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...
Bridging the gap between variational homogenization results and two-scale asymptotic averaging techniques on periodic network structures
Kropat, Erik; Meyer-Nieberg, Silja; Weber, Gerhard Wilhelm (American Institute of Mathematical Sciences (AIMS), 2017)
In modern material sciences and multi-scale physics homogenization approaches provide a global characterization of physical systems that depend on the topology of the underlying microgeometry. Purely formal approaches such as averaging techniques can be applied for an identification of the averaged system. For models in variational form, two-scale convergence for network functions can be used to derive the homogenized model. The sequence of solutions of the variational microcsopic models and the correspondi...
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 ...
AdS-plane wave and pp-wave solutions of generic gravity theories
GÜRSES, METİN; Sisman, Tahsin Cagri; Tekin, Bayram (2014-12-02)
We construct the anti-de Sitter-plane wave solutions of generic gravity theory built on the arbitrary powers of the Riemann tensor and its derivatives in analogy with the pp-wave solutions. In constructing the wave solutions of the generic theory, we show that the most general two-tensor built from the Riemann tensor and its derivatives can be written in terms of the traceless Ricci tensor. Quadratic gravity theory plays a major role; therefore, we revisit the wave solutions in this theory. As examples of o...
Citation Formats
M. Akhmet, “Almost periodicity in chaos,” Discontinuity, Nonlinearity, and Complexity, pp. 15–29, 2018, Accessed: 00, 2020. [Online]. Available: