Poincaré chaos and unpredictable functions
aa r X i v : . [ n li n . C D ] A ug Poincaré chaos and unpredictable functions
Marat Akhmet a, ∗ , Mehmet Onur Fen b a Department of Mathematics, Middle East Technical University, 06800 Ankara, Turkey b Basic Sciences Unit, TED University, 06420 Ankara, Turkey
Abstract
The results of this study are continuation of the research of Poincaré chaos initiated in papers (AkhmetM, Fen MO. Commun Nonlinear Sci Numer Simulat 2016;40:1–5; Akhmet M, Fen MO. Turk J Math,doi:10.3906/mat-1603-51, accepted). We focus on the construction of an unpredictable function, con-tinuous on the real axis. As auxiliary results, unpredictable orbits for the symbolic dynamics and thelogistic map are obtained. By shaping the unpredictable function as well as Poisson function we haveperformed the first step in the development of the theory of unpredictable solutions for differential anddiscrete equations. The results are preliminary ones for deep analysis of chaos existence in differentialand hybrid systems. Illustrative examples concerning unpredictable solutions of differential equationsare provided.
Keywords:
Poincaré chaos; Unpredictable function; Unpredictable solutions; Unpredictable sequence.
It is useless to say that the theory of dynamical systems is a research of oscillations, and the latestmotion of the theory is the Poisson stable trajectory [1]. In paper [2] inspired by chaos investigationwe have introduced a new type of oscillation, next to the Poisson stable one, and called the initialpoint for it the unpredictable point and the trajectory itself the unpredictable orbit . These noveltiesmake a connection of the homoclinic chaos and the latest types of chaos possible through the conceptof unpredictability , that is sensitivity assigned to a single orbit. Thus, the
Poincaré chaos concept hasbeen eventually shaped in our paper [2]. We have also determined the unpredictable function on thereal axis as an unpredictable point of the Bebutov dynamics in paper [3] to involve widely differentialand discrete equations to the chaos investigation. Nonetheless, we need a more precise description ofwhat one understands as unpredictable function. The present research is devoted to this constructiveduty. One can say that the analysis became productive, since we have learnt that the unpredictablefunctions can be bounded, and this is also true for Poisson functions, newly introduced in this paper. ∗ Corresponding Author Tel.: +90 312 210 5355, Fax: +90 312 210 2972, E-mail: [email protected] unpredictability as individual sensitivity for a motion. Thus, by issuing from the single point ofa trajectory we use it as the Ariadne’s thread to come to phenomenon, which we call as
Poincaré chaos in [2]. This phenomenon makes the all types of chaos closer, since it is another description of motionsin dynamics with homoclinic structure and from another side it admits ingredients similar to late chaostypes. That is, transitivity, sensitivity, frequent separation and proximality. Presence of infinitely manyperiodic motions in late definitions can be substituted by continuum of Poisson stable orbits. Our mainhopes are that this suggestions may bring research of chaos back to the theory of classical dynamicalsystems. The strong argument for this, is the fact that we introduced a new type of motions. That isthe already existing list of oscillations in dynamical systems from equilibrium to Poisson stable orbits isnow prolonged with unpredictable motions. This enlargement will give a push for the further extensionof dynamical systems theory. In applications, some properties and/or laws of dynamical systems can belost or ignored. For example, if one considers non-autonomous or non-smooth systems. Then we canapply unpredictable functions [3], a new type of oscillations which immediately follow almost periodicsolutions of differential equations in the row of bounded solutions. They can be investigated for any typeof equations, since by our results they can be treated by methods of qualitative theory of differential2quations.We utilize the topology of uniform convergence on any compact subset of the real axis to introducethe unpredictable functions. More precisely, the Bebutov dynamical system [8] has been applied. Ad-ditionally, using the same dynamics we have introduced Poisson functions. All these make our duty ofincorporating chaos investigation to theory of differential equations initiated in papers [9]-[14] and in thebook [15] seems to proceed in the correct way.The main goal of this paper is the construction of a concrete unpredictable function and its applicationto differential equations. To give the procedure we start with unpredictable sequences as motions ofsymbolic dynamics and the logistic map. Then an unpredictable function is determined as an improperconvolution integral with a relay function. Finally, we demonstrated in examples unpredictable functionsas solutions of differential equations.
