Homoclinic period blow-up in reversible and conservative systems
type: Item
instance of: scholarly article
title: Homoclinic period blow-up in reversible and conservative systems
zbMATH Open document ID: 0762.34023
DOI: 10.1007/BF00946632
author: B. Fiedler
author: A. Vanderbauwhede
published in: ZAMP. Zeitschrift für angewandte Mathematik und Physik
publication date: 1992-09-27T00:00:00Z
review text: It is shown that in conservative systems of autonomous ordinary differential equations each non-degenerate homoclinic orbit asymptotic to a hyperbolic equilibrium possesses an associated family of periodic orbits. The family is parametrized by the period, and the periodic orbits accumulate on the homoclinic orbit as the period tends to infinity. A similar result holds for symmetric orbits in reversible systems. The statements are proved by constructing a one-dimensional equation for the homoclinic solution, which vanishes identically for the special cases. The results extend earlier work by Devaney and Henrard and provide a positive answer to a conjecture of Strömgren.
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C25
Mathematics Subject Classification ID: 37G99
zbMATH DE Number: 63820
zbMATH Keywords: conservative systems of autonomous ordinary differential equations
zbMATH Keywords: homoclinic orbit
zbMATH Keywords: hyperbolic equilibrium
zbMATH Keywords: periodic orbits
zbMATH Keywords: symmetric orbits in reversible systems
zbMATH Keywords: conjecture of Strömgren
reviewed by: Alois Steindl
MaRDI profile type: Publication
cites work: Dichotomies in stability theory
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cites work: Homoclinic orbits in Hamiltonian systems
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cites work: Using Melnikov's method to solve Silnikov's problems
cites work: On Morse-Smale dynamical systems
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full work available at URL: https://doi.org/10.1007/bf00946632
OpenAlex ID: W2082741679
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description: scientific article; zbMATH DE number 63820