The Birth of a Hidden Attractor Through Boundary Crisis

Abstract

Hidden attractors have been reported in a significant number of dynamical systems, according to their definition related to the existence of saddle points. Since it has now become common knowledge that there are different types of saddle invariant sets besides a fixed saddle point, it is of great interest to investigate the subject. In this paper, a sudden appearance of a hidden chaotic attractor is observed with the variation of a control parameter in some two-dimensional mapping systems. In order to reveal the mechanism behind the sudden appearance of hidden attractors, the generalized cell mapping method with GPU parallel computing is employed in order to obtain very accurate invariant sets. It is found that hidden attractors are generated from an existing chaotic saddle through a boundary crisis, where a hidden chaotic attractor is touching a periodic or chaotic saddle on its basin boundary. Three examples of two-dimensional mapping systems are given to demonstrate the birth of hidden attractors through a boundary crisis.