In this paper, we provide the lower bound and upper bound of the maximum number of limit cycles Z(n) that planar piecewise linear differential systems with two zones separated by the curves <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1272_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="115" /> </InlineMediaObject> <EquationSource Format="TEX">$y=\pm x^3(x>0)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>y</mi> <mo>=</mo> <mo>±</mo> <msup> <mi>x</mi> <mn>3</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>></mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> under perturbation of arbitrary polynomials of x,  y with degree n can have, where <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1272_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="45" /> </InlineMediaObject> <EquationSource Format="TEX">$n\in \mathbb {N}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>∈</mo> <mi mathvariant="double-struck">N</mi> </mrow> </math></EquationSource> </InlineEquation>. By the first two order Melnikov functions, we achieve that <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1272_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="180" /> </InlineMediaObject> <EquationSource Format="TEX">$7\le Z(1) \le 12,~Z(2)=5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>7</mn> <mo>≤</mo> <mi>Z</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>≤</mo> <mn>12</mn> <mo>,</mo> <mspace width="3.33333pt" /> <mi>Z</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1272_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="80" /> </InlineMediaObject> <EquationSource Format="TEX">$Z(n)\ge 3n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>Z</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <mn>3</mn> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation> for any <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1272_Article_IEq5.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$n\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1272_Article_IEq6.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="117" /> </InlineMediaObject> <EquationSource Format="TEX">$Z(n)\le \frac{13n}{2}+11$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>Z</mi> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mo>≤</mo> <mfrac> <mrow> <mn>13</mn> <mi>n</mi> </mrow> <mn>2</mn> </mfrac> <mo>+</mo> <mn>11</mn> </mrow> </math></EquationSource> </InlineEquation> when n is even, while <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12346_2025_1272_Article_IEq7.gif" Format="GIF" Height="25" Rendition="HTML" Resolution="72" Type="Linedraw" Width="141" /> </InlineMediaObject> <EquationSource Format="TEX">$Z(n)\le \frac{13(n-1)}{2}+21$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>Z</mi> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mo>≤</mo> <mfrac> <mrow> <mn>13</mn> <mo stretchy="false">(</mo> <mi>n</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> <mo>+</mo> <mn>21</mn> </mrow> </math></EquationSource> </InlineEquation> when n is odd.

Limit Cycles in Planar Piecewise Linear Differential Systems with Multiple Switching Curves

期刊论文
发表日期：

2025年4月15日

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Ranran Jia,
Liqin Zhao

摘要

In this paper, we provide the lower bound and upper bound of the maximum number of limit cycles Z(n) that planar piecewise linear differential systems with two zones separated by the curves $y=\pm x^3(x>0)$ $y = \pm x^{3} (x > 0)$ under perturbation of arbitrary polynomials of x, y with degree n can have, where $n\in \mathbb {N}$ $n \in N$ . By the first two order Melnikov functions, we achieve that $7\le Z(1) \le 12,~Z(2)=5$ $7 \leq Z (1) \leq 12, Z (2) = 5$ , $Z(n)\ge 3n$ $Z (n) \geq 3 n$ for any $n\ge 3$ $n \geq 3$ , and $Z(n)\le \frac{13n}{2}+11$ $Z (n) \leq \frac{13 n}{2} + 11$ when n is even, while $Z(n)\le \frac{13(n-1)}{2}+21$ $Z (n) \leq \frac{13 (n - 1)}{2} + 21$ when n is odd.