Impulsive surfaces on dynamical systems

Impulsive surfaces on dynamical systems E. M. Bonottoa,1, M. C. Bortolanb, T. Caraballo∗,c,2, R. Collegarid,3 aInstituto de Ciˆencias Matem´aticas e de Computa¸c˜ao, Universidade de S˜ao Paulo - Campus de S˜ao Carlos, 13560-970, S˜ao Carlos SP, Brazil. bDepartamento de Matem´atica, Universidade Federal de Santa Catarina, Campus Trindade, 88040-900, Florian´opolis, Brazil. cDpto. Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Apdo. de Correos 1160, 41080 - Sevilla, Spain. dInstituto de Ciˆencias Matem´aticas e de Computa¸c˜ao, Universidade de S˜ao Paulo - Campus de S˜ao Carlos, 13560-970, S˜ao Carlos SP, Brazil. Abstract This work is devoted to the construction of impulsive sets in Rn. In the literature, there are many examples of impulsive dynamical systems whose impulsive sets are chosen in an abstract way, and in this paper we present sufficient conditions to characterize impulsive sets in Rnwhich satisfy some “tube conditions” and ensure a good behavior of the flow. Moreover, we present some examples to illustrate the theoretical results. Key words: Impulsive sets; hypersurfaces; dynamical systems. 2010 MSC: Primary 53A05; Secondary 34A37. 1. Introduction Oftentimes it is necessary to cause an abrupt change in the evolution of a real model in order to avoid an inconsistent result. One of the theories used to study such systems is the theory of impulsive differential equations - the reader may consult [13] for more details and [5], [6], [9], [14], [15] for some applications. ∗Corresponding author Email addresses: [email protected] (E. M. Bonotto), [email protected] (M. C. Bortolan), [email protected] (T. Caraballo), [email protected] (R. Collegari) 1Partially supported by FAPESP grant 2014/25970-5 and CNPq grant 307317/2013-7. 2Partially supported by FEDER and Ministerio de Econom´ıa y Competitividad (Spain) under grant MTM2015-63723-P, and Consejer´ıa de Innovaci´on, Ciencia y Empresa (Junta de Andaluc´ıa) under Proyecto de Excelencia P12-FQM-1492. 3Partially supported by FAPESP grants 2013/23933-2 and 2014/20691-0. Preprint submitted to Acta Mathematica Hungarica April 25, 2016 Recently there has been an interest in the theory of impulsive dynamical systems (IDS, for short) since it possesses a richer dynamics than the continuous case. In this new framework, the impulsive set is in the phase space and the impulses vary on time, and are not pre-assigned. The reader is referred to [2], [3], [1], [4], [7], [8], [10], [12] for more details. This impulsive set plays an important role to control the evolution of the system. For instance, in the study of an impulsive predator-prey model presented in [15], when the density of the prey hits a level which is the threshold with slight damage (this level is the impulsive set), then there are released its natural predators in order to control the environment (the addition of predators is represented by an impulse function). One of the first and crucial conditions we must have is that the impulsive set needs to be, in some sense, transversal to the flow, and moreover, in order to obtain important topological properties for impulsive systems, it also needs to satisfy what we call “tube conditions”, which is presented in Definition 4. Due to the difficulties to verify the tube condition for impulsive sets in general spaces, some examples in the literature consider abstract impulsive sets satisfying such condition. In this work, we present sufficient conditions for a class of impulsive sets to satisfy the tube condition in n-dimensional spaces, which is very important since we have plenty of applications in finite dimensional spaces. Some examples are included at the end of the paper. 