We characterize global centers (all solutions are periodic) of the piecewise linear equation <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$x'=a(t)|x| + b(t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>=</mo> <mi>a</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">|</mo> <mi>x</mi> <mo stretchy="false">|</mo> </mrow> <mo>+</mo> <mi>b</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> 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 <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$a(t)=P(h(t))h'(t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>P</mi> <mrow> <mo stretchy="false">(</mo> <mi>h</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <msup> <mi>h</mi> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$b(t)=Q(h(t))h'(t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>b</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>Q</mi> <mrow> <mo stretchy="false">(</mo> <mi>h</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <msup> <mi>h</mi> <mo>′</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.

Global Centers in Piecewise Linear Equations in the Cylinder

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2026年1月16日

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J. L. Bravo,
R. Trinidad-Forte

摘要

We characterize global centers (all solutions are periodic) of the piecewise linear equation $x'=a(t)|x| + b(t)$ $x^{'} = a (t) | x | + b (t)$ 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 $a(t)=P(h(t))h'(t)$ $a (t) = P (h (t)) h^{'} (t)$ , $b(t)=Q(h(t))h'(t)$ $b (t) = Q (h (t)) h^{'} (t)$ .