# Triangular function feedback control for chaotic systems featuring coexisting attractors

## TL;DR

This paper introduces a new trigonometric feedback method to control chaotic systems with multiple attractors, showing it can stabilize or create periodic behavior.

## Contribution

A novel trigonometric feedback control strategy for regulating Hopf bifurcation in high-dimensional chaotic systems is proposed.

## Key findings

- The system undergoes a supercritical Hopf bifurcation at d0=−(1+b), producing stable limit cycles.
- Numerical simulations confirm periodic oscillations at d = −1 and equilibrium convergence at d = −3.
- Phase portraits and Lyapunov exponents validate the suppression of chaotic dynamics.

## Abstract

Chaos has emerged as a significant area of research, with the control of chaotic systems being central to this field. This study proposes a novel trigonometric feedback control strategy to regulate Hopf bifurcation in a four-dimensional hyperchaotic system featuring coexisting attractors. By introducing a nonlinear controller dsin(x−xe), we establish the stability criteria for equilibrium points under the parameter space a>0, b>0, and 0<c<π. Theoretical analysis reveals that the system undergoes a supercritical Hopf bifurcation at d0=−(1+b), leading to the emergence of stable limit cycles. Numerical simulations validate the control efficacy: periodic oscillations are observed at d = −1, while equilibrium convergence is achieved at d = −3. Phase portrait analysis and Lyapunov exponent spectra confirm the suppression of chaotic dynamics. This work advances the theoretical framework for bifurcation control in high-dimensional chaotic systems and offers practical implications for secure communication applications.