Analytical and numerical study of the hidden boundary of practical stability: complex versus real Lorenz systems

TL;DR

This paper compares the stability boundaries of complex and real Lorenz systems using analytical and numerical methods, revealing hidden stability boundaries and the role of attractors and bifurcations.

Contribution

It introduces a detailed analysis of the global stability boundary for the complex Lorenz system, extending previous work on the real system with new analytical and numerical techniques.

Findings

Identification of the inner boundary of global stability using Lyapunov functions.

Analysis of homoclinic bifurcations in the complex Lorenz system.

Discovery of hidden attractors and transient chaos related to stability boundaries.

Abstract

This work presents the continuation of the recent article "The Lorenz system: hidden boundary of practical stability and the Lyapunov dimension", published in the Nonlinear Dynamics journal. In this work, in comparison with the results for classical real-valued Lorenz system (henceforward -- Lorenz system), the problem of analytical and numerical identification of the boundary of global stability for the complex-valued Lorenz system (henceforward -- complex Lorenz system) is studied. As in the case of the Lorenz system, to estimate the inner boundary of global stability the possibility of using the mathematical apparatus of Lyapunov functions (namely, the Barbashin-Krasovskii and LaSalle theorems) is demonstrated. For additional analysis of homoclinic bifurcations in complex Lorenz system a special analytical approach by Vladimirov is utilized. To outline the outer boundary of global stability and identify the so-called hidden boundary of global stability, possible birth of hidden attractors and transient chaotic sets is analyzed.