Homoclinic orbits, and self-excited and hidden attractors in a Lorenz-like system describing convective fluid motion

TL;DR

This paper analyzes a Lorenz-like system modeling convective fluid motion, demonstrating the existence of self-excited and hidden attractors, and providing analytical estimates of their Lyapunov dimensions.

Contribution

It introduces a detailed analysis of self-excited and hidden attractors in a Lorenz-like system, including the existence of homoclinic orbits and Lyapunov dimension estimates.

Findings

Existence of self-excited and hidden attractors for certain parameters

Application of the fishing principle to demonstrate homoclinic orbit

Analytical upper estimates of Lyapunov dimensions

Abstract

In this tutorial, we discuss self-excited and hidden attractors for systems of differential equations. We considered the example of a Lorenz-like system derived from the well-known Glukhovsky--Dolghansky and Rabinovich systems, to demonstrate the analysis of self-excited and hidden attractors and their characteristics. We applied the fishing principle to demonstrate the existence of a homoclinic orbit, proved the dissipativity and completeness of the system, and found absorbing and positively invariant sets. We have shown that this system has a self-excited attractor and a hidden attractor for certain parameters. The upper estimates of the Lyapunov dimension of self-excited and hidden attractors were obtained analytically.