Mehmet Onur Fen

Entrainment by Chaos

A new phenomenon, the entrainment of limit cycles by chaos, which results from the appearance of cyclic irregular behavior, is discussed. In this study, sensitivity is considered as the main ingredient of chaos to be captured, and the period-doubling cascade is chosen for extension. Theoretical results are supported by simulations and discussions regarding Chua’s oscillators, entrainment of toroidal attractors by chaos, synchronization, and controlling problems. It is demonstrated that the entrainment cannot be considered as generalized synchronization of chaotic systems.

Shunting inhibitory cellular neural networks with chaotic external inputs

Taking advantage of external inputs, it is shown that shunting inhibitory cellular neural networks behave chaotically. The analysis is based on the Li-Yorke definition of chaos. Appropriate illustrations which support the theoretical results are depicted.

Generation of cyclic/toroidal chaos by Hopfield neural networks

We discuss the appearance of cyclic and toroidal chaos in Hopfield neural networks. The theoretical results may strongly relate to investigations of brain activities performed by neurobiologists. As new phenomena, extension of chaos by entrainment of several limit cycles as well as the attraction of cyclic chaos by an equilibrium are discussed. Appropriate simulations that support the theoretical results are depicted. Stabilization of tori in a chaotic attractor is realized not only for neural networks, but also for differential equations theory, and this phenomenon has never been reported before in the literature. It is demonstrated that the proposed chaos generation technique cannot be considered as generalized synchronization.

Attraction of Li-Yorke chaos by retarded SICNNs

Abstract In the present study, dynamics of retarded shunting inhibitory cellular neural networks (SICNNs) is investigated with Li–Yorke chaotic external inputs and outputs. Within the scope of our results, we prove the presence of generalized synchronization in coupled retarded SICNNs, and confirm it by means of the auxiliary system approach. We have obtained more than just synchronization, as it is proved that the Li–Yorke chaos is extended with its ingredients, proximality and frequent separation, which have not been considered in the theory of synchronization at all. Our procedure is used to synchronize chains of unidirectionally coupled neural networks. The results may explain the high performance of brain functioning and can be extended by specific stability analysis methods. Illustrations supporting the results are depicted. For the first time in the literature, proximality and frequent separation features are demonstrated numerically for continuous-time dynamics.

Unpredictable points and chaos

Abstract It is revealed that a special kind of Poisson stable point, which we call an unpredictable point, gives rise to the existence of chaos in the quasi-minimal set. The existing definitions of chaos are formulated in sets of motions. This is the first time in the literature that description of chaos is initiated from a single motion. The theoretical results are exemplified by means of the symbolic dynamics.

Li-Yorke chaos in hybrid systems on a time scale

By using the reduction technique to impulsive differential equations [1], we rigorously prove the presence of chaos in dynamic equations on time scales (DETS). The results of the present study are based on the Li-Yorke definition of chaos. This is the first time in the literature that chaos is obtained for DETS. An illustrative example is presented by means of a Duffing equation on a time scale.

SICNNs with Li-Yorke chaotic outputs on a time scale

The existence of Li-Yorke chaos in the dynamics of shunting inhibitory cellular neural networks (SICNNs) on time scales is investigated. It is rigorously proved by taking advantage of external inputs that the outputs of SICNNs exhibit Li-Yorke chaos. The theoretical results are supported by simulations, and the controllability of chaos on the time scale is demonstrated by means of the Pyragas control technique. This is the first time in the literature that the existence as well as the control of chaos are provided for neural networks on time scales. HighlightsChaotic dynamics of SICNNs on time scales is considered.The results are based on the Li-Yorke definition of chaos.Controllability of the chaos on time scales is numerically demonstrated.Simulations which support the theoretical results are depicted.This is the first paper which considers chaos for neural networks on time scales.

Poincaré chaos and unpredictable functions

Abstract The results of this study are continuation of the research of Poincare chaos initiated in the papers (M. Akhmet and M.O. Fen, Commun Nonlinear Sci Numer Simulat 40 (2016) 1-5; M. Akhmet and M.O. Fen, Turk J Math, doi:10.3906/mat-1603-51, in press). We focus on the construction of an unpredictable function, continuous on the real axis. As auxiliary results, unpredictable orbits for the symbolic dynamics and the logistic map are obtained. By shaping the unpredictable function as well as Poisson function we have performed the first step in the development of the theory of unpredictable solutions for differential and discrete equations. The results are preliminary ones for deep analysis of chaos existence in differential and hybrid systems. Illustrative examples concerning unpredictable solutions of differential equations are provided.

Extension of Lorenz Unpredictability

It is found that Lorenz systems can be unidirectionally coupled such that the chaos expands from the drive system. This is true if the response system is not chaotic, but admits a global attractor, an equilibrium or a cycle. The extension of sensitivity and period-doubling cascade are theoretically proved, and the appearance of cyclic chaos as well as intermittency in interconnected Lorenz systems are demonstrated. A possible connection of our results with the global weather unpredictability is provided.

Chaotification of Impulsive Systems by Perturbations

In this paper, we present a new method for chaos generation in nonautonomous impulsive systems. We prove the presence of chaos in the sense of Li–Yorke by implementing chaotic perturbations. An impulsive Duffing oscillator is used to show the effectiveness of our technique, and simulations that support the theoretical results are depicted. Moreover, a procedure to stabilize the unstable periodic solutions is proposed.