Bifurcation and coalescence of a plethora of homoclinic orbits for a Hamiltonian system
type: Item
instance of: scholarly article
title: Bifurcation and coalescence of a plethora of homoclinic orbits for a Hamiltonian system
zbMATH Open document ID: 0854.34047
DOI: 10.1007/BF02218892
published in: Journal of Dynamics and Differential Equations
publication date: 1996-08-28T00:00:00Z
review text: The authors investgate the set of homoclinic solutions of the reversible Hamiltonian system \(u^{iv}+ Pu''+ u- u^2= 0\). It is shown rigorously that for \(P\leq - 2\) there is a unique homoclinic solution of the system, that solution is given, and on the zero energy surface its orbit coincides with the transverse intersection of the global stable and unstable manifold. At \(P= -2\) infinitely many families of homoclinic solutions are created which exist for \(P\in (- 2, -2+ \varepsilon)\) for some \(\varepsilon> 0\). Second the development of the set of symmetric solutions is monitored numerically as \(P\) increases from \(- 2\). It is observed that two branches extend from \(P= - 2\) to \(P= 2\). All other symmetric branches are in the form of closed loops with a turning point between \(P= -2\) and \(P= 2\). Numerically, it is observed that close to each turning point a bifurcation of a branch of nonsymmetrical homoclinic orbits occurs, which can be followed back to \(P= - 2\). Finally, the observed phenomena are explained in the language of geometric dynamical systems theory. It is shown how the different orbits are related to each other and how the highly complicated order of turning points can be related to the intersection of stable and unstable manifolds. This paper demonstrates nicely, how geometric dynamical systems theory can contribute to the understanding of very complicated dynamic phenomena.
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 34C23
zbMATH DE Number: 895800
zbMATH Keywords: homoclinic solutions
zbMATH Keywords: reversible Hamiltonian system
zbMATH Keywords: symmetric solutions
zbMATH Keywords: turning point
zbMATH Keywords: bifurcation
author: Boris Buffoni
author: Alan R. Champneys
author: John F. Toland
reviewed by: Alois Steindl
describes a project that uses: AUTO
MaRDI profile type: Publication
cites work: A theory of solitary water-waves in the presence of surface tension
cites work: Points of egress in problems of Hamiltonian dynamics
cites work: Homoclinic orbits in the dynamic phase-space analogy of an elastic strut
cites work: Q4712455
cites work: The Numerical Computation of Connecting Orbits in Dynamical Systems
cites work: Infinitely many large amplitude homoclinic orbits for a class of autonomous Hamiltonian systems
cites work: Periodic and homoclinic orbits for Lorentz-Lagranian systems via variational methods
cites work: Global existence of homoclinic and periodic orbits for a class of autonomous Hamiltonian systems
cites work: A plethora of solitary gravity-capillary water waves with nearly critical Bond and Froude numbers
cites work: SUBSIDIARY HOMOCLINIC ORBITS TO A SADDLE-FOCUS FOR REVERSIBLE SYSTEMS
cites work: Hunting for homoclinic orbits in reversible systems: A shooting technique
cites work: Bifurcation of a plethora of multi-modal homoclinic orbits for autonomous Hamiltonian systems
cites work: Homoclinic orbits in Hamiltonian systems
cites work: Q4749133
cites work: NUMERICAL ANALYSIS AND CONTROL OF BIFURCATION PROBLEMS (I): BIFURCATION IN FINITE DIMENSIONS
cites work: NUMERICAL ANALYSIS AND CONTROL OF BIFURCATION PROBLEMS (II): BIFURCATION IN INFINITE DIMENSIONS
cites work: Numerical Computation and Continuation of Invariant Manifolds Connecting Fixed Points
cites work: Solitary waves with damped oscillatory tails: An analysis of the fifth- order Korteweg-de Vries equation
cites work: Homoclinic, heteroclinic, and periodic orbits for a class of indefinite Hamiltonian systems
cites work: Structural localization phenomena and the dynamical phase-space analogy
cites work: Comparative lagrangian formulations for localized buckling
cites work: Q3486279
cites work: Perturbed homoclinic solutions in reversible 1:1 resonance vector fields
cites work: Q3485198
cites work: A topological shooting method and the existence of kinks of the extended Fisher-Kolmogorov equation
cites work: Q3717434
cites work: Global bifurcations and chaos. Analytical methods
full work available at URL: https://doi.org/10.1007/bf02218892
OpenAlex ID: W2042338802
description: scientific article; zbMATH DE number 895800