Unstable limit cycle estimation from small perturbations: a minimal-data-driven approach

Abstract While stable solutions of dynamical systems are generally regarded as more significant than unstable ones, the latter play a crucial role in the dynamical integrity of stable steady-states. In fact, the boundaries of basins of attraction are often formed by unstable saddle-type solutions and their stable manifolds. This study proposes a method for estimating unstable limit cycles surrounding stable equilibrium points. The method leverages the shape of trajectories converging towards the equilibrium. Trajectories from small perturbations near the equilibrium state are sufficient to provide an acceptable estimate of the unstable limit cycle. No mathematical model of the system dynamics is required for the computation, which relies on just a single trajectory in the phase space. Consequently, the method is computationally efficient and can potentially be implemented in real-world structures.