Let ( X, d ) be a metric space and π : T + × X → X, where T + is either the set of non-negative realnumbers or the set of non-negative integers, be a semi-flow on X, i.e., π (0 , x ) = x for all x ∈ X, π ( t, x ) iscontinuous in the pair of variables t and x, and π ( t , π ( t , x )) = π ( t + t , x ) for all t , t ∈ T + , x ∈ X. A point x ∈ X is called positively Poisson stable (stable P + ) if there exists a sequence { t n } satisfying t n → ∞ as n → ∞ such that lim n →∞ π ( t n , x ) = x [16]. For a given point x ∈ X, let us denote by Θ x theclosure of the trajectory T ( x ) = { π ( t, x ) : t ∈ T + } , i.e., Θ x = T ( x ) . The set Θ x is a quasi-minimal setif the point x is stable P + and T ( x ) is contained in a compact subset of X [16].It was demonstrated by Hilmy [17] that if the trajectory corresponding to a Poisson stable point x is contained in a compact subset of X and it is neither a rest point nor a cycle, then the quasi-minimalset contains an uncountable set of motions everywhere dense and Poisson stable. The following theoremcan be proved by adapting the technique given in [16, 17]. Theorem 2.1 ([2]) Suppose that x ∈ X is stable P + and T ( x ) is contained in a compact subset of X. If Θ x is neither a rest point nor a cycle, then it contains an uncountable set of motions everywhere denseand stable P + . The definitions of an unpredictable point and unpredictable trajectory are as follows.
Definition 2.1 ([2]) A point x ∈ X and the trajectory through it are unpredictable if there exist apositive number ǫ (the unpredictability constant) and sequences { t n } and { τ n } , both of which diverge toinfinity, such that lim n →∞ π ( t n , x ) = x and d ( π ( t n + τ n , x ) , π ( τ n , x )) ≥ ǫ for each n ∈ N . Markov [16] proved that a trajectory stable in both Poisson and Lyapunov (uniformly) senses mustbe an almost periodic one. Since Definition 2.1 implies instability, an unpredictable motion cannot be3lmost periodic. In particular, it is neither an equilibrium, nor a cycle.Based on unpredictable points, a new chaos definition was provided in the paper [2] as follows.
Definition 2.2 ([2]) The dynamics on the quasi-minimal set Θ x is called Poincaré chaotic if x is anunpredictable point. It is worth noting that Poincaré chaos admits properties similar to the ingredients of Devaney chaos[5]. In the paper [2], it was proved that if x is an unpredictable point, then the dynamics on Θ x is sensitive.That is, there exists a positive number e ǫ such that for each x ∈ Θ x and for each positive number δ there exist a point x ∈ Θ x and a positive number ¯ t such that d ( x , x ) < δ and d ( f (¯ t, x ) , f (¯ t, x )) ≥ e ǫ . Besides, since the trajectory T ( x ) is dense in Θ x , transitivity is also present in the dynamics. Moreover,according to Theorem 2.1, there exists a continuum of stable P + orbits in Θ x . In our previous paper [2]we gave the definition of Poincaré chaos for flows, but in this paper we give the definition for semi-flows,since the discussion in the paper [2] is valid also for the latter case.Let us denote by C ( R ) the set of continuous functions defined on R with values in R m , and assumethat C ( R ) has the topology of uniform convergence on compact sets, i.e., a sequence { h k } in C ( R ) is saidto converge to a limit h if for every compact set U ⊂ R the sequence of restrictions { h k | U } converges to { h | U } uniformly.One can define a metric ρ on C ( R ) as [8] ρ ( h , h ) = ∞ X k =1 − k ρ k ( h , h ) , (2.1)where h , h belong to C ( R ) and ρ k ( h , h ) = min (cid:26) , sup s ∈ [ − k,k ] k h ( s ) − h ( s ) k (cid:27) , k ∈ N . Let us define the mapping π : R + × C ( R ) → C ( R ) by π ( t, h ) = h t , where h t ( s ) = h ( t + s ) . Themapping π is a semi-flow on C ( R ) , and it is called the Bebutov dynamical system [8].Using the Bebutov dynamical system, we give the descriptions of a Poisson function and an unpre-dictable function in the next definitions. Definition 2.3
A Poisson function is a Poisson stable point of the Bebutov dynamical system.
Definition 2.4 ([3]) An unpredictable function is an unpredictable point of the Bebutov dynamical sys-tem.
