Beyond the Melnikov method: A computer assisted approach

详细信息    查看全文

• 作者：Maciej J. Capiński^a ; ¹ ; ^{maciej.capinski@agh.edu.pl" class="auth_mail" title="E-mail the corresponding author} ; Piotr Zgliczyński^b ; ² ; ^{umzglicz@cyf-kr.edu.pl" class="auth_mail" title="E-mail the corresponding author}
• 关键词：37D10 ; 58F15 ; 65G20
• 刊名：Journal of Differential Equations
• 出版年：2017
• 出版时间：5 January 2017
• 年：2017
• 卷：262
• 期：1
• 页码：365-417
• 全文大小：935 K

文摘

We present a Melnikov type approach for establishing transversal intersections of stable/unstable manifolds of perturbed normally hyperbolic invariant manifolds (NHIMs). The method is based on a new geometric proof of the normally hyperbolic invariant manifold theorem, which establishes the existence of a NHIM, together with its associated invariant manifolds and bounds on their first and second derivatives. We do not need to know the explicit formulas for the homoclinic orbits prior to the perturbation. We also do not need to compute any integrals along such homoclinics. All needed bounds are established using rigorous computer assisted numerics. Lastly, and most importantly, the method establishes intersections for an explicit range of parameters, and not only for perturbations that are ‘small enough’, as is the case in the classical Melnikov approach.