Global stabilization of fixed points using predictive control

Global stabilization of fixed points using predictive control Eduardo Liz and Daniel Franco Citation: Chaos 20, 023124 (2010); doi: 10.1063/1.3432558 View online: http://dx.doi.org/10.1063/1.3432558 View Table of Contents: http://scitation.aip.org/content/aip/journal/chaos/20/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Multi-rate optimal controller design for electromagnetic suspension systems via linear matrix inequality optimization J. Appl. Phys. 117, 17B506 (2015); 10.1063/1.4906588 Stability analysis for stochastic Volterra–Levin equations with Poisson jumps: Fixed point approach J. Math. Phys. 52, 042702 (2011); 10.1063/1.3573598 A hybrid local/global optimal control algorithm for dissipative systems with time-dependent targets: Formulation and application to relaxing adsorbates J. Chem. Phys. 128, 074104 (2008); 10.1063/1.2830709 Autonomous Reactor Control Using Model Based Predictive Control for Space Propulsion Applications AIP Conf. Proc. 746, 791 (2005); 10.1063/1.1867199 Neural optimal etch time controller for reactive ion etching J. Vac. Sci. Technol. B 16, 2707 (1998); 10.1116/1.590261 Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 150.214.182.15 On: Fri, 07 Oct 2016 07:47:03 Global stabilization of fixed points using predictive control Eduardo Liz1,a兲and Daniel Franco2,b兲 1Departamento de Matemática Aplicada II, E.T.S.I. Telecomunicación, Universidad de Vigo, Campus Marcosende, 36310 Vigo, Spain 2Departamento de Matemática Aplicada, Universidad Nacional de Educación a Distancia, Apartado de Correos 60149, 28080 Madrid, Spain 共Received 25 November 2009; accepted 3 May 2010; published online 17 June 2010兲 We analyze the global stability properties of some methods of predictive control. We particularly focus on the optimal control function introduced by de Sousa Vieira and Lichtenberg 关Phys. Rev. E 54, 1200 共1996兲兴. We rigorously prove that it is possible to use this method for the global stabilization of a discrete system xn+1=f共xn兲into a positive equilibrium for a class of maps commonly used in population dynamics. Moreover, the controlled system is globally stable for all values of the control parameter for which it is locally asymptotically stable. Our study highlights the difficulty of obtaining global stability results for other methods of predictive control, where higher iterations of fare used in the control scheme. © 2010 American Institute of Physics.关doi:10.1063/1.3432558兴 In many situations, the mechanisms of control of chaos and targeting seek not only to suppress any possible chaotic behavior, leading the system to a suitable equilibrium, but also to make its basin of attraction as large as possible. While local stability analysis is not difficult to address, in general, global stability results are often based on numerical observance. We tackle this problem for some methods of predictive control and succeed in proving a sharp result of global stabilization valid for a wide family of maps usually employed in the mathematical modeling of discrete systems. I. INTRODUCTION We discuss the issue of global stabilization of a chaotic dynamical system into an equilibrium employing predictionbased control 共PBC兲, that is, techniques for control of chaos that use predicted iterations of the system. As far as we know, these methods were introduced for discrete dynamical systems by Ushio and Yamamoto.1However, we point out that in the case of stabilization of fixed points, the same method was first suggested by de Sousa Vieira and Lichtenberg;2using a combination of a nonlinear delayed feedback control 共DFC兲method3and a technique of Socolar et al.,4they arrived at an “optimal” control technique that overcomes some admitted drawbacks of the DFC method. Some interesting generalizations of PBC methods were introduced by Polyak and Maslov.5,6 One of the advantages of predictive control methods is that conditions for local stabilization of fixed points in terms of the control parameter are easy to find.1,2However, the problem of global stabilization is more difficult to address, and, except for very particular maps,2,7only conclusions based on simulations can be found in literature so far.5,6As explained by Bocaletti et al. 