Replication of Period-Doubling Route to Chaos in Systems with Delay
aa r X i v : . [ n li n . C D ] F e b Replication of Period-Doubling Route to Chaos in Systems withDelay
Mehmet Onur Fen , ∗ , Fatma Tokmak Fen Department of Mathematics, TED University, 06420 Ankara, Turkey Department of Mathematics, Gazi University, 06500 Ankara, Turkey
Abstract
In this study, replication of period-doubling route to chaos in coupled systems with delay is considered.The replication of sensitivity is rigorously proved based on a novel definition. Period-doubling cascadesin coupled systems with delay are theoretically discussed. Examples with simulations supporting thetheoretical results concerning sensitivity and period-doubling cascade are provided.
Keywords:
Replication of chaos; Period-doubling cascade; Sensitivity; Systems with delay; Unidirec-tional coupling
Primary 34K23; Secondary 34K18
The occurrence of time delays is an issue that is crucial for nonlinear processes. In the case that timedelays are taken into consideration, delay differential equations can be utilized in modeling of suchprocesses. Infinite-dimensional dynamical systems can arise from delay differential equations [1, 2], andsuch equations can exhibit chaotic behavior, even in scalar case [3, 4]. Delay differential equations areuseful in various fields such as neural networks, secure communication, mechanics, robotics, medicine,biology, economics, and lasers [5]-[12]. Some open problems concerning differential equations with delaycan be found in the papers [13]-[15]. Another phenomenon that can be observed in dynamics of nonlinearsystems is period-doubling cascade, which is capable of giving rise to the emergence of chaos. Likewisesystems with delay, the presence of period-doubling cascades as well as chaos can be observed and haveapplications in a variety of scientific areas [16]-[23].Motivated by the effective scientific roles of chaos and systems with delay, in the present study, weconsider the replication of period-doubling route to chaos in unidirectionally coupled systems in whichthe secondary system is with delay. More precisely, we take into account the systems ∗ Corresponding Author. E-mail: [email protected], Tel: +90 312 585 0217 ′ ( t ) = F ( t, x ( t )) (1.1)and y ′ ( t ) = Ay ( t ) + G ( t, x ( t ) , y ( t − τ )) (1.2)where the functions F : R × R m → R m and G : R × R m × R n → R n are continuous in all of theirarguments, all eigenvalues of the matrix A ∈ R n × n have negative real parts, and τ is a positive number.Our purpose is to rigorously prove that system (1.2) exhibits chaotic motions, provided that the same istrue for system (1.1) under certain conditions that will be mentioned in the next section. We understandchaos as the presence of sensitivity, which can be considered as the main ingredient of chaos [24]-[26],and infinitely many unstable periodic solutions in a compact region.The existence of chaos in delay differential equations was demonstrated in the studies [3, 4, 27]. Theresults of the papers [3, 4] are based on the Li-Yorke definition of chaos [28, 29]. Utilizing the resultsobtained in their study [30], Lani-Wayda and Walther [27] constructed a delay differential equationpossessing chaotic behavior. On the other hand, global attractors for delay differential equations wereinvestigated in [31, 32]. The global attractiveness of a three-dimensional compact invariant set of adifferential equation modeling a system governed by delayed positive feedback and instantaneous dampingwas shown by Krisztin and Walther [31]. In addition to the presence of a global attractor in the dynamicsof a differential equation with state-dependent delay, one of its topological properties were investigatedby [32]. Moreover, results concerning periodic solutions of state-dependent delay differential equationswere presented by Kuang and Smith [33]. The technique provided in this study is different comparedto the papers [3, 4, 27] such that we take into account replication of sensitivity and period-doublingcascades in unidirectionally coupled systems.Regular inputs such as periodic, quasi-periodic, and almost periodic motions can lead to the formationof outputs of the same types in dynamics of certain types of differential equations [34, 35]. The main ideaof our investigation is the usage of chaotic motions as inputs in systems with delay, and it is demonstratedthat chaotic outputs are obtained. The inputs are supplied from solutions of another system possessingchaos. The reader is referred to the papers [36, 37] for some applications of input-output systems.The foundations of chaos generation in systems of differential equations by means of perturbations andimpulsive actions were laid by Akhmet [38]-[40]. An answer to the question whether continuous chaoticinputs generate chaotic outputs was given in the study [42] for systems without delay. It was rigorouslyproved in [42] that under certain conditions chaotic dynamics of a system of differential equations canbe replicated by another system under unidirectional coupling between them. Chaos in the senses of2evaney [41] and Li-Yorke [28] as well as period-doubling cascade were considered by Akhmet and Fen[42]. Moreover, the study [43] was concerned with replication of period-doubling route to chaos in systemswith impulsive actions. Systems with delay were not taken into account in the studies [42, 43]. Themain novelty of the present research is the consideration of replication of chaos problem for systems withdelay. Due to the presence of delay, a novel definition for replication of sensitivity is provided and thecontraction mapping principle is utilized for its verification. The obtained results are valid for systemswith arbitrary high dimensions. The book [44] comprises