the equilibrium points, we obtain the geometric structure of the global attractor as follows. 1. When ν+γ>β, the attractor Ahas a single point (N,0). 2. When ν+γ<β, the attractor Aconsists of two points, (N,0), and �N(ν+γ) β,νN�1 ν+γ−1 β��, and heteroclinic solutions between them. 2.3.3 Application to climate change: the Lorenz-84 model A model of atmospheric circulation was introduced by Lorenz in 1984, defined by a system of three nonlinear autonomous differential equations [58, 59, 80]. Let x represents the pole ward temperature gradient or the intensity of the westerly wind current, yand zare the strengths of cosine and sine phases of a chain of superposed waves transporting heat poleward, respectively. Then a modified Hadley circulation can be modeled by dx dt=−ax−y2−z2+aF,(2.17) dy dt=−y+xy−bxz+G,(2.18) dz dt=−z+bxy+xz,(2.19) where the coefficient a, if less than 1, allows the westerly wind current to damp less rapidly than the waves, the terms in brepresent the displacement of the waves due to interaction with the westerly wind, the terms in Fand Gare thermal forcings: Frepresents the symmetric cross-latitude heating contrast and Gaccounts for the asymmetric heating contrast between oceans and continents. Despite the simplicity of the Lorenz-84 model, it addresses many key applications in climate studies such as how the coexistence of two possible climates combined with variations of the solar heating causes seasons with inter-annual variability [6, 58, 59, 67], how the climate is affected by the interactions atmosphere and 30 2 Autonomous dynamical systems the oceans [7, 70], how the asymmetry between oceans and continents may result in complex behaviors of the system [62], etc. In addition to applications, the Lorenz- 84 model has also attracted much attentions from mathematicians because of certain interesting and subtle mathematical aspects of its underlying differential equations such as multistability, intransitivity and bifurcation [35]. Here we will focus on the Laypunov stability and existence of attractors for the Lorenz-84 model. Notice that the vector field of system (2.17)-(2.19) is continuously differentiable, which ensures existence and uniqueness of maximal solutions to the corresponding IVP. Moreover, once we prove that the solutions are absorbed by a ball centered at zero, then it follows immediately from Theorem 1.5 that all solutions are defined globally in time (more details will be provided in Chapter 3). Stability analysis The equilibrium points of system (2.17) – (2.19) can be calculated by solving a(F−x∗)(1 −2x∗+(1 +b2)(x∗)2)=G2,(2.20) (1 −x∗)G 1−2x∗+(1 +b2)(x∗)2=y∗,(2.21) bx∗G 1−2x∗+(1 +b2)(x∗)2=z∗.(2.22) When G=0, system (2.20) – (2.22) has only one solution, (F,0,0), and hence (2.17) – (2.19) possesses only one equilibrium, (F,0,0). To analyze its stability, we first shift the point (F,0,0) to the origin (0,0,0) by performing the change of variable ˜x=x−F. The resulting system reads d˜x dt=−a˜x−y2−z2,(2.23) dy dt=−y+(˜x+F)y−b(˜x+F)z,(2.24) dz dt=−z+b(˜x+F)y+(˜x+F)z.(2.25) Though we can use again the first approximation method to analyze the stability of (0,0,0), to present alternative methods we study the stability of the (0,0,0) for this system, by the Lypunov theory and Tchetaev’s theorem. 1. When F<1, the equilibrium (0,0,0) is globally asymptotically stable. Sketch of proof: Define the function V(˜x,y,z):=1 2(˜x2+y2+z2).(2.26) 2.3 Applications 31 Then V(0,0,0) =0 and V(˜x,y,z)>0 for all ( ˜x,y,z)∈R3\{(0,0,0)}, what implies that Vis positive definite in any ball centered in (0,0,0) (in particular, in the entire R3). Moreover, the derivative of Valong trajectories of system (2.17) – (2.19) satisfies ˙ V=−a˜x2−(1 −F)y2−(1 −F)z2 which implies that Vis negative definite when F<1. The conclusion follows immediately from Theorem 2.4. 2. When F>1, the equilibrium (0,0,0) is unstable. Sketch of proof: The previous Lyapunov function defined in (2.26) is no longer valid, but we can apply Tchetaev’s theorem. To this end, let us consider the function V(˜x,y,z):=1 2(−˜x2+y2+z2).(2.27) Then, straightforward computations give ˙ V=a˜x2+(F−1)y2+(F−1)z2+2˜x(y2+z2). Observing that lim (˜x,y,x)→(0,0,0) 2˜x(y2+z2) a˜x2+(F−1)y2+(F−1)z2=0, we can find a positive ρsuch that ������ 2˜x(y2+z2) a˜x2+(F−1)y2+(F−1)z2������ ≤1 2,for ( ˜x,y,z)∈B((0,0,0);ρ). Hence 2˜x(y2+z2)≥−1 2(a˜x2+(F−1)y2+(F−1)z2), which implies that ˙ V=a˜x2+(F−1)y2+(F−1)z2+2˜x(y2+z2) ≥1 2(a˜x2+(F−1)y2+(F−1)z2) for all ( ˜x,y,z)∈B((0,0,0);ρ). The instability of (0,0,0) then follows immediately from Tchetaev’s theorem. 3. When F=1, we cannot deduce any information from Theorem 2.3. However, since the Lyapunov function defined in (2.26) has a negative semi-definite ˙ V, the equilibrium point (0,0,0) is at least stable. Next we will apply the LaSalle invariance principle to prove that the equilibrium point is also attractive, and hence asymptotically stable. In fact, ˙ V=−a˜x2≤0 when F=1. Denote by Ethe set E:={(˜x,y,z)∈R3:˙ V(˜x,y,z)=0}={(0,y0,z0):y0,z0∈R}. 32 2 Autonomous dynamical systems Let us prove that (0,0,0) is the unique invariant subset of E. To this end, pick (0,y0,z0)∈E\{(0,0,0)}, and denote by ϕ(·) the corresponding solution of the IVP. Then we only need to observe that d dtϕ1(0) =−y2 0−z2 0<0. Since ϕ1(0) =0, then there exists ε>0 such that ϕ1(t)<0 for all t∈(0,ε). This proves that (0,0,0) is the only invariant subset of E, and therefore, the equilibrium point (0,0,0) is asymptotically stable. When G�0, the dynamical behavior of system (2.17) – (2.19) becomes complicated and exhibits chaotic attractors. Note that from (2.20) – (2.22) it is difficult to obtain an explicit expression of the equilibrium points on parameters a,b,Fand G. It is also difficult to determine how many equilibrium points there exist. Hence it is impossible to apply the Lyapunov method to obtain stability properties. However, we can still investigate the existence of global attractor. Global attractor System (2.17) – (2.19) possesses a global solution that generates an autonomous dynamical system ϕ(t,x0,y0,z0) (see Chapter 3 for a detailed proof). To prove the existence of an attractor, we first prove the existence of a compact absorbing set. In fact, it is sufficient to prove the existence of a bounded absorbing set, as its closure will give us a compact absorbing set. Let D⊂R3be a bounded set. Then, there exists k>0 such that |u0|≤kfor all d∈D. Then, for any u0=(x0,y0,z0)∈Dwe have d dt(x2+y2+z2)=2xdx dt+2ydy dt+2zdz dt =2�−ax2−y2−z2+aFx+Gy� ≤−ax2−y2−2z2+aF2+G2 ≤−µ1(x2+y2+z2)+aF2+G2, where µ1=min{a,1}, and hence x2(t)+y2(t)+z2(t)≤(x2 0+y2 0+z2 0)|u0|2e−µ1t+aF2+G2 µ1 (1 −e−µ1t) ≤k2e−µ1t+aF2+G2 µ1 . This implies that given any fixed positive ε, there exists T=T(ε,D)>such that x2(t)+y2(t)+z2(t)≤aF2+G2 µ1 +ε,for all t≥T.(2.28) 2.3 Applications 33 In fact, simple calculations yield that we can take T(ε,D)=−1 µ1log ε k2. Note that we can always take ksuch that k2>ε, so that −1 µ1log ε k2>0. The inequality (2.28) implies that Kε:=�(x,y,z)∈R3:x2+y2+z2≤aF2+G2 µ1 +ε� is a bounded absorbing set for ϕ. Hence by Theorem 2.7, we conclude immediately that the dynamical system generated by solutions to (2.17) – (2.18) possesses a global attractor A. When G=0, this attractor Aconsists of a single point, (F,0,0). When G�0, the geometric structure of Ais difficult to obtain. In fact, numerical simulation shows that the attractor Aexhibits chaotic behavior. 3.2 Nonautonomous Attractors 41 To illustrate the fundamental character of this distinction, consider the following simple nonautonomous initial value problem: dx dt=−ax+bsint,x(t0)=x0,t≥t0,(3.11) where aand bare positive constants. The solution mapping ϕ(t,t0,x0) is given explicitly by ϕ(t,t0,x0)=b(asint−cost) a2+1+�x0−b(asint0−cost0) a2+1�ea(t0−t).(3.12) Clearly for any x0, the forward limit limt→∞ ϕ(t,t0,x0) does not exist. But if we consider the difference between any two solutions corresponding to two different initial values x0and y0, it follows from an easy computation that ϕ(t,t0,x0)−ϕ(t,t0,y0)=(x0−y0)ea(t0−t). Hence although individually each solution does not have a limit as tgoes to +∞, all solutions behave similarly in the sense that the difference between any two solutions goes to zero when tgoes to +∞, i.e., all solutions approach the same target as time goes on. The goal is then to find a particular or special solution such that the trajectory of this solution describes the path that any other solution should approach during their evolution. It is easy to check that the function A(t):=b(asint−cost) a2+1(3.13) is such a particular solution to the IVP (3.11), that provides the long term information on the future evolution of the system, in the sense that any other solution will eventually approach it, i.e., lim t→∞ |ϕ(t,t0,x0)−A(t)|=0,for all t0,x0∈R. However, if we take limit with tfixed and the initial time t0goes to −∞ in the expression in (3.12), we obtain that lim t0→−∞ ϕ(t,t0,x0)=A(t),for all t,x0∈R.(3.14) The limit in (3.14) is said to be taken in the “pullback” sense, as we are solving a series of IVPs with the same initial value x0at retrospective initial times t0, and look at the solution at a current fixed time t. The concepts of the forward and pullback limit are illustrated in Figures 3.1 and 3.2 by solution trajectories of equation (3.11). In some occasions we can obtain special solutions to an underlying ODE that attract any other solution in both the pullback and the forward sense such as in example (3.11). But it is also possible that the attraction happens in either the pullback 42 3 Nonautonomous dynamical systems Fig. 3.1: Forward limit of solutions to equation (3.11) with t0fixed and t→ ∞. Fig. 3.2: Pullback limit of solutions to equation (3.11) with tfixed and t0→ −∞. or the forward sense, but not both. Taking limit in the pullback sense may result in obtaining special (particular) solutions of a nonautonomous ODE which are independent of the initial values x0. This fact is very important in the random cases, because such stationary solutions can be seen as a natural extension of the concept of equilibrium points in the random framework. The set A(t) defined in (3.13) can be further shown to be invariant under the dynamical system ϕ(or ϕ–invariant), i.e., ϕ(t,t0,A(t0)) =A(t) for every t≥t0.We have therefore constructed a family of limiting objects A(t), which in addition is a particular solution of the equation in (3.11). These objects exist in actual time trather than asymptotically in the future (t→∞), and convey the effect of the dissipation due to the term −ax. Note that A(t) is merely a time-dependent object that attracts solutions starting from all the initial data x0in the past. More generally, in the force-dissipative case, we can obtain, for all t, a collection of objects, A={A(t):t∈R}={A(t)}t∈Rthat depend on time t, by letting t0→ −∞. This collection Ais the so-called pullback attractor for processes. Each A(t) may be more complicated than a point, and attracts particular subsets of initial data taken in the asymptotic past. The rigorous definition of pullback attractors and forward attractors will be given below, following the definition of pullback and forward attraction. Recall that in the basic definition of attractor (Definition 2.6), the attractor for a dynamical system on Xattracts all bounded subsets of X. While for nonautonomous dynamical systems, the candidates for nonautonomous attractors are families of sets parameterized by time. Thus we will define the concept of attraction based on families of sets D={D(t)}t∈R, instead of a single set (which is the special case where D(t)≡D). Definition 3.5. Let ϕbe a process on Rd. A family of sets A={A(t)}t∈Ris said to 3.2 Nonautonomous Attractors 43 (i) pullback attract another family D={D(t)}t∈Rif lim t0→−∞ dist(ϕ(t,t0,D(t0)),A(t)) =0,for all t∈R; (ii) forward attract another family D={D(t)}t∈Rif lim t→∞ dist(ϕ(t,t0,D(t0)),A(t)) =0 for all t0∈R. The pullback attraction is said to be uniform if lim t0→−∞ sup t∈R dist(ϕ(t,t0,D(t0)),A(t)) =0,(3.15) and the forward attraction is said to be uniform if lim t→∞ sup t0∈R dist(ϕ(t,t0,D(t0)),A(t)) =0.