共Ref. 8, Sec 4.1兲, when talking about the problem of targeting, in many practical situations, it is important to drive most trajectories of a dynamical system to a periodic orbit that yields superior performance over the others according to some criteria. Thus, the problem of targeting consists not only of choosing an appropriate attractor but also making its basin of attraction as large as possible. Among the dynamical systems exhibiting chaos, a wellknown family is given by one-dimensional maps of the form xn+1 =f共xn兲,共1兲 where fis a unimodal function with a unique positive equilibrium K. Examples of such maps employed in the modeling of population dynamics are the quadratic map f共x兲 =rx共1−x兲, the exponential 共Ricker兲map f共x兲=xer共1−x兲, and the generalized Beverton–Holt map f共x兲=rx/共1+x ␥ 兲共see Ref. 9兲. Here, rand ␥ are positive parameters intrinsic to the model, such as the natural growth rate. It is worth noticing that, although these three maps can be considered discrete approximations of the continuous logistic equation, the Ricker and the Beverton–Holt models have more biological meaning 共see, e.g., Ref. 10兲. One important property, shared by these and other models, is that they exhibit chaos for some values of the parameters,11 but multistability is not possible; this means that even if they have an infinite number of periodic orbits, at most one of them can be attracting. In particular, if the unique positive equilibrium Kis asymptotically stable 关that is, 兩f⬘共K兲兩⬍1兴, then it is a global attractor. A very interesting mathematical derivation of this property is due to Singer,12 who used the Schwarzian derivative as a fundamental tool. In this setting, an important control problem consists of stabilizing the unique positive equilibrium, keeping the global property, that is, in such a way that all positive trajectories converge to the equilibrium. The aim of most of the proposed mechanisms of control of chaos consists of stabilizing one of the unstable periodic orbits 共UPOs兲embedded in a chaotic attractor either by adjusting a parameter of the model, or modifying the state variable, by introducing an a兲Author to whom correspondence should be addressed. Electronic mail: [email protected]. URL: http://www.dma.uvigo.es/ eliz. b兲Electronic mail: [email protected]. CHAOS 20, 023124 共2010兲 1054-1500/2010/20共2兲/023124/9/$30.00 © 2010 American Institute of Physics20, 023124-1 Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 150.214.182.15 On: Fri, 07 Oct 2016 07:47:03 external parameter that one can control to drive the chaotic system to a stable situation. The first strategy has its origin in the seminal work of Ott, Grebogi, and Yorke;13 although it was successfully applied in physics, it is difficult to implement in ecological systems 共for more comments and references, see Ref. 8, Sec. 2兲. The second group of methods seems to be the most appropriate when we try to manage biological systems since intrinsic parameters such as the growth rate are, in general, difficult to modify, while external parameters such as migration, culling, or enrichment are more easily controllable 共for further discussion and references see, e.g., Ref. 14兲. One of the methods suggested in this context is the constant feedback 共CF兲method,15 which consists of modifying the one-dimensional system 共1兲to xn+1 =f共xn兲−C,共2兲 where Cis a positive constant. Gueron16 proved analytically that for a general family of unimodal maps, it is always possible to find values of Cfor which Eq. 共2兲has a stable positive equilibrium. However, the region of attraction is usually small and many trajectories are driven to zero. Biologically, this means that only intermediate values of the population size survive after control, while the others go to extinction due to the Allee effect.10,17 The first author recently proved18 a result of global stabilization for the proportional feedback 共PF兲control method, which modifies Eq. 共1兲to xn+1 =f共 ␥ xn兲,共3兲 where ␥ 苸共0,1兲. In population dynamics, this means that a percentage of the population is removed by migration or harvesting. The other interesting strategy of control is based on a threshold mechanism. The idea of incorporating a selfregulatory threshold dynamics on a chaotic system goes back to the work of Sinha and Biswas19 and Glass and Zheng20 共see also Refs. 21 and 22兲. This strategy of control has also a clear interpretation in biological terms, corresponding to control measures such as culling of a stock population, hunting or catching of a managed population stock, or treatment of infectious diseases 共for more comments and further references, see Ref. 23兲. In this paper, we focus on global stabilization of system 共1兲using predictive feedback control. Methods of predictive control to stabilize an unstable