some applications of the replication of chaostechnique to neural networks, economics, and weather dynamics.In the literature, unidirectionally coupled chaotic systems have been considered within the scope ofsynchronization [45]-[47]. In the case of identical systems, synchronization occurs when asymptotic prox-imity of the states of the drive and response systems is valid [45]. For the presence of synchronization inthe dynamics of non-identical systems, the asymptotic proximity is considered with the help of a func-tional relation, which determines the phase space trajectory of the response system from the trajectoryof the drive [46]. The approach utilized in this study is different from synchronization of chaos sincethe coupled system (1.1)-(1.2) is not taken into account from the asymptotic point of view. For thatreason, following the terminology of paper [42], we call system (1.1) the generator and system (1.2) the replicator .The rest of the paper is organized as follows. In the next section, preliminary results and conditionson the coupled system (1.1)-(1.2), which are required for replication of sensitivity and the existenceof unstable periodic solutions, are provided. In Section 3, replication of sensitivity is theoreticallyinvestigated. Section 4, on the other hand, is concerned with replication of period-doubling cascade.Section 5 is devoted to examples in which the Lorenz system [24] and Duffing equation [48] are utilizedas generator systems. Finally, some concluding remarks are given in Section 6. Our main assumption on the generator system (1.1) is the existence of a nonempty set A of all solutionsof the system that are uniformly bounded on R . In this case there exists a compact set Λ ⊂ R m suchthat the trajectories of all solutions that belong to A lie inside Λ . We also assume that there exists apositive number T such that F ( t + T, x ) = F ( t, x ) and G ( t + T, x, y ) = G ( t, x, y ) (2.3)3or all t ∈ R , x ∈ R m , and y ∈ R n .Since we suppose that all eigenvalues of the matrix A in the replicator system (1.2) have negativereal parts, there exist numbers K ≥ and ω > such that (cid:13)(cid:13) e At (cid:13)(cid:13) ≤ Ke − ωt for all t ≥ .Throughout the paper, we make use of the usual Euclidean norm for vectors and the spectral normfor square matrices.The following conditions are required. (C1) There exists a positive number L F such that k F ( t, x ) − F ( t, e x ) k ≤ L F k x − e x k for all t ∈ R and x, e x ∈ Λ ; (C2) There exists a positive number L such that k G ( t, x, y ) − G ( t, e x, y ) k ≥ L k x − e x k for all t ∈ R , x, e x ∈ Λ , and y ∈ R n ; (C3) There exists a positive number L such that k G ( t, x, y ) − G ( t, e x, y ) k ≤ L k x − e x k for all t ∈ R , x, e x ∈ Λ , and y ∈ R n ; (C4) There exists a positive number L such that k G ( t, x, y ) − G ( t, x, e y ) k ≤ L k y − e y k for all t ∈ R , x ∈ Λ , and y, e y ∈ R n ; (C5) There exists a positive number M G such that sup t ∈ R ,x ∈ Λ ,y ∈ R n k G ( t, x, y ) k ≤ M G ; (C6) ω − KL e ωτ/ > .For the existence and uniqueness of the bounded solutions of system (1.2) the conditions ( C and ( C are utilized. Conditions ( C , ( C , ( C , and ( C , on the other hand, are required to showthe proximity of the bounded solutions of (1.2) on a closed interval with length τ in the verification ofreplication of sensitivity. Moreover, the condition ( C is used in the replication of sensitivity to show thedivergence of the bounded solutions, and the condition ( C is required also in their global exponentialstability.Suppose that the conditions ( C , ( C hold. For a fixed solution x ∈ A of system (1.1), it can beverified that a function y ( t ) which is bounded on the whole real axis is a solution of system (1.2) if andonly if the integral equation y ( t ) = t Z −∞ e A ( t − s ) G ( s, x ( s ) , y ( s − τ )) ds is satisfied. Denote by C the set of continuous functions ϕ : R → R n with k ϕ k ∞ ≤ M , where k ϕ k ∞ = sup t ∈ R k ϕ ( t ) k M = KM G ω . (2.4)Let us define the operator Γ on C through the equation Γ ϕ ( t ) = t Z −∞ e A ( t − s ) G ( s, x ( s ) , ϕ ( s − τ )) ds. If ϕ is a function in C , then one can confirm that k Γ ϕ k ∞ ≤ M , which yields Γ( C ) ⊆ C . Additionally,if ϕ , ϕ belong to C , then k Γ ϕ − Γ ϕ k ∞ ≤ KL ω k ϕ − ϕ k ∞ . Therefore, if ω − KL > , thenthe operator Γ is a contraction. For that reason, if the conditions ( C , ( C hold and the inequality ω − KL > is valid, then for each fixed solution x ∈ A of system (1.1), there exists a unique solution φ x of system (1.2) which is bounded on the whole real axis such that sup t ∈ R k φ x ( t ) k ≤ M and φ x ( t ) = t Z −∞ e A ( t − s ) G ( s, x ( s ) , φ x ( s − τ )) ds. (2.5)It is worth noting that the condition ( C implies the inequality ω − KL > . Moreover, if conditions ( C − ( C are satisfied, then for each x ∈ A the bounded solution φ x ( t ) of system (1.2) is globallyexponentially stable [49].To investigate the replication of sensitivity theoretically, we introduce the set of uniformly boundedfunctions B = { φ x ( t ) : x ∈ A } . (2.6)There is a one-to-one correspondence between the sets A and B under the condition ( C . In otherwords, for each solution x ∈ A of generator (1.1) there exists a unique bounded solution φ x ∈ B ofreplicator (1.2), and vice versa. The definition of sensitivity for system (1.1) is as follows.
Definition 3.1 [42]. System (1.1) is called sensitive if there exist positive numbers ǫ and ∆ such that or an arbitrary positive number δ and for each x ∈ A , there exist x ∈ A , t ∈ R , and an interval J ⊂ [ t , ∞ ) with a length no less than ∆ such that k x ( t ) − x ( t ) k < δ and k x ( t ) − x ( t ) k > ǫ for all t ∈ J . The next definition is concerned with the replication of sensitivity by systems with delay.
