Limit Cycle Analysis of 3-D Nonlinear systems

TL;DR

This paper analyzes the behavior of limit cycles in three-dimensional nonlinear systems using a method similar to centre manifold analysis, providing insights into how the limit cycle radius changes under certain conditions.

Contribution

It introduces a novel approach to study limit cycles in 3D nonlinear systems by extending centre manifold analysis and examining the radius variation through $mbda-$ equations.

Findings

Limit cycle radius increases under specific conditions.

Conversion to polar form aids in analysis.

Theoretical predictions are supported by an example.

Abstract

Considering Limit Cycles as one of the limits of Lienard equation, an analyis analogous to centre manifold analysis has been done for a $3-D$ nonlinear system exhibiting Limit Cycle. A rigorous study on radius of the Limit Cycle orbit has been done by considering $\lambda-\omega$ equations for the particular system and subsequently converting the system equations from cartesian to polar form. It has been shown through an analysis analogous to Centre Manifold Analysis and reduction of the system dynamics on a lower dimensional space, the Limit cycle radius undergoes an increment change. One example is provided to support the theoretical predictions.