T-periodic orbit of the discrete-time system 共1兲have the form1 xn+1 =g共xn,un兲, where the control input unis determined by the difference between the predicted state fT共xn兲and the current state xn, that is, un=fT共xn兲−xn. As usual, fTis defined as the Tth iteration of f, and it is used as a prediction of xn+T. For T=1 共stabilization of fixed points兲, the simplest scheme of PBC is written as xn+1 =f共xn兲− ␣ 共f共xn兲−xn兲,共4兲 where ␣ is a real control parameter. Equation 共4兲is exactly the control proposed by de Sousa Vieira and Lichtenberg 共Ref. 2, Sec. III兲to avoid some handicaps of the DFC method, such as the increase in dimensionality of the system3 and the so-called odd limitation number.24 Some additional good properties of the control law 共4兲 underlined in Ref. 2are that the fixed points of the controlled system are the same as those in the uncontrolled system, knowledge of the location of the UPO is not necessary, and it is easy to implement in the sense that the control term contains only the amplified versions of the input and output of the dynamical system. In our main result, we prove that for a family of unimodal maps, including the quadratic map, the Ricker map, and the generalized Beverton–Holt map, the optimal nonlinear control of de Sousa Vieira and Lichtenberg leads to the global stabilization of a chaotic system into its positive equilibrium. Moreover, this is true for all values of the parameters for which local stabilization is possible. We also discuss the case of more general methods of predictive control, showing that this result of global stability is no longer valid. This paper is organized as follows: in Sec. II, we state our main assumptions and recall some related results. Section III is devoted to establish the main global stability result. In Sec. IV, the issue of local and global stability is discussed for other PBC methods. Finally, in Sec. V, we discuss the main conclusions and directions for future research. The proofs and auxiliary results are placed in Appendixes A and B. II. STATEMENT OF THE PROBLEM First, we introduce some general properties and notation that will be used throughout the paper from now on. Denote by I=关0,b兴a real interval 共b=+⬁is allowed兲. The map f:I→Ithat defines the dynamical system 共1兲will be a C3 function satisfying the following properties: 共A1兲fhas only two fixed points: x=0 and x=K⬎0, with f⬘共0兲⬎1. 共A2兲fhas a unique critical point c⬍Kin such a way that f⬘共x兲⬎0 for all x苸共0,c兲,f⬘共x兲⬍0 for all x苸共c,b兲. 共A3兲共Sf兲共x兲⬍0 for all x⫽c, where 共Sf兲共x兲=f⵮共x兲 f⬘共x兲−3 2 冉 f⬙共x兲 f⬘共x兲 冊 2 is the Schwarzian derivative of f. 共A4兲f⬙共x兲⬍0 for all x苸共0,c兲. These assumptions are motivated by the fact that many maps usually employed in discrete models fulfill them. In particular, the quadratic map for all r⬎1, the exponential map for all r⬎0, and the generalized Beverton–Holt map for r⬎1 and ␥ ⱖ2共see Ref. 12兲. We notice that if fsatisfies 共A3兲and f⬙共0兲⬍0, then condition 共A4兲holds; otherwise, f⬘ would have a positive local minimum, which contradicts the maximum principle 共Ref. 12, Proposition 2.4兲. Thus, the family of maps satisfying conditions 共A1兲–共A4兲is essentially 023124-2 E. Liz and D. Franco Chaos 20, 023124 共2010兲 Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 150.214.182.15 On: Fri, 07 Oct 2016 07:47:03 the family of S-unimodal maps considered by Collet and Eckmann 共Ref. 25, Sec. II.4兲共see also Ref. 26, Sec. 5.3兲. Similar conditions were assumed for CF control in Refs. 16 and 17 and for PF in Ref. 18. Besides the above-mentioned maps, there are other functions satisfying conditions 共A1兲– 共A4兲, which do not come from population dynamics; an example is the sine map f共x兲=rsin共 ␲ x兲,0ⱕxⱕ1, with r苸共1/ ␲ ,1兲共Ref. 27,p.369兲. When used in population models, maps satisfying condition 共A4兲correspond to compensation models, where the per capita production is a decreasing function of the population density 关see, e.g., Ref. 28 共Sec. 1.2兲and Ref. 29 共Sec. 1.4兲兴. This condition is not satisfied by depensation models, for which the per capita production is smaller than one for low values of the population density. The latter models exhibit the so-called Allee effect, which means that if the initial population size is below a critical parameter, the population will die out. An example is the generalization of the quadratic map f共x兲=rx2共1−x兲共see Ref. 28兲. In Fig. 1, we plot the graph of this map with r=4; although it has a unique positive fixed point and a unique critical point 共maximum兲,it does not satisfy 共A4兲and exhibits the Allee