A New Approach in the Search for periodic orbits

In this paper we present a new approach for finding periodic orbits in dynamical systems modeled by differential equations. It is based on the Homotopy Analysis Method (HAM) but it differs from the usual way it is applied. We apply the HAM to construct approximations of a formal series solution of the equation. These approximations exist for any value of the frequency. They can be obtained by choosing a suitable linear operator and allowing the initial conditions to vary freely. Then we study the behavior of the obtained expressions as a function of the frequency. This procedure allows us to find those values of the frequency for which the series converges and therefore to find the periodic orbits. We show several examples of application of the proposed method. Mathematica and MATCONT were applied for all the calculations.