Mechanism Analysis of Hidden Chaos in a Generalized Vijayakumar System

In order to further discover the hidden chaotic attractor and its generating mechanism in the Vijayakumar system, we give a generalized system showing hidden chaotic attractors which are not from homoclinic orbit or heteroclinic orbit and consider Hopf bifurcations (codimension one and two) by first Lyapunov coefficient and second Lyapunov coefficient. The existence of periodic orbits is strictly proved theoretically. We have considered the problem of Hopf bifurcation in the chaotic system with hidden attractors, which will be helpful to reveal the intrinsic relationship between the local stability of equilibria and global complex dynamical behaviors of the chaotic system. Finally, numerical simulations are obtained for showing the correctness of theoretical results.

Automated summary

In this study, a generalized system is proposed to investigate the hidden chaotic attractors and their generating mechanism in the Vijayakumar system. Hopf bifurcations are considered using the first and second Lyapunov coefficients, and the existence of periodic orbits is theoretically proven. The findings of this study provide valuable insights into the relationship between the global complex dynamical behaviors of chaotic systems and the local stability of equilibria.