This chapter is devoted to the application of PSO in dynamical systems. The core subject of the chapter is the problem of detecting periodic orbits of nonlinear mappings. This problem is very interesting and significant, as the study of periodic orbits can reveal several crucial properties of a dynamical system. Traditional root-finding algorithms, such as the Newton-family methods, are widely applied on such problems. However, obstacles arise as soon as non-differentiable or discontinuous mappings come under investigation. In such cases, PSO has been shown to be a very useful and efficient alternative. The chapter aims at presenting fundamental ideas and specific application issues. We thoroughly discuss the transformation of the original problem to a corresponding global optimization task. The application of the deflection technique, presented in Chapter Five, for computing several periodic orbits is analyzed and the algorithm is illustrated on well known benchmark problems. Finally, we present and discuss a very significant application, i.e., the detection of periodic orbits in 3-dimensional galactic potentials.
ON POINTWISE DISCONTINUOUS MAPPINGS WITH VALUES IN REGULAR SPACES