2. Preliminaries We begin our study by briefly recalling the theory of impulsive dynamical systems. Let (X, d) be a metric space with metric d,R+be the set of nonnegative real numbers and N={1,2,3,...}be the set of natural numbers. A semigroup in Xis a family of functions {π(t): t⩾0}from Xto X, indexed on R+, satisfying the following conditions: i)π(0)x=xfor all x∈X; ii)π(t+s) = π(t)π(s) for all t, s ∈R+; 2 iii) the map R+×X∋(t, x)7→ π(t)xis continuous. Let {π(t): t⩾0}be a semigroup in X. For each D⊆Xand J⊆R+we define F(D, J) = [ t∈J π(t)−1(D).A point x∈Xis called an initial point if F(x, t) = ∅for all t > 0. Definition 1. An impulsive dynamical system (IDS, for short) (X, π, M, I)consists of a semigroup {π(t): t⩾0}on a metric space (X, d), a nonempty closed subset M⊆Xsuch that for every x∈Mthere exists ǫx>0 such that F(x, (0, ǫx)) ∩M=∅and [ t∈(0,ǫx) {π(t)x} ∩ M=∅,(1) and a continuous function I:M→X(its role will be specified later). The set M is called the impulsive set and the function Iis called impulsive function. Remark 2. Condition (1) means that the flow of the semigroup {π(t): t⩾0} is, in some sense, transversal to Mat any point of M. Thanks to condition (1) we may define the function φ:X→(0,+∞] by φ(x) =      s, if π(s)x∈Mand π(t)x /∈Mfor 0 < t < s, +∞,if π(t)x /∈Mfor all t > 0. (2) If π(t)x∈Mfor some t > 0, then the value φ(x) represents the smallest positive time such that the trajectory of xmeets M. In this case, we say that the point π(φ(x))xis the impulsive point of x. The impulsive positive trajectory of x∈Xby the IDS (X, π, M, I) is a map ˜π(·)xdefined in an interval Jx⊆R+, 0 ∈Jx, with values in Xgiven inductively by the following rule: if φ(x) = +∞then ˜π(t)x=π(t)xfor all t∈R+. However, if φ(x)<+∞then we denote x=x+ 0and define ˜π(·)xon [0, φ(x+ 0)] by ˜π(t)x=     π(t)x+ 0,if 0 ⩽t < φ(x+ 0), I(π(φ(x+ 0))x+ 0),if t=φ(x+ 0). 3 Now let s0=φ(x+ 0), x1=π(s0)x+ 0and x+ 1=I(π(s0)x+ 0). In this case s0<+∞and the process can go on, but now starting at x+ 1. This process ends after a finite number of steps if φ(x+ n) = +∞for some n∈N∪ {0}, or it may proceed indefinitely, if φ(x+ n)<+∞for all n∈Nand, in this case, ˜π(·)xis defined in the interval [0, T (x)), where T(x) = ∞ X i=0 si. The reader may see [3], [1], [7], [8], [12] for more details. Next, we present the concepts of tube and STC which allow us to obtain important topological properties for impulsive systems. Definition 3. Let {π(t): t⩾0}be a semigroup on X. A closed set Scontaining x∈Xis called a section through xif there exists λ > 0and a closed subset Lof Xsuch that: i)F(L, λ) = S; ii)F(L, [0,2λ]) contains a neighborhood of x; iii)F(L, ν)∩F(L, ζ) = ∅, if 0⩽ν < ζ ⩽2λ. The set F(L, [0,2λ]) is called a λ−tube (or simply tube)and the set Lis called abar. Definition 4. Let (X, π, M, I)be an IDS. We say that a point x∈Msatisfies the strong tube condition (STC), if there exists a section Sthrough xsuch that S=F(L, [0,2λ]) ∩M. We say that Msatisfies STC if each point x∈M satisfies STC. 3. The Main Results Let f:Rn→Rnbe a function such that the autonomous differential equation ˙x=f(x), x ∈Rn,(3) defines a dynamical system in Rn. In particular, we also assume that equation (3) satisfies conditions for the existence, uniqueness and extendability to the 4 whole real line of its solutions for all initial points in Rnas well as the continuous dependence on initial conditions. Let (Rn, π) be the dynamical system generated by system (3). Thus πis a group and F(x, t) = π−1(t)x=π(−t)xfor all x∈Rnand t⩾0. Moreover, the map R×Rn∋(t, x)7→ π(t)xis continuous. In the sequel, we present a class of impulsive dynamical systems (Rn, π, M, I) associated