Improved nested-layer particle swarm optimization-based bifurcation point detection for the parameter space containing various bifurcation points

Nested-layer particle swarm optimization (NLPSO) is a powerful method to detect bifurcation parameters in dynamical systems. Although NLPSO requires no carefully set initial system parameters, Lyapunov exponents, or the derivative of objective functions, the method can accurately detect bifurcation parameters. However, NLPSO has a serious problem in that it fails to detect a target bifurcation parameter when various types of bifurcation parameters with various periods coexist within the search parameter space. In this study, we clarify problems in detecting bifurcation parameters using conventional NLPSO and solve them by adding a penalty term and imposing a simple condition on the NLPSO objective functions. Using these extended objective functions, NLPSO accurately detected target bifurcation parameters.