Limit cycles of discontinuous piecewise quadratic and cubic polynomial perturbations of a linear center

TL;DR

This paper uses high-order averaging theory to compute and analyze the maximum number of limit cycles in discontinuous piecewise quadratic and cubic polynomial perturbations of a linear center, revealing they can have more limit cycles than continuous perturbations.

Contribution

It applies high-order averaging theory to discontinuous piecewise polynomial perturbations, providing new bounds on limit cycles and conditions for centers or foci at infinity.

Findings

Discontinuous quadratic and cubic perturbations can produce more limit cycles than continuous ones.

The maximum number of limit cycles is computed for orders 1 through 5.

Conditions for the existence of centers or foci at infinity are established.

Abstract

We apply the averaging theory of high order for computing the limit cycles of discontinuous piecewise quadratic and cubic polynomial perturbations of a linear center. These discontinuous piecewise differential systems are formed by two either quadratic, or cubic polynomial differential systems separated by a straight line. We compute the maximum number of limit cycles of these discontinuous piecewise polynomial perturbations of the linear center, which can be obtained by using the averaging theory of order $n$ for $n=1,2,3,4,5$. Of course these limit cycles bifurcate from the periodic orbits of the linear center. As it was expected, using the averaging theory of the same order, the results show that the discontinuous quadratic and cubic polynomial perturbations of the linear center have more limit cycles than the ones found for continuous and discontinuous linear perturbations. Moreover we provide sufficient and necessary conditions for the existence of a center or a focus at infinity if the discontinuous piecewise perturbations of the linear center are general quadratic polynomials or cubic quasi--homogenous polynomials.