It is clear that an unpredictable function is a Poisson function. According to Theorem
III. [8], amotion π ( t, h ) lies in a compact set if h is a bounded and uniformly continuous function. Therefore, anunpredictable function h determines Poincaré chaos in the Bebutov dynamical system if it is bounded and4niformly continuous. Moreover, any system of differential equations which admits uniformly continuousand bounded unpredictable solution has a Poincaré chaos. For differential equations, we say that asolution is an unpredictable one if it is uniformly continuous and bounded on the real axis.Let us consider the system x ′ ( t ) = Ax ( t ) + f ( x ( t )) + g ( t ) , (2.2)where all eigenvalues of the constant p × p matrix A have negative real parts, the function f : R p → R p is bounded, and the function g : R → R p is a uniformly continous and bounded. Since the eigenvalues ofthe matrix A have negative real parts, there exist positive numbers K and ω such that (cid:13)(cid:13) e At (cid:13)(cid:13) ≤ K e ωt for t ≥ [18].The presence of an unpredictable solution in the dynamics of (2.2) is mentioned in the next theorem. Theorem 2.2 ([3]) If g ( t ) is an unpredictable function and the function f ( x ) is Lipschitzian with asufficiently small Lipschitz constant L f such that K L f − ω < , then system (2.2) possesses a uniqueuniformly exponentially stable unpredictable solution. In the next two sections we will consider unpredictable functions whose domain consists of all integers,that is, unpredictable sequences . In this section, we will show the presence of an unpredictable point in the symbolic dynamics [5, 19] witha distinguishing feature.Let us consider the space Σ = { s = ( s s s . . . ) | s j = 0 or } of infinite sequences of ’s and ’swith the metric d ( s, t ) = ∞ X k =0 | s k − t k | k , where s = ( s s s . . . ) , t = ( t t t . . . ) ∈ Σ . The Bernoulli shift σ : Σ → Σ is defined as σ ( s s s . . . ) =( s s s . . . ) . The map σ is continuous and Σ is a compact metric space [5, 19].In the book [5], the sequence s ∗ = ( 0 1 |{z} blocks |
00 01 10 11 | {z } blocks |
000 001 010 011 . . . | {z } blocks | . . . ) , (3.3)was considered, which is constructed by successively listing all blocks of ’s and ’s of length n, thenlength n + 1 , etc. In the proof of the next lemma, an element s ∗∗ = ( s ∗∗ s ∗∗ s ∗∗ . . . ) of Σ will beconstructed in a similar way to s ∗ with the only difference that the order of the blocks will be chosen ina special way. An extension of the constructed sequence to the left hand side will also be provided.5 emma 3.1 For each increasing sequence { m n } of positive integers, there exist a sequence s ∗∗ ∈ Σ andsequences { α n } , { β n } of positive integers, both of which diverge to infinity, such that (i) d ( σ α n + r ( s ∗∗ ) , σ r ( s ∗∗ )) ≤ − m n , r = − n, − n + 1 , . . . , n, (ii) d ( σ α n + β n ( s ∗∗ ) , σ β n ( s ∗∗ )) ≥ for each n ∈ N . Proof.
Fix an arbitrary increasing sequence { m n } of positive integers. For each n ∈ N , define α n = P n + m n k =1 k k and β n = n + m n + 1 . Clearly, both of the sequences { α n } and { β n } diverge to infinity. Wewill construct a sequence s ∗∗ ∈ Σ such that the inequalities ( i ) and ( ii ) are valid.First of all, we choose the terms s ∗∗ , s ∗∗ , . . . , s ∗∗ α − by successively placing the blocks of ’s and ’sin an increasing length, starting from the blocks of length till the end of the ones with length m + 1 . The order of the blocks with the same length can be arbitrary without any repetitions. Let us take s ∗∗ α + k = s ∗∗ k for k = 0 , , . . . , m + 1 , i.e., the terms