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Persistence of Li-Yorke chaos in systems with relay Marat Akhmet1, Mehmet Onur Fen2,∗, Ardak Kashkynbayev1 1Department of Mathematics, Middle East Technical University, 06800 Ankara, Turkey 2Basic Sciences Unit, TED University, 06420 Ankara, Turkey Abstract It is rigorously proved that the chaotic dynamics of the non-smooth system with relay function is 7 1 persistentevenifa chaotic perturbationis applied. We considerchaosina modified Li-Yorkesense such 0 2 that infinitely many almost periodic motions take place in its basis. It is demonstrated that the system n under investigation possesses countable infinity of chaotic sets of solutions. Coupled Duffing oscillators a J are used to show the effectiveness of our technique, and simulations that support the theoretical results 9 2 are represented. Moreover, a chaos control procedure based on the Ott-Grebogi-Yorke algorithm is ] proposed to stabilize the unstable almost periodic motions embedded in the chaotic attractor. D C Keywords: Persistenceofchaos;Li-Yorkechaos;Almostperiodicmotions;Relaysystem;Chaoscontrol; . n i Duffing equation l n [ 1 v 5 1 Introduction 8 3 8 The word “persistence” is not popular for differential equations since it is not usual to say about per- 0 1. sistence of periodic solutions against periodic perturbations as well as other forms of regular motions 0 7 such as quasi-periodic and almost periodic solutions in a similar way. In the literature, these types of 1 problems have been investigated as a part of synchronization [1]-[3]. Chaos is not an exceptional term : v i in this row, and those results which we recognize as synchronization of chaos can be interpreted as a X r specific type of persistence of chaos [4]-[9]. The specification is characterized through an asymptotic a relation between solutions of coupled systems. In other words, persistence has not been considered by researchers explicitly, except under the mask of synchronization or entrainment [1]-[14]. In this study, we consider synchronization in its ultimately generalized form, without any additional asymptotic conditions, considering only ingredients of Li-Yorke chaos [15]. This is the main theoretical novelty of the present paper. An extension of the original definition of Li and Yorke [15] to dimensions greater than one was per- formedby Marotto[16]. It wasdemonstratedin [16] that a multidimensional continuouslydifferentiable map possesses Li-Yorke chaos if it has a snap-backrepeller. Moreover,generalizationsof Li-Yorke chaos ∗Corresponding Author. E-mail: [email protected],Tel: +903125850217 1 tomappingsinBanachspacesandcompletemetricspacescanbefoundin[17,18]. Besides,inthepaper [19], the Li-Yorke definition of chaos was modified in such a way that infinitely many periodic motions separated from the motions of the scrambled set are replaced with almost periodic ones. In the present paper, we will also consider the Li-Yorke chaos in this modified sense. Inpaper[20],wehaveconsideredunpredictabilityasaglobal phenomenoninweatherdynamicsonthe basisofconnectedLorenzsystems,andthisextensionofchaoswasperformedbymeansofperturbations ofLorenzsystemsbychaoticsolutionsoftheircounterparts. Oursuggestionmaybeakeytoexplainwhy