to (Rn, π) such that M satisfies STC. Since the impulsive set needs to be closed and, in some sense, transversal to the flow, from now on we will always assume that M⊂Rnis a hypersurface in Rnof class Ck(a closed (n−1)−dimensional submanifold of Rnof class Ck), k⩾1, satisfying the following transversality condition: for each p∈Mwe have h~np, f(p)i 6= 0,(T) where ~npdenotes the normal vector of Mat p, and h·,·i is the scalar product in Rn. Let S ⊂ Rnbe a manifold in Rn. A differentiable map g:S → Ris called asubmersion at a point p∈ S if the differential map dg(p): Rn→Ris a surjective linear map and gis called a submersion if it is a submersion at each point p∈ S. A point c∈Ris a regular value of gif for each point p∈g−1(c), gis a submersion at p. Next, we present an auxiliary result and the reader is referred to [11] for more details. Lemma 5. Let S ⊂ Rnbe a manifold of class Ckin Rn. Then for each p∈ S, there exist an open set V⊂Rn,p∈V, and a submersion g:V→Rof class Cksuch that S ∩ V=g−1(0). Moreover, TpS= ker dg(p)and S ∩ Vmay be assumed to be connected. Using the previous lemma and condition (T) we obtain the following result. Lemma 6. Let p∈M,V⊂Rnbe an open set with p∈Vand g:V→Rbe a submersion of class Cksuch that M∩V=g−1(0). Then f(p)/∈ker dg(p). In the sequel, we show that hypersurfaces in Rnsatisfying (T) are impulsive sets satisfying STC. Lemma 7 shows the condition of transversality of the flow and Theorem 9 gives us the STC. 5 Lemma 7. Given x∈M, there exists ǫx>0such that F(x, (0, ǫx)) ∩M=∅ and π((0, ǫx))x∩M=∅. Proof. Arguing by contradiction, suppose that there exists a sequence {tn}n∈N⊂R+,tn n→+∞ −→ 0, such that π(tn)x∈M,n∈N. By Lemma 5, there are an open set V⊂Rn, with x∈V, and a submersion β:V→Rof class Ck,k⩾1, such that V∩M=β−1(0). We can assume that there exists λ > 0 such that π([0, λ])x⊂Vand tn< λ, for all n∈N. Defining h: [0, λ]→R by h(t) = β(π(t)x), we have h(0) = h(tn) = 0 for all n∈Nand, by the Rolle Theorem, there exists τn∈(0, tn) such that h′(τn) = h∇β(π(τn)x), f(π(τn)x)i= 0, n ∈N. As n→+∞, we obtain h∇β(x), f(x)i= 0, which contradicts Lemma 6. Analogously, if there exists a sequence {tn}n∈N⊂R+,tn n→+∞ −→ 0, such that F(x, tn)∩M6=∅for all n∈N, then we can also assume that there exists λ > 0 such that F(x, [0, λ]) ⊂Vand tn< λ for all n∈N. For each n∈N, let yn∈F(x, tn)∩Mand define the mapping hn: [0, tn]→Rby hn(t) = β(π(t)yn). Using again the Rolle Theorem we obtain τn∈(0, tn) with h′ n(τn) = 0, n∈N, that yields h∇β(x), f(x)i= 0 which is a contradiction. Hence, the result is proved. Lemma 8. Let V⊂Rnbe an open set and β:V→Rbe a submersion such that β−1(0) = V∩M6=∅. If x∈V∩Mand t > 0are such that π((0, t])x⊂V\M and π([−t, 0))x⊂V\Mthen β(π(t)x)β(π(−t)x)<0. Proof. Let x∈V∩Mand t > 0 satisfying the statement of the lemma. First, we note that β(π(t)x)β(π(s)x)>0 for all 0 < s ⩽t. Indeed, if there exists 0< s < t such that β(π(t)x)β(π(s)x)<0, then one can obtain τ∈(s, t) such that β(π(τ)x) = 0, that is, π(τ)x∈Mwhich is a contradiction. Analogously, we have β(π(−t)x)β(π(−s)x)>0 for all 0 < s ⩽t. Now, suppose the opposite, i.e., that β(π(t)x)β(π(−t)x)>0. Then, either β(π([−t, 0) ∪(0, t])x)>0 or β(π([−t, 0) ∪(0, t])x)<0 which implies that h∇β(x), f(x)i= 0 and gives also a contradiction by Lemma 6. 