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

TL;DR

This paper introduces a constructive method for designing dissipative dynamical systems that produce stable periodic orbits with customizable transient behaviors and relaxation laws, such as exponential or power law.

Contribution

The paper presents a novel technique to engineer nonlinear dynamical systems with predetermined stable periodic orbits and specific relaxation dynamics.

Findings

Able to generate stable periodic orbits of various shapes

Can design systems with exponential or power law relaxation

Provides a systematic approach for controlling transient dynamics

Abstract

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.