the weather unpredictability is observed everywhere. This is true also for unpredictability and lack of forecastingineconomics[21,22]. Tocompletetheexplanationofweatherandeconomicalunpredictability asglobalphenomenabythe analysisofinterconnectedmodels,weneedtoargumentpersistenceofchaos ofamodelagainstchaoticperturbationsassolutionsofanothersimilarmodels. Fromthis pointofview, results of the present paper are very motivated. Another motivation of this study relies on the richness of a single chaotic model for motions, as a supply of infinitely many different periodic [15, 23], almost periodic [19] and even Poisson stable [24] motions. Of course, the diversity of motions is useless if one cannotcontrolchaos[25, 26]. In other words,chaospersistence means extensionofchaoscontrollability. In our former paper [27], persistence of chaos was considered in coupled Lorenz systems by tak- ing into account sensitivity and existence of infinitely many periodic motions embedded in the chaotic attractor. However, in the present paper, we consider chaos in the sense of Li-Yorke with countable infinity of almost periodic solutions in basis instead of periodic ones. In the present study, all results concerningtheexistenceofalmostperiodicmotionsaswellasLi-Yorkechaosarerigorouslyproved,anda more comprehensive theoreticaldiscussionis performedcomparedto [27]. The demonstrationof infinite number of Li-Yorke chaotic sets of solutions in the dynamics is another novelty of the present paper. Moreover, a numerical chaos control technique based on the Ott-Grebogi-Yorke (OGY) [25] algorithm is proposed for the stabilization of the unstable almost periodic motions. On the other hand, the paper [19] was concerned with the Li-Yorke chaotic dynamics of shunting inhibitory cellular neural networks with discontinuous external inputs. The concept of persistence of chaos was not considered in [19] at all. It was demonstrated in [19] that the chaotic structure of the discontinuity moments of the external inputs givesrise to the appearanceof chaos,and chaosdoes not takeplace in the dynamics either inthe case of regular discontinuity moments or in the absence of the discontinuous external inputs. On the contrary, in the present paper, a continuous chaotic perturbation is applied to a relay system which is already chaotic in the absence of perturbation, and it is proved that the chaotic structure is permanent in the dynamics regardless of the applied perturbation. As the source of chaotic perturbation we make use of solutions of another system of differential equations, but it is also possible to use any data which is known to be chaotic in the sense of Li-Yorke. 2 In the present study, we take into account the systems ′ x =F(x,t) (1.1) and ′ z =Az+f(z,t)+ν(t,ζ), (1.2) wheret R,thefunctionsF :Rm R Rm andf :Rn R Rn arecontinuousinalltheirarguments, ∈ × → × → f(z,t)isalmostperiodicintuniformlyforz Rn,A Rn×n isamatrixwhoseeigenvalueshavenegative ∈ ∈ real parts, and m , if ζ <t ζ , i Z, 0 2i 2i+1 ν(t,ζ)= ≤ ∈ (1.3) m1, if ζ2i−1 <t ζ2i, i Z, ≤ ∈ is a relay function in which m , m Rn with m = m . In (1.3), the sequence ζ = ζ , i Z, of 0 1 0 1 i ∈ 6 { } ∈ switching moments is defined through the equation ζ =τ +κ , (1.4) i i i where τj , j Z, is a family