Infinite emergent oscillations with locally stable equilibrium

The emergence of multistable states brings about significant unpredictability and complexity in nonlinear dynamical systems, which will induce undesired states and may lead to critical failures or misleading dynamics detection. In this paper, a simple five-term autonomous dissipative chaotic system with unique sine nonlinearity is presented. Notably, an infinite many coexisting attractors can emerge even if the system has a single Lyapunov and Jacobi stable equilibrium point. These attractors cannot be triggered by the initial trajectories near the small neighborhoods of the stable equilibrium point, leading to hidden extreme multistability. Bifurcation route, phase portraits, and basin of attraction confirm coexistence among different periodic and chaotic hidden oscillations. To reveal the boundary between the hidden oscillations and the locally stable space, Jacobi stability and instable boundary analysis using Kosambi–Cartan–Chern theory based on the principles of Finsler space theory are presented. Besides, a predefined-time controller is designed to stabilize the hidden oscillations, which has highlights in the independence between the initial conditions and the predefined settling time, as well as a single flexible adjusted parameter. Simulation experiments confirm the correctness of the theoretical analyses.