Limit cycles appearing from perturbations of cubic piecewise smooth center with double invariant real straight lines

TL;DR

This paper determines the exact number of limit cycles emerging from a perturbed piecewise smooth cubic system with double invariant lines, showing it depends linearly on polynomial degree and exceeds smooth system estimates.

Contribution

It provides a precise count of limit cycles for a specific class of piecewise smooth systems using first-order averaging theory, highlighting the influence of polynomial degree.

Findings

Limit cycles count depends linearly on polynomial degree n.

Number of limit cycles is at least twice that of smooth systems.

Exact limit cycle number is derived for the perturbed system.

Abstract

This paper investigates the exact number of limit cycles given by the averaging theory of first order for the piecewise smooth integrable non-Hamiltonian system \begin{eqnarray*} (\dot{x},\ \dot{y})=\begin{cases} (-y(x+a)^2+\varepsilon f^+(x,y),\ x(x+a)^2+\varepsilon g^+(x,y)),\ \ x\geq0,\\ (-y(x+b)^2+\varepsilon f^-(x,y),\ x(x+b)^2+\varepsilon g^-(x,y)),\ ~ \, x<0,\\ \end{cases}\end{eqnarray*} where $ab\neq 0$, $0<|\varepsilon|\ll 1$, and $f^\pm(x,y)$ and $g^\pm(x,y)$ are polynomials of degree $n$. It is proved that the exact number of limit cycles emerging from the period annulus surrounding the origin is linear depending on $n$ and it is at least twice the associated estimation of smooth systems.