Homoclinic Points Calculation Method With Particle Swarm Optimization

Abstract

This paper proposes a novel algorithm to accurately calculate the coordinates of homoclinic points observed in discrete-time dynamical systems. The proposed method is based on the particle swarm optimization method. Compared with the current methods, the proposed methodology has the advantages of not requiring the careful selection of the initial conditions and not requiring information related to the derivative of the objective function. It is shown that the proposed method can successfully obtain the homoclinic points of a system, even if the system parameters are close to those that describe the homoclinic bifurcation sets of the system; this is achieved via the construction of an efficient objective function that depends on the Euclidean distance between the points in each manifold. We apply the developed method to two-and three-dimensional discrete-time dynamical systems and demonstrate the validity of the algorithm via the numerical work. The reliability of the proposed algorithm was achieved by evaluating a metric based on the success rate of the method.INDEX TERMS particle swarm optimization, homoclinic points, homoclinic tangency, discrete dynamical systems, global bifurcation.