Definition 3.2
System (1.2) replicates the sensitivity of system (1.1) if there exist positive numbers ǫ and ∆ such that for an arbitrary positive number δ and for each bounded solution φ x ∈ B , there exista bounded solution φ x ∈ B , t ∈ R , and an interval e J ⊂ [ t , ∞ ) with a length no less than ∆ such that sup t ∈ [ t − τ,t ] k φ x ( t ) − φ x ( t ) k < δ and k φ x ( t ) − φ x ( t ) k > ǫ for all t ∈ e J . The main result of the present section is provided in the next theorem. In the proof of the theorem,first of all, utilizing the initial proximity of two solutions of system (1.1) in A we estimate the distancebetween them backward in time. Then, this estimation is used to verify the initial proximity of thebounded solution of system (1.2) on an interval of length τ by means of the contraction mapping principle.Finally, an equicontinuous family of functions is constructed based on the equicontinuity of both A andthe bounded solutions of system (1.2), and it is use to show the divergence of the bounded solutions of(1.2). The provided proof technique makes it possible to determine the numbers ǫ and ∆ mentioned inDefinition 3.2. Theorem 3.1
Assume that the conditions ( C − ( C are valid. If system (1.1) is sensitive, thensystem (1.2) replicates the sensitivity of (1.1). Proof.
Fix an arbitrary positive number δ and a bounded solution φ x ∈ B of system (1.2). Let usdenote R = 2 KM ωω − KL e ωτ/ , (3.7)where M is the number defined by (2.4), and R = KL ω − KL . (3.8)The numbers R and R are positive by condition ( C . Suppose that δ = γδ e − NL F , where γ is apositive number such that γ < R + R (3.9)and N is a positive number satisfying the inequality N ≥ τ + 2 ω ln (cid:18) γδ (cid:19) . (3.10)6ince system (1.1) is sensitive, there exist positive numbers ǫ and ∆ such that k x ( t ) − x ( t ) k < δ (3.11)and k x ( t ) − x ( t ) k > ǫ , t ∈ J, (3.12)for some x ∈ A , t ∈ R , and for some interval J ⊂ [ t , ∞ ) with a length no less than ∆ .Making use of the relation x ( t ) − x ( t ) = x ( t ) − x ( t ) + t Z t ( F ( s, x ( s )) − F ( s, x ( s ))) ds, it can be verified for t ∈ [ t − N, t ] that k x ( t ) − x ( t ) k ≤ k x ( t ) − x ( t ) k + (cid:12)(cid:12)(cid:12)(cid:12) t Z t L F k x ( s ) − x ( s ) k ds (cid:12)(cid:12)(cid:12)(cid:12) , Applying the Gronwall-Bellman inequality [50] we obtain that k x ( t ) − x ( t ) k ≤ k x ( t ) − x ( t ) k e L F | t − t | , t ∈ [ t − N, t ] . Therefore, k x ( t ) − x ( t ) k < γδ for t ∈ [ t − N, t ] in accordance with the inequality (3.11).One can confirm that the function ψ ( t ) = φ x ( t ) − φ x ( t ) is a solution of the system ψ ′ ( t ) = Aψ ( t ) + G ( t, x ( t ) , ψ ( t − τ ) + φ x ( t − τ )) − G ( t, x ( t ) , φ x ( t − τ )) . (3.13)For t ≥ t − N , ψ ( t ) satisfies the integral equation ψ ( t ) = e A ( t − t + N ) ( φ x ( t − N ) − φ x ( t − N ))+ t Z t − N e A ( t − s ) ( G ( s, x ( s ) , ψ ( s − τ ) + φ x ( s − τ )) − G ( s, x ( s ) , φ x ( s − τ ))) ds. Let us denote by H the set of continuous functions ψ ( t ) defined on R such that k ψ ( t ) k ≤ R e − ω ( t − t + N ) / + R γδ (3.14)for t − N − τ ≤ t ≤ t , where the numbers R and R are respectively defined by the equations (3.7) and73.8), and k ψ k ∞ ≤ K (cid:18) M + M G ω (cid:19) in which k ψ k ∞ = sup t ∈ R k ψ ( t ) k . Define an operator Π on H throughthe equation Π ψ ( t ) = φ x ( t ) − φ x ( t ) , t < t − N,e A ( t − t + N ) ( φ x ( t − N ) − φ x ( t − N ))+ t Z t − N e A ( t − s ) ( G ( s, x ( s ) , ψ ( s − τ ) + φ x ( s − τ )) − G ( s, x ( s ) , φ x ( s − τ ))) ds, t ≥ t − N. First of all, we will show that
