Period functions and critical periods of piecewise linear system

Abstract

In this article, we study a rich and complex phenomena of planar piecewise linear systems having two domains of which the separation curves are not straight. We show that for positive integers n and m and non-negative integers n₁, n₂, ..., nₘ, there exist two types of piecewise linear systems: one has a period annulus possessing exactly n critical periods; the other has m period annuli possessing exactly n₁, n₂, ..., nₘ critical periods. Moreover, an algebraic curve can be chosen as the separation line in system of the first type.