(3.16) Remark 3.3. In general the pullback and forward attraction are independent concepts. However, when the attraction is uniform, then it is straightforward to prove that both concepts are equivalent. Observe that in the motivating example (3.11), the convergence of solutions onto A(t) defined in (3.13) is both pullback and forward. To illustrate the difference between pullback and forward attractions, consider the following IVPs: du dt=2tu+1,u(t0)=u0,(3.17) dv dt=−2tv+1,v(t0)=v0.(3.18) whose solutions are given explicitly by u(t;t0,u0)=u0et2−t2 0+et2�t t0 e−s2ds v(t;t0,v0)=v0e−(t2−t2 0)+e−t2�t t0 es2ds. It is clear that u(t;t0,u0)−→ +∞as t→+∞, u(t;t0,u0)−→ et2�t −∞ e−s2ds as t0→ −∞, and hence u(·;·,·) possesses a limit in the pullback sense but not in the forward sense. On the other hand, 44 3 Nonautonomous dynamical systems 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 t 0 50 100 150 200 250 300 u u0=0 u0=1 u0=2 u0=3 u0=4 (a) Solution trajectories of Eq. (3.17) starting from t0=0 and u0=0,1,2,3,4. -10 -8 -6 -4 -2 0 2 t 0 2 4 6 8 10 12 14 u t0=-1 t0=-1.5 t0=-2 t0=-5 t0=-10 (b) Solution trajectories of Eq. (3.17) starting from t0=−1,−1.5,−2,−5,−10 and u0= 4. 0 0.5 1 1.5 2 2.5 3 t 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 v v0=0 v0=1 v0=2 v0=3 v0=4 (c) Solution trajectories of Eq. (3.18) starting from t0=0 and u0=0,1,2,3,4. -100 -90 -80 -70 -60 -50 -40 -30 -20 -10 0 1 2 3 4 5 6 t v 10306 t0=-30 t0=-50 t0=-75 t0=-100 (d) Solution trajectories of Eq. (3.18) starting from t0=−30,−50,−75,−100 and v0= 4. Fig. 3.3: Forward and pullback trajectories of Eqs. (3.17) and (3.18). 3.2 Nonautonomous Attractors 45 v(t;t0,v0)−→ 0 as t→+∞, v(t;t0,v0)−→ +∞as t0→ −∞. Hence vis convergent to 0 in the forward sense, but is not convergent in the pullback sense (see Fig. 3.3). Next we will define the concept of pullback and forward attractors. Notice that the family of sets Din Definition 3.5 is the key to define attractors; different characterizations of Dwill give different attractors. In the general abstract framework for attractors, we should consider the attraction (by the attractor) of a universe (denoted by U) of nonautonomous sets, i.e., a collection of families D={D(t)}t∈Rthat satisfies some specific property. For example, Ucan be the collection of all families Dthat satisfies lim t→+∞e−tsup d∈D(t) |d|=0. However, all the applications in this chapter requires no more than the attraction (by the attractor) of families of sets which are formed by the same bounded set for all t, i.e., D={D(t)≡D0}t∈R,where D0is a bounded set. Hence at this point we will simply restrict our definition of attractors based on families of nonempty bounded subsets of X, D={D(t):t∈R}where D(t)is a bounded subset of X. Later in Chapter 4 we will generalize the definition of attractors by using the concept of universe. Definition 3.6. A family of sets A={A(t)}t∈Ris said to be a (global) pullback attractor for the process ϕ, if (i) A(t) is compact for all t∈R, (ii) ϕ(t,t0,A(t0)) =A(t) for all t≥t0, (iii) Apullback attracts all families of bounded subsets of X, i.e., lim t0→−∞ dist(ϕ(t,t0,D(t0)),A(t)) =0 for any fixed t∈R. The pullback attractor is said to be uniform if the attraction property is uniform in time as established in (3.15). Definition 3.7. A family of sets A={A(t)}t∈Ris said to be a (global) forward attractor for the process ϕ, if (i) A(t) is compact for all t∈R, (ii) ϕ(t,t0,A(t0)) =A(t) for all t≥t0, (iii) Aforward attracts all families of bounded subsets of X, i.e., lim t→∞ dist(ϕ(t,t0,D(t0)),A(t)) =0 for any fixed t0∈R. 46 3 Nonautonomous dynamical systems The forward attractor is said to be uniform if the attraction property is uniform in time as established in (3.16). Remark 3.4. When the attraction is uniform then pullback and forward attraction are equivalent. However in general the forward limit defining a nonautonomous forward attractor is different from the pullback limit. In fact, the forward limit is taken asymptotically in the future whereas the pullback limit is taken asymptotically in the past. Moreover, the limiting objects obtained forward in time do not have the same dynamical meaning at current time as the limiting objects obtained pullback in time. For a more complete discussion about differences between pullback and forward attractions, and in particular how to construct forward attractors the reader is referred to the papers [47, 50]. The concept of attractor is closely related to that of absorbing sets, which is defined below. Definition 3.8. A family B={B(t)}t∈Rof nonempty subsets of Xis said to be (pullback) absorbing for the process ϕif for each t∈Rand every family D={D(t):t∈R} of nonempty bounded subsets of X, there exists TD(t)>0 such that ϕ(t,t0,D(t0)) ⊂B(t) for allt0≤t−TD(t). The absorption is uniform if TD(t) does not depend on the time variable t. The existence of compact absorbing sets for a nonautonomous dynamical system is the key to obtain the existence of pullback attractors, as stated in the following theorem due to Caraballo et al. (see [11] for more details). Theorem 3.1. Suppose that the process ϕ(·,·,·)has a family B={B(t)}t∈Rof nonempty compact subsets of X which is pullback absorbing for ϕ(·,·,·). Then ϕ(·,·,·)has a global pullback attractor A={A(t):t∈R}, whose component subsets are defined by A(t)=Λ(B,t):=� s≤t      � τ≤s ϕ(t,τ,B(τ))      ,t∈R. Moreover, Ais minimal in the sense that if ˜ A={˜ A(t)}t∈Ris a family of closed sets such that limτ→−∞ dist(ϕ(t,τ,B(τ)),˜ A(t)) =0,then A(t)⊂˜ A(t). The pullback attractor given in Theorem 3.1 is minimal with respect to set inclusion (see [29]). However it may not be unique in general (see [10]). For example, consider the following nonautonomous ODE dx(t) dt=(−α+max{t,0})x(t),(3.19) which generates a process by setting ϕ(t,t0,x0)=x(t;t0,x0),where x(·;t0,x0) denotes the unique solution to equation (3.19) satisfying x(t0;t0,x0)=x0. Let a(t)≤0 and b(t)≥0 be any entire solutions to equation (3.19) defined for all t∈R, then the family 3.2 Nonautonomous Attractors 47 A={A(t)}t∈Rdefined by A(t)=[a(t),b(t)] satisfies all the conditions in Definition 3.6, and thus is a global pullback attractor for the process ϕ. Similar to the autonomous case, in order to guarantee the uniqueness of global pullback attractors, we need to impose additional hypotheses on the attractor such as assuming the family {A(t)}t∈Ris uniformly bounded (i.e., there exists a bounded set B⊂Xsuch that A(t)⊂Bfor all t∈R). It is not difficult to check in example (3.19) that the unique uniformly bounded attractor is given by A(t)={0},for all t∈R. In fact, if we assume in addition in Theorem 3.1 that Ais uniformly bounded, i.e., ∪t∈RA(t) is bounded, which means that there exists a bounded set B⊂Xsuch that A(t)⊂Bfor all t∈R, then Ais the unique pullback attractor with all properties defined in Definition 3.6. A sufficient condition to ensure the uniform boundedness of Ais that the family of compact absorbing sets in Theorem 3.1 is uniformly bounded. The existence and uniqueness of pullback attractors can be summarized in the following theorem. Theorem 3.2. Suppose that the process ϕ(·,·,·)possesses a uniformly bounded family B={B(t)}t∈Rof nonempty compact subsets of X which is pullback absorbing for ϕ(·,·,·). Then ϕ(·,·,·)has a unique global pullback attractor A={A(t):t∈R}, whose component subsets are defined by A(t)=Λ(B,t):=� s≤t      � τ≤s ϕ(t,τ,B(τ))      ,t∈R. Remark 3.5. There are other conditions to ensure the uniqueness of the pullback attractor, such as requiring the attractor to belong to a certain class of set valued functions which are attracted by the attractor (see [23]). Remark 3.6. If the family Bin Theorem 3.1 and 3.2 is in addition ϕ-positively invariant, then the components of the pullback attractor A={A(t):t∈R}are determined by A(t)=� t0≤t ϕ(t,t0,B(t0))for each t∈R. A pullback attractor consists of entire solutions, i.e., functions ξ:R→Rsuch that ξ(t)=ϕ(t,t0,ξ(t0)) for all (t,t0)∈R2 ≥. In special cases it consists of a single entire solution. We will next state a theorem to obtain a pullback attractor that consists of a single entire solution in the finite dimensional space Rd. Before that we first define the following property that is required in the theorem. Definition 3.9. A nonautonomous dynamical system ϕis said to satisfy a uniform strictly contracting property if for each R>0, there exist positive constants Kand αsuch that |ϕ(t,t0,x0)−ϕ(t,t0,y0)|2≤Ke−α(t−t0)·|x0−y0|2(3.20) for all (t,t0)∈R2 ≥and x0,y0∈B(0;R), the closed ball in Rdcentered at the origin with radius R>0. 48 3 Nonautonomous dynamical systems This uniform strictly contracting property along with the existence of a pullback absorbing set ensure the existence of a global attractor that consists of singleton sets (i.e., a single entire solution) in both the forward and pullback senses, as stated in the following theorem. Theorem 3.3. Suppose that a process ϕon Rdis uniform strictly contracting on aϕ-positively invariant pullback absorbing family B={B(t):t∈R}of nonempty compact subsets of Rd. Then the process ϕhas a unique global forward and pullback attractor A={A(t):t∈R}with component sets consisting of singleton sets, i.e., A(t) ={ξ∗(t)}for each t ∈R, where ξ∗is an entire solution of the process. The proof of Theorem 3.3 is done by constructing an appropriate Cauchy sequence which converges to a unique limit. The reader is referred to [48, 49] for more details about the proof. 