×

zbMATH — the first resource for mathematics

A variational approach to homoclinic orbits in Hamiltonian systems. (English) Zbl 0731.34050
The Hamiltonian system of ordinary differential equations \[ \dot x=J\nabla_ xH(t,x), \tag{1} \] where \(J\) denotes the \(2n\times 2n\) matrix \(J=\begin{pmatrix} 0 & I \\ -I & 0 \end{pmatrix}\) with \(J^*=J^{-1}=-J\), \(\nabla_ x H(t,x)\) denotes the vector \(\partial H(t,x)/\partial x\), \(1\leq i\leq 2n\), and the map \(H: \mathbb R\times \mathbb R^{2n}\to \mathbb R\) belongs to the class \(\alpha C^ 2\), are considered in this paper.
The main purpose is to study the so-called doubly asymptotic solutions. We recall that if \(x\) is a periodic solution of the system (1), another solution \(z\) will be called doubly asymptotic to \(x\) if \(| z(t)-x(t)| \to 0\) when \(t\to\pm\infty\).
The author simplifies the problem by applying Floquet’s theory to the linearized system around \(x\dot y=JH''(t,x(t))y\) using a variational approach for the solution.

MSC:
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
58E30 Variational principles in infinite-dimensional spaces
34C25 Periodic solutions to ordinary differential equations
70H05 Hamilton’s equations
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal.14, 349-381 (1973) · Zbl 0273.49063
[2] Aubin, J.P., Ekeland, I.: Applied nonlinear analysis. New York: Wiley 1984 · Zbl 0641.47066
[3] Clarke, F.: Periodic solutions of Hamiltonian inclusions. J. Differ. Equations40, 1-6 (1981) · Zbl 0461.34030
[4] Ekeland, I.: Periodic solutions of Hamilton’s equations and a theorem of P. Rabinowitz. J. Differ. Equations34, 523-534 (1979) · Zbl 0446.70019
[5] Ekeland, I.: Convexity methods in Hamiltonian systems. Berlin Heidelberg New York: Springer 1989 · Zbl 0742.58047
[6] Greenspan, B.D., Holmes, P.J.: Homoclinic orbits, subharmonics and global bifurcations in forced oscillations. In: Barenblatt, G.I., Iooss, G., Joseph, D.D. (eds.) Nonlinear dynamics and turbulence. Boston London Melburne: Pitman 1983 · Zbl 0532.58019
[7] Hofer, H., Wysocki, K.: First order elliptic systems and the existence of homoclinic orbits in Hamiltonian systems. Preprint Rutgers University, 1989 · Zbl 0702.34039
[8] Lions, P.L.: Solutions of Hartree-Fock equations for Coulomb systems. Preprint CEREMADE n0 8607, Paris, 1988 · Zbl 0693.35047
[9] Lions, P.L.: The concentration-compactness principle in the calculus of variations. Rev. Mat. Iberoam.1, 145-201 (1985) · Zbl 0704.49005
[10] Melnikov, V.K.: On the stability of the center for periodic perturbations. Trans. Mosc. Math. Soc.12, 1-57 (1963)
[11] Moser, J.: Stable and random motions in dynamical systems. Princeton: Princeton University Press 1973 · Zbl 0271.70009
[12] Moser, J.: New aspects in the theory of stability of Hamiltonian systems. Commun. Pure Appl. Math.11, 81-114 (1958) · Zbl 0082.40801
[13] Poincaré, H.: Les méthodes nouvelles de la mécanique céleste. Paris: Gauthier-Villars 1899
[14] Rabinowitz, P.H.: Periodic and heteroclinic orbits for a periodic Hamiltonian system. Preprint University of Wisconsin-Madison, 1988 · Zbl 0701.58023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.