Computing periodic orbits of nondifferentiable/discontinuous mappings through particle swarm optimization

Periodic orbits of nonlinear mappings play a central role in the study of dynamical systems. Traditional root finding algorithms, such as the Newton–family algorithms, have been widely applied for the detection of periodic orbits. However, in the case of discontinuous/nondifferentiable mappings and mappings with poorly behaved partial derivatives, this approach is not valid. In such cases, stochastic optimization algorithms have proved to be a valuable tool. In this paper, a new approach for computing periodic orbits through Particle Swarm Optimization, is introduced. The results indicate that the algorithm is robust and efficient. Moreover, the method can be combined with established techniques, such as Deflection, to detect several periodic orbits of a mapping. Finally, the minor effort which is required to implement the proposed approach, renders it an efficient alternative for computing periodic orbits of nonlinear mappings.