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Nested-layer particle swarm optimization (NLPSO) is a powerful method to detect bifurcation parameters in discrete-time dynamical systems. Although NLPSO requires no carefully set initial system parameters, Lyapunov exponents or derivation of objective functions, the method can quickly and accurately detect the bifurcation parameter. Previous studies have proven the effectiveness of NLPSO for discrete-time dynamical systems, but they have not demonstrated the effectiveness of continuous dynamical systems. This study proposes an NLPSO-based method to detect bifurcation parameters in non-autonomous continuous-time dynamical systems and applies the method to the Duffing equation. By adding Poincaré maps computation to the algorithm, the NLPSO accurately detected period-doubling and saddlenode bifurcation parameters in the non-autonomous dynamical systems and the discrete-time dynamical systems, without a change in objective functions.

Nested-layer particle swarm optimization method for bifurcation point detection in non-autonomous systems