Hidden Strange Nonchaotic Attractors

Marius-F Danca,Nikolay Kuznetsov

doi:10.3390/math9060652

Abstract

In this paper, it is found numerically that the previously found hidden chaotic attractors of the Rabinovich–Fabrikant system actually present the characteristics of strange nonchaotic attractors. For a range of the bifurcation parameter, the hidden attractor is manifestly fractal with aperiodic dynamics, and even the finite-time largest Lyapunov exponent, a measure of trajectory separation with nearby initial conditions, is negative. To verify these characteristics numerically, the finite-time Lyapunov exponents, ‘0-1’ test, power spectra density, and recurrence plot are used. Beside the considered hidden strange nonchaotic attractor, a self-excited chaotic attractor and a quasiperiodic attractor of the Rabinovich–Fabrikant system are comparatively analyzed.

Highlights

In the last three decades, strange nonchaotic attractors (SNAs) have attracted much attention from both theoretical and experimental points of view, reportedly arising in many physically relevant situations
The first experimental observation of an SNA was in a magnetoelastic ribbon [9]
Strange nonchaotic attractors occur in all dissipative dynamical systems that exhibit the period-doubling route to chaos, where the attractors formed at the accumulation points are fractal sets with zero Lyapunov exponents

Summary

Introduction

In the last three decades, strange nonchaotic attractors (SNAs) have attracted much attention from both theoretical and experimental points of view, reportedly arising in many physically relevant situations. Strange nonchaotic attractors occur in all dissipative dynamical systems that exhibit the period-doubling route to chaos, where the attractors formed at the accumulation points are fractal sets with zero Lyapunov exponents. Such attractors are, not physically observable because the set of parameter values for them to arise has a Lebesgue measure of zero in the parameter space [13]. While SE attractors can be localized numerically with some standard computational procedures by starting from a point in a neighborhood of unstable equilibrium, the basins of hidden attractors, which can be chaotic (HCAs) or non-chaotic (e.g., stable cycles), are not connected with equilibria.

Hidden Attractors in RF System

SNAs of the RF System

Conclusions and Discussion