A chameleon system with a cosine function: bifurcation analysis, multistability, and offset boosting

Abstract This paper proposes a novel three-dimensional chameleon system derived from the classical Duffing oscillator. The innovation lies in transforming an externally driven Duffing system into an internally modulated autonomous system through a coupling between the driving force and system state. This coupling mechanism leads to a chameleon system with rich multistable dynamics. Through rigorous mathematical analysis, we establish the system’s basic properties, including dissipativity, local stability and a supercritical Hopf bifurcation. Using a bifurcation diagram, for small values of parameter c , we detect that c = 0 serves as a critical boundary between hidden and self-excited chaotic regimes. Using a continuation diagram, we find that a supercritical Hopf bifurcation occurs at c = 0 followed by a period-doubling route to self-excited chaos as c increases. When c = 0, by varying two independent parameters, we conduct comprehensive bifurcation analysis, revealing intricate dynamical behaviors. A remarkable discovery is the system’s ability to exhibit four nested hidden chaotic attractors, with their chaotic intensity (characterized by maximum Lyapunov exponents) monotonically decreasing from 0.1044 to 0.0226 as the distance from the equilibrium point increases. Additionally, we introduce an offset boosting mechanism that enables controlled shifting of attractors - a feature with potential applications in chaos-based engineering systems. The system is validated through numerical simulations and physical implementation using an electronic circuit that employs MC33078 and OPA2134 operational amplifiers with an 8th-order polynomial approximation of the cosine function.