3.2.2 Nonautonomous attractors for skew product flows Let (θ,ψ) be a nonautonomous dynamical system over the metric spaces P(a base space or a parameter space) and X(the phase space) with corresponding metrics dP and dX, respectively. Consider its associated skew product semiflow π:R+ 0×P×X→ P×Xdefined by π(t,(p,x)) :=(θt(p),ψ(t,p,x)). Then as it was shown in Subsection 3.1.2, πis an autonomous semi-dynamical system on the extended product space X=P×X. Consequently the theory of global attractors for autonomous semi-dynamical systems in Chapter 2 can be directly applied here. In fact, a global attractor for πis a nonempty compact subset Aof Xthat is π–invariant, i.e., π(t,A)=Afor all t≥0 and lim t→+∞distX(π(t,D),A)=0, for each bounded subset Dof X. Comparing to the global attractor for processes, the global attractor for skew product flows is valuable only if it provides us with some more specific information on the dynamics of an underlying system on the phase space X, as this is the space where the evolution of any dynamical system takes place. In fact, when Pis compact and θ–invariant, the global attractor Aof the autonomous semi-dynamical systems πcan be rewritten in the form A=� p∈P {p}×A(p),(3.21) where A(p) is a compact subset of Xfor each p∈P. More precisely, A(p)={x∈X: (p,x)∈A}, i.e., A(p) is the projection of the global attractor Aon its second com- 3.2 Nonautonomous Attractors 49 ponent. Therefore, this family of subsets {A(p)}p∈Pof the phase space Xserves as a good candidate to provide information on the dynamics of an underlying system. In addition, observe that the invariance property of the global attractor Aimplies that π(t,A)=Afor all t≥0. In fact, it follows directly from (3.21) that A=� ˜p∈P {˜p}×A(˜p)=� p∈P {θt(p)}×ψ(t,p,A(p)) =π(t,A), which implies that ψ(t,p,A(p)) =A(θt(p)) for all t≥0 and p∈P. This confirms that the family of sets {A(p)}p∈Pof the phase space Xcan better describe the dynamics of a system than the global attractor on the extended space X, since Xalso includes the base (or parameter) space Pas a component, which usually does not have a physical meaning similar to the state space component. For these reasons, in addition to the global attractor for skew product flows, we also want to define other types of attractors consisting of families of compact subsets of the phase space X, that are similar to the forward and pullback attractors for processes. Definition 3.10. Let π=(θ,ψ) be a skew product flow on the metric space P×X.A pullback attractor for πis a nonautonomous set A={A(p)⊂X}p∈Psuch that (i) A(p) is a nonempty, compact set for every p∈P; (ii) Ais invariant for the cocycle, i.e., ψ(t,p,A(p)) =A(θt(p)) for all t≥0 and p∈P; (iii) pullback attracts the bounded subsets of X, i.e., lim t→∞ distX(ψ(t,θ−t(p),D),A(p)) =0 (3.22) holds for every nonempty bounded subset Dof Xand p∈P. Definition 3.11. Let π=(θ,ψ) be a skew product flow on the metric space P×X.A forward attractor for πis a nonautonomous set A={A(p)⊂X}p∈Psuch that (i) A(p) is a nonempty, compact set for every p∈P; (ii) Ais invariant for the cocycle, i.e., ψ(t,p,A(p)) =A(θt(p)) for all t≥0 and p∈P; (iii) forward attracts the bounded subsets of X, i.e., lim t→+∞distX(ψ(t,p,D),A(θt(p))) =0 (3.23) holds for every nonempty bounded subset Dof Xand p∈P. Remark 3.7. As in the case of attractors for processes, there is no clear relationship between the existence of forward and pullback attractors for skew product flows; one can exist while the other does not. See, for instance, Kloeden and Lorenz [47, 50] for a more detailed discussion. If we assume further that the limits taken in (3.22) and (3.23) are uniform, in the sense that 50 3 Nonautonomous dynamical systems lim t→+∞sup p∈P distX(ψ(t,θ−t(p),D),A(p)) =0 (pullback), lim t→+∞sup p∈P distX(ψ(t,p,D),A(θt(p))) =0 (forward), then we obtain another type of attractor, called uniform pullback attractor and uniform forward attractor, respectively. Similar to the idea of equivalence between uniform pullback and forward attractions for processes, when an attractor for the skew product flow is uniform (either pullback or forward), it will be both uniform pullback and uniform forward, so one can simply refer to it as a uniform attractor. For more details regarding relationships among different types of nonautonomous attractors, the reader is referred to [24] and [51]. We next illustrate how to obtain attractors for the skew product flow. Consider again the system (3.11) which can be written as dx dt=−ax+p0(t),x(t0)=x0,(t,t0)∈R2 +,(3.24) where p0(t):=bsint. In order to set the problem in the skew product framework, we need to define (1) the base (parameter) space P, (2) the driving dynamical system θt, and (3) the cocycle ψ. To this end, we define Pby P={p0(t+·):0≤t≤2π}, define θt:P→Pby θt(p)(·)=p(t+·) for all p∈P, and define ψ(t,p,x0) by the solution to the IVP (3.24), which can be calculated explicitly to be ψ(t,p,x0)=x0e−a(t−t0)+e−at �t t0 easp(s)ds.(3.25) At this point to estimate the pullback limit, we need to obtain first the expression for ψ(t,θ−t(p),x0) and then evaluate the expression at the limit of t→∞. By (3.25) we obtain that ψ(t,θ−t(p),x0)=x0e−a(t−t0)+e−at �t t0 easθ−t(p)(s)ds =x0e−a(t−t0)+e−at �t t0 easp(s−t)ds =x0e−a(t−t0)+�0 t0−t eaτp(τ)dτ.(3.26) Taking the limit of (3.26) at t→∞yields lim t→+∞ψ(t,θ−t(p),x0)=�0 −∞ easp(s)ds. 3.3 Applications 57 In addition, these component subsets have the form Aα=�α∗ l(t),α∗ u(t)�, where α∗ l(t) and α∗ u(t) are entire bounded solutions of the Riccati equation. All other bounded entire solutions of the Riccati equation (3.32) lie between α∗ l(t) and α∗ u(t). We can use differential inequalities to obtain bounds on these entire solutions. Indeed, denoting β∗=r2 r1+r2 ,and γ∗=r2 r1+r2+dM , it is not difficult to prove that A(t)=�α∗ l(t),α∗ u(t)�⊂�γ∗,β∗�. To investigate the case where the pullback attractor consists of a single entire solution, we need to find conditions under which α∗ l(t)≡α∗ u(t),t∈R. Suppose that they are not equal and consider their difference ∆α(t)=α∗ u(t)−α∗ l(t). Then d∆α(t) dt=D(t)�α∗ u(t)+α∗ l(t)�∆α(t)−(D(t)+r1+r2)∆α(t) ≤dM·2α∗ u(t)∆α(t)−(dm+r1+r2)∆α(t) ≤�2dMr2 r1+r2 −dm−r1−r2�∆α(t). Thus 0≤∆α(t)≤e�2dMr2 r1+r2−dm−r1−r2�(t−t0)∆α(t0)→0 as t→∞, (as well as when t0→ −∞) provided 2dMr2 r1+r2 −dm−r1−r2<0,(3.38) which is equivalent to 2dMr2<dm(r1+r2)+(r1+r2)2. To conclude, the pullback attractor for the Riccati ODE (3.37) consists of a singleton entire solution, α∗(t), provided (3.38) holds. Moreover, α∗(t) is also asymptotically stable in the forward sense. Remark 3.9. Since dm<dM, this holds, e.g., if dM(r2−r1)<(r1+r2)2. Inequality (3.38) essentially puts a restriction on the width of the interval in which D(t) can take its values, unless r1>r2. Now we can analyze the global dynamics of x(t) and z(t). 58 3 Nonautonomous dynamical systems Suppose that α∗(t) is the unique entire solution in the pullback attractor of the Riccati ODE (3.37). Then α∗(t)∈�γ∗,β∗�⊂(0,1) for all t∈R. Moreover, for t sufficiently large, x(t) and z(t) components of system (3.35)–(3.37) satisfy dx(t) dt=D(t)(I−x(t)) −ax(t) λ+x(t)z(t)+bνα∗(t)z(t),(3.39) dz(t) dt=−νz(t)−D(t)α∗(t)z(t)+cx(t) λ+x(t)z(t).(3.40) System (3.39)–(3.40) has a steady state equilibrium (I,0). Hence (I,0,α∗(t)) is a nonautonomous “equilibrium” solution of the system (3.35)–(3.37). Theorem 3.8. Assume that D :R→[dm,dM], with 0<dm<dM<∞, is continuous, a ≥c, b ∈(0,1) and ν>0. Then, system (3.39) -(3.40) possesses a pullback attractor A={A(t):t∈R}inside the nonnegative quadrant. Moreover, (i) When ν+dmγ∗>c, the axial steady state solution (I,0) is asymptotically stable in the nonnegative quadrant and the pullback attractor Ahas a singleton component subset A(t)= {(I,0)}for all t ∈R. (ii) When ν+dMβ∗<cdmI λ(a−c+ν+dM−bνβ∗)+dmI, the pullback attractor Aalso contains points strictly inside the positive quadrant in addition to the point {(I,0)}. Variable nutrition input rate Now we assume that the nutrition input value Ican vary continuously with time while the consumption rate Dis a constant. Similarly we assume that Iis bounded with positive values, in particular, I(t)∈[im,iM] for all t∈R, where 0 <im≤iM< ∞. When Dis a constant, Ivaries in time and there are no delays in time, the system (3.30) - (3.32) with Utaking the form (3.33) becomes dx(t) dt=D(I(t)−x(t))−ax(t) λ+x(t)(y1+y2)+bνy1(t),(3.41) dy1(t) dt=−(ν+D)y1(t)+cx(t) λ+x(t)y1(t)−r1y1(t)+r2y2(t),(3.42) dy2(t) dt=−νy2(t)+cx(t) λ+x(t)y2(t)+r1y1(t)−r2y2(t).(3.43) 3.3 Applications 59 Lemma 3.2. Suppose that �x0,y1,0,y2,0�∈R3 +. Then, all solutions to system (3.41)– (3.43) with initial value (x(t0),y1(t0),y2(t0)) =�x0,y1,0,y2,0�are (i) nonnegative for all t >t0; (ii) uniformly bounded in R3 +. Moreover, the nonautonomous dynamical system on R3 +generated by the system of ODES (3.41)–(3.43) has a pullback attractor A={A(t):t∈R}in R3 +. Proof. Similar to the proof of Lemma 3.1. � Using the new variables z(t) and α(t) defined as in (3.34), equations (3.41)–(3.43) become dx(t) dt=D(I(t)−x(t)) −ax(t) λ+x(t)z(t)+bνα(t)z(t),(3.44) dz(t) dt=−νz(t)−Dα(t)z(t)+cx(t) λ+x(t)z(t),(3.45) dα(t) dt=−Dα(t)(1 −α(t)) −r1α(t)+r2(1 −α(t)).(3.46) Equation (3.46) has a unique steady state solution α∗=D+r1+r2−�(D+r1+r2)2−4Dr2 2D which is asymptotically stable on (0,1). Hence, when t→∞, replacing α(t) by α∗ in equations (3.44) and (3.45), we have dx(t) dt=D(I(t)−x(t)) −ax(t) λ+x(t)z(t)+bνα∗z(t),(3.47) dz(t) dt=−νz(t)−Dα∗z(t)+cx(t) λ+x(t)z(t).(3.48) For more details of the long term dynamics of the solutions to (3.47) - (3.48) we establish the following theorem. Theorem 3.9. Assume that I :R→[im,iM], with 0<im<iM<∞, is continuous, a≥c, b ∈(0,1) and ν>0. Then system (3.47) -(3.48) has a pullback attractor A={A(t):t∈R}inside the nonnegative quadrant. Moreover, (i) when ν+Dα∗>c,the entire solution (w∗(t),0) is asymptotically stable in R2 + where w∗(t)=De−Dt �t −∞ I(s)eDsds, and the pullback attractor Ahas a singleton component subset A(t)={(w∗(t),0)} for all t ∈R, 60 3 Nonautonomous dynamical systems (ii) when ν+Dα∗<cDiM λ(a−c+ν−bνα∗+D)+DiM the pullback attractor Aalso contains points strictly inside the positive quadrant in addition to the set {(w∗(t),0)}. Over-yielding in nonautonomous chemostats An interesting feature related to nonautonomous chemostats and also other nonautonomous population models in general, is the so called over-yielding effect produced by temporal variation of one or more parameters in the model. For a given amount of nutrient that is fed in a chemostat during a given period of time T, one can compare the biomass production over the time period taking into account the way the amount of nutrient is distributed over the time period. It is said that there exists a biomass over-yielding when a time varying input produces more biomass than a constant input. To illustrate the effect of over-yielding in nonautonomous chemostats, we consider the chemostat model with wall growth and variable inputs (3.30)-(3.32), rewritten also with the new variables as (3.35)-(3.37). When D(t)=Dis constant and I(·) is a non-constant T-periodic function with 1 T�t+T t I(s)ds=¯ I, a periodic solution of system (3.47) - (3.48) has to fulfill the equations 0=D(¯ I−¯x)−a1 T�t+T t U(x(s))z(s)ds+bγα∗¯z,(3.49) 0=−(γ+Dα∗)¯z+c1 T�t+T t U(x(s))z(s)ds,(3.50) where ¯x,¯zdenote the average values of the variables x(·), z(·) over the period T. Combining equations (3.49) and (3.50), one obtains the relation D(¯ I−¯x)=�a(γ+Dα∗) c−bγα∗�¯z.(3.51) One can also write from equation (3.48) 0=1 T�t+T t z�(s) z(s)ds=−(γ+Dα∗)+c1 T�t+T t U(x(s))ds. As the function U(·) as defined in (3.33) is concave and increasing, one deduces the inequality ¯x>x∗, where x∗stands for the steady state of the variable x(·) with the constant input I(t)=¯ I. Similarly, x∗satisfies the equality cU(x∗)=γ+Dα∗. One 3.3 Applications 61 can then compare the corresponding biomass variables, by using equation (3.51), to obtain �bγα∗−a(γ+Dα∗) c�(¯z−z∗)>0. We conclude that an over-yielding occurs when the condition bcγα∗>a(γ+Dα∗) (3.52) is fulfilled. One can see that the nutrient recycling of the dead biomass (bγ�0) is essential to obtain an over-yielding. In the special case where the wall growth is neglected, our system (3.30) - (3.32) reduces to the system of two ODEs dx(t) dt =D(t)[I(t)−x(t)]−ax(t) λ+x(t)y(t),(3.53) dy(t) dt =−D(t)y(t)+ax(t) λ+x(t)y(t).(3.54) Assume now that I(·)=Iis constant and D(·) is a non-constant T-periodic function with 1 T�t+T t D(s)ds =¯ D. From equations (3.53)-(3.54) a periodic solution has to fulfill I=x(t)+y(t) (3.55) 0=1 T�t+T t y�(s) y(s)ds=−¯ D+a1 T�t+T t U(x(s))ds.(3.56) From equation (3.56), one obtains, as before, the inequality ¯x>x∗and thus ¯y<y ∗. Consequently over-yielding never occurs. Remark 3.10. For the chemostat model with wall growth and periodic D(·), it is still unknown whether an over-yielding is possible or not, although numerical simulations tend to show that it is not. For more general time varying inputs (i.e. not necessarily periodic), one can also study the influence of the variations of the inputs on the characteristics of the pullback attractor as can be observed in our previous theorems 3.8 and 3.9. 