Post-Double Hopf Bifurcation Dynamics and Adaptive Synchronization of a Hyperchaotic System

TL;DR

This paper analyzes the complex dynamics of a four-dimensional hyperchaotic system near double Hopf bifurcations and proposes an adaptive control method for synchronization, supported by numerical simulations.

Contribution

It provides a detailed bifurcation analysis of a hyperchaotic system and introduces an adaptive synchronization scheme with demonstrated effectiveness.

Findings

Identification of periodic and torus regimes in the system.

Development of an adaptive control scheme for hyperchaotic synchronization.

Numerical validation of the control method's effectiveness.

Abstract

In this paper a four-dimensional hyperchaotic system with only one equilibrium is considered and its double Hopf bifurcations are investigated. The general post-bifurcation and stability analysis are carried out using the normal form of the system obtained via the method of multiple scales. The dynamics of the orbits predicted through the normal form comprises possible regimes of periodic solutions, two-period tori, and three-period tori in parameter space. Moreover, we show how the hyperchaotic synchronization of this system can be realized via an adaptive control scheme. Numerical simulations are included to show the effectiveness of the designed control.