A variational approach to homoclinic orbits in Hamiltonian systems. (English) Zbl 0731.34050

Coti Zelati, Vittorio; Ekeland, Ivar; Séré, Eric

Math. Ann. 288, No. 1, 133-160 (1990).

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.

Reviewer: I. G. Rozett (Samarkand)

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

Keywords:

Hamiltonian system; doubly asymptotic solutions; periodic solution; Floquet’s theory

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 · doi:10.1016/0022-1236(73)90051-7
[2]	Aubin, J. P.; Ekeland, I., Applied nonlinear analysis (1984), New York: Wiley, New York · Zbl 0641.47066
[3]	Clarke, F., Periodic solutions of Hamiltonian inclusions, J. Differ. Equations, 40, 1-6 (1981) · Zbl 0461.34030 · doi:10.1016/0022-0396(81)90007-3
[4]	Ekeland, I., Periodic solutions of Hamilton’s equations and a theorem of P. Rabinowitz, J. Differ. Equations, 34, 523-534 (1979) · Zbl 0446.70019 · doi:10.1016/0022-0396(79)90034-2
[5]	Ekeland, I., Convexity methods in Hamiltonian systems (1989), Berlin Heidelberg New York: Springer, Berlin Heidelberg New York
[6]	Greenspan, B. D.; Holmes, P. J.; Barenblatt, G. I.; Iooss, G.; Joseph, D. D., Homoclinic orbits, subharmonics and global bifurcations in forced oscillations, Nonlinear dynamics and turbulence (1983), Boston London Melburne: Pitman, Boston London Melburne · 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 n^0 8607, Paris, 1988
[9]	Lions, P. L., The concentration-compactness principle in the calculus of variations, Rev. Mat. Iberoam., 1, 145-201 (1985) · Zbl 0704.49005 · doi:10.4171/RMI/6
[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 (1973), Princeton: Princeton University Press, Princeton · 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 · doi:10.1002/cpa.3160110105
[13]	Poincaré, H., Les méthodes nouvelles de la mécanique céleste (1899), Paris: Gauthier-Villars, Paris · JFM 30.0834.08 · doi:10.1007/BF02742713
[14]	Rabinowitz, P.H.: Periodic and heteroclinic orbits for a periodic Hamiltonian system. Preprint University of Wisconsin-Madison, 1988 · Zbl 0701.58023