Non-conservative perturbations of homoclinic snaking scenarios

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• 作者：Jü ; rgen Knobloch<sup> ; sup><sup>juergen.knobloch@tu-ilmenau.de" class="auth_mail" title="E-mail the corresponding authorsup> ; Martin Vielitz
• 关键词：37C29 ; 37G20 ; 37G40 ; 37J20
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：5 January 2016
• 年：2016
• 卷：260
• 期：1
• 页码：517-566
• 全文大小：718 K

文摘

Homoclinic snaking refers to the continuation of homoclinic orbits to an equilibrium E near a heteroclinic cycle connecting E and a periodic orbit P. Typically homoclinic snaking appears in one-parameter families of reversible, conservative systems.

sp0110">Here we discuss perturbations of this scenario which are both non-reversible and non-conservative. We treat this problem analytically in the spirit of the work <span id="bbr0030">[3]span>. The continuation of homoclinic orbits happens with respect to both the original continuation parameter 渭 and the perturbation parameter 位. The continuation curves are parametrised by the dwelling time L of the homoclinic orbit near P  . It turns out that <span id="mmlsi1" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039615004593&_mathId=si1.gif&_user=111111111&_pii=S0022039615004593&_rdoc=1&_issn=00220396&md5=5c35dfde518bbfa0907c44610d33c250" title="Click to view the MathML source">位(L)span><span class="mathContainer hidden"><span class="mathCode">scroll">位stretchy="false">(Lstretchy="false">)span>span>span> tends to zero while the 渭 vs. L diagram displays isolas or criss-cross snaking curves in a neighbourhood of the original snakes-and-ladder structure.

sp0120">In the course of our studies we adapt both Fenichel coordinates near P and the analysis of Shilnikov problems near P to the present situation.