Graphical Structure of Attraction Basins of Hidden Chaotic Attractors: The Rabinovich–Fabrikant System

Abstract

For systems with hidden attractors and unstable equilibria, the property that hidden attractors are not connected with unstable equilibria is now accepted as one of their main characteristics. To the best of our knowledge this property has not been explored using realtime interactive three-dimensions graphics. Aided by advanced computer graphic analysis, in this paper, we explore this characteristic of a particular nonlinear system with very rich and unusual dynamics, the Rabinovich-Fabrikant system. It is shown that there exists a neighborhood of one of the unstable equilibria within which the initial conditions do not lead to the considered hidden chaotic attractor, but to one of the stable equilibria or are divergent. The trajectories starting from any neighborhood of the other unstable equlibria are attracted either by the stable equilibria, or are divergent.

Hidden and self-exited attractors in a heterogeneous Cournot oligopoly model

“…Following Definition 1, the algorithm used to detect numerically hidden and self-excited attractors of the considered system (1) is presented in the diagram in Fig 2 . In systems in spaces with higher dimension with unstable equilibria, the attraction basins are chosen usually as planar sections containing unstable equilibria. The case of three-dimensional neighborhoods of the Fabrikant-Rabinovich system is treated in [8]. As the diagram shows, the main steps in finding hidden attractors bases on testing if the analyzed attractor has initial conditions within no matter how small neighborhoods of all unstable equilibria (X * 0 , X * 1 ).…”

Coupled Discrete Fractional-Order Logistic Maps

“…On the other hand, from a computational point of view, based on the complexity or simplicity in finding a basin of attraction in the phase space, it is natural to consider the following classification of attractors: self-excited attractors, which can be revealed numerically by integrating the systems with initial conditions within small neighborhoods of unstable equilibria, and hidden attractors, which have the basins of attraction not connected with any equilibria [19][20][21][22][23]. Examples of hidden attractors in continuous-time FO systems exist in some classical systems, such as the Rabinovich-Fabrikant system [24][25][26], Hopfield neuronal system [27], economic system [28], hyperchaotic discontinuous system [29], and so on [30].…”

A New Circumscribed Self‐Excited Spherical Strange Attractor

“…Among other topics, one of the new-brand categories of studying the nonlinear system's dynamic is hidden attractors [49][50][51]. Based on the studies declared in [52][53][54], generally, two principal categories can be defined for the system's attractors: self-excited and hidden attractors. A self-excited attractor is a kind of attractor that possesses an unstable equilibrium within its basin of attraction.…”