3.3.2 Nonautonomous SIR model Here we release the assumption that the total population is a constant, i.e., N(t)= S(t)+I(t)+R(t) can vary (either deterministically or randomly) with time. This can be achieved in various ways, but here we consider the simplest situation, where 62 3 Nonautonomous dynamical systems the reproduction spam is modeled by a time-dependent function, Λ(t). To this end, consider a temporal forcing term given by a continuous function Λ(t):R→Rwith positive bounded values Λ(t)∈[Λ1,Λ2],∀t∈R,0<Λ1≤Λ2. With this time-varying force, system (2.12) – (2.14) becomes dS dt=Λ(t)−β NSI−νS,(3.57) dI dt=β NSI−νI−γI,(3.58) dR dt=γI−νR.(3.59) For simplicity write u(t):=(S(t),I(t),R(t)) and u0=(S0,I0,R0). Lemma 3.3. Suppose that u0=(S0,I0,R0)∈R3 +. Then, the solution to system (3.57)–(3.59) with initial value u(t0)=(S(t0),I(t0),R(t0)) =u0is defined globally in time and is nonnegative for all t ≥t0. Proof. The proof follows a similar argument to the proof of Lemma 3.1. � Thanks to Lemma 3.3 we can define a process on R3 +by ϕ(t,t0,u0)=u(t;t0,u0),for u0∈R3 +and t≥t0,(3.60) where u(t;t0,u0) is the solution to (3.57)–(3.59) corresponding to the initial value u0. Next we prove that this process possesses a pullback attractor with respect to bounded subsets. Lemma 3.4. Assume 2ν>max{γ,1}and γ>β. Then, for any initial value u0∈R3 +, the corresponding solution to (3.57)–(3.59) satisfies |u(t;t0,u0)|2≤e−κ(t−t0)|u0|2+e−κt�t −∞ eκsΛ2(s)ds for all t ≥t0, provided κ=min{2ν−γ,2ν−1,2ν+γ−β}>0. Moreover, the family D0={D0(t):t∈R}given by D0(t)=B(0,ρκ(t)) ∩R3 +,where ρκ(t)is the nonnegative constant ρ2 κ(t)=1+νe−κt�t −∞ eκsΛ2(s)ds, is a pullback absorbing family for the process ϕ. 3.3 Applications 63 Proof. First, multiplying (3.57) by S, (3.58) by I, (3.59) by Rand summing them up we obtain d dt|u(t)|2=2Λ(t)S−2ν|u(t)|2+2γIR−2γI2−2β�S2I N−SI2 N�. Since 2Λ(t)S≤Λ2(t)+S2,2γIR ≤γI2+γR2, on account that 0 ≤S N,I N≤1,we deduce d dt|u(t)|2+κ|u(t)|2≤Λ2(t)≤Λ2 2. Multiplying by eκtand integrating between t0and t eκt|u(t)|2≤eκt0|u0|+�t t0 eκsΛ2(s)ds ≤eκt0|u0|+�t −∞ eκsΛ2(s)ds. Let D⊂R3 +be bounded. Then there exists d>0 such that |u0|≤dfor all u0∈D. By Lemma 3.4 we deduce for t0≤tand u0∈D, |ϕ(t,t0,u0)|2≤e−κteκt0|u0|2+e−κt�t −∞ eκsΛ2(s)ds ≤e−κteκt0d2+e−κt�t −∞ eκsΛ2(s)ds. Denoting T(t,D)=κ−1log(eκtd−2),then |ϕ(t,t0,u0)|2≤1+e−κt�t −∞ eκsΛ2(s)ds, for all t0≤T(t,D) and all u0∈D. Consequently, the family D0is pullback absorbing for the process ϕ.� These results enable us to conclude with the existence of the pullback attractor for our nonautonomous SIR model. Theorem 3.10. Under the assumptions in Lemma 3.4 the process ϕpreviously defined in R3 +possesses a pullback attractor A. 64 3 Nonautonomous dynamical systems 3.3.3 Nonautonomous Lorenz-84 model In the simple Lorenz-84 model (2.17) – (2.19), the forcing terms Fand Gare assumed to be constant. To make the model more realistic, we allow the forcing terms to vary over time, resulting in the following nonautonomous Lorenz model: dx dt=−ax−y2−z2+aF(t),(3.61) dy dt=−y+xy−bxz+G(t),(3.62) dz dt=−z+bxy+xz,(3.63) The nonautonomous terms F(·) and G(·) can be periodic, almost periodic or even belong to a type of functions more general than almost periodic, as in [46]. To illustrate the theory of attractors for the process formalism, we will consider very general nonautonomous terms. Indeed, we will assume that the mappings F,G:R�→ Rare continuously differentiable and satisfy �t −∞ elsF2(s)ds<+∞,�t −∞ elsG2(s)ds<+∞,∀t∈R, where l:=min{a,1}. First we need to construct the process generated by (3.61)-(3.63). It is easy to check that the right hand side of system (3.61)-(3.63) defines a vector field which is continuously differentiable. Thus, the standard theory of existence and uniqueness of local solutions for systems of ordinary differential equations guarantees the existence of local solutions for the initial value problems associated to (3.61)-(3.63). The next theorem will ensure that the local solutions can be extended to a global one defined for all t∈R. For simplicity, denote by u(t):=(x(t),y(t),z(t)) and u0=(x0,y0,z0) Lemma 3.5. Assume that a >0and b ∈R. Then for any initial condition u0∈R3 and any initial time t0∈R, the solution u(·):=u(·;t0,u0)of the IVP associated to (3.61)-(3.63) satisfies |u(t;t0,u0)|2≤e−l(t−t0)|u0|2+ae−lt �t −∞ elsF2(s)ds+e−lt �t −∞ elsG2(s)ds, (3.64) for all t ≥t0, where l :=min{a,1}. Proof. We immediately deduce that d dt|u(t)|2=−2(ax2+y2+z2)+2axF(t)+2yG(t), and, consequently, d dt|u(t)|2+l|u(t)|2≤aF2(t)+G2(t). (3.65) 3.3 Applications 65 Multiplying (3.65) by elt, we obtain that d dt�elt |u(t)|2�≤aeltF2(t)+eltG2(t). Integrating between t0and t elt |u(t)|2≤elt0|u0|2+a�t t0 elsF2(s)ds+�t t0 elsG2(s)ds ≤elt0|u0|2+a�t −∞ elsF2(s)ds+�t −∞ elsG2(s)ds, whence (3.64) follows. � Thanks to Lemma 3.5 we can define a process on R3by ϕ(t,t0,u0)=u(t;t0,u0) for u0∈R3and t≥t0.(3.66) Considering now the universe of fixed nonempty bounded subsets of R3, we will be able to prove that there exists a pullback absorbing family for the process ϕ(·,·,·). Theorem 3.11. Assume that a >0and b ∈R. Let F,G satisfy (4.69). Then, the family D0={D0(t):t∈R}defined by D0(t)=B(0,ρl(t)), where ρl(t)is given by ρ2 l(t)=1+ae−lt �t −∞ elsF2(s)ds+e−lt �t −∞ elsG2(s)ds, ∀t∈R, is a pullback absorbing family for the process ϕ(·,·,·). Furthermore, there exists the pullback attractor for the process ϕ. Proof. Let D⊂R3be bounded. Then, there exists d>0 such that |u0|≤dfor all u0∈D. Thanks to Lemma 4.2, we deduce that for every t0≤tand any u0∈D, |ϕ(t,t0,u0)|2≤e−ltelt0|u0|2+ae−lt �t −∞ elsF2(s)ds+e−lt �t −∞ elsG2(s)ds ≤e−ltelt0d2+ae−lt �t −∞ elsF2(s)ds+e−lt �t −∞ elsG2(s)ds. If we set T(t,D):=l−1log(eltd−2), we have |ϕ(t,t0,u0)|2≤1+ae−lt �t −∞ elsF2(s)ds+e−lt �t −∞ elsG2(s)ds, for all t0≤T(t,D) and for all u0∈D. Consequently the family D0={D0(t):t∈R}defined by D0(t)= B(0,ρl(t))is pullback absorbing for the process ϕ, and thanks to Theorem 3.1 the process defined in (3.66) possesses a pullback attractor. � 66 3 Nonautonomous dynamical systems Remark 3.11. A more detailed analysis concerning the structure of the pullback attractor for the nonautonomous Lorenz-84 model can be found in [1]. More precisely, one can see how the properties of the nonautonomous terms (periodicity, almost periodicity or a property called more general than almost periodicity by Kloeden and Rodrigues in [46]) in the model are inherited by the pullback attractor. Also an estimate of the Hausdorffdimension of the pullback attractor is constructed in [1], for which the skew product flow formalism has to be used. However, we do not include more details in this work due to the page limit. The reader is referred to the paper [1] for more details. 4.2 Formulation of Random Dynamical System and Random Attractor 73 Similar to the theory of nonautonomous pullback attractors, the existence of a random attractor relies on the existence of a pullback absorbing set, which can be defined following the same manner as the definition of pullback absorbing sets for a cocycle (see Definition 3.8). Definition 4.6. A random set K∈Bis said to be pullback absorbing for the universe Bif for every B∈Band ω∈Ω, there exists T(B,ω)>0 such that ψ(t,θ−t(ω),B(θ−t(ω))) ⊂K(ω),for all t≥T(B,ω). In a general phase space, an absorbing compact set is clearly an attracting compact set while the opposite is not necessarily true. However, for the special space X=Rd considered in all of the applications in this book, a compact attracting set is also a compact absorbing set. In fact, the closure of an ε-neighborhood of a compact attracting set, is absorbing and compact. Now we are ready to establish a necessary and sufficient condition ensuring the existence of random pullback attractor. Theorem 4.1. (Existence of random attractor: necessary &sufficient condition) Let (θ,ψ)be an RDS on X and let Bbe a family of random sets. Then, there exists a random pullback attractor Afor Bif and only if there exists a compact random set K which is attracting for B. Furthermore, there exists a unique minimal random pullback attractor ABfor B. The above theorem presents an elegant result, however from the application point of view, it is more effective to have a sufficient condition which ensures the existence of random attractors, stated as follows. Theorem 4.2. (Existence of random attractor: sufficient condition) Let (θ,ψ)be an RDS on X and let Bbe a family of random sets. Assume that there exists a compact random set K which is absorbing for B. Then, there exists a random pullback attractor A for B. Moreover, this is the unique minimal random pullback attractor AB for B. Analogous results ensuring the existence of a random forward attractor are still not available up to date. However, it is possible to develop a different theory in which a certain type of pullback attractor is also forward attractor, and vice versa. Thanks to the θt-invariance of the probability measure P, for any ε>0 P{ω∈Ω: dist((ψ(t,θ−t(ω),B(θ−t(ω))),A(ω)≥ε} =P{ω∈Ω: dist((ψ(t,ω,B(ω)),A(θtω)≥ε}. Since P–almost sure convergence implies convergence in probability, then a random pullback attractor also converges forward, but only in the weaker sense of convergence in probability. The same argument states that a random forward attractor is also a pullback attractor, but only in probability. This gives rise to a new type of random attractors which are also forward attractors and are termed as “attractors in 74 4 Random dynamical systems probability” or “weak attractors”. The reader is referred to [5] for more details on this topic. The random attractor, while exists, is closely related to the universe Bof random sets to be attracted. For this reason, one RDS can have different pullback random attractors, each of which associated to a different universe. It is then interesting to discover if there is any relationship among these attractors. One interesting result along this line can be found in [26], stated as follows. Theorem 4.3. Let (θ,ψ)be an RDS on X and let Bbe the family of all compact deterministic subsets of X (i.e., if D ∈Bthen D(ω)=D0, which is a compact set, for all ω∈Ω). Assume that there exists a compact deterministic set K which is attracting for B. Then, there exists a random pullback attractor Afor B, and this attractor is unique in the sense that if ˜ Ais a random pullback attractor for every compact determinist set, then A=˜ A,P-a.s. Furthermore, every random compact, invariant set is a subset of A,P-almost surely. As an immediate consequence we deduce that if Bis an arbitrary universe of random sets for which an RDS (θ,ψ) possesses a random attractor AB, then as this random set is invariant, we have that AB⊂A,P-a.s. Futhermore, if the universe Bcontains every compact deterministic set, then AB=A,P-a.s. An interesting point to be highlighted here is that the large class of random attractors which can exist associated to different universes can be reduced to just one unique attractor. In summary, if an RDS possesses an attractor for the universe of compact deterministic sets, then for any other universe Bof random sets there are only three possibilities: (a) There exists no random pullback attractor for the universe B. (b) There exists a random pullback attractor ABfor the universe Band is unique, then AB=A,P-a.s. (when Bcontains the compact deterministic sets) (c) There exists a random pullback attractor ABfor the universe Bbut is not unique, then AB⊂A,P-a.s. 4.2.1 Some properties of the random attractor In this section we will discuss some important properties of the random pullback attractor for an RDS (θ,ψ) on the phase space X. First of all, the random attractor associated to the universe of compact deterministic subsets of Xwill be called the global random attractor in what follows and this is the largest random pullback attractor that an RDS may have, at light of the arguments stated in the previous subsection. Recall first the definition of ω-limit set of a random set B, analogous to the ωlimit set defined in the nonautonomous framework, is given by Γ(B,ω)=� T≥0� t≥T ψ(t,θ−t(ω),B(θ−t(ω)).