effect. Condition 共A3兲is a more technical assumption. The Schwarzian derivative was first introduced into the study of one-dimensional dynamical systems by Singer,12 and it became a valuable tool 共see, e.g., Refs. 25 and 26兲. Although it is difficult to give a biological interpretation of it, it is a remarkable fact 共already observed by Singer兲that the most usual maps employed in the modeling of population dynamics satisfy condition 共A3兲. We recall12 that if fsatisfies conditions 共A1兲and 共A3兲, and f⬘共K兲ⱖ−1, then Kis a global attractor of system 共1兲. We notice that, for ␣ 苸关0,1兴, the method 共4兲is welldefined in the sense that, since f:I→I, the modified map F ␣ 共x兲ªf共x兲− ␣ 共f共x兲−x兲= ␣ x+共1− ␣ 兲f共x兲 also maps Iinto Ibecause F ␣ 共x兲is a convex combination of xand f共x兲. This property does not hold for general DFC and PBC methods. If we think of population models, this is important because negative values of the population do not make sense. We limit our discussion to this range of values of the control parameter ␣ . The method 共4兲was applied in Ref. 2to the quadratic map, with r⬎3关to ensure that Kis unstable for f共x兲 =rx共1−x兲兴. The authors observed that the positive fixed point K=1−1/ris locally stable for the controlled system if ␣ ⬎共r−3兲/共r−1兲ª ␣ 0. Moreover, the equilibrium Kis a global attractor for system 共4兲for all ␣ 苸共 ␣ 0,1兲. However, as pointed out later by McGuire et al.,7the quadratic map is a very particular case since the modified map F ␣ is still quadratic. Hence, it trivially shares the global stability properties of f. For general unimodal maps, function F ␣ does not inherit properties 共A2兲and 共A3兲; usually, the shape of F ␣ is no longer one-humped for ␣ ⬎0. For example, the exponential map f共x兲=xer共1−x兲is bimodal for 0⬍ ␣ ⬍er−2/共1+er−2兲and increasing for er−2/共1+er−2兲⬍ ␣ ⬍1.30 Thus, although numerical experiments suggest that Kis still globally attracting for F ␣ when 兩F ␣ ⬘共K兲兩⬍1, an analytic proof of this result is not available. Our main result in this paper fills this gap for the family of maps satisfying conditions 共A1兲–共A4兲. III. GLOBAL STABILIZATION In this section, we state our main result. As before, we denote F ␣ 共x兲= ␣ x+共1− ␣ 兲f共x兲. We begin with a preliminary result, which is of independent interest. Proposition 1: Assume that f⬘共x兲⬍0and F ␣ ⬘共x兲⬍0for all x in an interval J傺I. If 共Sf兲共x兲⬍0for all x苸J, then 共SF ␣ 兲共x兲⬍0for all x苸J and all ␣ 苸共0,1兲. As an immediate consequence of Proposition 1 and Theorem 5 in Ref. 31, we get the following corollary. Corollary 1: If f satisfies conditions (A1)–(A3), and f⬘共K兲⬍−1, then the family of maps 兵F ␣ : ␣ 苸共0,1兲其 undergoes a period-doubling bifurcation at ␣ 0=共f⬘共K兲+1兲/ 共f⬘共K兲−1兲, where F ␣ 0 ⬘共K兲=−1. Corollary 1 solves an open problem posed in Ref. 7. There, the authors proposed to investigate the conditions under which period doubling occurs for general unimodal maps, which are being stabilized at a fixed point using control 共4兲. They showed that if fhas a negative Schwarzian derivative, then the same property is inherited by F ␣ when the positive feedback parameter ␣ is sufficiently small. Corollary 1 proves that period doubling occurs regardless of the value of ␣ 苸共0,1兲necessary for stabilization. Next we formulate our main result. Theorem 1: Assume that f satisfies conditions (A1)–A4), and define F ␣ 共x兲=f共x兲− ␣ 共f共x兲−x兲.IfF ␣ ⬘共K兲ⱖ−1, then the positive equilibrium K is a global attractor of Eq. (4) on 共0,b兲. The proofs of Proposition 1 and Theorem 1 can be found in Appendix A. As an example, consider the Ricker map f共x兲=xer共1−x兲, which satisfies conditions 共A1兲–共A4兲for all r⬎0. The positive equilibrium K=1 is globally stable for rⱕ2, then it undergoes a typical sequence of period-doubling bifurcations, and it becomes chaotic for r⬎2.6924.11 According to Corollary 1 and Theorem 1, for all r⬎2, the family of maps 兵F ␣ 共x兲= ␣ x+共1− ␣ 兲xer共1−x兲: ␣ 苸关0,1兲其 0.0 0. 2 0. 4 0.6 0.8 1 .0 0.0 0.2 0.4 0.6 0.8 1 . 0 FIG. 1. 共Color online兲Graph of the map f共x兲=4x2共1−x兲and the line y=x. The iterates of initial points x0⬍1/2 converge to zero 共Allee effect兲. 023124-3 Global stabilization Chaos 20, 023124 共2010兲 Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 150.214.182.15 On: Fri, 07 Oct 2016 07:47:03 experiences a period-halving bifurcation at ␣ =共r−2兲/rin such a way that the positive equilibrium is globally attracting for the controlled system xn+1 =xner共1−xn兲− ␣ 共xner共1−xn兲−xn兲共5兲 for all ␣ 苸关共r−2兲/r,1兲. We show the bifurcation diagram for r=3 in Fig. 2. The equilibrium is stabilized in a periodhalving bifurcation at ␣ =1/3共see the vertical dashed line兲. Theorem 1 ensures that, for ␣ ⱖ1/3, all orbits starting at x0⬎0 converge to the positive equilibrium K=1. Next we give a numerical validation of Theorem 1 by means of a statistical approach applied to Eq. 