Symmetry, Hopf bifurcation, and offset boosting in a novel chameleon system

Chameleon systems are dynamical systems that exhibit either self-excited or hidden oscillations depending on the parameter values. This paper presents a comprehensive investigation of a quadratic chameleon system, including an analysis of its symmetry, dissipation, local stability, Hopf bifurcation, and various chaotic dynamics as the control parameters $(\mu ,a,c)$ vary. Here, $\mu$ serves as the dissipation parameter in the $y$ direction. Bifurcation analysis for four scenarios with $\mu =0$ was performed, revealing the emergence of various dynamical phenomena under different parameter settings. Offset boosting means introducing a constant into one of the state variables of the system for boosting the variable to a different level. Additionally, hidden chaotic bistability with offset boosting was exhibited by varying $\mu$. The parameter $\mu$ serves as both the Hopf bifurcation parameter and the offset boosting parameter, while the other parameters $(a,c)$ also play critical roles as control parameters, resulting in period-doubling routes to self-excited or hidden chaotic attractors. These findings enrich our understanding of nonlinear dynamics in quadratic chameleon systems.