Birth of limit cycles for a class of continuous and discontinuous differential systems in (d+2)-dimension

TL;DR

This paper investigates the maximum number of limit cycles bifurcating from periodic orbits in a high-dimensional reversible differential system, using averaging theory, with results for polynomial and piecewise polynomial perturbations.

Contribution

It provides explicit formulas for the maximum number of bifurcating limit cycles in both polynomial and discontinuous polynomial perturbations of a high-dimensional reversible system.

Findings

Maximum limit cycles in polynomial case: n^d(n-1)/2

Maximum limit cycles in discontinuous case: n^{d+1}

Results extend bifurcation analysis to high-dimensional systems

Abstract

The orbits of the reversible differential system $\dot{x}=-y$, $\dot{y}=x$, $\dot{z}=0$, with $x,y \in R$ and $z\in R^d$, are periodic with the exception of the equilibrium points $(0,0, z)$. We compute the maximum number of limit cycles which bifurcate from the periodic orbits of the system $\dot{x}=-y$, $\dot{y}=x$, $\dot{z}=0$, using the averaging theory of first order, when this system is perturbed, first inside the class of all polynomial differential systems of degree $n$, and second inside the class of all discontinuous piecewise polynomial differential systems of degree $n$ with two pieces, one in $y> 0$ and the other in $y<0$. In the first case this maximum number is $n^d(n-1)/2$, and in the second is $n^{d+1}$.