Center boundaries for planar piecewise-smooth differential equations with two zones

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• 作者：Claudio A. Buzzi^a ; ^{buzzi@ibilce.unesp.br" class="auth_mail" title="E-mail the corresponding author} ; Rubens Pazim^b ; ^{pazim@ufmt.br" class="auth_mail" title="E-mail the corresponding author} ; Set Pé ; rez-Gonzá ; lez^a ; ^{setperez@mat.uab.cat" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Piecewise linear differential system ; Limit cycle ; Non-smooth differential system
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 January 2017
• 年：2017
• 卷：445
• 期：1
• 页码：631-649
• 全文大小：571 K

文摘

This paper is concerned with 1-parameter families of periodic solutions of piecewise smooth planar vector fields, when they behave like a center of smooth vector fields. We are interested in finding a separation boundary for a given pair of smooth systems in such a way that the discontinuous system, formed by the pair of smooth systems, has a continuum of periodic orbits. In this case we call the separation boundary as a center boundary  . We prove that given a pair of systems that share a hyperbolic focus singularity class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303468&_mathId=si1.gif&_user=111111111&_pii=S0022247X16303468&_rdoc=1&_issn=0022247X&md5=aea6137fc72dc24d84aca28dffc17059" title="Click to view the MathML source">p₀class="mathContainer hidden">class="mathCode">$p_0$, with the same orientation and opposite stability, and a ray class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303468&_mathId=si2.gif&_user=111111111&_pii=S0022247X16303468&_rdoc=1&_issn=0022247X&md5=e59bdf22ab33153e4022fdff2859d468" title="Click to view the MathML source">Σ₀class="mathContainer hidden">class="mathCode">${\mathrm{\Sigma }}_0$ with endpoint at the singularity class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303468&_mathId=si1.gif&_user=111111111&_pii=S0022247X16303468&_rdoc=1&_issn=0022247X&md5=aea6137fc72dc24d84aca28dffc17059" title="Click to view the MathML source">p₀class="mathContainer hidden">class="mathCode">$p_0$, we can find a smooth manifold Ω such that class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303468&_mathId=si3.gif&_user=111111111&_pii=S0022247X16303468&_rdoc=1&_issn=0022247X&md5=870cba26bd6c74c0620168988e9caec1" title="Click to view the MathML source">Σ₀∪{p₀}∪Ωclass="mathContainer hidden">class="mathCode">${\mathrm{\Sigma }}_0\cup \{p_0\}\cup \mathrm{\Omega }$ is a center boundary. The maximum number of such manifolds satisfying these conditions is five. Moreover, this upper bound is reached.