of the first block ( s ∗∗ α s ∗∗ α +1 · · · s ∗∗ α + m +1 ) of length m + 2 is chosen the same as the first m + 2 terms of the sequence s ∗∗ . Moreover, we take the secondblock of length m + 2 in a such a way that its first term s ∗∗ α + β is different from s ∗∗ β . After that wecontinue placing the remaining blocks of length m + 2 and the ones with length greater than m + 2 tillthe last block of length m + 2 , and again, the blocks of the same length can be in any order withoutrepetitions.For each n ≥ , we set the last block of length m n + n such that s ∗∗ α n − k = s ∗∗ α k − k for each k =1 , , . . . , n − . Then, the terms of the first block of length m n + n +1 are constituted by taking s ∗∗ α n + k = s ∗∗ k ,k = 0 , , . . . , m n + n. Moreover, the second block of length m n + n + 1 is chosen such that s ∗∗ α n + β n = s ∗∗ β n . Lastly, the remaining blocks of ’s and ’s are successively placed similar to the case mentioned aboveso that the lengths of the blocks in s ∗∗ are in an increasing order and there are no repetitions withinthe blocks of the same length. By this way the construction of the sequence s ∗∗ ∈ Σ is completed. Wefix the extension of the sequence s ∗∗ to the left by choosing σ − k ( s ∗∗ ) = s ∗∗ α k − k , k ∈ N . For each n ∈ N , we have σ α n − n ( s ∗∗ ) k = σ − n ( s ∗∗ ) k , k = 0 , , . . . , m n + 2 n and σ α n + β n ( s ∗∗ ) = σ β n ( s ∗∗ ) so that theinequalities ( i ) and ( ii ) are valid. (cid:3) The technique presented in [5] can be used to show that the trajectory T ( s ∗∗ ) = (cid:8) σ i ( s ∗∗ ) : i ∈ Z (cid:9) isdense in Σ , i.e., Θ s ∗∗ = Σ . By Lemma 2.2 in [2] any sequence σ i ( s ∗∗ ) , i ∈ Z , is an unpredictable point ofthe Bernoulli dynamics on Σ , and Σ is a quasi-minimal set. It implies from the last theorem that T ( s ∗∗ ) is an unpredictable function on Z , i.e., an unpredictable sequence. According to Theorem 3.1 presented in[2], the dynamics on Σ is Poincaré chaotic. Moreover, there are infinitely many unpredictable sequencesin the set. 6 An unpredictable solution of the logistic map
In this section, we will demonstrate the presence of an unpredictable solution of the equation η n +1 = F µ ( η n ) , (4.4)where F µ ( s ) = µs (1 − s ) is the logistic map.The result is provided in the next theorem. Theorem 4.1
For each µ ∈ [3 + (2 / / , and sequence of positive numbers δ n → , there exists asolution { η n } , n ∈ Z , of equation (4.4) such that (i) | η i n + r − η r | < δ n , r = − h n, − h n + 1 , . . . , h n, (ii) | η i n + j n − η j n | ≥ ǫ for each n ∈ N , where ǫ is a positive number, h > is a natural number, and { i n } , { j n } are integer valued sequencesboth of which diverge to infinity. Proof.
Fix µ ∈ [3 + (2 / / , and a sequence { δ n } of positive real numbers with δ n → as n → ∞ . Take a neighborhood U ⊂ [0 , of the point − /µ. According to Theorem of paper [20], thereexist a natural number h > and a Cantor set Λ ⊂ U such that the map F h µ on Λ is topologicallyconjugate to the Bernoulli shift σ on Σ . Therefore, there exists a homeomorphism S : Σ → Λ such that S ◦ σ = F h µ ◦ S. Since S is uniformly continuous on Σ , for each n ∈ N , there exists a number δ n > such that for each s , s ∈ Σ with d ( s , s ) < δ n , we have (cid:12)(cid:12) η − η (cid:12)(cid:12) < δ n /µ h − , where η = S ( s ) ,η = S ( s ) . Let { m n } be an increasing sequence of natural numbers such that − m n < δ n for each n ∈ N . Accord-ing to Lemma 3.1, there exist a sequence s ∗∗ ∈ Σ and sequences { α n } , { β n } both of which diverge to in-finity such that d ( σ α n + r ( s ∗∗ ) , σ r ( s ∗∗ )) ≤ − m n , r = − n, − n + 1 , . . . , n, and d ( σ α n + β n ( s ∗∗ ) , σ β n ( s ∗∗ )) ≥ for each n ∈ N . Now, let { η n } , n ∈ Z , be the solution of (4.4) with η h k = S ( σ k ( s ∗∗ )) , k ∈ Z . Since the inequality | F µ ( u ) − F µ ( u ) | ≤ µ | u − u | is valid for every u , u ∈ [0 , , we have for each n ∈ N that | η i n + r − η r | <δ n , r = − h n, − h n + 1 , . . . , h n, where i n = h α n . Besides, using the arguments presented in [21], onecan verify the existence of a positive number ǫ such that | η i n + j n − η j n | ≥ ǫ , n ∈ N , where j n = h β n . (cid:3) By the topological equivalence and results on Σ of the last section, one can make several observationsfrom the proved theorem. Any number η n , n ∈ Z , is an unpredictable point of the logistic map dynamics,and the Cantor set Λ mentioned in the proof of Theorem 4.1 is a quasi-minimal set. Moreover, thesequence { η n } is unpredictable. By Theorem 3.1 mentioned in [2], the dynamics on the quasi-minimal7et is Poincaré chaotic, and there are infinitely many unpredictable sequences in the set. The lastobservation will be applied in the next section to construct an unpredictable function. In this part of the paper, we will provide an example of an unpredictable function benefiting from thedynamics of the logistic map (4.4).Let us fix two different points d and d in R p , and suppose that γ is a positive number. Take a se-quence { k n } of positive integers satisfying both of the inequalities − k n ≤ n and e − γk n ≤ γ k d − d k n for each n ∈ N . Fix µ ∈ [3 + (2 / / , and a sequence { δ n } of positive numbers such that δ n ≤ k d − d k nk n , n ∈ N . In a similar way to the items ( i ) and ( ii ) of Theorem 4.1, one can verify thatthere exist a positive number ǫ , a sequence { i n } of even positive integers, a sequence { j n } of positiveintegers, and a solution { η n } , n ∈ Z , of the logistic map (4.4) such that the inequalities | η i n + r − η r | ≤ δ n , r = − k n , − k n + 1 , . . . , k n − , (5.5)and | η i n + j n − η j n | ≥ ǫ (5.6)hold for each n ∈ N . It is easy to observe that the constructed sequence { η n } is unpredictable and consequently, it generatesa quasi-minimal set and Poincaré chaos similar to that of the last section.Now, consider the function φ : R → R p defined as φ ( t ) = t Z −∞ e − γ ( t − s ) ν ( s ) ds, (5.7)where the function ν ( t ) is defined as ν ( t ) = d , if ζ j < t ≤ ζ j +1 , j ∈ Z ,d , if ζ j − < t ≤ ζ j , j ∈ Z , and the sequence { ζ j } , j ∈ Z , of switching moments is defined through the equation ζ j = j + η j for each j, in which { η j } is the solution of (4.4) satisfying (5.5) and (5.6). The function φ ( t ) is bounded such that sup t ∈ R k φ ( t ) k ≤ max {k d k , k d k} γ . Moreover, φ ( t ) is uniformly continuous since its derivative is bounded.In the proof of the following theorem, we will denote by [ ( a, b ] the oriented interval such that [ ( a, b ] = a, b ] if a < b and [ ( a, b ] = ( b, a ] if a > b. Theorem 5.1
The function φ ( t ) is unpredictable. Proof.
First of all, we will show that ρ ( φ i n , φ ) → as n → ∞ , where ρ is the metric defined by equation(2.1). Let us fix an arbitrary natural number n. The functions φ ( i n + s ) and φ ( s ) satisfy the equation φ ( i n + s ) − φ ( s ) = s Z −∞ e − γ ( s − u ) ( ν ( i n + u ) − ν ( u )) du. It is worth noting that for each r ∈ Z both of the points ζ r and ζ i n + r − i n belong to the interval ( r, r + 1) . Moreover, k ν ( i n + s ) − ν ( s ) k = k d − d k for s ∈ ∞ [ r = −∞ \ ( ζ r , ζ i n + r − i n ] , and k ν ( i n + s ) − ν ( s ) k = 0 , otherwise.Since for each r = − k n , − k n + 1 , . . . , k