Codimension-two bifurcation parameter detection strategy with nested-layer particle swarm optimization

Abstract

Nested-layer particle swarm optimization (NLPSO) is a powerful bifurcation parameter detection method that requires no careful initialization, second-order partial derivative, or Lyapunov exponents. Previous studies have proven the effectiveness of NLPSO for codimension-one bifurcation, but not for codimension-two bifurcation phenomena. This study proposes a novel objective function for NLPSO for detecting six types of bifurcations, including three codimension-one bifurcations and three codimension-two bifurcations. By setting three parameters according to predetermined laws in each target bifurcation parameter, the modified NLPSO with the proposed objective function accurately detected both codimension-one and codimension-two bifurcation parameters in discrete-time dynamical systems and non-autonomous systems, without a change in the search algorithm. Furthermore, especially for detecting bifurcation parameters located at the ends of the Neimark-Sacker bifurcation curve, the modified NLPSO with the proposed objective function significantly reduced the computational amount using different codimension-one and codimension-two bifurcation searches.