of equipotentially almost periodic sequences and the sequence κ , i ∈ { i} n o κ [0,1], is a solution of the logistic map 0 ∈ κ =G (κ ), (1.5) i+1 µ i where G (s)=µs(1 s) and µ is a parameter. Here, τj =τ τ for each integers i and j. µ − i i+j − i The presence of Li-Yorke chaos in the dynamics of (1.1) is one of our main assumptions. We fix a valueofµ between3.84and4suchthatthe map(1.5)ischaoticinthe senseofLi-Yorke[15]. Forsucha valueofthe parameter,the interval[0,1]isinvariantunder the iterationsof(1.5)[28]. Aninterpretation of the relay system (1.2) from the economic point of view can be found in [21]. According to the results of papers [19, 29], one can confirm that system (1.2) is Li-Yorke chaotic under certain conditions, which will be given in the next section. We establish a unidirectional coupling between the systems (1.1) and (1.2) to set up the following system, ′ y =Ay+f(y,t)+ν(t,ζ)+h(x(t)), (1.6) where x(t) is a solution of (1.1), and h : Rm Rn is a continuous function. Our purpose is to prove → 3 rigorously that the dynamics of system (1.6) is Li-Yorke chaotic. In other words, we will show that the chaos of (1.2) is persistent even if it is perturbed with the solutions of system (1.1). Sufficientconditionsonsystems(1.1),(1.2)and(1.6)forthepersistenceofchaos,andthedescriptions concerning almost periodicity and Li-Yorke chaos are provided in the next section. 2 Preliminaries Throughout the paper, we will make use of the usual Euclidean norm for vectors and the norm induced by the Euclidean norm for matrices [30]. In our theoretical discussions, we will make use of the concept of Li-Yorke chaotic set of functions [19, 31, 32]. The description of the concept is as follows. Suppose that Γ Rp is a bounded set, and denote by ⊂ = ψ(t) ψ :R Γ is continuous (2.7) H { | → } a collection of uniformly bounded functions. Acoupleoffunctions ψ(t),ψ(t) iscalledproximalifforarbitrarysmallǫ>0andarbitrary ∈H×H largeE >0, there exists a(cid:0)ninterval(cid:1)J with a lengthno less than E suchthat ψ(t) ψ(t) <ǫ for each − t J. On the other hand, we say that the couple ψ(t),ψ(t) is freq(cid:13)uently (ǫ ,∆(cid:13)) separated ∈ ∈H×H (cid:13) 0 (cid:13) − if there exist positive numbers ǫ ,∆ and infinitely(cid:0)many disj(cid:1)oint intervals with lengths no less than ∆ 0 such that ψ(t) ψ(t) > ǫ for each t from these intervals. We call the couple ψ(t),ψ(t) 0 − ∈ H×H a Li-Yorke(cid:13)pair if it is(cid:13)proximal and frequently (ǫ ,∆)-separated for some positiv(cid:0)e numbers(cid:1)ǫ and ∆. (cid:13) (cid:13) 0 0 Moreover,a set is called a scrambled set if does not contain any almost periodic function and S ⊂H S each couple of different functions inside is a Li-Yorke pair. S×S is called a Li-Yorke chaotic set if: (i) admits a countably infinite subset of almost periodic H H functions; (ii) there exists an uncountable scrambledset ; (iii) for any function ψ(t) and any S ⊂H ∈S almostperiodicfunctionψ(t) ,thepair ψ(t),ψ(t) isfrequently(ǫ ,∆) separatedforsomepositive 0 ∈H − numbers ǫ and ∆. (cid:0) (cid:1) 0 Remark 2.1 The criterion for the existence of a countably infinite subset of almost periodic functions in a Li-Yorke chaotic set can bereplaced with theexistenceof a countably infinitesubsetof quasi-periodic or periodic