n − the distance between the points ζ r and ζ i n + r − i n areat most δ n , one can verify for each s ∈ [ − k n , k n ] that k φ ( i n + s ) − φ ( s ) k ≤ − k n Z −∞ e − γ ( s − u ) k ν ( i n + u ) − ν ( u ) k du + k n − X r = − k n (cid:12)(cid:12)(cid:12)(cid:12) ζ r Z ζ in + r − i n k ν ( i n + u ) − ν ( u ) k du (cid:12)(cid:12)(cid:12)(cid:12) ≤ k d − d k γ e − γk n + 3 k n δ n k d − d k≤ n . Hence, we have ρ ( φ i n , φ ) = ∞ X k =1 − k ρ k ( φ i n , φ ) ≤ k n X k =1 − k ρ k ( φ i n , φ ) + 2 − k n ≤ n . The last inequality implies that ρ ( φ i n , φ ) → as n → ∞ , i.e., φ ( t ) is a Poisson function.Now, let us show the existence of a positive number ǫ satisfying ǫ → as ǫ → such that ρ ( φ i n + j n , φ j n ) ≥ ǫ for each n ∈ N . For a fixed natural number n, using the equations φ ( i n + j n + s ) = e − γs φ ( i n + j n ) + s Z e − γ ( s − u ) ν ( i n + j n + u ) du and φ ( j n + s ) = e − γs φ ( j n ) + s Z e − γ ( s − u ) ν ( j n + u ) du,
9e obtain that k φ ( i n + j n + 1) − φ ( j n + 1) k ≥ (cid:13)(cid:13)(cid:13)(cid:13) ζ jn − j n Z ζ in + jn − i n − j n e − γ (1 − u ) ( d − d ) du (cid:13)(cid:13)(cid:13)(cid:13) − e − γ k φ ( i n + j n ) − φ ( j n ) k≥ e γǫ − γe γ k d − d k − e − γ k φ ( i n + j n ) − φ ( j n ) k . Therefore, it can be verified for each k ∈ N that sup s ∈ [ − k,k ] k φ ( i n + j n + s ) − φ ( j n + s ) k ≥ ( e γǫ − k d − d k γ (1 + e γ ) . (5.8)Let us denote ǫ = min (cid:26) , ( e γǫ − k d − d k γ (1 + e γ ) (cid:27) . It can be confirmed by means of inequality (5.8) that ρ k ( φ i n + j n , φ j n ) ≥ ǫ , k ∈ N . Thus, ρ ( φ i n + j n , φ j n ) ≥ ǫ for each n ∈ N . Consequently, the function φ ( t ) is unpredictable. (cid:3) One of the possible ways useful for applications to generate unpredictable functions from a given oneis provided in the next theorem.
Theorem 5.2
Let φ : R → H be an unpredictable function, where H is a bounded subset of R p . If h : H → R q is a function such that there exist positive numbers L and L satisfying L k u − u k ≤k h ( u ) − h ( u ) k ≤ L k u − u k for all u, u ∈ H , then the function ψ : R → R q defined as ψ ( t ) = h ( φ ( t )) is also unpredictable. Proof.
Since φ ( t ) is an unpredictable function, there exist a positive number ǫ and sequences { t n } and { τ n } , both of which diverge to infinity, such that lim n →∞ ρ ( φ t n , φ ) = 0 and ρ ( φ t n + τ n , φ τ n ) ≥ ǫ for each n ∈ N . Firstly, we will show that lim n →∞ ρ ( ψ t n , ψ ) = 0 . Fix an arbitrary positive number ǫ, and let us denote α = max { , L } . There exists a natural number n such that for all n ≥ n the inequality ρ ( φ t n , φ ) < ǫ/α is valid. For each k ∈ N , one can confirm that ρ k ( ψ t n , ψ ) = min (cid:26) , sup s ∈ [ − k,k ] k h ( φ ( t n + s )) − h ( φ ( s )) k (cid:27) ≤ min (cid:26) , L sup s ∈ [ − k,k ] k φ ( t n + s ) − φ ( s ) k (cid:27) ≤ αρ k ( φ t n , φ ) . Therefore, it can be verified for each n ≥ n that ρ ( ψ t n , ψ ) ≤ αρ ( φ t n , φ ) < ǫ. lim n →∞ ρ ( ψ t n , ψ ) = 0 . Next, we will show the existence of a positive number ǫ such that ρ ( ψ t n + τ n , ψ τ n ) ≥ ǫ for each n ∈ N . Denote β = min { , L } . For each k ∈ N , we have that ρ k ( ψ t n + τ n , ψ τ n ) = min (cid:26) , sup s ∈ [ − k,k ] k h ( φ ( t n + τ n + s )) − h ( φ ( τ n + s )) k (cid:27) ≥ min (cid:26) , L sup s ∈ [ − k,k ] k φ ( t n + τ n + s ) − φ ( τ n + s ) k (cid:27) ≥ βρ k ( φ t n + τ n , φ τ n ) . Thus, the inequality ρ ( ψ t n + τ n , ψ τ n ) ≥ βρ ( φ t n + τ n , φ τ n ) ≥ ǫ holds for each n ∈ N , where ǫ = βǫ . Consequently, the function ψ ( t ) is unpredictable. (cid:3) A corollary of Theorem 5.2 is as follows.
