Some Lattice Models with Hyperbolic Chaotic Attractors

Abstract

Examples of one-dimensional lattice systems are considered, in which patterns of different spatial scales arise alternately, so that the spatial phase over a full cycle undergo transformation according to expanding circle map that implies occurrence of Smale -Williams attractors in the multidimensional state space. These models can serve as a basis for design electronic generators of robust chaos within a paradigm of coupled cellular networks. One of the examples is a mechanical pendulum system interesting and demonstrative for research and educational experimental studies. 2 S. P. Kuznetsov ______________________________ CHEBOKSARY, 2-6 JUNE, 2019 ______________________________ the three-dimensional state space, but such attractors can occur in spaces of higher dimension too.Physical examples of systems with attractors of Smale -Williams type can be constructed using oscillators residing in states of excitation and inhibition alternately, while the angular variable has a sense of the oscillator phase [3]. Another approach is based on treatment of patterns arising in an active medium, say, that for Turing structures or standing waves, and the angular variable is a spatial phase [4,5,6]. A disadvantage of the first approach is that it requires, as a rule, a use of rather complex external driving for parameter modulation, combining low-frequency and high-frequency components. Within the second approach, instead of high-frequency modulation a spatial non-homogeneity is introduced that is effortless. A disadvantage is a need of exploiting systems of infinite dimension of the state space, which complicates mathematical description and practical implementations.An interesting seems application of the second approach for organizing hyperbolic chaos in finite-dimensional systems, namely, in lattices of cells, whose dynamics are governed by ordinary differential equations. In this concern, it is worth mentioning a paradigm of cellular neural networks (CNN) on a base of electronic components designed as arrays of cells arranged in space [7]. These systems were suggested, particularly, for parallel data processing, as an alternative to traditional computational approaches. As one of the directions, application of CNN was considered for analog modeling of complex space-time dynamics including Turing structures, spiral patterns, turbulence.This article discusses three models of one-dimensional cell arrays, which can inspire design of CNN generating rough chaos.

Coexistence of Thread and Sheet Chaotic Attractors for Three-Dimensional Lozi Map

“…This may be significant for possible applications [55]. It turned out, however, that hyperbolic chaos is not widespread in real-world systems, and its implementation requires special efforts [56].…”