(4.3) 4.2 Formulation of Random Dynamical System and Random Attractor 75 The set defined by (4.3) is also characterized as the set of points x∈Xsuch that there exists sequences tn→+∞,bn∈B(θ−tn(ω)), n∈Nsuch that x=lim n→∞ ψ(tn,θ−tn(ω),bn). The ω−limit set Γ(B,ω) define in (4.3) is always forward invariant, but not strictly invariant in general. A particular case that Γ(B,ω) is also strict invariant happens when Bis attracted by some compact random set. We next elaborate some properties of the structure of the global random attractor. Unfortunately, it is impossible to make the following contents less abstract. But to provide necessary information to the readers who are keen to get a deeper understanding of the global random attractor, we still include the most typical results whose proofs can be found in Crauel [26, 27]. Note skipping the following results will not affect the understanding of the applications to be studied later. Theorem 4.4. Let (θ,ψ)be an RDS on the phase space X, and assume that there exists the global random attractor ω�→ A(ω). Then, (i) Ais measurable with respect to the past of ψ, i.e., for every x ∈X, the mapping ω�→ dist(x,A(ω)) is measurable with respect to the σ-algebra F−=σ{ψ(s,θ−t(ω),x):x∈X,0≤s≤t}⊂F. (ii) Γ(B,ω)⊂A(ω),P-a.s., for every random set B attracted by A, in particular, for every compact deterministic set B. (iii) A(ω)=∪Γ(K,ω), where the union is taken over all compact deterministic subsets K ⊂X. (v) the global random attractor is connected P-a.s. (vi) if θis ergodic (see e.g. Arnold [3]) then A(ω)=Γ(K,ω)for every compact deterministic K ⊂X satisfying that P{A(ω)⊂K}>0. In order to characterize a random attractor for a universe B, the following result was established in Crauel [27]. Theorem 4.5. Let (θ,ψ)be an RDS on the phase space X, and assume that there exists a random pullback attractor A(ω)for the universe B. Then (i) the omega limit set Γ(B,ω)of any B ∈Bis a subset of the attractor, i.e. Γ(B,ω)⊂ A(ω). (ii) the minimal attractor for Bis given by AB(ω)=� B∈B Γ(B,ω). Furthermore, when the RDS satisfies a strictly uniformly contracting property, i.e., there exists K >0such that �ψ(t,ω,x0)−ψ(t,ω,y0)�X≤e−Kt �x0−y0�X 76 4 Random dynamical systems for all t ≥0,ω∈Ωand x0,y0∈X, then the random attractor consists of singleton subsets A(ω)={A(ω)}. 4.2.2 Generation of random dynamical systems The theory of random dynamical systems allows us to analyze the asymptotic behavior of stochastic models based on every realization of the system rather than statistically in the mean or mean square sense. Hence the main concern for applying the theory of RDS to a stochastic model is to know when the stochastic model generates an RDS. To make the presentation less abstract, we will develop the theory in X=Rd, which is sufficient for the study of stochastic models formed by systems of differential equations, either random differential equations or stochastic differential equations. The underlying theory, indeed, could be carried out in a more general metric space if necessary. Two types of stochastic models will be discussed. In the first type of models, the noise appears as random parameters expressed as continuous, temporally varying, and bounded functions of the flow θtω. The resulting systems are random differential equations in the form of (4.2), that have been used in biological and ecological applications. In the second type of models, the noise is included as a random perturbation to the deterministic models, expressed by terms involving white noise. The resulting systems are then Itˆ o or Stratonovich stochastic differential equations which have been widely used in engineering applications. In this book we mainly focus on stochastic differential equations with special structured white noise, e.g., additive or multiplicative white noise, that can be transformed into random differential equations with random parameters in the form of (4.2). More precisely, we will study stochastic models that can be described directly or indirectly by the formulation dx(t) dt=f(θtω,x),(4.4) where θ={θt}t∈Ris a measurable and measure-preserving dynamical system acting on the probability space (Ω,F,P). The generation of an RDS by the solutions to an IVP associated to (4.4) can be carried out as follows. Denote by x(t;t0,ω,x0) the global solution to the IVP dx(t) dt=f(θtω,x),x(t0)=x0 for t≥t0,x0∈X, and define ψ(t,ω,x0):=x(t;0,ω,x0).(4.5) Then, it is straightforward to check that ψ(t−t0,θt0ω,x0)=x(t−t0;0,θt0ω,x0)=x(t;t0,ω,x0), 4.2 Formulation of Random Dynamical System and Random Attractor 77 and thus ψ(t,θt0ω,x0)=x(t+t0;t0,ω,x0). Observe that in the cocycle expression ψ(t,θt0ω,x0) the variable trepresents the elapsed time since the initial time t0. Assuming that the vector field fsatisfies suitable conditions to ensure the existence and uniqueness of a global solution, the definition for ψin (4.5) allows us to verify that all conditions in Definition 4.2 are fulfilled, and therefore an RDS has been successfully generated. The next goal is to show how a stochastic differential equation can be transformed into a random differential equation of the type (4.2). Moreover, we want to show that the RDS generated by the solution to the random differential equation after transformation is homeomorphic to the RDS generated by the solution to the stochastic equation before transformation. To this end, a brief review of the required knowledge on stochastic differential equations is provided below. 4.2.3 A brief introduction to stochastic differential equations Let (Wt)t∈Rbe a standard Brownian motion or one dimensional Wiener process defined on the (canonical, if necessary) probability space (Ω,F,P). A traditional way to model random perturbation to a deterministic system dx(t) dt=f(t,x),t,x∈R,(4.6) is to include a white noise term in the system and consider the stochastic system dx dt=f(t,x)+σ(t,x)dWt dt,t,x∈R.(4.7) Note that the derivative of the Brownian motion, often referred to as the white noise, is only a formal notation (as Wtis nowhere differentiable) although it is commonly used in engineering applications. Equation (4.7) can be presented more rigorously as dx(t)=f(t,x)dt+σ(t,x)dWt,t,x∈R. A more general inclusion of the noise, e.g., dx dt=f�t,x,dWt dt�,t,x∈R, can be approximated by a system in the form (4.7) by using the Taylor formula. The differential equation (4.7) is only meaningful while being written as the following integral equation x(t)=x(0) +�t 0 f(s,x(s))ds+�t 0 σ(s,x(s))dWs,t≥0.(4.8) 78 4 Random dynamical systems The first integral in (4.8) is a regular deterministic integral and exists, for instance, when fis continuous. But the second integral in (4.8), which is a stochastic integral with respect to the Wiener process, is completely different. Recall that for every fixed ω∈Ω, the path t∈R�→ Wt(ω) is continuous, but is not of bounded variation on any time interval [t1,t2]. Thus although every sample path of Wtis continuous, it cannot be used as an integrator function to define any Riemann-Stieltjes integral with respect to the Wiener process. In fact a stochastic integral has to be defined as a limit in L2(Ω,F,P;R). Detailed construction of stochastic integrals can be found in many references in probability, e.g., [2, 64]; in what follows we only discuss the difficulties in defining the stochastic integrals. These difficulties make the random dynamical system approach, while applicable, a powerful alternative to the probabilistic approaches which rely heavily on stochastic integrals. For simplicity of exposition, consider the integral �T 0WsdWsand suppose that it can be calculated by using Riemann-Stieltjes sums. If this integral made sense, and could be defined pathwise, then, at least formally its value should be �T 0 WsdWs=1 2�W2 T−W2 0�=1 2W2 T.(4.9) Choose a sequence of partitions (∆n) of [0,T], ∆n={0=tn 0<tn 1< ... < tn n=T}, such that δn:=max 0≤k≤n−1(tn k+1−tn k)n→∞ −→ 0. Let a∈[0,1], denote by τn k:=atn k+(1 −a)tn k−1, and consider the corresponding Riemann-Stieltjes sums Sn= n � k=1 Wτn k(Wtn k−Wtn k−1).(4.10) Then, by using the decomposition Sn=W2 T 2−1 2 n � k=1 (Wtn k−Wtn k−1)2+ n � k=1 (Wτn k−Wtn k−1)2 + n � k=1 (Wtn k−Wτn k)(Wτn k−Wtn k−1), it is not difficult to check that lim n→∞ Sn=W2 T 2−(1 −2a)T 2in L2(Ω,F,P;R), or equivalently 4.2 Formulation of Random Dynamical System and Random Attractor 79 lim n→∞ �Ω������� Sn−      W2 T 2−(1 −2a)T 2     ������� 2 dP(ω)=0. Therefore defining �T 0WsdWsas a limit in mean square is dependent on the choice of the intermediate point τn k, and, consequently, dependent on the value of a. For each a∈[0,1], denote by Ia(t) the stochastic integral induced by the “Riemann-Stieltjes sums” (4.10) corresponding to τn k:=atn k+(1 −a)tn k−1. Then we have Ia(t)=W2 t 2−(1 −2a)t 2, which achieves different values for different choice of a. This property also holds true for a general integrand f(t,Wt) instead of just Wt. When a=1/2, i.e., the middle point in each interval of the partition is chosen, the resulting integral is called a Stratonovich stochastic integral (Stratonovich integral in short), with the integrator usually denoted by ◦dWt. It is worth mentioning that the differentiation properties of Stratonovich integrals follow the classical rules in calculus. When a=0, i.e., the left point of each interval of partition is chosen, the resulting integral is called Itˆo stochastic integral (Itˆ o integral in short), with the integrator denoted simply by dWt. The classical rules of calculus do not hold any more for Itˆ o integrals; the differentiation of Itˆ o integrals requires Itˆ o’s Calculus. But on the other hand, Itˆ o integrals possess nice properties concerning measurability, such as the martingale property, and the expectation of any Itˆ o integral is always 0. Note that general Itˆ o and Stratonovich integrals over finite time can be converted between each other according to the formula �T t0 f(t,Xt)◦dWt=�T t0 f(t,Xt)dWt+1 2�T t0 ∂f ∂X(t,Xt)f(t,Xt)dt, which guarantees that both types of integrals provide solutions with equivalent behaviors within finite period of time. However, this is not necessarily true for long time behavior, i.e., when T→∞or t0→ −∞. For example, one SDE with Stratonovich integrals can be stable while an otherwise identical SDE with Itˆ o integrals is not, and vice versa. This will be discussed further in Section 4.4. The readers who are interested in a rigorous construction of the Itˆ o integral are referred to the books by Arnold [2] or Oksendal [64]. We next introduce the Itˆ o formula, the stochastic calculus counterpart of the chain rule in classical deterministic calculus, that allows the conversion between an Itˆ o SDE and a Stratonovich SDE. Let (Ω,F,P) be a complete probability space, and let {Wt}t∈Rbe a one dimensional two-sided Wiener process. Denote by {Ft}t∈Rthe family of σ–algebras given by the filtration Ft=σ{Ws:s≤t}. Denote by Lp Ft(0,T), p=1,2 the linear space of measurable processes φ: [0,T]×Ω→Rsuch that for each t∈[0,T] the mapping φ(t,·) is Ft-measurable (i.e., Ft-adapted), and �T 0|φ(s)|pds<∞P-a.s. Then we have the following result. 