functions. The presence of Li-Yorke chaos with a basis consisting of infinitely many almost periodic motions in the dynamics of system (1.1) is one of our main assumptions. In other words, we suppose that system (1.1) possesses a set A of uniformly bounded solutions which is chaotic in the Li-Yorke sense. In this 4 case, there exists a compact region Λ Rm such that the trajectories of all solutions that belong to A ⊂ is inside Λ. An example of such a system was provided in [19] with a theoretical discussion. By means of the assumption on the matrix A, one can confirm the existence of positive numbers K and α such that eAt Ke−αt, t 0. In the remaining parts of the paper, Θ will stand for the set of k k ≤ ≥ all sequences ζ = ζ , i Z, generated by equation (1.4). Since the value of µ in (1.5) is fixed between i { } ∈ 3.84 and 4 so that the map is chaotic in the sense of Li-Yorke, the map (1.5) possesses a periodic orbit with period p for each natural number p [15]. We will denote by Θ the countably infinite set of all P ⊂ sequences ζ = ζ , i Z, generated by (1.4) in which κ is a periodic solution of (1.5). i i { } ∈ { } The following assumptions are required: (C1) There exists a positive number M such that sup f(y,t) M ; f f y∈Rn,t∈Rk k≤ (C2) There exists a positive number L < α/K such that f(y ,t) f(y ,t)) L y y for all f 1 2 f 1 2 k − k ≤ k − k y , y Rn, t R; 1 2 ∈ ∈ (C3) There exists a positive number ζ such that ζ ζ ζ for each ζ = ζ Θ and i Z; i+1 i i − ≥ { }∈ ∈ (C4) There exists a positive number L such that h(x ) h(x ) L x x for all x ,x Λ; 1 1 2 1 1 2 1 2 k − k≤ k − k ∈ (C5) There exists a positive number L such that h(x ) h(x ) L x x for all x ,x Λ; 2 1 2 2 1 2 1 2 k − k≥ k − k ∈ (C6) There exists a positive number M such that sup F(x,t) M . F F x∈Λ,t∈Rk k≤ Under the conditions (C1) (C3), one can confirmusing the results of papers [19, 29] that the relay − system (1.2) is Li-Yorke chaotic with infinitely many almost periodic motions in basis for the values of the parameter µ between 3.84 and 4. We refer the reader to [32] for further information about the dynamics of relay systems. Under the same conditions, for a given solution x(t) A of (1.1) and a ∈ sequenceζ Θ,system(1.6)possessesa uniquesolutionφ (t) whichis boundedonthe wholerealaxis x,ζ ∈ [33], and this solution satisfies the relation t φ (t)= eA(t−s)[f(φ (s),s)+ν(s,ζ)+h(x(s))]ds. (2.8) x,ζ x,ζ −∞ Z For a fixed sequence ζ Θ and a fixed solution x(t) A, the bounded solution φ (t) attracts all x,ζ ∈ ∈ other solutions of (1.6) such that the inequality φx,ζ(t) yx,ζ(t) K φx,ζ(t0) y0 e(KLf−α)(t−t0) is k − k≤ k − k satisfied for t t , where y (t) is a solution of (1.6) with y (t )=y for some y Rn and t R. 0 x,ζ x,ζ 0 0 0 0 ≥ ∈ ∈ To provide a theoretical discussion for the persistence of chaos, for each sequence ζ Θ, let us ∈ introduce the set B consisting of all bounded solutions φ (t) of system (1.6) in which x(t) belongs to ζ x,ζ A. K One can confirm that sup y(t) M for all y(t) B and η Θ, where M = (m+M +M ), ζ f h t∈Rk k ≤ ∈ ∈ α m=max m , m , and M =max h(x) . 0 1 h {k k k k} x∈Λ k k 5 The next section is devoted to the almost periodic solutions of system (1.6). 