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 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.

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Techniques are developed to find all periodic solutions in the simple pendulum by means of the homotopy analysis method (HAM). This involves the solution of the equations of motion in two different coordinate representations. Expressions are obtained for the cycles and periods of oscillations with a high degree of accuracy in the whole range of amplitudes. Moreover, the convergence of the method is easily checked. The aim of this work is to show how the dynamics of a simple example of oscillatory systems may be studied globally with the HAM and to incentivize the interest of advanced undergraduate students in this type of techniques.

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The competitive threshold linear networks have been recently developed and are typical examples of nonsmooth systems that can be easily constructed. Due to their flexibility for manipulation, they are used in several applications, but their dynamics (both local and global) are not completely understood. In this work, we take some recently developed threshold systems and by a simple modification in the parameter space, we obtain new global dynamic behavior. Heteroclinic cycles and other remarkable scenarios of global bifurcation are reported.