80 4 Random dynamical systems Theorem 4.6. (Itˆo’s formula) Let X0be a random variable which is F0-measurable, b∈L1 Ft(0,T)and σ∈L2 Ft(0,T). Denote by Xtthe stochastic process {Ft}-measurable for any t ∈[0,T]defined by Xt=X0+�t 0 b(s)ds+�t 0 σ(s)dWsfor all t ∈[0,T],(4.11) where the integral of b is in the pathwise sense. Let Y(t,x)∈C1,2([0,T]×R)be a real-valued function which is differentiable in t and twice differentiable in x. Then Y(t,Xt)=Y(0,X0)+�t 0 Ys(s,Xs)ds+�t 0 Yx(s,Xs)b(s)ds(4.12) +�t 0 Yx(s,Xs)σ(s)dWs+1 2�t 0 Yxx(s,Xs)σ2(s)ds for all t ∈[0,T]. Remark 4.3. Note that the expression for the Itˆ o formula (4.12) contains an additional term 1 2�t 0Yxx(s,Xs)σ2(s)ds comparing to the stochastic integral otherwise calculated by the deterministic chain rule of differentiation. This additional term, however, does not appear if the stochastic equation (4.11) is in the Stratonovich sense, i.e., Xt=X0+�t 0 b(s)ds+�t 0 σ(s)◦dWsfor all t∈[0,T]. We next recall the definitions of stochastic differential equations and their solutions. For T>0, assume that the mappings b: [0,T]×R→Rand σ: [0,T]×R→R be measurable, and the random variable ξis F0-measurable with values in R. A (Itˆ o) stochastic differential equations is properly interpreted as        dX(t)=b(t,X(t))dt+σ(t,X(t))dWt,in [0,T], X(0) =ξ.(4.13) Definition 4.7. A stochastic process {X(t)}t∈[0,T]defined on the probability space (Ω,F,P) is said to be a solution to (4.13) if it is a continuous and {Ft}-adapted process such that (i) {b(t,X(t))}t∈[0,T]∈L1 Ft(0,T) and {σ(t,X(t))}t∈[0,T]∈L2 Ft(0,T), (ii) it holds X(t)=ξ+�t 0 b(s,X(s))ds+�t 0 σ(s,X(s))dWsfor all t∈[0,T],P−a.s. A Stratonovich stochastic differential equations is properly interpreted as        dX(t)=b(t,X(t))dt+σ(t,X(t)) ◦dWt,in [0,T], X(0) =ξ,(4.14) 4.2 Formulation of Random Dynamical System and Random Attractor 81 with an equivalent integral formulation X(t)=ξ+�t 0 b(s,X(s))ds+�t 0 σ(s,X(s)) ◦dWsfor all t∈[0,T].P−a.s.(4.15) If σ(t,X) is differentiable with respect its second variable X, then the Stratonovich SDE (4.15) is equivalent to the following Itˆ o SDE (see, e.g., [78]) X(t)=ξ+�t 0 b(s,X(s))ds+1 2�t 0 σX(s,X(s))σ(s,X(s))ds+�t 0 σ(s,X(s))dWs. (4.16) The contents presented above cover SDEs in general forms and thus required some background in probability theory. The SDE applications considered in this book, however, are much simpler. As the emphasis of this chapter is the theory of random dynamical systems, we only focus on the SDEs whose solutions generate RDS. In particular, we will consider SDEs with additive white noise such as dx=f(x)dt+adWt,a∈R, or SDEs with linear multiplicative noise such as dx=f(x)dt+σxdWt,σ∈R.(4.17) By applying the formula (4.16) we can easily derive that the Itˆ o SDE (4.17) is equivalent to the following Stratonovich SDE dx=�f(x)−σ2 2x�dt+σx◦dWt.(4.18) Thus, when studying SDEs with linear multiplicative noise, either Itˆ o’s or Stratonovich’s formulation could be adopted. 4.2.4 Global asymptotic behavior of SDEs: conjugation of RDS In this section we discuss the global asymptotic behavior of random dynamical systems generated by stochastic differential equations. The local asymptotic behavior in the neighborhood of equilibria has been extensively studied in the literature (see e.g., [57, 60, 61]), but will not be commented here. We are interested in analyzing the global behavior of SDEs by the theory of random dynamical systems. The “prerequisite” for applying the theory of RDS to study SDEs is to show that solutions to an underlying SDE generate an RDS. The main technique is to transform the SDE into an equivalent random differential equation by a homeomorphism, which is referred to as the conjugation of random dynamical systems. The challenges of using this transformation technique lie in the fact that an explicit expres- 82 4 Random dynamical systems sion for such a homeomorphism is needed, but difficult or impossible to construct explicitly for specific SDEs system from real world applications. In other words, although we can prove that such a homeomorphism exists, an explicit expression which allows us to work directly with the solutions of both equations (random and stochastic) may not be available. For more details about the abstract aspects of this statement, the reader is referred to the paper by Imkeller and Schmalfuß [40]. Remark 4.4. The conjugation of RDS may work for more general SDEs than just the SDEs with additive or linear multiplicative noise. But for stochastic partial differential equations (SPDEs) the existence of conjugation is unknown for most cases except the special SPDEs with additive or linear multiplicative noise. We next introduce one special type of stochastic process, the Ornstein-Uhlenbeck process, that has been widely used in the literature to obtain conjugation of RDS. Consider the one-dimensional stochastic differential equation dz(t)=−λz(t)dt+dWt(4.19) for some λ>0. Equation (4.19) has a special stationary solution known as the stationary Ornstein-Uhlenbeck (OU) process. The OU process has several nice properties proved by Caraballo et al. in [20], which have been extensively used in latter studies of conjugation of RDS. The properties are stated in the following Proposition. Proposition 4.1. Let λbe a positive number. There exists a {θt}t∈R-invariant subset Ω∈Fof Ω=C0(R,R)(set of continuous functions which vanish at zero) of full measure such that for ω∈Ω, (i) lim t→±∞ |ω(t)| t=0; (ii) the random variable given by z∗(ω):=−λ�0 −∞ eλτω(τ)dτ is well defined; (iii) the mapping (t,ω)�→ z∗(θtω)=−λ�0 −∞ eλτθtω(τ)dτ =−λ�0 −∞ eλτω(t+τ)dτ+ω(t) (4.20) is a stationary solution of (4.19) with continuous trajectories satisfying 4.3 Applications 89 It can be proved that equation (4.37) has a pullback attractor Aγ={Aγ(ω)}ω∈Ω with its component subsets given by Aγ(ω)=� t≥0 γ(t;θ−tω,[0,1]). These component subsets have the form Aγ(θtω)=[γl(θtω),γu(θtω)], where γl(θtω) and γu(θtω) are entire bounded solutions of equation (4.37). All other bounded entire solutions of (4.37) lie between these two. We next estimate bounds of these entire solutions by using differential inequalities. On the one hand, since γ(t)≤1 and D(θtω)>0, we have γ�(t)=D(θtω)(γ2(t)−γ(t)) −(r1+r2)γ(t)+r2 ≤−(r1+r2)γ(t)+r2. On the other hand, γ�(t)=D(θtω)γ2(t)−(D(θtω)+r1+r2)γ(t)+r2 ≥−(d(1 +ε)+r1+r2)γ(t)+r2. Let α(t) and β(t) satisfy α�(t)=−(d(1 +ε)+r1+r2)α(t)+r2,α(0) =γ(0),(4.38) β�(t)=−(r1+r2)β(t)+r2,β(0) =γ(0).(4.39) Then α(t)≤γ(t)≤β(t) and [γl(θtω),γu(θtω)] ⊆[α∗,β∗], where α∗=r2 r1+r2+d(1 +ε),β∗=r2 r1+r2 (4.40) are asymptotically stable steady states for (4.38) and (4.39) respectively. In summary, Aγ(ω)=[γl(ω),γu(ω)] ⊆�α∗,β∗�. We provide the sufficient condition for Ato consist of only a single entire solution in the next theorem. Theorem 4.8. The pullback attractor Aassociated to the random dynamical system γ(t,ω,·)generated by (4.37) consists of a single entire solution, denoted by γ∗(θtω), provided 2r2d(1 +ε)<(r1+r2+d(1 −ε))(r1+r2). Note that this sufficient condition is equivalent to dε<(r1+r2)2+d(r1+r2) r1+3r2 , 90 4 Random dynamical systems which essentially represents the restriction on the magnitude of noise on D. The long term dynamics of x(t) and y(t) are given in the following theorem. Theorem 4.9. Given a ≥c, 0<b<1,ν>0, assume that D(θtω)and I(θtω)are continuous and essentially bounded, with d(1−ε)≤D(θtω)≤d(1+ε)and i(1−ε)≤ I(θtω)≤i(1 +ε). Then, system (4.35)-(4.36) has a pullback attractor A={A(ω): ω∈Ω}inside the nonnegative quadrant. Moreover, letting x∗(ω)=�0 −∞ D(θsω)I(θsω)e−�0 sD(θτω)dτds,(4.41) (i) the pullback attractor Ahas a singleton component subset A(ω)={(x∗(ω),0)} provided ν+d(1 −ε)α∗≥c, (ii)the pullback attractor Aalso contains points strictly inside the positive quadrant in addition to the singleton solution {(x∗(ω),0)}provided ac2di(1 −ε)2>�mc(aν+ad(1 +ε)−cbνβ∗)+acdi(1 −ε)2�·(ν+d(1 +ε)β∗). (4.42) 4.3.2 Random and stochastic SIR In this section, a random and a stochastic version of the SIR model analyzed in the previous chapters will be considered. In particular, we will first consider an SIR model described by a system of random differential equations whose coefficients contain real noise perturbation, i.e., bounded and continuous functions of θtω. Then we consider an SIR model described by a system of stochastic differential equations obtained from perturbing the deterministic SIR model by a white noise. Differences between the SDE model and the RDE model will be commented from modeling and analysis point of view. The techniques on construction of conjugated random dynamical systems introduced in Subsection 4.2.4 will be illustrated by the SDE SIR system. For a more detailed analysis the reader is referred to [12]. Random SIR with real noise The total population S+I+Rwas assumed to be a constant in the autonomous SIR model (2.12) – (2.14). This is due to the same constant birth rate and death rate for all the groups, which is not realistic. Here we will release this assumption and assume a total random reproduction spam, modeled by Λ(θtω) with Λ(θtω)∈Λ[1 −ε,1+ε],Λ>0,ε∈(0,1).(4.43) This assumption results in a system of random differential equations: 4.3 Applications 91 dS dt(t,ω)=Λ(θtω)−νS(t)−βS(t)I(t) N(t),(4.44) dI dt(t,ω)=−(ν+γ)I(t)+βS(t)I(t) N(t),(4.45) dR dt(t,ω)=γI(t)−νR(t),(4.46) where ν,β,γare positive constants which represent the death rate, the infection rate and the removal rate, respectively, and N(t)=S(t)+I(t)+R(t) is the total population. Remark 4.8. A more general model could be considered, in which more parameters are modeled to be random. But for simplicity of exposition, we only discuss the simplest RDE system (4.44) – (4.46). The analysis to be adopted, however, can be applied to a general RDE system with more random parameters. First of all we prove that solutions corresponding to nonnegative initial conditions remain nonnegative, stated as below. Lemma 4.1. The set R3 +={(S,I,R)∈R3:S≥0,I≥0,R≥0}is positively invariant for the system (4.44)–(4.46), for each fixed ω∈Ω. Proof. It is not difficult to check that the vector field, at the boundary of R3 +, points toward the inside of R3 +. In fact, on the plane S=0 we have S�>0, the plane I=0 is invariant since on it we have I�=0 while on R=0 we have R�≥0. The positive S-semi axis is invariant, and in addition we have: dS(t,ω) dt=Λ(θtω)−νS(t), which can be solved to obtain explicitly S(t,ω)=S0e−ν(t−t0)+e−νt�t t0 Λ(θsω)eνsds.(4.47) Notice that last term in (4.47) is bounded, Λ(1 −ε)νe−νt�t t0 eνsds≤e−νt�t t0 Λ(θsω)eνsds≤Λ(1 +ε)νe−νt�t t0 eνsds, i.e., Λ(1 −ε)(1 −e−ν(t−t0))≤e−νt�t t0 Λ(θsω)eνsds≤Λ(1 +ε)(1 −e−ν(t−t0)). Thus if the system starts on the positive R−semi axis, the solution will enter the plane I=0, while on the positive I−semi axis S�>0 and R�>0. � Observe that replacing ωby θ−tωin (4.47) and setting t0=0 gives 92 4 Random dynamical systems S(t;ω,S0)=S0e−νt+�0 −t Λ(θpω)eνpdp,(4.48) which, for t→∞, pullback converges to S∗(ω):=�0 −∞ Λ(θpω)eνpdp.(4.49) Note that S∗(ω) is positive and bounded. In fact, for any ω∈Ω, we have: Λ(1 −ε)≤S∗(ω)≤Λ(1 +ε).