Bifurcation analysis and chaotic dynamics of the generalized chameleon system

Abstract Chameleon chaotic systems possess the unique nature to exhibit a wide range of chaotic attractors through the adjustment of system parameters, enabling the generation of numerous chaotic signals suitable for various applications. This paper is devoted to studying the zero-Hopf bifurcation with characterize its dynamical behavior of the generalized Chameleon system. The local stability of equilibrium points is analyzed, as well as the existence of transcritical bifurcation at the origin. For the proposed system, a zero-Hopf equilibrium point at the origin is characterized at the origin. By using the averaging theory of first order, a limit cycle can be bifurcated from the zero-Hopf equilibrium located at the origin. Numerical simulations are employed to illustrate the system’s rich dynamical behavior, including the periodic orbits and chaotic attractors. Phase portraits, bifurcation diagrams, and Lyapunov exponents are utilized to characterize the transitions between different dynamical behaviors. The results reveal that subtle parameter variations can induce dramatic qualitative changes in the system’s long-term behavior, offering insights into the interplay between bifurcations and chaos in three-dimensional nonlinear systems.