Limit Cycles in a Chaotic System with a Four-Wing Attractor Through Hopf Bifurcation

This study presents an analytical investigation of small-amplitude limit cycles and Hopf bifurcations in a three-dimensional system with a four-wing attractor characterized by four quadratic nonlinearities. The analysis focuses on local codimension-1 and codimension-2 bifurcations emerging at various equilibrium points. Employing the projection method, we compute the first and second Lyapunov coefficients to determine the nature of the bifurcations. Furthermore, we derive necessary and sufficient conditions for the existence of centers on a local center manifold. To corroborate the analytical results and explore the intricate dynamical behavior of the system, the study is complemented by numerical simulations, including bifurcation diagrams that illustrate the local bifurcation structures.