Derivation of Local Bifurcation Points in Autonomous Systems by Using Particle Swarm Optimization

Abstract

This study extends the particle swarm optimization (PSO)-based bifurcation points derivation method to the autonomous systems. PSO avoids the need to carefully set the initial parameters of the dynamical system or to differentiate the objective function. In addition, it can derive the bifurcation points quickly and accurately, and its efficacy has previously been demonstrated for discrete dynamical systems and non-autonomous systems but not for autonomous systems. This paper proposes an extended PSO-based method for autonomous systems by incorporating the computation of a Poincaré map and a bisection method in the algorithm. To validate the effectiveness of the proposed method, it was applied to deriving the local bifurcation points of a three-dimensional autonomous system to confirm and demonstrate its effectiveness.