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

The Hamiltonian system of ordinary differential equations

x ˙=J x H(t,x),(1)

where J denotes the 2n×2n matrix J=0I-I0 with J * =J -1 =-J, x H(t,x) denotes the vector H(t,x)/x, 1i2n, and the map H:× 2n belongs to the class αC 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)|0 when t±.

The author simplifies the problem by applying Floquet’s theory to the linearized system around xy ˙=JH '' (t,x(t))y using a variational approach for the solution.

34C37Homoclinic and heteroclinic solutions of ODE
37J45Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods
58E30Variational principles on infinite-dimensional spaces
34C25Periodic solutions of ODE
70H05Hamilton’s equations
[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. New York: Wiley 1984
[3]Clarke, F.: Periodic solutions of Hamiltonian inclusions. J. Differ. Equations40, 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. Equations34, 523-534 (1979) · Zbl 0446.70019 · doi:10.1016/0022-0396(79)90034-2
[5]Ekeland, I.: Convexity methods in Hamiltonian systems. Berlin Heidelberg New York: Springer 1989
[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
[7]Hofer, H., Wysocki, K.: First order elliptic systems and the existence of homoclinic orbits in Hamiltonian systems. Preprint Rutgers University, 1989
[8]Lions, P.L.: Solutions of Hartree-Fock equations for Coulomb systems. Preprint CEREMADE n0 8607, Paris, 1988
[9]Lions, P.L.: The concentration-compactness principle in the calculus of variations. Rev. Mat. Iberoam.1, 145-201 (1985)
[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
[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. Paris: Gauthier-Villars 1899
[14]Rabinowitz, P.H.: Periodic and heteroclinic orbits for a periodic Hamiltonian system. Preprint University of Wisconsin-Madison, 1988