Computing the number, location, and stability of fixed points of Poincare maps

A numerical method is presented to compute the number of fixed points of Poincare maps of either autonomous or nonautonomous ordinary differential equations, The method consists of three concepts: the Poincare map, the second map constructed from the Poincare map, and topological degree, The topological degree calculated from the second map is equal to the number of fixed points of the Poincare map in a given domain of a Poincare section, Thus the computation procedure is simply to compute the topological degree of the second map, The combined use of this method and Newton's iterative method gives the location and stability of all the fixed points in the domain.