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Homoclinic Solutions of Differential Equations

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An Introduction to Minimax Theorems and Their Applications to Differential Equations

Part of the book series: Nonconvex Optimization and Its Applications ((NOIA,volume 52))

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Abstract

In recent years, starting with works of Bolotin [Bol], Coti-Zelati, Ekeland and Séré [CZES], Coti-Zelati & Rabinowitz [CZR1], [CZR2], Rabinowitz [Ra4], variational methods have been applied to study the existence of homoclinic and heteroclinic solutions of second-order equations and Hamiltonian systems. The search for homoclinic and heteroclinic solutions is a classical problem, originated from the work of Poincaré and has been developed from several points of view. Existence of homoclinic solutions can be obtained by analyzing the intersection properties of the stable and unstable manifolds of the fixed points. There is a standard method to find infinitely nearby homoclinics provided that the stable and unstable manifolds of a fixed point intersect transversally. For this approach we refer the reader to Moser [Mo], Guckenheimer & Holmes [GH], Wiggins [Wig].

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Authors and Affiliations

  1. ISEG, Universidade Técnica de Lisboa, Portugal

    Maria do Rosário Grossinho

  2. CMAF, Universidade de Lisboa, Portugal

    Maria do Rosário Grossinho

  3. University of Rousse, Bulgaria

    Stepan Agop Tersian

Authors
  1. Maria do Rosário Grossinho
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  2. Stepan Agop Tersian
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do Rosário Grossinho, M., Tersian, S.A. (2001). Homoclinic Solutions of Differential Equations. In: An Introduction to Minimax Theorems and Their Applications to Differential Equations. Nonconvex Optimization and Its Applications, vol 52. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3308-2_7

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  • DOI: https://doi.org/10.1007/978-1-4757-3308-2_7

  • Publisher Name: Springer, Boston, MA

  • Print ISBN: 978-1-4419-4849-6

  • Online ISBN: 978-1-4757-3308-2

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