Corollary 5.1 If φ : R → H is an unpredictable function, where H is a bounded subset of R p , thenthe function ψ : R → R p defined as ψ ( t ) = P φ ( t ) , where P is a constant, nonsingular, p × p matrix, isalso an unpredictable function. Proof.
The function h : H → R p defined as h ( u ) = P u satisfies the inequality L k u − u k ≤ k h ( u ) − h ( u ) k ≤ L k u − u k , for u , u ∈ H with L = 1 / (cid:13)(cid:13) P − (cid:13)(cid:13) and L = k P k . Therefore, by Theorem 5.2, the function ψ ( t ) isunpredictable. (cid:3) In the next section, the existence of Poincaré chaos in the dynamics of differential equations will bepresented.
Consider the differential equation x ′ ( t ) = − x ( t ) + ν ( t ) , (6.9)where the function ν ( t ) is defined as ν ( t ) = . , if ζ j < t ≤ ζ j +1 , j ∈ Z , − . , if ζ j − < t ≤ ζ j , j ∈ Z . (6.10)11n (6.10), the sequence { ζ j } is defined as ζ j = j + η j , j ∈ Z , and { η j } is the unpredictable sequencedetermined in Section 5 for the map (4.4) with µ = 3 . . According to Theorem 5.1, ψ ( t ) = t Z −∞ e − t − s ) / ν ( s ) ds is a globally asymptotically stable unpredictable solution of (6.9). We represent a solution of (6.9)corresponding to the initial data x ( ζ ) = 0 . , ζ = 0 . in Figure 1. The choice of the coefficient µ = 3 . and the initial value ζ = 0 . is approved by the shadowing analysis in paper [22]. The simulation seenin Figure 1 supports the result of Theorem 5.1 such that the equation (6.9) behaves chaotically. t x Figure 1: Chaotic behavior in equation (6.9). The figure confirms that Poincaré chaos takes place in thedynamics of equation (6.9).Next, we will demonstrate the chaotic behavior of a multidimensional system of differential equations.Let us take into account the system x ′ ( t ) = Ax ( t ) + ν ( t ) , (6.11)where x ( t ) = ( x ( t ) , x ( t ) , x ( t )) ∈ R , A = − − − − − − , and the function ν : R → R is definedas ν ( t ) = ( − , , , if ζ j < t ≤ ζ j +1 , j ∈ Z , (3 , , − , if ζ j − < t ≤ ζ j , j ∈ Z . (6.12)Similarly to equation (6.10), in (6.12), the sequence { ζ j } of switching moments is defined through theequation ζ j = j + η j , where { η j } is the unpredictable sequence determined in Section 5 for the map (4.4)with µ = 3 . . By means of the transformation y = P − x, where P = − − − , system (6.11) can be12ritten as y ′ ( t ) = Dy ( t ) + ν ( t ) , (6.13)where D = − − − , and ν ( t ) = (0 , , − , if ζ j < t ≤ ζ j +1 , j ∈ Z , (1 , , , if ζ j − < t ≤ ζ j , j ∈ Z . System (6.13) admits an unpredictable solution ψ ( t ) in accordance with Theorem 5.1. Therefore,Corollary 5.1 implies that ψ ( t ) = P ψ ( t ) is an unpredictable solution of (6.11).To demonstrate the chaotic behavior, we depict in Figure 2 the x − coordinate of the solution of (6.11)corresponding to the initial data x ( ζ ) = 0 . , x ( ζ ) = 0 . , x ( ζ ) = 0 . , ζ = 0 . . The coefficient µ = 3 . and the initial value ζ = 0 . are considered for shadowing in [22]. The chaotic behavior isalso valid in the remaining coordinates, which are not just pictured here. Moreover, Figure 3 shows the − dimensional chaotic trajectory of the same solution. It is worth noting that the chaotic solutions of(6.11) take place inside the compact region H = (cid:8) ( x , x , x ) ∈ R | − ≤ x ≤ . , . ≤ x ≤ , − . ≤ x ≤ . (cid:9) . (6.14)Figures 2 and 3 support the results of Theorem 5.1 and Corollary 5.1 such that the represented solutionbehaves chaotically. t x Figure 2: The x − coordinate of the chaotic solution of system (6.11).Now, we will demonstrate the extension of unpredictable solutions and Poincaré chaos. For thatpurpose, we consider the system z ′ ( t ) = Bz ( t ) + f ( z ( t )) + h ( ψ ( t )) , (6.15)13 x x x Figure 3: Chaotic trajectory of system (6.11). The figure reveals the presence of Poincaré chaos in system(6.11).where z ( t ) = ( z ( t ) , z ( t ) , z ( t )) ∈ R , B = − − − / − / − , and ψ ( t ) is the unpredictablesolution of (6.11). In system (6.15), the function f : R → R , f ( u ) = ( f ( u ) , f ( u ) , f ( u )) , is defined as f ( u ) = 0 .
