Piecewise linear differential systems with an algebraic line of separation

Abstract

We study the number of limit cycles of planar piecewise linear differential systems separated by a branch of an algebraic curve. We show that for each $n\in\mathbb{N}$ there exist piecewise linear differential systems separated by an algebraic curve of degree $n$ having [n/2] hyperbolic limit cycles. Moreover, when n=2,3, we study in more detail the problem, considering a perturbation of a center and constructing examples with 4 and 5 limit cycles, respectively. These results follow by proving that the set of functions generating the first order averaged function associated to the problem is an extended complete Chebyshev system in a suitable interval.

Keywords

• piecewise linear differential system
• algebraic separation
• limit cycle
• ect-system