Topological Horseshoe Theory Research Articles

CHAOS IN SIMPLE CELLULAR NEURAL NETWORKS WITH CONNECTION MATRICES SATISFYING DALE'S RULE

In this paper, it is shown that chaos can take place in simple three-dimensional cellular neural networks with connection matrices satisfying Dale's rule. In addition, a rigorous computer-assisted verification of chaoticity in these cellular neural networks is given by virtue of topological horseshoe theory.

Chaotic dynamics in a class of three dimensional Glass networks

In this paper, we study chaotic dynamics of a class of three-dimensional Glass networks with different decay constants, illustrate how the horseshoe is generated, and present a rigorous computer-assisted verification of chaoticity by virtue of interval analysis and topological horseshoe theory.

A computer-assisted verification of hyperchaos in the Saito hysteresis chaos generator

This paper presents a computer-assisted verification of hyperchaos in the well-known Saito hysteresis chaos generator (SHCG) by virtue of topological horseshoe theory. By means of interval analysis we find two disjoint compact subsets in a carefully chosen 3D cross section that can guarantee the existence of a topological horseshoe for the corresponding third-return Poincare map. Numerical studies show that the Poincare map expands in two directions. It justifiably indicates that there exists hyperchaos in the SHCG.

HORSESHOES IN A NEW SWITCHING CIRCUIT VIA WIEN-BRIDGE OSCILLATOR

In this paper we revisit a switching circuit designed by the authors and present a theoretical analysis on the existence of chaos in this circuit. For the ordinary differential equations describing this circuit, we give a computer-aided proof in terms of cross-section and Poincare map, by applying a modern theory of topological horseshoes theory to the obtained Poincare map, that this map is semiconjugate to the two-shift map. This implies that the corresponding differential equations exhibit chaos.

A planar topological horseshoe theory with applications to computer verifications of chaos

In this paper, we present an elementary theory on the existence and robustness of horseshoes under perturbations in terms of stability of crossing. The framework is developed in the setting of 2D Euclidean space but can be generalized to metric spaces. As an application, we give a rigorous verification of the existence of a horseshoe in the Ikeda map.

A rigorous verification of chaos in an inertial two-neuron system

In this paper a rigorous verification for existence of horseshoe embedded in the attractor of a two-neurons system obtained is presented. The arguments are given in a spirit of computer-assisted proof by using topological horseshoe theory and elementary symbolic dynamics.