Π ( H ) ⊆ H . Suppose that ψ ( t ) belongs to H . If t − N ≤ t ≤ t ,then it can be obtained using inequality (3.14) that k Π ψ ( t ) k ≤ (cid:13)(cid:13)(cid:13) e A ( t − t + N ) (cid:13)(cid:13)(cid:13) k φ x ( t − N ) − φ x ( t − N ) k + t Z t − N (cid:13)(cid:13)(cid:13) e A ( t − s ) (cid:13)(cid:13)(cid:13) k G ( s, x ( s ) , ψ ( s − τ ) + φ x ( s − τ )) − G ( s, x ( s ) , φ x ( s − τ )) k ds + t Z t − N (cid:13)(cid:13)(cid:13) e A ( t − s ) (cid:13)(cid:13)(cid:13) k G ( s, x ( s ) , φ x ( s − τ )) − G ( s, x ( s ) , φ x ( s − τ )) k ds ≤ KM e − ω ( t − t + N ) + t Z t − N KL e − ω ( t − s ) (cid:16) R e − ω ( s − τ − t + N ) / + R γδ (cid:17) ds + t Z t − N KL γδ e − ω ( t − s ) ds< K (cid:18) M + L R e ωτ/ ω (cid:19) e − ω ( t − t + N ) / + Kγδ ω ( L + L R )= R e − ω ( t − t + N ) / + R γδ . Additionally, since M < R the inequality k Π ψ ( t ) k < R e − ω ( t − t + N ) / + R γδ is also valid for t − N − τ ≤ t < t − N . On the other hand, it can be confirmed that k Π ψ k ∞ ≤ K (cid:18) M + M G ω (cid:19) .Thus, Π ( H ) ⊆ H .Now, our purpose is to verify that the operator Π is a contraction. Let ψ ( t ) and ψ ( t ) be functionsthat belong to H . The inequality k Π ψ ( t ) − Π ψ ( t ) k ≤ t Z t − N (cid:13)(cid:13)(cid:13) e A ( t − s ) (cid:13)(cid:13)(cid:13) (cid:13)(cid:13) G ( s, x ( s ) , ψ ( s − τ ) + φ x ( s − τ )) − G ( s, x ( s ) , ψ ( s − τ ) + φ x ( s − τ )) (cid:13)(cid:13) ds< KL ω (cid:16) − e − ω ( t − t + N ) (cid:17) k ψ − ψ k ∞
8s valid for t ≥ t − N . Moreover, k Π ψ ( t ) − Π ψ ( t ) k = 0 for t < t − N . Hence, k Π ψ − Π ψ k ∞ ≤ KL ω k ψ − ψ k ∞ , and the operator Π is a contraction since the inequality KL ω < holds by condition ( C .According to the uniqueness of solutions of system (3.13), the function ψ ( t ) = φ x ( t ) − φ x ( t ) is theunique fixed point of the operator Π . Let us denote ψ ( t ) = φ x ( t ) − φ x ( t ) , t < t − N,e A ( t − t + N ) ( φ x ( t − N ) − φ x ( t − N )) , t ≥ t − N, which belongs to H . The sequence of functions { ψ k ( t ) } , where ψ k +1 ( t ) = Π ψ k ( t ) , k = 0 , , , . . . , converges to φ x ( t ) − φ x ( t ) on R . Thus, k φ x ( t ) − φ x ( t ) k ≤ R e − ω ( t − t + N ) / + R γδ for t − N − τ ≤ t ≤ t . Using the inequalities (3.9) and (3.10), one can confirm for t − τ ≤ t ≤ t that k φ x ( t ) − φ x ( t ) k ≤ R e − ω ( − τ + N ) / + R γδ ≤ ( R + R ) γδ < δ . Hence, we have sup t ∈ [ t − τ,t ] k φ x ( t ) − φ x ( t ) k < δ .Next, we will show the existence of positive numbers ǫ and ∆ such that k φ x ( t ) − φ x ( t ) k > ǫ for all t ∈ e J , where e J ⊂ [ t , ∞ ) is an interval with length ∆ .Let M F = sup t ∈ R ,x ∈ Λ k F ( t, x ) k . Both of the sets A and B = { φ x ( t − τ ) : x ∈ A } are equicontinuousfamilies on R since sup t ∈ R k x ′ ( t ) k ≤ M F and sup t ∈ R k φ ′ x ( t − τ ) k ≤ k A k M + M G for each solution x ∈ A ofsystem (1.1).Suppose that G ( t, x, y ) = ( G ( t, x, y ) , G ( t, x, y ) , . . . , G n ( t, x, y )) , where G i ( t, x, y ) , i = 1 , , . . . , n ,are real valued functions. Let us denote Λ = { y ∈ R n : k y k ≤ M } and define the function G : R × Λ × Λ × Λ → R n by G ( t, x , x , y ) = G ( t, x , y ) − G ( t, x , y ) . Due to the periodicity of the function G ( t, x, y ) in t , the function G ( t, x , x , y ) is uniformly continuous on R × Λ × Λ × Λ . Therefore, the setof functions F = { G i ( t, x ( t ) , φ x ( t − τ )) − G i ( t, x ( t ) , φ x ( t − τ )) : 1 ≤ i ≤ n, x, x ∈ A } is an equicontinuous family on R . Thus, there exists a positive number ξ < ∆ , which is independent of9he functions x ( t ) and x ( t ) , such that for each t , t ∈ R with | t − t | < ξ , the inequality (cid:12)(cid:12) ( G i ( t , x ( t ) , φ x ( t − τ )) − G i ( t , x ( t ) , φ x ( t − τ ))) − ( G i ( t , x ( t ) , φ x ( t − τ )) − G i ( t , x ( t ) , φ x ( t − τ ))) (cid:12)(cid:12) < L ǫ √ n (3.15)holds for each i = 1 , , . . . , n .Let us denote by η the midpoint of the interval J and set α = η − ξ/ . There exists an integer j , ≤ j ≤ n , such that | G j ( η, x ( η ) , φ x ( η − τ )) − G j ( η, x ( η ) , φ x ( η − τ )) |≥ √ n k G ( η, x ( η ) , φ x ( η − τ )) − G ( η, x ( η ) , φ x ( η − τ )) k . We obtain by means of the condition ( C and inequality (3.12) that | G j ( η, x ( η ) , φ x ( η − τ )) − G j ( η, x ( η ) , φ x ( η − τ )) | ≥ L √ n k x ( η ) − x ( η ) k > L ǫ √ n . One can confirm in accordance with inequality (3.15) that | G j ( t, x ( t ) , φ x ( t − τ )) − G j ( t, x ( t ) , φ x ( t − τ )) | > | G j ( η, x ( η ) , φ x ( η − τ )) − G j ( η, x ( η ) , φ x ( η − τ )) | − L ǫ √ n> L ǫ √ n (3.16)for all t ∈ [ α, α + ξ ] .There exist numbers s k ∈ [ α, α + ξ ] , k = 1 , , . . . , n , such that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) α + ξ Z α ( G ( s, x ( s ) , φ x ( s − τ )) − G ( s, x ( s ) , φ x ( s − τ ))) ds (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = " n X k =1 α + ξ Z α ( G k ( s, x ( s ) , φ x ( s − τ )) − G k ( s, x ( s ) , φ x ( s − τ ))) ds ! / = ξ " n X k =1 ( G k ( s k , x ( s k ) , φ x ( s k − τ )) − G k ( s k , x ( s k ) , φ x ( s k − τ ))) / . (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) α + ξ Z α ( G ( s, x ( s ) , φ x ( s − τ )) − G ( s, x ( s ) , φ x ( s − τ ))) ds (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≥ ξ | G j ( s j , x ( s j ) , φ x ( s j − τ )) − G j ( s j , x ( s j ) , φ x ( s j − τ )) | > ξL ǫ √ n . Utilizing the equation φ x ( t ) − φ x ( t ) = φ x ( α ) − φ x ( α ) + t Z α A ( φ x ( s ) − φ x ( s )) ds + t Z α ( G ( s, x ( s ) , φ x ( s − τ )) − G ( s, x ( s ) , φ x ( s − τ ))) ds it can be deduced that k φ x ( α + ξ ) − φ x ( α + ξ ) k ≥ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) α + ξ Z α ( G ( s, x ( s ) , φ x ( s − τ )) − G ( s, x ( s ) , φ x ( s − τ ))) ds (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − k φ x ( α ) − φ x ( α ) k − α + ξ Z α k A k k φ x ( s ) − φ x ( s ) k ds − α + ξ Z α k G ( s, x ( s ) , φ x ( s − τ )) − G ( s, x ( s ) , φ x ( s − τ )) k ds> ξL ǫ √ n − (1 + ξ k A k + ξL ) max t ∈ [ α − τ,α + ξ ] k φ x ( t ) − φ x ( t ) k . Hence, max t ∈ [ α − τ,α + ξ ] k φ x ( t ) − φ x ( t ) k > ξL ǫ ξ k A k + ξL ) √ n . Suppose that max t ∈ [ α − τ,α + ξ ] k φ x ( t ) − φ x ( t ) k = k φ x ( λ ) − φ x ( λ ) k , where α − τ ≤ λ ≤ α + ξ . Let us denote ǫ = ξL ǫ ξ k A k + ξL ) √ n and ∆ = ξL ǫ k A k M + M G ) (2 + ξ k A k + ξL ) √ n . t ∈ e J , where e J = (cid:2) λ − ∆ / , λ + ∆ / (cid:3) , we have that k φ x ( t ) − φ x ( t ) k ≥ k φ x ( λ ) − φ x ( λ ) k − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t Z λ k A k k φ x ( s ) − φ x ( s ) k ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t Z λ k G ( s, x ( s ) , φ x ( s − τ )) − G ( s, x ( s ) , φ x ( s − τ )) k ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) > ξL ǫ ξ k A k + ξL ) √ n − ∆ ( k A k M + M G ) . Thus, k φ x ( t ) − φ x ( t ) k > ǫ for all t ∈ e J . Consequently, system (1.2) replicates the sensitivity of system(1.1). (cid:3) It is worth noting that if an autonomous system is utilized instead of the non-autonomous system(1.1) as the generator, then the result obtained in Theorem 3.1 is also valid with the counterpart ofcondition ( C . This is illustrated in the first example provided in Section 5.The next section is devoted to the replication of period-doubling cascade. Let us consider the system x ′ ( t ) = H ( t, x ( t ) , µ ) , (4.17)where µ is a real parameter and the function H : R × R m × R → R m , which is continuous in all of itsarguments, satisfies the equation H ( t + T, x, µ ) = H ( t, x, µ ) for all t ∈ R , x ∈ R m , and µ ∈ R . Wesuppose that there exists a finite value µ ∞ of the parameter µ such that the function F ( t, x ) on theright-hand side of system (1.1) is equal to H ( t, x, µ ∞ ) .System (1.1) is said to admit a period-doubling cascade [51, 52, 53] if there exists a sequence { µ j } , µ j → µ ∞ as j → ∞ , of period-doubling bifurcation values such that system (4.17) undergoes a period-doubling bifurcation as the parameter µ increases or decreases through each µ j , i.e., for each j ∈ N a newstable periodic solution with period p j T appears in the dynamics of (4.17) for some positive integer p , and the preceding p j − T -periodic solution loses its stability. Therefore, at the parameter value µ = µ ∞ there exist infinitely many unstable periodic solutions of system (4.17), and hence of system(1.1), all lying in a bounded region.We say that system (1.2) replicates the period-doubling cascade of system (1.1) if for each periodicsolution x ∈ A of (1.1) system (1.2) admits a periodic solution with the same period.The one-to-one correspondence between the periodic solutions of systems (1.1) and (1.2) is mentionedin the following lemma. 12 emma 4.1 Suppose that the conditions ( C , ( C , ( C hold and ω − KL > . Then x ∈ A is a k T -periodic solution of the generator system (1.1) for some positive integer k if and only if the boundedsolution φ x ∈ B of the replicator system (1.2) is k T -periodic. Proof.
