Creation of hidden n-scroll Lorenz-like attractors

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Abstract

Compared with the recently reported hidden two-scroll Lorenz-like attractors in symmetric quadratic and sub-quadratic Lorenz-like dynamical systems, little seems to be concerned with the generation of hidden n-scroll (n ∈ N) attractors as far as one knows, especially the one whose number of scrolls equals the one of equilibria. To achieve this target, we first constructed a new asymmetric quadratic Lorenz-like analogue and seized hidden single-scroll Lorenz-like attractors, which were also created through the collapse of pseudo singularly degenerate heteroclinic cycles. Then, utilizing the fractal process or rotation symmetry, the proposed system may exhibit hidden n-scroll Lorenz-like attractors coexisting with n stable equilibria, and two examples of hidden two/three-scroll Lorenz-like attractors coexisting with two/three stable equilibria were given. In addition, the existence of a single heteroclinic orbit was proved with the help of suitable Lyapunov functions. The obtained results may not only generalize the second part of Hilbert’s 16th problem (i.e., the degree may determine the geometrical structure of strange attractors), but also provide reference for practical application.

Keywords

extension of the second part of Hilbert’s 16th problem / hidden n-scroll Lorenz-like attractor / asymmetric pseudo singularly degenerate heteroclinic cycle / heteroclinic orbit / Lyapunov function

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