共5兲with r=3; this method not only illustrates our main result but also provides significant rates of convergence, not dependent on the initial condition. First we choose randomly 2000 initial conditions in the interval 关0,2.5兴, which is invariant and attracting for the map f. See Fig. 3, where we show the distribution of this sample and its evolution after 15 iterations of f. Since the uncontrolled system is chaotic, no periodic pattern is appreciated. We notice that, in Figs. 3–8,ndenotes the number of iterations. Figure 4shows the distribution of the random sample after 15 iterations of the controlled system for ␣ =0.35, which is close to the bifurcation point. It can be observed that the iterates of all initial conditions approach the positive equilibrium. The convergence is faster for larger values of ␣ , as it is shown for ␣ =0.4 and the same number of iterations. Next we estimate the probability density function of the random variable provided by the sample we chose using a kernel density estimation. We use the kernel density approximation ␾ 共x兲=1 hm兺 i=1 m k 冉 x−xi h 冊 , where 共xi兲i=1 mis the sample, kis the standard Gaussian function k 冉 x−xi h 冊 =1 冑2 ␲ e−共x−xi兲2/2h2, and his a smoothing bandwidth. We use h= 冉 max i=1 m 兵xi其− min i=1 m 兵xi其 冊 冒4. Our simulations with different samples of random initial conditions show that, when evolved under the equations of the controlled system 共5兲, the approximation of the probability density function converges to a delta function located at the positive fixed point. In Fig. 5, we represent the initial density function and its evolution after 50 iterations of the uncontrolled system 关Eq. 共5兲with ␣ =0兴. As observed before, there is no periodic pattern. The evolution of the density function for ␣ =0.35 共close to the bifurcation point ␣ =1/3兲after 15 and 50 iterations is shown in Fig. 6. The convergence to the delta function is faster for larger values of ␣ , as shown in Fig. 7for ␣ =0.4: after only 15 iterations, the density function is highly concentrated at the positive equilibrium point K=1. 0.0 0.1 0.2 0.3 0.4 0.0 0.5 1.0 1.5 2.0 2 .5 xn α 1/3 FIG. 2. 共Color online兲Bifurcation diagram for the controlled system 共5兲 with r=3 and ␣ as the bifurcation parameter. The value ␣ =0 corresponds to the uncontrolled 共chaotic兲system; the equilibrium becomes globally asymptotically stable after the bifurcation point ␣ =1/3共vertical dashed line兲. 500 1000 1500 200 0 0 .5 1.0 1.5 2.0 2.5 n= 0 0 500 1000 1500 200 0 0 .5 1.0 1.5 2.0 2.5 α=0, n=15 ( b ) ( a) FIG. 3. 共Color online兲Distribution of a sample of 2000 random initial conditions in the interval 关0,2.5兴, and the evolution of the uncontrolled system 共 ␣ =0兲after 15 iterations. 023124-4 E. Liz and D. Franco Chaos 20, 023124 共2010兲 Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 150.214.182.15 On: Fri, 07 Oct 2016 07:47:03 To further illustrate this numerical validation, we show in Fig. 8the time series of the solution of Eq. 共5兲, with r=3 and initial point x0=0.5, for ␣ =0.35 and ␣ =0.4. IV. OTHER METHODS OF PREDICTIVE CONTROL The global control attained with the method of de Sousa Vieira and Lichtenberg has the restriction that, in general, the perturbation required to stabilize all trajectories around the positive equilibrium is relatively large. In this regard, we emphasize that, while targeting problems usually require small perturbations, there are situations where large perturbations are more appropriate. For example, this is the case of interventions in population dynamics, where the control problem can be seen as a strategy for sustainable development 共e.g., by culling or enrichment兲.14,23,32,33 In order to reduce the control perturbation, an interesting generalization of PBC methods was introduced by Polyak and Maslov.5,6Their main idea consists of using higher iterations of the map fas predicted states. To stabilize a fixed point, this method has the form xn+1 =f共xn兲+共−1兲m+1␧共fm+1共xn兲−fm共xn兲兲,共6兲 where mis a positive integer and ␧⬎0. Notice that the case m=0 is equivalent to Eq. 共4兲, with the usual notation f0共x兲ªx. Now, assume that fis a map satisfying conditions 共A1兲– 共A4兲and Kis an unstable equilibrium of Eq. 共1兲. Then, f⬘共K兲⬍−1, and it is easy to check that Kis asymptotically stable for Eq. 