Connecting curve: A new tool for locating hidden attractors.

Abstract

Attractors in nonlinear dynamical systems can be categorized as self-excited attractors and hidden attractors. In contrast to self-excited attractors, which can be located by the standard numerical computational method, hidden attractors are hard to detect due to the fact that its basin of attraction is away from the proximity to equilibrium. In multistable systems, many attractors, including self-excited and hidden ones, co-exist, which makes locating each different oscillation more difficult. Hidden attractors are frequently connected to rare or abnormal oscillations in the system and often lead to unpredicted behaviors in many engineering applications, and, thus, the research in locating such attractors is considerably significant. Previous work has proposed several methods for locating hidden attractors but these methods all have their limitations. For example, one of the methods suggests that perpetual points are useful in locating hidden and co-existing attractors, while an in-depth examination suggests that they are insufficient in finding hidden attractors. In this study, we propose that the method of connecting curves, which is a collection of points of analytical inflection including both perpetual points and fixed points, is more reliable to search for hidden attractors. We analyze several dynamical systems using the connecting curve, and the results demonstrate that it can be used to locate hidden and co-existing oscillations.