Constructive approach to limiting periodic orbits with exponential and power law dynamics

In dynamical systems limit cycles arise as a result of a Hopf bifurcation, after a control parameter has crossed its critical value. In this study we present a constructive method to produce dissipative dynamics which lead to stable periodic orbits as time grows, with predesigned transient dynamics. Depending on the construction method a) the limiting orbit can be a regular circle, an ellipse or a more complex closed orbit and b) the approach to the limiting orbit can follow an exponential law or a power law. This technique allows to design nonlinear models of dynamical systems with desired (exponential or power law) relaxation properties.