For n-dimensional piecewise-smooth systems separated by a hyperplane, we derive the arbitrary-order Melnikov-like functions by the Lyapunov–Schmidt reduction to investigate the crossing limit cycles bifurcating from a family of crossing periodic orbits, which intersects the hyperplane as an m-dimensional manifold with <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="332_2025_10142_Article_IEq1.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="78" /> </InlineMediaObject> <EquationSource Format="TEX">$m \le n-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>≤</mo> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. Our results generalize previous publications in aspects of order from 1 to any order, dimension of intersection manifold from <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="332_2025_10142_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="77" /> </InlineMediaObject> <EquationSource Format="TEX">$m=n-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>=</mo> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> to <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="332_2025_10142_Article_IEq3.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="78" /> </InlineMediaObject> <EquationSource Format="TEX">$m\le n-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>≤</mo> <mi>n</mi> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, and expression of intersection manifold from coordinate form to parameter form as well as the period of this family of crossing periodic orbits. As application, we obtain the number of crossing limit cycles bifurcating from a family of crossing periodic orbits for some 3-dimensional piecewise-smooth systems investigated in previous publications by our main theorems.

Melnikov-Like Function of Arbitrary Order for n-Dimensional Piecewise-Smooth Differential Systems

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发表日期：

2025年3月8日

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Yingying Zheng,
Tao Li,
Xingwu Chen

摘要

For n-dimensional piecewise-smooth systems separated by a hyperplane, we derive the arbitrary-order Melnikov-like functions by the Lyapunov–Schmidt reduction to investigate the crossing limit cycles bifurcating from a family of crossing periodic orbits, which intersects the hyperplane as an m-dimensional manifold with $m \le n-1$ $m \leq n - 1$ . Our results generalize previous publications in aspects of order from 1 to any order, dimension of intersection manifold from $m=n-1$ $m = n - 1$ to $m\le n-1$ $m \leq n - 1$ , and expression of intersection manifold from coordinate form to parameter form as well as the period of this family of crossing periodic orbits. As application, we obtain the number of crossing limit cycles bifurcating from a family of crossing periodic orbits for some 3-dimensional piecewise-smooth systems investigated in previous publications by our main theorems.