Finite-time Lyapunov dimension and hidden attractor of the Rabinovich system

TL;DR

This paper investigates the Rabinovich system's complex dynamics, identifying hidden and self-excited attractors, and develops methods for estimating their finite-time Lyapunov dimensions through analytical and numerical techniques.

Contribution

It introduces a new approach for localizing hidden attractors and estimates their finite-time Lyapunov dimensions, advancing understanding of multistability in the Rabinovich system.

Findings

Hidden attractors can be localized using continuation and perpetual points.

A new concept of finite-time Lyapunov dimension is developed for numerical analysis.

Various estimates of the Lyapunov dimension for attractors are provided.

Abstract

The Rabinovich system, describing the process of interaction between waves in plasma, is considered. It is shown that the Rabinovich system can exhibit a {hidden attractor} in the case of multistability as well as a classical {self-excited attractor}. The hidden attractor in this system can be localized by analytical-numerical methods based on the {continuation} and {perpetual points}. For numerical study of the attractors' dimension the concept of {finite-time Lyapunov dimension} is developed. A conjecture on the Lyapunov dimension of self-excited attractors and the notion of {exact Lyapunov dimension} are discussed. A comparative survey on the computation of the finite-time Lyapunov exponents by different algorithms is presented and an approach for a reliable numerical estimation of the finite-time Lyapunov dimension is suggested. Various estimates of the finite-time Lyapunov dimension for the hidden attractor and hidden transient chaotic set in the case of multistability are given.