Non-gradient-based simultaneous strategy for bifurcation parameter detection

Abstract

This paper proposes a non-gradient-based simultaneous strategy for detecting bifurcation parameters in dynamical systems. The proposed method uses a single optimization algorithm with two interdependent objective functions, one for a periodic condition and the other for a bifurcation condition. In addition, a novel approach to easily detect a two-parameter bifurcation diagram is presented. A comparison between two population-based search algorithms, particle swarm optimization (PSO) and differential evolution (DE), shows that DE outperforms PSO in bifurcation parameter search. This work provides the first simultaneous strategy capable of directly detecting a two-parameter bifurcation diagram without requiring careful initialization, regardless of the stability of the periodic point. The proposed algorithm is simple, easy to understand, and computationally efficient, making it a powerful and widely applicable tool for accurate bifurcation parameter detection.