(4.50) The generation of an RDS by the solutions to (4.44) – (4.46) is stated as follows. Theorem 4.10. For any ω∈Ω, any t0∈Rand any initial data u0=(S(t0),I(t0),R(t0)) ∈ R3 +, system (4.44) –(4.46) admits a unique nonnegative and bounded solution u(·;t0,ω,u0)∈C([t0,+∞),R3 +)with u(t0;t0,ω,u0)=u0, provided that (4.43) is satisfied. Moreover the solution generates a random dynamical system ψ(t,ω,·)defined as ψ(t,ω,u0)=u(t;0,ω,u0),∀t≥0,u0∈R3 +,ω∈Ω. Proof. The system can be rewritten as du(t) dt=f(u,θtω), where f(u,θtω) is vector-valued function composed of the right hand sides of (4.44)- (4.46). Since Λ(θtω) is continuous with respect to t, the function f(·,θtω)∈C(R3 +× [t0,+∞),R3 +) and is continuously differentiable with respect to u. Then, by classical results of ODEs system (4.44)-(4.46) possesses a unique local solution. Summing the equations of the system we arrive at dN dt(t,ω)=Λ(θtω)−νN(t), whose solution satisfying N(t0)=N0is given by N(t;t0,ω,N0)=N0e−ν(t−t0)+e−νt�t t0 Λ(θsω)eνsds.(4.51) As in the proof of the previous lemma we have Λ(1−ε)+[N0−Λ(1−ε)]e−ν(t−t0)≤N(t;t0,ω,N0)≤Λ(1+ε)+[N0−Λ(1+ε)]e−ν(t−t0), (4.52) which implies that the solutions are bounded. Furthermore, the forward and backward limits of N(t;t0,ω,N0) satisfy, respectively, lim t→+∞N(t;t0,ω,N0)∈[Λ(1 −ε),Λ(1 +ε)],∀t0∈R, 4.3 Applications 93 and lim t0→−∞ N(t;t0,ω,N0)∈[Λ(1 −ε),Λ(1 +ε)],∀t∈R. Then the local solution can be extended to a global solution u(·;t0,ω,u0)∈C1([t0,∞),R3 +). It is clear that u(t+t0;t0,ω,u0)=u(t;0,θt0ω,u0), for all t0∈R,t≥0, ω∈Ω,u0∈R3 +. Then we can define a map ψ(t,ω,·) that is a random dynamical system by ψ(t,ω,u0)=u(t;0,ω,u0),∀t≥0,u0∈R3 +,ω∈Ω. � Theorem 4.11. For each ω∈Ωthere exists a tempered bounded closed random absorbing set K(ω)∈D(R3 +)of the random dynamical system {ψ(t,ω,·)}t≥0,ω∈Ωsuch that for any B ∈D(R3 +)and each ω∈Ωthere exists a TB(ω)>0such that ψ(t,θ−tω,B(θ−tω)) ⊂K(ω),∀t≥TB(ω). Moreover, for any 0<η<Λ(1 −ε), the set K(ω)can be chosen as the deterministic set Kη:={(S,I,R)∈R3 +:Λ(1 −ε)−η≤S+I+R≤Λ(1 +ε)+η}, for all ω∈Ω. Proof. Using (4.51) and (4.52) we deduce that N�(t,ω)≤0 on N=Λ(1 +ε)+η, and N�(t,ω)≥0 on N=Λ(1−ε)−ηfor all η∈[0,Λ(1−ε)). Then Kηis positively invariant for η∈[0,Λ(1 −ε)). Suppose that N0≥Λ(1 +ε)+η(the other case is similar), then N(t;t0,ω,N0)≤N0(ω)e−ν(t−t0)+Λ(1 +ε)[1 −e−ν(t−t0)]. Replacing ωby θ−tωgives N(t;θ−tω,N0(θ−tω)) ≤sup N0∈B(θ−tω) N0e−ν(t−t0)+Λ(1 +ε)[1 −e−ν(t−t0)].(4.53) Hence there exists a time TB(ω) such that for t>TB(ω), ψ(t,θ−tω,u0)∈Kηfor all u0∈B(θ−tω). That is, the set Kηis compact absorbing for all η∈(0,Λ(1 −ε)), and absorbs all tempered random sets of R3 +and in particular its bounded sets. � As a result of Theorem 4.10 and Theorem 4.53 we obtain Theorem 4.12. The random dynamical system generated by system (4.44)–(4.46) possesses a global random attractor. The reader is referred to Caraballo and Colucci [12] for additional analysis on the structure of the random attractor and individual dynamics of the three components. 94 4 Random dynamical systems Stochastic SIR with white noise Similar to the random system (4.44) – (4.46), we assume that the production spam parameter, Λ, is perturbed by noise. Instead of real noise, we now consider the effect of white noise. There are different ways to introduce the white noise into an ODE system, such as perturbing one or more parameters in an ODE system by an additive noise. Such types of perturbation, though widely used in the literature, are not always legitimate. For example assuming that the parameter Λis perturbed by some additive white noise σ˙ W(t), then the system (4.44) – (4.46) would become dS=�Λ−νS(t)−βS(t)I(t) N(t)�dt+σdWt,(4.54) dI=�−(ν+γ)I(t)+βS(t)I(t) N(t)�dt,(4.55) dR=�γI(t)−νR(t)�dt.(4.56) and it is straightforward check that equation (4.54) does not preserve the positiveness of the variable Sand, as a consequence, the population of Smay become negative and hence the model is not realistic. A proper choice of white noise perturbation should at least ensures the nonnegativeness of solutions while population models are considered. Though not fully justified, to illustrate the conjugation of RDS, we introduce multiplicative white noise of Stratonovich type for equations (4.54) – (4.56) to obtain dS=�Λ−νS(t)−βS(t)I(t) N(t)�dt+σS◦dW(t),(4.57) dI=�−(ν+γ)I(t)+βS(t)I(t) N(t)�dt+σI◦dW(t),(4.58) dR=�γI(t)−νR(t)�dt+σR◦dW(t),(4.59) For simplicity, we assume that the noise intensity is the same in all the equations but it does not make a substantial difference if we consider a different noise intensity in each equation. The same technique still apply, only with more complicated calculations. Following similar arguments used for the random SIR model, solutions corresponding to nonnegative initial data will remain nonnegative, ensured by the multiplicative type of noise structure. We will next transform the stochastic equations (4.57) – (4.59) into random differential equations random coefficients but without white noise. To this end, we consider the Ornstein-Uhlenbeck process z∗(θtω)=− 0 � −∞ esθtω(s)ds,t∈R,ω∈Ω0,(4.60) 4.3 Applications 95 which is the stationary solution to (4.19) for λ=1. According to the techniques introduced in the subsection 4.2.4, perform the following change of variables ˜ S(t)=S(t)e−σz∗(θtω),˜ I(t)=I(t)e−σz∗(θtω),˜ R(t)=R(t)e−σz∗(θtω). Then system (4.57) – (4.59) become d˜ S dt(t,ω)=Λe−σz∗−ν˜ S(t)−β˜ S(t)˜ I(t) ˜ N(t)+σ˜ Sz ∗,(4.61) d˜ I dt(t,ω)=−(ν+γ)˜ I(t)+β˜ S(t)˜ I(t) ˜ N(t)+σ˜ Iz∗,(4.62) d˜ R dt(t,ω)=γ˜ I(t)−ν˜ R(t)+σ˜ Rz∗.(4.63) Note that adding equations (4.61) – (4.63) results in d˜ N dt(t,ω)=Λe−σz∗−ν˜ N(t)+σ˜ Nz∗=Λe−σz∗−(ν−σz∗)˜ N.(4.64) The equation (4.64) has a nontrivial random solution that is both forward and pullback attracting. In fact, for any initial datum N0we have: N(t;ω,N0)=N0e−�t 0[ν−σz∗(θsω)]ds+�t 0 Λe−σz∗(θsω)e�t s[ν−σz∗(θτω)]dτds, and replacing ωby θ−tωwe obtain N(t;θ−tω,N0)=N0e−�0 −t[ν−σz∗(θpω)]dp+�0 −t Λe−σz∗(θpω)e−�0 p[ν−σz∗(θqω)]dqdp, which pullback converges to N∗(ω)=Λ�0 −∞ e−σz∗(θpω)e−�0 p[ν−σz∗(θqω)]dqdp(4.65) as t→+∞. Observe that N∗(ω) is well defined thanks to Proposition 4.1 because the integrand behaves like eνpfor p→ −∞ and z∗(θpω) p,1 p�0 p z∗(θqω)dq→0,for p→ −∞. Some comments are made below on differences between the random SIR model and the stochastic SIR model. 96 4 Random dynamical systems •The stochastic differential system with white noise (4.57) – (4.59) can be transformed to an equivalent random differential system with unbounded random coefficients, while the random system with real noise (4.54) – (4.56) has bounded random coefficients. •The differential equation describing the behavior of the total population N(t)= S(t)+I(t)+R(t) is similar in both cases and provides us with a stationary process that pullback and forward attracting any other solution. However, an important difference is that in the random system (4.54) – (4.56), this stationary process is bounded, what allows for a posterior control of this process to validate the underlying model. While in the stochastic system (4.57) – (4.59) we are not able to obtain such posterior estimates and the corresponding additional information. •The system with real noise, (4.54) – (4.56), preserves the form and structure of the underlying model. In other words, if we replace the random parameter Λ(θtω) by another random parameter with a special structure, the model is not a priori affected. While in the stochastic system (4.54) – (4.56) if there is a change in the noise structure, the underlying model may need to be redeveloped, even just to preserve the positiveness of solutions. 4.3.3 Stochastic Lorenz models In the simplest autonomous Lorenz-84 model (2.17) – (2.19), the forcing terms Fand Gare assumed to be constant. To make the model more realistic, we allowed the forcing terms to vary with respect to time, resulting in a nonautonomous Lorenz model studied in Chapter 3. Now we will study the model in which the forcing terms may vary randomly in time, described by the following system of RDEs with real noise: dx dt=−ax−y2−z2+aF(θtω),(4.66) dy dt=−y+xy−bxz+G(θtω),(4.67) dz dt=−z+bxy+xz.(4.68) In the system (4.66) – (4.68), the time dependence of the forcing terms is defined by a metric dynamical system (θt)t∈Ron the probability space (Ω,F,P). By analysis analog to those performed in the nonautonomous Lorenz model (2.17) – (2.19), the existence of a random attractor for the RDS generated by the system (4.66) – (4.68) can also be obtained. To this end we assume that the mappings F,G:Ω�→ Rare measurable and for every fixed ω∈Ω, the mappings t∈R�→ F(θtω) and t∈R�→ G(θtω) are continuous, and satisfy �t −∞ elsF2(θsω)ds<+∞,�t −∞ elsG2(θsω)ds<+∞∀t∈R,ω∈Ω(4.69) 4.3 Applications 97 where l:=min{a,1}. The first step is to construct the cocycle generated by (4.66)-(4.68). It is straightforward to check that the right hand sides of equations (4.66)-(4.68) define a vector field which is continuous with respect to tand locally Lipschitz with respect to the other variables. Thus, the existence and uniqueness of maximal solutions to the initial value problems associated to (4.66)-(4.68) is guaranteed by the results in Chapter 1. The next theorem ensures that the maximal solutions are also defined globally in time. For simplicity, denote by u(t):=(x(t),y(t),z(t)) and u0:=(x0,y0,z0). Lemma 4.2. Assume that a >0and b ∈R. Then for any initial condition u0∈R3 and any initial time t0∈R, the solution u(·):=u(·,t0,ω,u0)of the IVP associated to (4.66)-(4.68) satisfies |u(t,t0,ω,u0)|2≤e−l(t−t0)|u0|2+ae−lt �t −∞ elsF2(θsω)ds+e−lt �t −∞ elsG2(θsω)ds, (4.70) for all t ≥t0, where l :=min{a,1}. Proof. We immediately deduce that d dt|u(t)|2=−2(ax2+y2+z2)+2axF(θtω)+2yG(θtω), and, consequently, d dt|u(t)|2+l|u(t)|2≤aF2(θtω)+G2(θtω). (4.71) Multiplying (4.71) by elt, we obtain that d dt�elt |u(t)|2�≤aeltF2(θtω)+eltG2(θtω). Integrating between t0and t elt |u(t)|2≤elt0|u0|2+a�t t0 elsF2(θsω)ds+�t t0 elsG2(θsω)ds (4.72) ≤elt0|u0|2+a�t −∞ elsF2(θsω)ds+�t −∞ elsG2(θsω)ds, whence (4.70) follows. � Thanks to Lemma 4.2 we can define the cocycle on R3by ψ(t,ω,u0)=u(t;0,ω,u0) for u0∈R3,t≥0. Considering now the universe of tempered random subsets of R3, we will be able to prove that there exists a pullback absorbing family for the cocycle ψ(·,·,·). 