We consider a piecewise-smooth 2-dimensional system <Equation ID="Equ93"> <EquationSource Format="TEX">$\begin{aligned} \dot{\vec {x}}=\vec {f}(\vec {x})+\varepsilon \vec {g}(t,\vec {x},\varepsilon ) \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mover accent="true"> <mover accent="true"> <mi>x</mi> <mo stretchy="false">→</mo> </mover> <mo>˙</mo> </mover> <mo>=</mo> <mover accent="true"> <mi>f</mi> <mo stretchy="false">→</mo> </mover> <mrow> <mo stretchy="false">(</mo> <mover accent="true"> <mi>x</mi> <mo stretchy="false">→</mo> </mover> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>ε</mi> <mover accent="true"> <mi>g</mi> <mo stretchy="false">→</mo> </mover> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mover accent="true"> <mi>x</mi> <mo stretchy="false">→</mo> </mover> <mo>,</mo> <mi>ε</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\varepsilon >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> is a small parameter and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\vec {f}$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>f</mi> <mo stretchy="false">→</mo> </mover> </math></EquationSource> </InlineEquation> is discontinuous along a curve <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Omega ^0$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="normal">Ω</mi> <mn>0</mn> </msup> </math></EquationSource> </InlineEquation>. We assume that <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\vec {0}$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mn>0</mn> <mo stretchy="false">→</mo> </mover> </math></EquationSource> </InlineEquation> is a critical point for any <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\varepsilon \ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, and that for <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\varepsilon =0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ε</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> the system admits a trajectory <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\vec {\gamma }(t)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi>γ</mi> <mo stretchy="false">→</mo> </mover> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> homoclinic to <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\vec {0}$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mn>0</mn> <mo stretchy="false">→</mo> </mover> </math></EquationSource> </InlineEquation> and crossing transversely <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\Omega ^0$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="normal">Ω</mi> <mn>0</mn> </msup> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\vec {\gamma }(0)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover accent="true"> <mi>γ</mi> <mo stretchy="false">→</mo> </mover> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. In a previous paper, we have shown that, also in an n-dimensional setting, the classical Melnikov condition is enough to guarantee the persistence of the homoclinic orbit to perturbations, but more recently we have found an open condition, a geometric obstruction which is not possible in the smooth case, which prevents chaos for 2-dimensional systems when <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\vec {g}$</EquationSource> <EquationSource Format="MATHML"><math> <mover accent="true"> <mi>g</mi> <mo stretchy="false">→</mo> </mover> </math></EquationSource> </InlineEquation> is periodic in t. In this paper, we show that when this obstruction is removed we have chaos as in the smooth case. The proofs involve a new construction of the set from which the chaotic pattern originates. The results are illustrated by examples.

A New Construction for Melnikov Chaos in Piecewise-Smooth Planar Systems

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2025年12月1日

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Alessandro Calamai,
Matteo Franca,
Michal Pospíšil

摘要

We consider a piecewise-smooth 2-dimensional system $\begin{aligned} \dot{\vec {x}}=\vec {f}(\vec {x})+\varepsilon \vec {g}(t,\vec {x},\varepsilon ) \end{aligned}$ $\begin{matrix} \dot{\vec{x}} = \vec{f} (\vec{x}) + ε \vec{g} (t, \vec{x}, ε) \end{matrix}$ where $\varepsilon >0$ $ε > 0$ is a small parameter and $\vec {f}$ $\vec{f}$ is discontinuous along a curve $\Omega ^0$ $Ω^{0}$ . We assume that $\vec {0}$ $\vec{0}$ is a critical point for any $\varepsilon \ge 0$ $ε \geq 0$ , and that for $\varepsilon =0$ $ε = 0$ the system admits a trajectory $\vec {\gamma }(t)$ $\vec{γ} (t)$ homoclinic to $\vec {0}$ $\vec{0}$ and crossing transversely $\Omega ^0$ $Ω^{0}$ in $\vec {\gamma }(0)$ $\vec{γ} (0)$ . In a previous paper, we have shown that, also in an n-dimensional setting, the classical Melnikov condition is enough to guarantee the persistence of the homoclinic orbit to perturbations, but more recently we have found an open condition, a geometric obstruction which is not possible in the smooth case, which prevents chaos for 2-dimensional systems when $\vec {g}$ $\vec{g}$ is periodic in t. In this paper, we show that when this obstruction is removed we have chaos as in the smooth case. The proofs involve a new construction of the set from which the chaotic pattern originates. The results are illustrated by examples.