Symmetry, Hopf bifurcation, and offset boosting in a novel chameleon system - Jie Liu; Bo Sang; Lihua Fan; Chun Wang; Xueqing Liu; Ning Wang; Irfan Ahmad (AIMS Mathematics)
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 $ (μ, a, c) $ vary. Here, $ μ $ serves as the dissipation parameter in the $ y $ direction. Bifurcation analysis for four scenarios with $ μ = 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 $ μ $. The parameter $ μ $ 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.