Research on Hopf Bifurcation and Stability of Heterogeneous Lorenz System with Single Time Delay

TL;DR

This paper analyzes the stability and Hopf bifurcation conditions of heterogeneous Lorenz systems with a single time delay, providing theoretical insights and simulation verification relevant for secure communication applications.

Contribution

It investigates how different delay positions affect stability and bifurcation in heterogeneous Lorenz systems, offering new theoretical conditions and simulation validation.

Findings

Identified conditions for zero equilibrium stability.

Determined critical time delay for Hopf bifurcation.

Validated results through simulations.

Abstract

Time-delay chaotic systems refer to the hyperchaotic systems with multiple positive Lyapunov exponents. It is characterized by more complex dynamics and a wider range of applications as compared to those non-time-delay chaotic systems. In a three-dimensional general Lorenz chaotic system, time delays can be applied at different positions to build multiple heterogeneous Lorenz systems with a single time delay. Despite the same equilibrium point for multiple heterogeneous Lorenz systems with single time delay, their stability and Hopf bifurcation conditions are different due to the difference in time delay position. In this paper, the theory of nonlinear dynamics is applied to investigate the stability of the heterogeneous single-time-delay Lorenz system at the zero equilibrium point and the conditions required for the occurrence of Hopf bifurcation. First of all, the equilibrium point of each heterogeneous Lorenz system is calculated, so as to determine the condition that only zero equilibrium point exists. Then, an analysis is conducted on the distribution of the corresponding characteristic equation roots at the zero equilibrium point of the system to obtain the critical point of time delay at which the system is asymptotically stable at the zero equilibrium point and the Hopf bifurcation. Finally, mathematical software is applied to carry out simulation verification. Heterogeneous Lorenz systems with time delay have potential applications in secure communication and other fields.