Graphical structure of attraction basins of hidden attractors: the Rabinovich-Fabrikant system

TL;DR

This paper uses advanced 3D computer graphics to explore the structure of attraction basins of hidden attractors in the Rabinovich-Fabrikant system, revealing how initial conditions relate to stable and unstable equilibria.

Contribution

It provides the first visual and graphical analysis of attraction basin structures of hidden attractors in a complex nonlinear system.

Findings

Hidden attractors are not connected with unstable equilibria.

Neighborhoods of unstable equilibria lead to stable equilibria or divergence.

3D visualization reveals the structure of attraction basins.

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