Existence and uniqueness of limit cycles in a class of second order ODE's with inseparable mixed terms

TL;DR

This paper establishes conditions for the existence and uniqueness of limit cycles in a class of second order ODEs with mixed terms, showing they attract all non-constant solutions and extending previous results.

Contribution

It extends prior work to nonlinear g(x), proving limit cycle uniqueness and attraction in a broader class of second order ODEs with inseparable mixed terms.

Findings

Proved uniqueness of limit cycles under mild conditions.

Showed limit cycles attract all non-constant solutions.

Extended previous results to nonlinear g(x).

Abstract

We prove a uniqueness result for limit cycles of the second order ODE $\ddot x + \dot x \phi(x,\dot x) + g(x) = 0$. Under mild additional conditions, we show that such a limit cycle attracts every non-constant solution. As a special case, we prove limit cycle's uniqueness for an ODE studied in \cite{ETA} as a model of pedestrians' walk. This paper is an extension to equations with a non-linear $g(x)$ of the results presented in \cite{S}.