共6兲if ␧⬎ 冉 −1 f⬘共K兲 冊 mf⬘共K兲+1 f⬘共K兲−1 ª␧m. Then, the bifurcation point ␣ 0defined in the statement of Corollary 1 for Eq. 共4兲is replaced by ␧m=共−1/f⬘共K兲兲m ␣ 0 when control 共6兲is applied. For example, for the chaotic exponential map f共x兲=xe3共1−x兲, we have f⬘共K兲=2 and ␣ 0=1/3. Thus, ␧m= ␣ 0/2m, which means that the perturbation of fin Eq. 共6兲can be made as small as desired, choosing a sufficiently big value of m. Some disadvantages of this control method are that 共a兲it is computationally more expensive than Eq. 共4兲, especially for large m, and 共b兲it is not ensured 共see the example below兲 that the modified map 0 500 1000 1500 200 0 0.5 1.0 1.5 2.0 2.5 α= 0 .35, n= 1 5 0 500 1000 1500 200 0 0.5 1.0 1.5 2.0 2.5 α=0.4, n=15 (b) (a) FIG. 4. 共Color online兲Evolution of the random sample of 2000 initial conditions in the interval 关0,2.5兴after 15 iterations of the controlled system for ␣ =0.35 and ␣ =0.4. 0.5 1.0 1.5 2.0 2.5 0.1 0.2 0.3 n= 0 φ(x) x 0.5 1.0 1.5 2.0 2.5 0.05 0.10 0.15 0.20 0.25 0.30 0.35 x φ (x) α=0, n=50 ( b ) ( a) FIG. 5. 共Color online兲Approximation of the probability density function for the random variable corresponding to a sample of 2000 random initial conditions in the interval 关0,2.5兴, and its evolution after 50 iterations under the uncontrolled system 共5兲with r=3 and ␣ =0. 023124-5 Global stabilization Chaos 20, 023124 共2010兲 Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 150.214.182.15 On: Fri, 07 Oct 2016 07:47:03 F␧,m共x兲=f共x兲+共−1兲m+1␧共fm+1共x兲−fm共x兲兲 is well-defined for mⱖ2. We notice that the scheme for m=1 is still well-defined for ␧苸关0,1兴because F␧,1共x兲=f共x兲+␧共f2共x兲−f共x兲兲 =共1−␧兲f共x兲+␧f2共x兲 is a convex combination of f共x兲and f2共x兲. A natural question is whether or not the global stability result valid for m=0 still holds for Eq. 共6兲when m⬎0. In Ref. 6, it is claimed that this is an expected property, confirmed by simulation results using the quadratic family. However, we show that, in general, the asymptotic stability of the equilibrium is only a local property. Indeed, even when the equilibrium is locally stable, system 共6兲can have other stable periodic orbits, including new fixed points. For example, for m=2 and the chaotic quadratic map f共x兲=3.9x共1−x兲, the fixed point K=0.7435 is locally stable for Eq. 共6兲when ␧⬎␧2=0.086. However, for any ␧arbitrarily close to ␧2, there is a new fixed point K1in such a way that all orbits starting at a point x0⬍K1converge to zero. This means that the control method induces an Allee effect. In Fig. 9共a兲, we plot the function F␧,2共x兲=f共x兲−␧共f3共x兲−f2共x兲兲, with ␧=0.1 共solid line兲; for the sake of comparison, we also include the graph of function f共x兲=3.9x共1−x兲corresponding to the uncontrolled system 共dashed line兲and the line y=x.A magnification of the graph of F␧,2 between 0 and 0.05 is included in Fig. 9共b兲to emphasize the new positive fixed point K1⬇0.016 06 and the Allee effect. In Appendix B, we prove that a global stability result holds for m=1 in Eq. 共6兲when fis the quadratic map, but this is an exception. In general, even for m=1, the conclusions of Corollary 1 and Theorem 1 are no longer valid. Consider m=1 in Eq. 共6兲, that is, xn+1 =f共xn兲+␧共f2共xn兲−f共xn兲兲,共7兲 with f共x兲=xe3共1−x兲. The family of maps F␧,1共x兲=f共x兲 +␧共f2共x兲−f共x兲兲 undergoes a period-doubling bifurcation at ␧=1/6⬇0.166, where F␧,1 ⬘共1兲=−1. The fixed point K=1 is locally stable for ␧⬎1/6, but this equilibrium coexists with an attracting 2-cycle until this cycle disappears in a tangent bifurcation at ␧⬇0.1958. After this value, the positive fixed point seems to be globally attracting, but only until a new fixed point appears at ␧⬇0.4605. In Fig. 10, we show the bifurcation diagram for ␧苸关0,0.3兴. The unstable fixed point is represented by a dashed line between ␧=0 and ␧=1/6. The other dashed lines represent an unstable 2-cycle that disappears 共with the attracting 2-cycle兲after a saddle-node bifurcation at ␧⬇0.1958. 0.5 1.0 1.5 2.0 2.5 1 2 3 4 α= 0 . 3 5, n= 1 5 φ(x) x 0.5 1.0 1.5 2.0 2.5 5 10 15 20 25 30 α=0.35, n=50 φ (x) x (b) (a) FIG. 6. 共Color online兲Evolution of the approximation of the probability density function for the random variable corresponding to a sample of 2000 random initial conditions in the interval 关0,2.5兴after 15 and 50 iterations under the controlled system 共5兲with r=3 and ␣ =0.35. 0.5 1.0 1.5 2.0 2.5 0.5 1.0 1.5 2.0 2.5 3.0 3.5 α= 0 . 4 , n=5 φ (x) x 0.5 1.0 1.5 2.0 2.5 5 10 15 20 25 30 α=0.4, n=15 φ(x) x ( b) ( a) FIG. 7. 