98 4 Random dynamical systems Theorem 4.13. Assume a >0and b ∈R. Let F,G satisfy (4.69). Then, the family D0={D0(ω):ω∈Ω}defined by D0(ω)=B(0,ρl(ω)), where ρl(ω)is given by ρ2 l(ω)=1+a�0 −∞ elsF2(θsω)ds+�0 −∞ elsG2(θsω)ds, ∀t∈R,(4.73) is a pullback absorbing family for the cocycle ψ(·,·,·). Furthermore, there exists a pullback random attractor for ψ. Proof. Let D={D(ω):ω∈Ω}be a tempered random set. Then, lim t→∞ e−βtsup u0∈D(θ−tω) |u0|=0,for all β>0; Then, pick u0∈D(ω). Thanks to Lemma 4.2, we can deduce that |ψ(t,ω,u0)|2≤e−lt |u0(ω)|2+ae−lt �t −∞ elsF2(θsω)ds+e−lt �t −∞ elsG2(θsω)ds ≤e−lt|u0(ω)|2+ae−lt �t −∞ elsF2(θsω)ds+e−lt �t −∞ elsG2(θsω)ds. Replacing ωby θ−tω, and performing a suitable change of variable in the integrals results in |ψ(t,θ−tω,u0(θ−tω))|2≤e−lt|u0(θ−tω)|2+ae−lt �t −∞ elsF2(θs−tω)ds +e−lt �t −∞ elsG2(θs−tω)ds ≤e−lt|u0(θ−tω)|2+a�0 −∞ elsF2(θsω)ds+�0 −∞ elsG2(θsω)ds Taking limit when tgoes to +∞and on account of the temperedness of the family D, we easily obtain the pullback absorption of the family D0and therefore the existence of the pullback random attractor is proved. � Remark 4.9. Although we have considered a random version for the Lorenz-84 model so that we have coherent autonomous, nonautonomous and random variations of the same system, we would like to mention that in the paper [73], Schmalfuß considered a different stochastic Lorenz model. More precisely, a stochastic version with multiplicative linear Stratonovich noise is analyzed. The technique made use of the transformation from a stochastic differential equation into a random differential system by mean of the Ornstein-Uhlenbeck process as we have done in the previous stochastic SIR model. Moreover, the study of the finite dimensionality of the random attractor was carried out in the paper [73], which went beyond the objectives of the present work. References 105 [18] Tom´ as Caraballo, Xiaoying Han, Peter E. Kloeden, and Alain Rapaport. Dynamics of nonautonomous chemostat models. In Continuous and distributed systems. II, volume 30 of Stud. Syst. Decis. Control, pages 103–120. Springer, Cham, 2015. [19] Tom´ as Caraballo, Juan C. Jara, Jos´ e A. Langa, and Zhenxin Liu. Morse decomposition of attractors for non-autonomous dynamical systems. Adv. Nonlinear Stud., 13(2):309–329, 2013. [20] Tom´ as Caraballo, Peter E. Kloeden, and Bj¨ orn Schmalfuß. Exponentially stable stationary solutions for stochastic evolution equations and their perturbation. Appl. Math. Optim., 50(3):183–207, 2004. [21] C. Castaing and M. Valadier. Convex analysis and measurable multifunctions. Lecture Notes in Mathematics, Vol. 580. Springer-Verlag, Berlin-New York, 1977. [22] D. N. Cheban. Global attractors of infinite-dimensional nonautonomous dynamical systems. I. Izv. Akad. Nauk Respub. Moldova Mat., (3):42–55, 106, 109, 1997. [23] D. N. Cheban and D. S. Fakeeh. Global attractors of the dynamical systems without uniqueness. Sigma, Kishinev, 1994. [24] D. N. Cheban, P. E. Kloeden, and B. Schmalfuß. The relationship between pullback, forward and global attractors of nonautonomous dynamical systems. Nonlinear Dyn. Syst. Theory, 2(2):125–144, 2002. [25] Earl A. Coddington and Norman Levinson. Theory of ordinary differential equations. McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. [26] Hans Crauel. Global random attractors are uniquely determined by attracting deterministic compact sets. Ann. Mat. Pura Appl. (4), 176:57–72, 1999. [27] Hans Crauel. Random point attractors versus random set attractors. J. London Math. Soc. (2), 63(2):413–427, 2001. [28] Hans Crauel, Arnaud Debussche, and Franco Flandoli. Random attractors. J. Dynam. Differential Equations, 9(2):307–341, 1997. [29] Hans Crauel and Franco Flandoli. Attractors for random dynamical systems. Probab. Theory Related Fields, 100(3):365–393, 1994. [30] Hans Crauel and Franco Flandoli. Additive noise destroys a pitchfork bifurcation. J. Dynam. Differential Equations, 10(2):259–274, 1998. [31] Hans Crauel and Peter E. Kloeden. Nonautonomous and random attractors. Jahresber. Dtsch. Math.-Ver., 117(3):173–206, 2015. [32] A. Cunningham and R. M. Nisbet. Transients and oscillations in continuous culture. In Mathematics in microbiology, pages 77–103. Academic Press, London, 1983. [33] G. D’Ans, P. V. Kokotovic, and D. Gottlieb. A nonlinear regulator problem for a model of biological waster treatment. IEEE Transactions on Automatic Control, AC-16:341–347, 1971. [34] A. G. Fredrickson and Gregory Stephanopoulos. Microbial competition. Science, 213(4511):972–979, 1981. 106 4 Random dynamical systems [35] Joana G. Freire, Cristian Bonatto, Carlos C. DaCamara, and Jason A. C. Gallas. Multistability, phase diagrams, and intransitivity in the Lorenz-84 loworder atmospheric circulation model. Chaos, 18(3):033121, 8, 2008. [36] R. Freter. Mechanisms that control the microflora in the large intestine. Human Intestinal Microflora in Health and Disease (ed. Hentges, D. J. Academic Press), New York, pages 33–54, 1983. [37] R. Freter. An understanding of clonization of the large intestine requires mathematical analysis. Microecology and Therapy, 16:147–155, 1986. [38] Jack K. Hale. Stability and gradient dynamical systems. Rev. Mat. Complut., 17(1):7–57, 2004. [39] Philip Hartman. Ordinary differential equations, volume 38 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. Corrected reprint of the second (1982) edition [Birkh¨ auser, Boston, MA; MR0658490 (83e:34002)], With a foreword by Peter Bates. [40] Peter Imkeller and Bj¨ orn Schmalfuss. The conjugacy of stochastic and random differential equations and the existence of global attractors. J. Dynam. Differential Equations, 13(2):215–249, 2001. [41] H. W. Jannash and R. T. Mateles. Experimental bacterial ecology studies in continuous culture. Advances in Microbial Physiology, 11:165–212, 1974. [42] W. O. Kermack and McKendrick A. G. A contribution to the mathematical theory of epidemics. Proc. Royal Soc. London, 115:700–721, 1927. [43] W. O. Kermack and McKendrick A. G. A contribution to the mathematical theory of epidemics. Proc. Royal Soc. London, 138:55–83, 1932. [44] W. O. Kermack and McKendrick A. G. A contribution to the mathematical theory of epidemics. Proc. Royal Soc. London, 141:94–112, 1933. [45] P. E. Kloeden. Lyapunov functions for cocycle attractors in nonautonomous difference equations. Bul. Acad. S¸tiint¸e Repub. Mold. Mat., (1):32–42, 150, 153, 1998. [46] P. E. Kloeden and H. M. Rodrigues. Dynamics of a class of ODEs more general than almost periodic. Nonlinear Anal., 74(7):2695–2719, 2011. [47] Peter Kloeden and Thomas Lorenz. Pullback and forward attractors of nonautonomous difference equations. In Difference equations, discrete dynamical systems and applications, volume 150 of Springer Proc. Math. Stat., pages 37–48. Springer, Cham, 2015. [48] Peter E. Kloeden and Thomas Lorenz. Mean-square random dynamical systems. J. Differential Equations, 253(5):1422–1438, 2012. [49] Peter E. Kloeden and Thomas Lorenz. Pullback incremental attraction. Nonauton. Dyn. Syst., 1:53–60, 2014. [50] Peter E. Kloeden and Thomas Lorenz. Construction of nonautonomous forward attractors. Proc. Amer. Math. Soc., 144(1):259–268, 2016. [51] Peter E. Kloeden and Martin Rasmussen. Nonautonomous dynamical systems, volume 176 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2011. References 107 [52] J. W. M. La Riviere. Microbial ecology of liquid waste. Advances in Microbial Ecology, 1:215–259, 1977. [53] Jos´ e A. Langa, James C. Robinson, Antonio Su´ arez, and Alejandro Vidal- L´ opez. The stability of attractors for non-autonomous perturbations of gradient-like systems. J. Differential Equations, 234(2):607–625, 2007. [54] J. P. LaSalle. The stability of dynamical systems. Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1976. With an appendix: “Limiting equations and stability of nonautonomous ordinary differential equations” by Z. Artstein, Regional Conference Series in Applied Mathematics. [55] G. A. Leonov and N. V. Kuznetsov. Time-varying linearization and the Perron effects. Internat. J. Bifur. Chaos Appl. Sci. Engrg., 17(4):1079–1107, 2007. [56] Bruce R. Levin and Frank M. Stewart. The population biology of bacterial plasmids: a priori conditions for the existence of mobilizable nonconjugative factors. Genetics, 94(2):425–443, 1980. [57] Kai Liu. Stability of infinite dimensional stochastic differential equations with applications, volume 135 of Chapman &Hall/CRC Monographs and Surveys in Pure and Applied Mathematics. Chapman & Hall/CRC, Boca Raton, FL, 2006. [58] Edward N. Lorenz. Irregularity: a fundamental property of the atmosphere. Tellus, 36A:98–110, 1984. [59] Edward N. Lorenz. Can chaos and intransitivity lead to interannual variability? Tellus, pages 378–389, 1990. [60] X. Mao. Stability of stochastic differential equations with respect to semimartingales, volume 251 of Pitman Research Notes in Mathematics Series. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1991. [61] Xuerong Mao. Exponential stability of stochastic differential equations, volume 182 of Monographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker, Inc., New York, 1994. [62] C. Masoller, A. C. Sicardi Schifino, and Lilia Romanelli. Regular and chaotic behavior in the new Lorenz system. Phys. Lett. A, 167(2):185–190, 1992. [63] Richard K. Miller and Anthony N. Michel. Ordinary differential equations. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York- London, 1982. [64] Bernt Oksendal. Stochastic differential equations. Universitext. Springer- Verlag, Berlin, fifth edition, 1998. An introduction with applications. [65] Y. Orlov. Extended invariance principle for nonautonomous switched systems. IEEE Trans. Automat. Control, 48(8):1448–1452, 2003. [66] Lawrence Perko. Differential equations and dynamical systems, volume 7 of Texts in Applied Mathematics. Springer-Verlag, New York, third edition, 2001. [67] Roger A. Pielke and Xubin Zeng. Long=term variability of climate. J. Atmos. Sci., 51:155–159, 1994. [68] James C. Robinson. Infinite-dimensional dynamical systems. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2001. An introduction to dissipative parabolic PDEs and the theory of global attractors. 108 4 Random dynamical systems [69] James C. Robinson. An introduction to ordinary differential equations. Cambridge University Press, Cambridge, 2004. [70] Paul J. Roebber. Climate variability in a low-order coupled atmosphere ocean mode. Tellus, 47A:473–494, 1995. [71] Shankar Sastry. Nonlinear systems, volume 10 of Interdisciplinary Applied Mathematics. Springer-Verlag, New York, 1999. Analysis, stability, and control. [72] Bj¨ orn Schmalfuss. Backward cocycles and attractors of stochastic differential equations. In International Seminar on Applied Mathematics, Nonlinear Dynamics: Attractor Approximation and Global Behaviour, pages 185–192. 1992. [73] Bj¨ orn Schmalfuß. The random attractor of the stochastic Lorenz system. Z. Angew. Math. Phys., 48(6):951–975, 1997. [74] Hal L. Smith. Monotone dynamical systems, volume 41 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1995. An introduction to the theory of competitive and cooperative systems. [75] Hal L. Smith and Paul Waltman. The theory of the chemostat, volume 13 of Cambridge Studies in Mathematical Biology. Cambridge University Press, Cambridge, 1995. Dynamics of microbial competition. [76] Vadrevu Sree Hari Rao and Ponnada Raja Sekhara Rao. Dynamic models and control of biological systems. Springer, New York, 2009. [77] G. Stephanopoulos and G. Lapidus. Chemostat dynamics of plasmid-bearing plasmid-free mixed recombinant cultues. Chemical Engineering Science, 43:49–57, 1988. [78] R. L. Stratonovich. A new representation for stochastic integrals and equations. SIAM J. Control, 4:362–371, 1966. [79] P. A. Taylor and J. L. Williams. Theoretical studies on the coexistence of competing species under continuous flow conditions. Canadian Journal of Microbiology, 21:90–98, 1975. [80] Tam´ as T´ el and M´ arton Gruiz. Chaotic dynamics: an introduction based on classical mechanics. Cambridge, 2006. [81] Gerald Teschl. Ordinary differential equations and dynamical systems, volume 140 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2012. [82] H. Veldcamp. Ecological studies with the chemostat. Advances in Microbial Ecology, 1:59–95, 1977. [83] Paul Waltman. Competition models in population biology, volume 45 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1983. [84] Paul Waltman, Stephen P. Hubbell, and Sze Bi Hsu. Theoretical and experimental investigations of microbial competition in continuous culture. In Modeling and differential equations in biology (Conf., Southern Illinois Univ., Carbondale, Ill., 1978), volume 58 of Lecture Notes in Pure and Appl. Math., pages 107–152. Dekker, New York, 1980.