Poincaré chaos and unpredictable functions

Akhmet, Marat; Fen, Mehmet Onur

doi:10.1016/j.cnsns.2016.12.015

Cited by 63 publications

(88 citation statements)

References 19 publications

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“…The concept of the unpredictable functions was introduced in the paper [2]. The description of such functions relies on the dynamics of unpredictable points, which were presented in the study [1] for the first time in the literature.…”

Section: Introductionmentioning

confidence: 99%

Unpredictable solutions of linear differential and discrete equations

Akhmet¹,

Fen²,

Tleubergenova³

et al. 2019

Turk J Math

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In this study, the existence and uniqueness of the unpredictable solution for a non-homogeneous linear system of ordinary differential equations is considered. The hyperbolic case is under discussion. New properties of unpredictable functions are discovered. The presence of the solutions confirms the existence of Poincaré chaos. Simulations illustrating the chaos are provided. existence of unpredictable solutions simultaneously means the presence of Poincaré chaos, i.e., unpredictable solutions are "irregular". This makes the subject attractive for applications. Finally, we consider new properties of the functions. Sufficient conditions are provided such that a linear transformation of an unpredictable function is unpredictable, and it is proved that the sum of an unpredictable function and a periodic function is an unpredictable function.Ke α(t−s) g + (s + t n ) − g + (s) ds ≤ d t Ke α(t−s) ξds+

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Section: Introductionmentioning

confidence: 99%

Unpredictable solutions of linear differential and discrete equations

Akhmet¹,

Fen²,

Tleubergenova³

et al. 2019

Turk J Math

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“…According to the result of [9], a bounded sequence � is a solution of equation (1) if and only if the following relation is satisfied…”

Section: Resultsmentioning

confidence: 99%

“…Proof. By contraction mapping theorem there exists the fixed point of the operator Π which is a bounded solution of the system (1) and it satisfies the inequality || || � � Now, we prove the unpredictability of the solution (1). The coordinates of the sequence satisfy the relation (1), that is…”

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confidence: 90%

“…Definition 1. [1,2] A bounded sequence � � , � � �, in � � is called unpredictable if there exist a positive number � � and sequences � � , � � , � � �, of positive integers both of which diverge to infinity such that �� → � as � → � for each � in bounded intervals of integers and �� for each � � �.…”

Section: Introductionmentioning

confidence: 99%

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Strongly unpredictable solutions of difference equations

Akhmet

Tleubergenova²,

Zhamanshin³

2019

Int. j. math. phys.

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It so happens that the line of oscillations in the classical theory of dynamical systems, which is founded by H.Poincar´e and G.Birkhoff was broken at Poisson stable motions. The next oscillations were considered as actors of chaotic processes. This article discusses the new type of oscillations, unpredictable sequences, the presence of which proves the existence of Poincare chaos. The sequence is defined as an unpredictable function on the set of integers. The results continue the description of chaos which isinitiated from a single motion, an unpredictable one. To demonstrate the effectiveness of the approach, the existence and uniqueness of the unpredictable solution for a quasilinear difference equation are proved. An example with numerical simulations is presented to illustrate the theoretical results. Since unpredictability is request for all coordinates of solutions, the concept of strong unpredictability can be useful for investigation of neural networks, brain activity, robotics, where complexity is related to optimization and effectiveness.

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“…under the condition .C1/, that is, kF .y 1 / F .y 2 /k Ä L F ky 1 y 2 k for all y 1 ; y 2 2 R p . Moreover, according to Corollary 5.1 [3], the function G.t / is unpredictable if and only if g.t / is unpredictable.…”

Section: Definitionmentioning

confidence: 99%