共Color online兲Evolution of the approximation of the probability density function for the random variable corresponding to a sample of 2000 random initial conditions in the interval 关0,2.5兴after 5 and 15 iterations under the controlled system 共5兲with r=3 and ␣ =0.4. 023124-6 E. Liz and D. Franco Chaos 20, 023124 共2010兲 Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 150.214.182.15 On: Fri, 07 Oct 2016 07:47:03 V. CONCLUSIONS AND DIRECTIONS FOR FUTURE RESEARCH We have analyzed the global stability properties of some methods of predictive control. We particularly focused on the optimal control function introduced by de Sousa Vieira and Lichtenberg.2Based on their analysis of the logistic map, they affirmed that this method has good global stabilizing properties, thus decreasing the sensitivity to noise. Our main result 共Theorem 1兲comes up with a rigorous analytic proof of this observation for a family of maps commonly used in modeling of discrete-time systems. Moreover, we have given a positive answer to some questions related to the preservation of the period-doubling route to chaos in the controlled system posed in Ref. 7. We have reviewed as well the local and global properties of stabilization of fixed points of more general PBC methods. An important property of these methods is that they do not increase the dimension of the system; thus, the study of local stability is easier than in other control techniques. However, our study highlights the difficulty of obtaining global stability results when using higher iterations of the map f in the control scheme. de Sousa Vieira and Lichtenberg proposed a generalization of their optimal method to stabilize UPOs with prime period T⬎1 by using the map F ␣ ,T共x兲=fT共x兲− ␣ 共fT共x兲−x兲. Assume that we aim to stabilize an UPO of period T=2mof a chaotic map fusing this method. An interesting problem to study is whether or not it is possible to get a globally stable T-periodic orbit in the sense that it attracts all initial conditions, which are not an element of the other UPOs. Numerical simulations suggest that this is true for T=2 and fsatis- 0 10203040506 0 0.4 0.6 0.8 1.0 1.2 1.4 1.6 xn α=0.35 n 0 10203040506 0 0.4 0.6 0.8 1.0 1.2 1.4 1.6 α=0.4 xn n ( b ) (a) FIG. 8. 共Color online兲Time series for the controlled system 共5兲with r=3; the values of the control parameter are ␣ =0.35 and ␣ =0.4. In both cases, the initial point is x0=0.5, and 60 iterations were made. 0 . 0 0.2 0.4 0.6 0.8 1.0 0 .0 0 .2 0 .4 0 .6 0 .8 1.0 (a) 0.00 0.01 0.02 0.03 0.04 0.0 5 0.00 0.01 0.02 0.03 0.04 0.05 ( b ) K 1 FIG. 9. 共Color online兲共a兲Graphs of functions f共x兲=3.9x共1−x兲共dashed line兲 and F␧,2共x兲=f共x兲−␧共f3共x兲−f2共x兲兲, with ␧=0.1 共solid line兲. The fixed points are obtained as the intersections with the line y=x.共b兲Magnification of the graph of F␧,2 between 0 and 0.05; the fixed point K1⬇0.016 06 induces an Allee effect: all initial conditions below K1are driven to zero under successive iterations of F␧,2. 0.00 0.05 0.10 0.15 0.20 0.25 0.3 0 0.0 0.5 1.0 1.5 2.0 2.5 xn ε FIG. 10. 共Color online兲Bifurcation diagram for the controlled system 共7兲 with f共x兲=xe3共1−x兲and ␧as the bifurcation parameter. The fixed point K =1 is locally asymptotically stable but not globally stable between 0.166 and 0.1958. The dashed lines indicate instability. 023124-7 Global stabilization Chaos 20, 023124 共2010兲 Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 150.214.182.15 On: Fri, 07 Oct 2016 07:47:03 fying conditions 共A1兲–共A4兲; to find a proof of this result would be a nice improvement of Theorem 1. However, the answer to this problem is, in general, negative for T⬎2. For example, the quadratic map f共x兲=4x共1−x兲has two distinct four-periodic orbits with the same multiplier ␮ =−16; thus, global stabilization of one of them is not possible with this control scheme. An additional application of the new results proved in this paper is that the controlled map F ␣ 共x兲= ␣ x+共1− ␣ 兲f共x兲is used as a model in population dynamics when a probability of surviving the reproductive season 共iteroparous populations兲is assumed. If fmeans the nonlinear density growth of the population, the model xn+1 =F ␣ 共xn兲= ␣ xn+共1− ␣ 兲f共xn兲共8兲 assumes that a fraction ␣ of energy is invested into adult survivorship rather than reproduction 共for more details, see Refs. 10 and 30兲. Thus, Theorem 1 shows that even if fis chaotic and the survivorship parameter ␣ is large enough, the population is stabilized into the positive equilibrium regardless the initial size of the population. We notice that organisms might evolve into such strategy 共survival rather than reproduction兲under bad environmental conditions; this