3 Existence of almost periodic solutions Let σ , i Z, be a sequence in Rn. An integer p is an ǫ almost period of the sequence σ , if the i i { } ∈ − { } inequality σ σ < ǫ holds for all i Z. On the other hand, a set D R is said to be relatively i+p i k − k ∈ ⊂ dense if there exists a number l >0 such that r,r+l D = for all r R. Moreover, σ is almost i ∩ 6 ∅ ∈ { } periodic, if for any ǫ>0, there exists a relative(cid:2)ly dense(cid:3) set of its ǫ almost periods. − Let us denote ξj = ξ ξ for any integers i and j. We call the family of sequences ξj , j Z, i i+j − i { i} ∈ equipotentially almost periodic if for an arbitrary ǫ > 0, there exists a relatively dense set of ǫ almost − periods, common for all sequences ξj , j Z [34]. { i} ∈ Acontinuousfunctionϕ:R Rn issaidtobealmostperiodicifforanyǫ>0thereexistsl >0such → thatforanyintervalwithlengthlthereexistsanumberωinthisintervalsatisfying ϕ(t+ω) ϕ(t) <ǫ k − k for all t R [33]-[35]. ∈ The following modified version of an assertion from [36] is needed for the proof of the main theorem of the present section. Lemma 3.1 Suppose that ϕ : R Rn is a continuous almost periodic function and ξj , j Z, is a → i ∈ n o family of equipotentially almost periodic sequences. Then, for arbitrary η >0 and 0<θ <η, there exist relatively dense sets of real numbers Ω and even integers Q such that (i) ϕ(t+ω) ϕ(t) <η, t R; k − k ∈ (ii) ξq ω <θ, i Z, ω Ω, q Q. | i − | ∈ ∈ ∈ The existence of almost periodic solutions in system (1.6) is considered in the following theorem. Theorem 3.1 Suppose that conditions (C1) (C4) are valid. If the sequence ζ = ζ , i Z, belongs to i − { } ∈ and x(t) is an almost periodic solution of (1.1), then the bounded solution φ (t) is the unique almost x,ζ P periodic solution of (1.6). Proof. Let us denote by C the set of all almost periodic functions ψ : R Rn satisfying ψ M, 0 → k k0 ≤ where ψ =sup ψ(t) . Define the operator Π on C through the equation k k0 t∈Rk k 0 t Πψ(t)= eA(t−s)(f(ψ(s),s)+v(s,ζ)+h(x(s)))ds. −∞ Z First of all, we will show that Π(C ) C . Let ψ be an element of C . One can easily verify that 0 0 0 ⊆ Πψ M. k k0 ≤ 6 In order to show that Πψ(t) is almost periodic, let us fix an arbitrary positive number ǫ and set 1+L +L 2 m m f 1 0 1 H =K + k − k . 0 α 1 e−αζ (cid:18) − (cid:19) ζ ǫ Since Πψ is uniformly continuous, there exists a positive number η satisfying η < and η 5 ≤ 3H ǫ 0 such that Πψ(t ) Πψ(t ) < whenever t t <4η. 1 2 1 2 k − k 3 | − | According to Lemma A.3 [19], ζj , j Z, is a family of equipotentially almost periodic sequences, i ∈ since τj ,j Z, isafamilyofeqnuipootentiallyalmostperiodicsequencesandthe sequence κ ,i Z, i ∈ { i} ∈ n o is periodic. Let θ be a number with 0 < θ < η, and consider the numbers ω Ω and q Q as in ∈ ∈ Lemma 3.1 such that (i) ψ(t+ω) ψ(t) < η, t R; (ii) f(z,t+ω) f(z,t) < η, z Rn, t Z, k − k ∈ k − k ∈ ∈ (iii) x(t+ω) x(t) <η, t R; (iv) ζq ω <θ, i Z. k − k ∈ | i − | ∈ Foranyk Z,itcanbeverifiedthatifs (ζ +θ,ζ θ),thenν(s+ω,ζ) ν(s,ζ)=0.Therefore, k k+1 ∈ ∈ − − for each t from the intervals (ζ +η,ζ η), i Z, one can confirm that i i+1 − ∈ ǫ Πψ(t+ω) Πψ(t) <H η . 0 k − k ≤ 3 Supposethatt (ζe η,ζe+η)forsomei Z.Becauseη issufficientlysmallsuchthat5η <ζ,t+3η ∈ i− i ∈ belongs to the interval (ζei+η,ζei+1−η) so tehat ǫ Πψ(t+ω+3η) Πψ(t+3η) < . k − k 3 Thus, we have that Πψ(t+ω) Πψ(t) Πψ(t+ω) Πψ(t+ω+3η) k − k≤k − k + Πψ(t+ω+3η) Πψ(t+3η) + Πψ(t+3η) Πψ(t) k − k k − k <ǫ. Therefore, Πψ(t+ω) Πψ(t) <ǫforallt R.AccordinglyΠψ(t)isalmostperiodicandΠ(C ) C . 