Unstable Limit Cycles Estimation from Small Perturbations: A Model-Free Approach

TL;DR

This paper introduces a rapid, model-free method to estimate unstable limit cycles in dynamical systems using only small perturbation trajectories, aiding in understanding system stability boundaries.

Contribution

The proposed approach estimates unstable limit cycles without requiring a mathematical model, using only a single trajectory from small perturbations, enabling quick and practical analysis.

Findings

Method effectively estimates unstable limit cycles.

Requires only a single trajectory, no system model needed.

Potential for real-time structural stability assessment.

Abstract

Although stable solutions of dynamical systems are typically considered more important than unstable ones, unstable solutions have a critical role in the dynamical integrity of stable solutions. In fact, usually, basins of attraction boundaries are composed of unstable solutions and their stable manifolds. This study proposes a method for estimating unstable limit cycles surrounding stable equilibrium points. The method exploits the shape of the decrement of trajectories converging towards the equilibrium. For the method, trajectories obtained from small perturbations from the equilibrium state are sufficient to estimate the unstable limit cycle roughly. No mathematical model of the system dynamics is needed for the computation, which requires only a single trajectory in the phase space. As such, the method is computationally very rapid and potentially implementable in real structures.