means that a population whose fecundity encodes for chaos can stabilize itself at its equilibrium density. For a nice discussion on the evolutionary advantage of self-control, see Ref. 34. As stated there, this makes a difference between ecological systems and physical or chemical systems. Regarding the application of PBC methods to population problems, an important drawback is that their implementation requires full knowledge of the model equation. Although the exact form of the original map is difficult to obtain, in some cases good census data are available 共see, e.g., Ref. 35, where census data of a freshwater fish population were collected every fourth month over 20 years兲; this allows one to derive a stock-recruitment relationship that fits well into some known function, such as the Ricker map. Then, the model can be validated, and the control method could be implemented to stabilize the population. On the other hand, while another admitted drawback in the application of predictive control in the process industries is that a quick computation upon observation must be performed 共see, e.g., Ref. 36兲, in the setting of population dynamics, the time scales are different; for instance, in a fish farm, the control 共4兲can provide an assessment on whether next season harvesting should be incremented or, conversely, some enrichment is needed to stabilize the stock size. We mention that the control method 共4兲can help not only to stabilize the population but also to avoid extinction 共see Ref. 30兲. For further discussion and references on the application of control methods to population problems, especially to prevent extinction, we refer to Ref. 33. Related results on global stability in Eq. 共8兲for particular choices of f were proved in Refs. 37 and 38. ACKNOWLEDGMENTS The authors sincerely thank the insightful critique of three anonymous reviewers of this paper, which significantly helped to improve it. We are greatly indebted to Dr. János Karsai from the University of Szeged 共Hungary兲for very helpful discussions. This research was supported in part by the Spanish Ministry of Science and Innovation and FEDER 共Grant No. MTM2007-60679兲. APPENDIX A: PROOF OF THE MAIN RESULT Proof of Proposition 1: Since f⬘共x兲⬍0 and 共Sf兲共x兲⬍0 for all x苸J, we have 2f⵮共x兲 f⬘共x兲⬍3 冉 f⬙共x兲 f⬘共x兲 冊 2 ⇒2f⵮共x兲⬎3共f⬙共x兲兲2 f⬘共x兲 for all x苸J. Using that F ␣ ⬘共x兲⬍0onJ, the previous inequality, and the obvious relations F ␣ ⬘共x兲= ␣ +共1− ␣ 兲f⬘共x兲,F ␣ ⬙共x兲=共1− ␣ 兲f⬙共x兲, F ␣ ⵮共x兲=共1− ␣ 兲f⵮共x兲, we get, for all x苸J, 2F ␣ ⵮共x兲=2共1− ␣ 兲f⵮共x兲⬎3共1− ␣ 兲共f⬙共x兲兲2 f⬘共x兲 =3 共F ␣ ⬙共x兲兲2 F ␣ ⬘共x兲− ␣ ⬎3共F ␣ ⬙共x兲兲2 F ␣ ⬘共x兲. This inequality implies that 共SF ␣ 兲共x兲⬍0. To use Proposition 1 to prove our main result, we need the following auxiliary result from Ref. 39. Lemma 1: (Reference 39, Corollary 2.9) Let g:共0,b兲 →关0,b兴be a continuous map with a unique fixed point K such that 共g共x兲−x兲共x−K兲⬍0for all x⫽K. Assume that there are points 0ⱕa⬍K⬍dⱕb such that the restriction of g to 共a,d兲has at most one turning point and (whenever it makes sense) g共x兲ⱕg共a兲for every xⱕa and g共x兲ⱖg共d兲for every xⱖd. If g is decreasing at K, assume additionally that 共Sg兲 ⫻共x兲⬍0for all x苸共a,d兲except at most one critical point of g, and −1ⱕg⬘共K兲⬍0. Then K is a global attractor of g. Proof of Theorem 1: Notice that f⬘共0兲⬎1 implies that F ␣ ⬘共0兲⬎1 for all ␣ 苸关0,1兲. On the other hand, F ␣ ⬘共x兲 =0⇔f⬘共x兲=− ␣ /共1− ␣ 兲. If F ␣ has no critical points, then it is strictly increasing, and therefore Kis a global attractor because F ␣ 共x兲⬎xfor x苸共0,K兲and F ␣ 共x兲⬍xfor x苸共K,b兲. Notice that this happens when f⬘共x兲⬎− ␣ /共1− ␣ 兲for all x苸共0,b兲. Next, since 共Sf兲共x兲⬍0 for all x⫽c,fcan have at most one inflexion point in 共c,b兲. First we consider the case when fhas no inflexion points in 共c,b兲. Then f⬘is strictly decreasing in 共c,b兲, and there is at most one point c1⬎csuch that F ␣ ⬘共c1兲=0. Clearly, c1is a local maximum of F ␣ since F ␣ ⬘共x兲⬎0on共0,c1兲and F ␣ ⬘共x兲⬍0on共c1,b兲.Ifc1⬎K, then F ␣ ⬘共K兲⬎0, and it follows that Kis globally attracting. Thus, we can assume that c⬍c1⬍K. Since F ␣ 共x兲ⱕF ␣ 共c1兲for all x苸共0,c1兴and, by Proposition 1, 共SF ␣ 兲共x兲⬍0on共c1,b兲,an application of Lemma 1 with a=c1,d=bproves that Kis a global attractor of F ␣ . It remains to consider the case when fhas an inflexion point ␥ in 共c,b兲. It is clear that f⬘attains a global minimum 023124-8 E. Liz and D. Franco Chaos 20, 023124 共2010兲 Reuse of AIP Publishing content is subject to the terms at: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 150.214.182.15 On: Fri, 07 Oct 2016 07:47:03