0 0 k − k ∈ ⊆ Now, let ψ and ψ be elements of C . Then, 1 2 0 t KL Πψ (t) Πψ (t) KL e−α(t−s) ψ (s) ψ (s) ds f ψ ψ . k 1 − 2 k≤ f k 1 − 2 k ≤ α k 1− 2k0 −∞ Z KL KL Hence, Πψ Πψ f ψ ψ . Since f < 1, the operator Π : C C is contractive k 1− 2k0 ≤ α k 1− 2k0 α 0 → 0 according to condition (C2). Consequently, the bounded solution φ (t) is the unique almost periodic x,ζ solution of system (1.6). (cid:3) We will consider the chaotic dynamics of system (1.6) in the next section. 7 4 Persistence of chaos The followinglemmas, whichareconcernedwith the proximalityandfrequentseparationfeaturesofthe bounded solutions of (1.6), are needed for the proof of the main theorem of the present section. Lemma 4.1 Supposethatconditions(C1) (C4)arevalid. Ifacoupleoffunctions(x(t),x(t)) A A − ∈ × is proximal, then the same is true for the couple (φx,ζ(t),φxe,ζ(t)) Bζ Bζ for any sequence ζ Θ. ∈ × e ∈ Lemma 4.2 Suppose that conditions (C1) (C3), (C5) and (C6) are valid. If a couple of functions − (x(t),x(t)) A A is frequently(ǫ ,∆) separated for some positive numbersǫ and ∆, then thereexist 0 0 ∈ × − positive numbers ǫ1 and ∆ such that the couple of functions (φx,ζ(t),φxe,ζ(t)) Bζ Bζ is frequently e ∈ × (ǫ ,∆) separated for any sequence ζ Θ. 1 − ∈ The proofs of Lemma 4.1 and Lemma 4.2 are provided in the Appendix. The main result of the present section is mentioned in the next theorem. Theorem 4.1 Suppose that the conditions (C1) (C6) are valid. If ζ , then B is a Li-Yorke ζ − ∈ P chaotic set. Proof. Since the set A is Li-Yorke chaotic, there exists a countably infinite set A of almost 1 AP ⊂ periodicsolutions. Fixanarbitrarysequenceζ ,andletusdenoteby ζ thesubsetofB consisting ∈P AP2 ζ of bounded solutions φ (t) of (1.6) such that x(t) . According to Theorem 3.1, the elements of x,ζ 1 ∈ AP ζ are the almost periodic solutions of system(1.6). It canbe verifiedusing condition (C5) that ζ AP2 AP2 is also countably infinite. Now, suppose that A is an uncountable scrambled set. Let us define the set of functions 1 S ⊂ ζ = φ (t): x(t) . The set ζ B is uncountable, and it does not contain any almost S2 { x,ζ ∈S1} S2 ⊂ ζ periodic solutions in accordance with condition (C5). Since each couple of different functions inside is a Li-Yorke pair, Lemma 4.1 and Lemma 4.2 together imply that the same is true for each 1 1 S ×S coupleofdifferentfunctions inside ζ ζ. Hence, ζ isascrambledset. Besides,onecanconfirmusing S2×S2 S2 Lemma4.2onemoretimethateachcoupleoffunctionsinside ζ ζ isfrequently ǫ ,∆ separated S2×AP2 1 − for some positive numbers ǫ and ∆. Consequently, B is a Li-Yorke chaotic set for e(cid:0)ach ζ(cid:1) . (cid:3) 1 ζ ∈P Remark 4.1 In Theorem 4.1, we demonstrate that for each fixed sequence ζ , system (1.6) admits ∈ P a Li-Yorke chaotic set B . It can be easily shown that B B = whenever ζ and η are different ζ ζ η ∩ ∅ sequences in . Therefore, there are countably infinite Li-Yorke chaotic sets in the dynamics of (1.6). P An illustrative example which supports the theoretical results is presented in the next section based on Duffing oscillators. 