Attractors and bifurcations of chaotic systems

Abstract

The hidden bifurcation idea was discovered by the core idea of the Leonov and Kuznetsov method for searching hidden attractors (i.e., homotopy and numerical continuation) differently in order to uncover hidden bifurcations governed by a homotopy parameter ɛ while keeping the numbers of spirals. This idea was first discovered by Menacer et al. In 2016, in the multispiral Chua system, The first part of this thesis is devoted to providing a basic understanding of dynamic systems and chaos, followed by an introduction to the hidden attractors, history, and definitions. An effective procedure for the numerical localization of hidden attractors in multidimensional dynamical systems has been presented by Leonov et Kuznetsov. In this part, we end with the study of hidden attractors in the Chua system. The second part of the analysis consists of first, hidden modalities ofspirals of chaotic attractor via saturated function series and numerical results. Before reaching the asymptotic attractor which possesses an even number of spirals, these latter are generated one after one until they reach their maximum number, matching the value fixed by ɛ. Then, we end up by symmetries in hidden bifurcation routes to multiscroll chaotic attractors generated by saturated function series. The method to find such hidden bifurcation routes (HBR) depends upon two parameters.