First suppose that x ∈ A is a k T -periodic solution of the generator system (1.1). Using theintegral equation (2.5) we obtain that k φ x ( t + k T ) − φ x ( t ) k ≤ t Z −∞ (cid:13)(cid:13)(cid:13) e A ( t − s ) (cid:13)(cid:13)(cid:13) k G ( s, x ( s ) , φ x ( s + k T − τ )) − G ( s, x ( s ) , φ x ( s − τ )) k ds ≤ KL ω sup t ∈ R k φ x ( t + k T ) − φ x ( t ) k . The last inequality implies that sup t ∈ R k φ x ( t + k T ) − φ x ( t ) k = 0 . Thus, φ x ( t ) is k T -periodic.Conversely, let us assume that φ x ∈ B is k T -periodic. Then we have G ( t, x ( t ) , φ x ( t − τ )) = G ( t, x ( t + k T ) , φ x ( t − τ )) for all t ∈ R . Using the last equation and condition ( C , one can confirm that x ∈ A is k T -periodic. (cid:3) It is worth noting that if x ∈ A is an unstable periodic solution of system (1.1), then the periodicsolution ( x, φ x ) ∈ A × B of the coupled system (1.1)-(1.2) is also unstable.The following theorem can be proved by using Lemma 4.1. Theorem 4.1
Suppose that the conditions ( C − ( C hold. If system (1.1) admits a period-doublingcascade, then system (1.2) replicates the period-doubling cascade of (1.1). The result of Theorem 4.1 is also valid in the case that the generator system is an autonomous onewith the counterpart of condition ( C and the periods of its periodic solutions appearing in the cascadeand the number T satisfying (2.3) are commensurable.A corollary of Theorem 4.1 is as follows. Corollary 4.1
Suppose that the conditions ( C − ( C hold. If system (1.1) admits a period-doublingcascade, then the same is true for the coupled system (1.1)-(1.2). It is worth noting that the coupled system (1.1)-(1.2) possesses exactly the same sequence of period-doubling bifurcation values with the generator system (1.1) under the conditions of Theorem 4.1. Forthat reason the Feigenbaum universality [51] holds also for the coupled system (1.1)-(1.2) provided thatit is valid for (1.1). 13
Examples
Two illustrative examples that support the theoretical results are provided in this section. The replicationof sensitivity is discussed in the first example, and the second one is concerned with replication of period-doubling route to chaos.
Let us consider the Lorenz system [24, 54] x ′ ( t ) = − x ( t ) + 10 x ( t ) x ′ ( t ) = − x ( t ) x ( t ) + 28 x ( t ) − x ( t ) (5.18) x ′ ( t ) = x ( t ) x ( t ) − x ( t ) . It was demonstrated by Tucker [55] that system (5.18) admits a chaotic attractor.In this example, we use the Lorenz system (5.18) as the generator, and as the replicator we take intoaccount the system y ′ ( t ) = − . y ( t ) + 0 . y ( t − . . x ( t ) y ′ ( t ) = − . y ( t ) + 0 . y ( t − . − . x ( t ) (5.19) y ′ ( t ) = − . y ( t ) + 0 . x ( t ) + 0 . t, where ( x ( t ) , x ( t ) , x ( t )) is a solution of system (5.18).System (5.19) is in the form of (1.2) with A = diag ( − . , − . , − . ,G ( t, x , x , x , y , y , y ) = (0 . y + 1 . x , . y − . x , . x + 0 . t ) , and τ = 0 . .The conditions of Theorem 3.1 are satisfied for the coupled system (5.18)-(5.19) with K = 1 , ω = 2 . , L = 0 . , L = 1 . , L = 0 . , and accordingly, system (5.19) replicates the sensitivity of the Lorenzsystem (5.18).In order to illustrate the replication of sensitivity, we depict in Figure 1 the trajectories of two initiallynearby solutions of system (5.19) in which initially nearby solutions of (5.18) that eventually diverge areutilized. Let us consider the constant functions u ( t ) = − . , u ( t ) = 3 . , u ( t ) = 4 . , v ( t ) = − . , v ( t ) = 3 . , and v ( t ) = 4 . . Using the solution ( x ( t ) , x ( t ) , x ( t )) of (5.18) with x (0) = 7 . ,14 (0) = 2 . , x (0) = 33 . in (5.19), we obtain the trajectory shown in blue corresponding to the initialdata y ( t ) = u ( t ) , y ( t ) = u ( t ) , y ( t ) = u ( t ) , t ∈ [ − . , . On the other hand, the trajectory in redrepresents the solution of (5.19) corresponding to y ( t ) = v ( t ) , y ( t ) = v ( t ) , y ( t ) = v ( t ) , t ∈ [ − . , ,when the solution ( x ( t ) , x ( t ) , x ( t )) of (5.18) with x (0) = 7 . , x (0) = 2 . , x (0) = 33 . is utilizedin (5.19). The time interval [0 , . is used in the simulation. Figure 1 confirms the result of Theorem3.1 such that the trajectories in blue and red eventually diverge even if they are nearby on the interval [ − . , . y y y -10 204050 0 010 -2020 -40 Figure 1: Replication of sensitivity by system (5.19). The figure supports the result of Theorem 3.1 suchthat two trajectories of the replicator system (5.19) which are nearby on the interval [ − . , eventuallydiverge.Next, to demonstrate the chaotic behavior of system (5.19), using the solution ( x ( t ) , x ( t ) , x ( t )) of (5.18) with x (0) = 7 . , x (0) = 2 . , x (0) = 33 . one more time, we represent in Figure 2 thetime series of the y -coordinate of (5.19) corresponding to the initial data y ( t ) = u ( t ) , y ( t ) = u ( t ) , y ( t ) = u ( t ) for t ∈ [ − . , . The irregularity seen in Figure 2 manifests the replication of chaos. t -4-2024 y Figure 2: Time series of the y -coordinate of the coupled system (5.18)-(5.19). The figure reveals thechaotic behavior in the dynamics of the replicator system (5.19).15 .2 Example 2 Let us take into account the Duffing equation x ′′ ( t ) + 0 . x ′ ( t ) + x ( t ) = µ cos t, (5.20)where µ is a parameter. It was shown by Sato et al. [48] that equation (5.20) displays period-doublingbifurcations and leads to chaos at µ = µ ∞ ≡ .Using the new variables x ( t ) = x ( t ) and x ( t ) = x ′ ( t ) , equation (5.20) can be rewritten as the system x ′ ( t ) = x ( t ) x ′ ( t ) = − . x ( t ) − x ( t ) + µ cos t. (5.21)One can confirm that the chaotic attractor of system (5.21) with µ = µ ∞ takes place inside thecompact region Λ = (cid:8) ( x , x ) ∈ R : | x | ≤ . , | x | ≤ (cid:9) . Next, we consider the system with delay y ′ ( t ) = − y ( t ) + y ( t ) + 1 . x ( t ) − . x ( t ) + 0 . ty ′ ( t ) = − . y ( t ) − y ( t ) + 0 .
14 arctan( y ( t − . . x ( t ) , (5.22)where ( x ( t ) , x ( t )) is a solution of system (5.21). The system (5.21)-(5.22) is a unidirectionally coupledone in which (5.21) is the generator and (5.22) is the replicator.Systems (5.21) and (5.22) are respectively in the forms of (1.1) and (1.2), where F ( t, x , x ) = (cid:0) x , − . x − x + µ cos t (cid:1) ,A = − − . − ,G ( t, x , x , y , y ) = (cid:0) . x − . x + 0 . t, .
14 arctan y + 0 . x (cid:1) , and τ = 0 . . The eigenvalues of the matrix A are −
52 + 12 i and − − i . Let us denote P = . − . . e At = e − t/ P cos (cid:0) t (cid:1) − sin (cid:0) t (cid:1) sin (cid:0) t (cid:1) cos (cid:0) t (cid:1) P − , it can be verified that (cid:13)(cid:13) e At (cid:13)(cid:13) ≤ Ke − ωt for all t ≥ , where K = k P k (cid:13)(cid:13) P − (cid:13)(cid:13) ≈ . and ω = 2 . .The conditions ( C − ( C are valid for systems (5.21) and (5.22) with L F = 90 . , L = 0 . , L = 2 . , L = 0 . , and M G = 15 . . According to our theoretical results, system (5.22)replicates the period-doubling cascade of system (5.21), and the coupled system (5.21)-(5.22) is chaoticat the parameter value µ = µ ∞ .Figure 3 depicts the projections of periodic and irregular orbits of the coupled system (5.21)-(5.22)on the y − y plane. The projections of period- , period- , and period- orbits are shown in Figure3, (a), (b), and (c), respectively. The values . , . , and . of the parameter µ are respectivelyused in Figure 3, (a), (b), and (c). Figure 3, (d), on the other hand, represents the projection of theirregular orbit for µ = 40 corresponding to the initial data x ( t ) = u ( t ) , x ( t ) = u ( t ) , y ( t ) = u ( t ) , y ( t ) = u ( t ) for − . ≤ t ≤ , where u ( t ) = 1 . , u ( t ) = − . , u ( t ) = 1 . , and u ( t ) = − . areconstant functions. The time series of the y -coordinate of the solution of the coupled system (5.21)-(5.22) corresponding to the same initial data and the same value of µ that are utilized in Figure 3, (d) isshown in Figure 4. Figures 3 and 4 manifest that system (5.22) replicates the period-doubling cascadeof (5.21).Figure 3: Projections of periodic and irregular orbits of the coupled system (5.21)-(5.22) on the y − y plane. (a) Period- orbit. (b) Period- orbit. (c) Period- orbit. (d) Irregular orbit. The values . , . , . , and of the parameter µ are respectively used in (a), (b), (c), and (d). The figure revealsthe replication of period-doubling route to chaos. 17
20 40 60 80 100 120 t -202 y Figure 4: Irregular behavior in the dynamics of the replicator system (5.22). The figure shows the timeseries of the y -coordinate of the coupled system (5.21)-(5.22) with µ = 40 corresponding to the initialdata x ( t ) = u ( t ) , x ( t ) = u ( t ) , y ( t ) = u ( t ) , y ( t ) = u ( t ) for − . ≤ t ≤ , where u ( t ) = 1 . , u ( t ) = − . , u ( t ) = 1 . , and u ( t ) = − . . This paper is devoted to replication of chaos for unidirectionally coupled systems in which the replicatoris a system with delay. It is rigorously proved that the replicator exhibits dynamics similar to the one ofthe generator system, which is the source of chaotic motions. The results are based on the replicationof sensitivity and the existence of infinitely many unstable periodic solutions in a compact region. Dueto the presence of delay, a novel definition as well as a more complicated proof for the replication ofsensitivity are provided compared to the paper [42]. Using the technique presented in this paper it ispossible to obtain high dimensional systems with delay which possess chaotic motions. The obtainedtheoretical results may be applied to various fields such as neural networks, secure communication,robotics, medicine, biology, economics, and lasers in which dynamics are described through differentialequations with delay [5, 6, 7, 8, 11, 12].
References [1] J. Hale, Functional Differential Equations, Springer-Verlag, New York, 1971.[2] H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences,Springer, New York, 2011.[3] H-O. Walther, Homoclinic solution and chaos in ˙ x ( t ) = f ( x ( t − , Nonlinear Anal. 5 (1981) 775–788.[4] U. an der Heiden, H-O. Walther, Existence of chaos in control systems with delayed feedback, J.Differ. Equ. 47 (1983) 273–295.[5] S. Lakshmanan, M. Prakash, C. P. Lim, R. Rakkiyappan, P. Balasubramaniam, S. Nahavandi,Synchronization of an inertial neural network with time-varying delays and its application to securecommunication, IEEE T. Neur. Net. Lear. 29 (2018) 195–207.186] J. M. Daly, Y. Ma, S. L. Waslander, Coordinated landing of a quadrotor on a skid-steered groundvehicle in the presence of time delays, Auton. Robot. 38 (2015) 179–191.[7] A. Zenati, M. Chakir, M. Tadjine, M. Denai, Analysis of leukaemic cells dynamics with multi-stagematuration process using a new non-linear positive model with distributed time-delay, IET ControlTheory Appl. 13 (8) (2019) 3052–3064.[8] F. Conti, R. A. V. Gorder, The role of network structure and time delay in a metapopulationWilson-Cowan model, J. Theor. Biol. 477 (2019) 1–13.[9] U. an der Heiden, Delays in physiological systems, J. Math. Biology, 8 (1979) 345–364.[10] A. Longtin, M. Milton, Complex oscillations in the human pupil light reflex using nonlinear delay-differential equations, Bull. Math. Biol. 51 (1989) 605–624.[11] M. Szydłowski, Time-to-build in dynamics of economic models I: Kalecki’s model, Chaos Soliton.Fract. 14 (2002) 697–703.[12] T. Erneux, D. Lenstra, Synchronization of mutually delay-coupled quantum cascade lasers withdistinct pump strengths, Photonics, 6 (4) (2019) 125–138.[13] H.-O. Walther, Topics in delay differential equations, Jahresber Dtsch. Math.-Ver. 116 (2014) 87–114.[14] T. Krisztin, Global dynamics of delay differential equations, Period Math. Hung. 56 (2008) 83–95.[15] J. Diblík, M. Kúdelčíková, M. Ružičková, Positive solutions to delayed differential equations of thesecond-order, Appl. Math. Lett. 94 (2019) 52–58.[16] S. Houri, M. Asano, H. Yamaguchi, N. Yoshimura, Y. Koike, L. Minati, Generic rotating-frame-basedapproach to chaos generation in nonlinear micro- and nanoelectromechanical system resonators, Phy.Rev. Lett. 125 (2020) 174301.[17] E. Mosekilde, E. Reimer Larsen, J. D. Sterman, J. Skovhus Thomsen, Nonlinear mode-interactionin the macroeconomy, Ann. Oper. Res. 37 (1992) 185–215.[18] C. K. Volos, V.-T. Pham, H. E. Nistazakis, I. N. Stouboulos, A dream that has come true: Chaosfrom a nonlinear circuit with a real memristor, Int. J. Bifurcat. Chaos 30 (2020) 2030036.[19] T. B. Simpson, J. M. Liu, A. Gavrielides, V. Kovanis, P. M. Alsing, Period-doubling route to chaosin a semiconductor laser subject to optical injection, Appl. Phys. Lett. 64 (1994) 3539.[20] C. Grebogi, J. A. Yorke, The Impact of Chaos on Science and Society, United Nations UniversityPress, Tokyo, 1997. 1921] H. Korn, P. Faure, Is there chaos in the brain? II. Experimental evidence and related models, C. R.Biologies, 326 (2003) 787–840.[22] K. M. Cuomo, A. V. Oppenheim, Circuit implementation of synchronized chaos with applicationsto communications, Phys. Rev. Lett. 71 (1993) 65–68.[23] H. A. Ndofor, F. Fabian, J. G. Michel, Chaos in industry environments, IEEE Trans. Eng. Manag.65 (2018) 191–203.[24] E. N. Lorenz, Deterministic nonperiodic flow, J. Atmos. Sci. 20 (1963) 130–141.[25] S. Wiggins, Global Bifurcation and Chaos: Analytical Methods, Springer-Verlag, New York, Berlin,1988.[26] C. Robinson, Dynamical systems: Stability, Symbolic Dynamics, and Chaos, CRC Press, BocaRaton, 1995.[27] B. Lani-Wayda, H.-O. Walther, Chaotic motion generated by delayed negative feedback Part II:Construction of nonlinearities, Math. Nachr. 180 (1996) 141–211.[28] T. Y. Li, J. A. Yorke, Period three implies chaos, Am. Math. Mon. 82 (1975) 985–992.[29] F. M. Marotto, Snap-back repellers imply chaos in R nn