Global Centers in Piecewise linear Differential Equations in the Cylinder

TL;DR

The paper addresses when the scalar piecewise linear ODE $x' = a(t)|x| + b(t)$ has a global center, i.e., all solutions are periodic. It develops a framework combining zero-dimensional Abelian integrals and the Laurent-trigonometric polynomial correspondence to reduce the problem to a composition structure: a global center occurs iff there exist $P,Q$ and a trig polynomial $H$ with $a(t)=P(H(t))H'(t)$ and $b(t)=Q(H(t))H'(t)$. The core contribution is proving that, under a generic hypothesis ensuring two simple zeros of the solution for a range of initial data, the integrals $A(t)=\\int_0^t a(s) ds$ and $B(t)=\\int_0^t b(s) ds$ must factor through a common function, ultimately yielding the composition form that characterizes all global centers. This links the global-center property in piecewise-linear dynamics to composition centers familiar from Abel equations, enabling explicit structure and a clear criterion for the existence of a global center.

Abstract

We characterize global centers (all solutions are periodic) of the piecewise linear equation $x'=a(t)|x| + b(t)$ when the coefficients $a,b$ are trigonometric polynomials, under some generic hypotheses. We prove that the global centers are those determined by the composition condition on $a,b$. That is, the equation has a global center if and only if there exist polynomials $P, Q$ and a trigonometric polynomial $h$ such that $a(t)=P(h(t))h'(t)$, $b(t)=Q(h(t))h'(t)$.

Theorems & Definitions (17)

• Theorem 2
• Proposition 3
• Proposition 4
• proof
• Lemma 5
• proof
• Corollary 6
• proof
• Proposition 7: Tineo
• Corollary 8