6 Theorem 9. The set M⊂Rnis an impulsive set which satisfies STC. Proof. By Lemma 7 the closed set Mis an impulsive set. Now let us show that Msatisfies STC. Let x∈Mbe given. By Lemma 5 there exist an open set V⊂Rnwith x∈Vand a submersion β:V→Rsuch that V∩M= β−1(0) is connected. Using the same argument as in Lemma 7, we may obtain a compact set S⊂V∩M,x∈S, and λ > 0 be such that π((0,2λ])S⊂V\M and F(S, (0,2λ]) ⊂V\M. Define L=π(λ)S. We are going to show that F(L, [0,2λ]) is a λ-tube through x. It is clear that F(L, λ) = S. Now, suppose by contradiction that F(L, [0,2λ]) does not contain a neighborhood of x, that is, there exists a sequence xn n→+∞ −→ xsuch that xn/∈F(L, [0,2λ]) for all n∈N. By Lemma 8, there exists 0 < s < λ such that β(π(s)x)β(π(−s)x)<0, consequently there is n0∈Nsuch that π([−s, s])xn⊂Vand β(π(s)xn)β(π(−s)xn)<0 for all n≥n0. Thus, for n≥n0, there is τn∈(−s, s) such that β(π(τn)xn) = 0, that is, π(τn)xn∈M, which gives a contradiction. To finish the proof, let us show that F(L, ν)∩F(L, µ) = ∅, 0 ⩽ν < µ ⩽2λ. Suppose, again by contradiction, that there exists y∈F(L, ν)∩F(L, µ) for some 0 ⩽ν < µ ⩽2λ. Then a=π(ν−λ)y∈Sand b=π(µ−λ)y∈S⊂M. Consequently, π(µ−ν)a=b∈M and this is a contradiction since a∈Sand π((0,2λ])S∩M=∅as we have π((0,2λ])S⊂V\M, which concludes the proof. Corollary 10. Let M⊂Rnbe a hypersurface in Rnsatisfying (T) and I:M→ Rnbe a continuous function. Then (Rn, π, M, I)is an IDS with Msatisfying STC. Example 11. In [4, Example 3.2], the authors study the uniform stability of the system          ˙x=−αy +yz −x3 ˙y=x−βxz −γy3 ˙z=θxy −z3, (4) subject to the impulsive condition I:M→R3, where α, β, γ, θ > 0, βα > 1, Mis an impulsive set and Iis an impulse function. The set M 7 was taken arbitrary and satisfying the condition STC. It is natural to ask for a concrete impulsive set satisfying STC. For instance, if M={(x, y, z)∈ R3:x2+αy2+αβ−1 θz2−r2= 0}, where r > 0 is a constant, then Mis an impulsive set satisfying STC. In fact, let g:R3→Rbe given by g(x, y, z) = x2+αy2+αβ −1 θz2−r2. Since 0 is a regular value of g, then it is well known that M=g−1(0) is a hypersurface in R3. Let F(x, y, z) = (−αy +yz −x3, x −βxz −γy3, θxy −z3), (x, y, z)∈R3. Since h∇g(x, y, z), F(x, y, z)i=−2x4−2αγy4−2(αβ −1) θz4<0 for all (x, y, z)∈M, then Msatisfies condition (T). Thus, by Theorem 9, the hypersurface M=g−1(0) is an impulsive set which satisfies STC. Note that [ n∈N(x, y, z)∈R3:x2+αy2+αβ −1 θz2−n2= 0 is also an impulsive set satisfying STC. Example 12. Let (Rn, π) be a dynamical system given by the system ˙x=f(x), where f∈C1(Rn,Rn) and xifi(x)<0 for all x∈Rn,x6= 0, i= 1,2,...,n. Let a1,...,an, r ∈R+\ {0}and α1,...,αn∈Nbe even numbers. Let M={(x1,...,xn)∈Rn:a1xα1 1+...+anxαn n=r}. If I:M→Rnis a continuous map, then (Rn, π, M, I) is an IDS with Msatisfying STC. Example 13. In [2, Example 4.8], the authors analyze the global attractor for the system ˙x=−x, ˙y=−y, in R2subject to an impulsive function I:M→ I(M), where M={(x, y)∈R2:x2+y2= 1}. It was just mentioned that M satisfies STC. Using Example 12, we may see easily that Msatisfies STC. 4. Conclusion The characterization of impulsive sets that satisfy STC is very important in the theory of impulsive dynamical systems. The function φ:X→(0,+∞] 8 defined in (2) is continuous in X\Mwhen Msatisfies STC, see [7, Theorem 3.8]. The continuity of this function is essential to obtain results about invariance, stability, recursive properties, existence of attractors, etc., as the reader can see in [2], [3], [4], [7], [8], [10], [12]. Acknowledgements. We would like to thank the anonymous referee for the helpful suggestions which allowed us to improved the presentation of our paper. References [1] E. M. Bonotto. Flows of characteristic 0+in impulsive semidynamical systems. J. Math. Anal. Appl., 332(1):81–96, 2007. [2] E. M. Bonotto, M. C. Bortolan, A. N. Carvalho, and R. Czaja. Global attractors for impulsive dynamical systems—a precompact approach. J. Differential Equations, 259(7):2602–2625, 2015. [3] E. M. Bonotto and M. Federson. 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