Almost periodicity in chaos

Periodicity plays a significant role in the chaos theory from the beginning since the skeleton of chaos can consist of infinitely many unstable periodic motions. This is true for chaos in the sense of Devaney [1], Li-Yorke [2] and the one obtained through period-doubling cascade [3]. Countable number of periodic orbits exist in any neighborhood of a structurally stable Poincaré homoclinic orbit, which can be considered as a criterion for the presence of complex dynamics [4]-[6]. It was certified by Shilnikov [7] and Seifert [8] that it is possible to replace periodic solutions by Poisson stable or almost periodic motions in a chaotic attractor. Despite the fact that the idea of replacing periodic solutions by other types of regular motions is attractive, very few results have been obtained on the subject. The present study contributes to the chaos theory in that direction. In this paper, we take into account chaos both through a cascade of almost periodic solutions and in the sense of Li-Yorke such that the original Li-Yorke definition is modified by replacing infinitely many periodic motions with almost periodic ones, which are separated from the motions of the scrambled set. The theoretical results are valid for systems with arbitrary high dimensions. Formation of the chaos is exemplified by means of unidirectionally coupled Duffing oscillators. The controllability of the extended chaos is demonstrated numerically by means of the Ott-Grebogi-Yorke [9] control technique. In particular, the stabilization of tori is illustrated.
Discontinuity, Nonlinearity, and Complexity


Perturbed Li-Yorke homoclinic chaos
Akhmet, Marat; Fen, Mehmet Onur; Kashkynbayev, Ardak (University of Szeged, 2018-01-01)
It is rigorously proved that a Li-Yorke chaotic perturbation of a system with a homoclinic orbit creates chaos along each periodic trajectory. The structure of the chaos is investigated, and the existence of infinitely many almost periodic orbits out of the scrambled sets is revealed. Ott-Grebogi-Yorke and Pyragas control methods are utilized to stabilize almost periodic motions. A Duffing oscillator is considered to show the effectiveness of our technique, and simulations that support the theoretical resul...
Persistence of Li-Yorke chaos in systems with relay
Akhmet, Marat; Kashkynbayev, Ardak (University of Szeged, 2017-01-01)
It is rigorously proved that the chaotic dynamics of the non-smooth system with relay function is persistent even if a chaotic perturbation is applied. We consider chaos in a modified Li-Yorke sense such that there are infinitely many almost periodic motions embedded in the chaotic attractor. It is demonstrated that the system under investigation possesses countable infinity of chaotic sets of solutions. An example that supports the theoretical results is represented. Moreover, a chaos control procedure bas...
Almost periodic solutions of the linear differential equation with piecewise constant argument
Akhmet, Marat (2009-10-01)
The paper is concerned with the existence and stability of almost periodic solutions of linear systems with piecewise constant argument where t∈R, x ∈ Rn [·] is the greatest integer function. The Wexler inequality [1]-[4] for the Cauchy's matrix is used. The results can be easily extended for the quasilinear case. A new technique of investigation of equations with piecewise argument, based on an integral representation formula, is proposed. Copyright © 2009 Watam Press.
Strongly unpredictable solutions of difference equations
Akhmet, Marat; Tleubergenova, M.; Zhamanshin, A. (2019-01-01)
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 whic...
All timelike supersymmetric solutions of three-dimensional half-maximal supergravity
DEĞER, NİHAT SADIK; Moutsopoulos, George; Samtleben, Henning; Sarıoğlu, Bahtiyar Özgür (2015-06-22)
We first classify all supersymmetric solutions of the 3-dimensional half-maximal ungauged supergravity that possess a timelike Killing vector by considering their identification under the complexification of the local symmetry of the theory. It is found that only solutions that preserve 16/2(n), 1 <= n <= 3 real supersymmetries are allowed. We then classify supersymmetric solutions under the real local symmetry of the theory and we are able to solve the equations of motion for all of them. It is shown that ...
Citation Formats
M. Akhmet, “Almost periodicity in chaos,” Discontinuity, Nonlinearity, and Complexity, pp. 15–29, 2018, Accessed: 00, 2020. [Online]. Available: