Approximating hidden chaotic attractors via parameter switching

Abstract

In this paper, the problem of approximating hidden chaotic attractors of a general class of nonlinear systems is investigated. The parameter switching (PS) algorithm is utilized, which switches the control parameter within a given set of values with the initial value problem numerically solved. The PS-generated attractor approximates the attractor obtained by averaging the control parameter with the switched values, which represents the hidden chaotic attractor. The hidden chaotic attractors of a generalized Lorenz system and the Rabinovich-Fabrikant system are simulated for illustration. In Refs. 1–3, it is proved that the attractors of a chaotic system, considered as the unique numerical approximations of the underlying ω-limit sets (see, e.g., Ref. 31) after neglecting sufficiently long transients, can be numerically approximated by switching the control parameter in some deterministic or random manner, while the underlying initial value problem (IVP) is numerically integrated with the parameter switching (PS) algorithm. The attractors, whose basins of attractions are not connected with equilibria, are called hidden attractors, while the attractors for which the trajectories starting from a point in a neighborhood of an unstable equilibrium are attracted by some attractor are called self-excited attractors.4–6 In this paper, we prove analytically and verified numerically that the PS algorithm can be used to approximate any desired hidden attractors of a class of general systems, which model systems such as Lorenz, Chen, and Rössler.