Hopf bifurcation analysis of the generalized Lorenz system with time delayed feedback control

TL;DR

This paper investigates how nonlinear delayed feedback control can regulate chaos in the generalized Lorenz system by analyzing Hopf bifurcations and deriving formulas for bifurcating periodic solutions.

Contribution

It introduces a feedback method with delay to control chaos and provides explicit formulas for bifurcation characteristics using advanced mathematical techniques.

Findings

Delay influences system dynamics and bifurcation points

Explicit formulas for bifurcation direction and stability

Control of chaos through feedback delay adjustment

Abstract

In this work we propose a feedback approach to regulate the chaotic behavior of the whole family of the generalized Lorenz system, by designing a nonlinear delayed feedback control. We first study the effect of the delay on the dynamics of the system and we investigate the existence of Hopf bifurcations. Then, by using the center manifold reduction technique and the normal form theory, we derive the explicit formulas for the direction, stability and period of the periodic solutions bifurcating from the steady state at certain critical values of the delay.