Number of limit cycles for planar systems with invariant algebraic curves

TL;DR

This paper develops a general method to prove the nonexistence of limit cycles outside invariant algebraic curves in planar polynomial systems, leading to new bounds and conditions for the number of limit cycles in various quadratic and cubic systems.

Contribution

It introduces a unified approach to determine when limit cycles cannot exist outside invariant algebraic curves, providing algebraic conditions and finiteness results for quadratic and cubic systems.

Findings

Proves finiteness of limit cycles depending on degrees of invariant algebraic curves.

Establishes algebraic conditions for nonexistence of limit cycles in parametric families.

Provides a new proof that quadratic systems with an invariant parabola have at most one limit cycle.

Abstract

For planar polynomials systems the existence of an invariant algebraic curve limits the number of limit cycles not contained in this curve. We present a general approach to prove non existence of periodic orbits not contained in this given algebraic curve. When the method is applied to parametric families of polynomial systems that have limit cycles for some values of the parameters, our result leads to effective algebraic conditions on the parameters that force non existence of the periodic orbits. As applications we consider several families of quadratic systems: the ones having some quadratic invariant algebraic curve, the known ones having an algebraic limit cycle, a family having a cubic invariant algebraic curve and other ones. For any quadratic system with two invariant algebraic curves we prove a finiteness result for its number of limit cycles that only depends on the degrees of these curves. We also consider some families of cubic systems having either a quadratic or a cubic invariant algebraic curve and a family of Lienard systems. We also give a new and simple proof of the known fact that quadratic systems with an invariant parabola have at most one limit cycle. In fact, what we show is that this result is a consequence of the similar result for quadratic systems with an invariant straight line.