8 5 An example Consider the forced Duffing equation x′′+1.5x′+4x+0.02x3 =cost+ν (t,ζ), (5.9) 1 where 0.5, if ζ <t ζ , i Z, 2i 2i+1 ν (t,ζ)= ≤ ∈ (5.10) 1 2.9, if ζ2i−1 <t ζ2i, i Z, ≤ ∈ is a relay function. The sequence ζ = ζ, i Z, of switching moments is defined through the equation i { } ∈ ζ =1.05i+κ (5.11) i i in which the sequence κ , κ [0,1], is a solution of the logistic map (1.5). Clearly, ζ ζ 0.05 i 0 i+1 i { } ∈ − ≥ for each i Z. ∈ Using the variables x =x and x =x′, one can write (5.9) as a system in the following form, 1 2 x′ =x , 1 2 (5.12) x′ = 4x 1.5x 0.02x3+cost+ν (t,ζ). 2 − 1− 2− 1 1 0 1 The matrix of coefficients corresponding to the linear part of (5.12) admits the eigen-   4 1.5  − −  3 √5   values i . The coefficient 0.02 of the nonlinear term in (5.12) is sufficiently small in absolute −4 ± 4 − value such that for each periodic orbit κ , i Z, of the map (1.5), system (5.12) possesses a unique i { } ∈ quasi-periodic solution since the periods of the functions cost and ν (t,ζ) are incommensurable. In 1 what follows, we will make use of the value µ=3.9 so that system (5.12) possesses Li-Yorke chaos with infinitely many quasi-periodic motions in basis [19, 29]. In order to show the chaotic behavior of system (5.12), in Figure 1 we depict the x coordinate of 1 − the solution with the initial data ζ = 0.56, x (t ) = 0.24, x (t ) = 0.17 and t = 0.56. The simulation 0 1 0 2 0 0 result seen in Figure 1 reveals the presence of chaos in the dynamics of (5.12). One can numerically verify that the chaotic solutions of (5.12) take place inside the compact region Λ= (x ,x ) R2 : 0.4 x 1.25, 1.3 x 1.3 . (5.13) 1 2 1 2 ∈ − ≤ ≤ − ≤ ≤ (cid:8) (cid:9) 9 1 1 0.5 x 0 0 50 100 150 200 250 t Figure 1: Chaotic behavior in the x coordinate of system (5.12). 1 − Next, we take into account the Duffing equation z′′+3.5z′+2.5z 0.01z3 = 1.5cos(πt)+ν (t,ζ). (5.14) 2 − − In equation (5.14), the relay function ν (t,ζ) is defined as 2 1.7, if ζ <t ζ , i Z, 2i 2i+1 ν (t,ζ)= ≤ ∈ (5.15) 2  0.4, if ζ2i−1 <t ζ2i, i Z, − ≤ ∈  in which the sequence ζ = ζ , i Z, of switching moments is defined in the same way as in equation i { } ∈ (5.11). Under the variables z = z and z = z′, one can confirm that equation (5.14) is equivalent to the 1 2 system z′ =z , 1 2 (5.16) z′ = 2.5z 3.5z +0.01z3 1.5cos(πt)+ν (t,ζ). 2 − 1− 2 1− 2 0 1 0 System(5.16)isintheformof(1.2)withA= andf(z ,z ,t)= .   1 2   2.5 3.5 0.01z3 0.5cos(πt)  − −   1−  It can be verified that the eigenvalues of the matrix A are 1 and 5/2. Moreover, the inequality − − eAt Ke−αt holdsforallt 0withK =5.0695andα=1. Thecoefficient0.01ofthenonlinearterm k k≤ ≥ in (5.16) is sufficiently small such that according to the results of [29] system (5.16) admits Li-Yorke chaos,but this time the basis of the chaos consists of infinitely many periodic motions since the periods of the functions cos(πt) and ν (t,ζ) are commensurable for each periodic orbit κ , i Z, of (1.5). 2 i { } ∈ Figure 2 represents the z coordinate of the solution of (5.16) corresponding to the initial data 1 − ζ = 0.56, z (t ) = 0.03, z (t ) = 0.32, where t = 0.56. One can observe in Figure 2 that Li–Yorke 0 1 0 2 0 0 − chaos takes place in the dynamics of system (5.16). Now, to demonstrate the persistence of chaos, we establish a unidirectional coupling between the 10