03 sin u , f ( u ) = 0 . u for | u | ≤ , f ( u ) = 0 . for | u | > , f ( u ) = 0 .
06 tanh u , and thefunction h : R → R , h ( u ) = ( h ( u ) , h ( u ) , h ( u )) , is defined as h ( u ) = 2 arctan( u ) , h ( u ) = u +0 . u ,h ( u ) = 0 . u , where u = ( u , u , u ) . The eigenvalues of the matrix B are − / , − ± i, and e Bt = P e Jt P − , where J = − / − −
10 1 − and P = − / / . One can verify that (cid:13)(cid:13) e Bt (cid:13)(cid:13) ≤ K e − ωt , t ≥ , with K = k P k (cid:13)(cid:13) P − (cid:13)(cid:13) ≈ . and ω = 2 . The function f ( u ) is bounded and it satisfies the Lipschitz condition k f ( u ) − f ( u ) k ≤ L f k u − u k , u, u ∈ R , with L f = 0 . such that the inequality K L f − ω < is valid.On the other hand, the function h ( u ) satisfies the conditions of Theorem 5.2 with L = 0 . and L = 2 . inside the region H defined by (6.14) so that h ( x ( t )) is an unpredictable function if x ( t ) is an unpredictable function. Therefore, according to Theorem 2.2, system (6.15) possesses a uniqueuniformly exponentially stable unpredictable solution.Let us denote by θ ( t ) the solution of (6.11) whose trajectory is shown in Figure 3. To demonstratethe extension of Poincaré chaos, we take into account the system z ′ ( t ) = Bz ( t ) + f ( z ( t )) + h ( θ ( t )) . (6.16)14e depict in Figure 4 the z − coordinate of the solution of (6.16) with z ( ζ ) = 0 . , z ( ζ ) = 0 . ,z ( ζ ) = − . , where ζ = 0 . . Moreover, the trajectory of the same solution is represented in Figure 5.The simulations shown in Figures 4 and 5 support Theorem 2.2 such that the Poincaré chaos of system(6.11) is extended by (6.16). t z Figure 4: The z − coordinate of the chaotic solution of system (6.16). The figure manifests the presenceof Poincaré chaos in system (6.16). z z z Figure 5: Chaotic trajectory of system (6.16). It is seen in the figure that the Poincaré chaos of system(6.11) is extended by (6.16).
In our previous paper [2] we developed the concept of Poisson stable points to unpredictable ones, whichimply the Poincaré chaos in the quasi-minimal set. This new definition makes homoclinic chaos and thelate descriptions of the phenomenon be closer to each other such that a unified background of chaos can beobtained in the future. Besides, in article [3], an unpredictable function was defined as an unpredictablepoint of the Bebutov dynamical system. In the present paper, we have obtained samples of unpredictablefunctions and sequences, which are in the basis of Poincaré chaos. The unpredictable sequences areutilized in the construction of piecewise continuous functions, and such functions in their own turn areutilized for the construction of continuous unpredictable functions. Thus, one can claim that the basics15f the new theory of unpredictable functions have been lied in the present research. Automatically, theresults concerning the analyses of functions and sequences make it possible to formulate new problems ofthe existence of unpredictable solutions for differential equations of different types as well as for discreteand hybrid systems of equations, similar to the results for periodic, almost periodic and other types ofsolutions. These are all strong arguments for the insertion of chaos research to the theory of differentialequations. In addition to the role of the present paper for the theory of differential equations, theconcept of unpredictable points and orbits introduced in our studies [2, 3] and the additional results ofthe present study will bring the chaos research to the scope of the classical theory of dynamical systems.Moreover, not less importantly, these concepts extend the boundaries of the theory of dynamical systemssignificantly, since we are dealing with a new type of motions, which are behind and next to Poisson stabletrajectories. From another point of view, our study requests the development of techniques to determinewhether a point is an unpredictable one in concrete dynamics. For that purpose one can apply theresearch results which exist for the indication of Poisson stable points [4]. One more interesting studydepending on the present results can be performed if one tries to find a numerical approach for therecognition of unpredictable points.