Existence of homoclinic solutions to periodic orbits in a center manifold
type: Item
instance of: scholarly article
title: Existence of homoclinic solutions to periodic orbits in a center manifold
zbMATH Open document ID: 1059.37046
author: Vittorio Coti Zelati
author: Marta Macrì
published in: Journal of Differential Equations
publication date: 2004-08-19T00:00:00Z
review text: The authors study a Hamiltonian system with Lagrangian \[ L(x,\dot{x},q,\dot{q})=\tfrac{1}{2}(\dot{x}^2 - x^2)+ \tfrac{1}{2}\dot{q}^2 + (1+\delta(x))V(q), \] where \(V\) is \(2\pi\)-periodic, nonnegative and \(V''(0)=0\) while \(\delta\) and \(\delta'\) satisfy some boundedness conditions. The point \(q=x=0\) is a saddle-center equilibrium with a two-dimensional center manifold \(\{q=\dot{q}=0\}\) foliated by periodic orbits. The main result states that if there is no homoclinic orbit to the equilibrium, then there exists an orbit which is homoclinic to some periodic orbit. In addition, the phase difference of the asymptotic phases at \(t=-\infty\) and \(t=+\infty\) is contained in an arbitrary interval \([\alpha,\beta]\subseteq [0,2\pi]\). In particular, this implies the existence of infinitely many homoclinic orbits to periodic orbits in the center manifold. The proof is based on variational techniques: Using the Palais-Smale property of some suitable functional, periodic solutions with large period are constructed. The homoclinic orbits are then found as limits of such periodic orbits.
reviewed by: Jörg Härterich
Mathematics Subject Classification ID: 37J45
Mathematics Subject Classification ID: 37C29
Mathematics Subject Classification ID: 49J35
Mathematics Subject Classification ID: 34C37
zbMATH DE Number: 2092988
zbMATH Keywords: variational method
zbMATH Keywords: saddle-center equilibrium
zbMATH Keywords: Hamiltonian system
MaRDI profile type: Publication
full work available at URL: j.jde.2004.03.030
OpenAlex ID: W2167841308
cites work: Homoclinic orbits in families of hypersurfaces with hyperbolic periodic orbits
cites work: Homoclinic orbit to a center manifold
cites work: Q4463906
cites work: Drift in phase space: A new variational mechanism with optimal diffusion time
cites work: A functional analysis approach to Arnold diffusion
cites work: The existence of homoclinic motions
cites work: Homoclinic trajectories of invariant sets of Hamiltonian systems
cites work: A variational construction of chaotic trajectories for a reversible Hamiltonian system
cites work: A variational approach to chaotic dynamics in periodically forced nonlinear oscillators
cites work: Irregular dynamics and homoclinic orbits to Hamiltonian saddle centers
cites work: On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer-type problem set on ℝN
cites work: PERIODIC AND HOMOCLINIC ORBITS IN A TWO-PARAMETER UNFOLDING OF A HAMILTONIAN SYSTEM WITH A HOMOCLINIC ORBIT TO A SADDLE-CENTER
cites work: Q3978425
cites work: Cascades of homoclinic orbits to, and chaos near, a Hamiltonian saddle- center
cites work: Periodic and heteroclinic orbits for a periodic Hamiltonian system
cites work: Heteroclinics for a reversible Hamiltonian system. II
cites work: Heteroclinics for a reversible Hamiltonian system
cites work: Q4714152
DOI: 10.1016/J.JDE.2004.03.030
description: scientific article; zbMATH DE number 2092988