## Homoclinic solutions of Hamiltonian systems with symmetry.(English)Zbl 0944.37030

Consider a Hamiltonian system with a Hamiltonian of the following form: $H(z,t)=\tfrac{1}{2}Az\cdot z+F(z,t).$ The authors show that if (i) the spectrum of the matrix $$JA$$ (where $$J$$ is the standard symplectic matrix) does not intersect the imaginary axis; (ii) $$F$$ is invariant under the action of a compact Lie group; and (iii) $$F$$ is superquadratic at zero and infinity, then the system has infinitely many geometrically distinct homoclinic solutions.

### MSC:

 37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010) 37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
Full Text:

### References:

 [1] Arioli, G.; Gazzola, F.; Terracini, S., Multibump periodic motions of an infinite lattice of particles, Math. Z., 223, 627-642, (1996) · Zbl 0871.34028 [2] Bartsch, T., A simple proof of the degree formula for Z/p-equivariant maps, Math. Z., 212, 285-292, (1992) · Zbl 0790.55003 [3] Bartsch, T., Topological methods for variational problems with symmetries, (1993), Springer-Verlag New York/Berlin · Zbl 0789.58001 [4] Bartsch, T.; Clapp, M., Critical point theory for indefinite functionals with symmetries, J. funct. anal., 138, 107-136, (1996) · Zbl 0853.58027 [5] Benci, V., On critical point theory for indefinite functionals in the presence of symmetries, Trans. amer. math. soc., 274, 533-572, (1982) · Zbl 0504.58014 [6] Bourbaki, N., Algebra I, (1989), Springer-Verlag New York/Berlin · Zbl 0623.18008 [7] Bröcker, T.; Dieck, T.tom, Representation of compact Lie groups, (1985), Springer-Verlag New York/Berlin [8] Zelati, V.Coti; Ekeland, I.; Séré, E., A variational approach to homoclinic orbits in Hamiltonian systems, Math. ann., 288, 133-160, (1990) · Zbl 0731.34050 [9] Coti Zelati, V.; Rabinowitz, P.H., Homoclinic orbits for second order Hamiltonian systems possessing superquadratic potentials, J. amer. math. soc., 4, 693-727, (1991) · Zbl 0744.34045 [10] Zelati, V.Coti; Rabinowitz, P.H., Homoclinic type solutions for a semilinear elliptic PDE on rn, Comm. pure appl. math., 45, 1217-1269, (1992) · Zbl 0785.35029 [11] Dieck, T.tom, Transformation groups, (1987), de Gruyter Berlin [12] Ding, Y.H.; Girardi, M., Infinitely many homoclinic orbits of a Hamiltonian system with symmetry, Nonl. anal. TMA, 38, 391-415, (1999) · Zbl 0938.37034 [13] Y. H. Ding, and, M. Willem, Homoclinic orbits of a Hamiltonian system, ZAMP, in press. · Zbl 0997.37041 [14] Hofer, H.; Wysocki, K., First order elliptic systems and the existence of homoclinic orbits in Hamiltonian systems, Math. ann., 288, 483-503, (1990) · Zbl 0702.34039 [15] Kryszewski, W.; Szulkin, A., Generalized linking theorem with an application to semilinear Schrödinger equation, Adv. differential equations, 3, 441-472, (1998) · Zbl 0947.35061 [16] Mawhin, J.; Willem, M., Critical point theory and Hamiltonian systems, (1989), Springer-Verlag New York/Berlin · Zbl 0676.58017 [17] Rabinowitz, P.H., Minimax methods in critical point theory with applications to differential equations, Amer. math. soc., 65, (1986), Providence · Zbl 0609.58002 [18] Séré, E., Existence of infinitely many homoclinic orbits in Hamiltonian systems, Math. Z., 209, 27-42, (1992) · Zbl 0725.58017 [19] Séré, E., Looking for the Bernoulli shift, Ann. inst. H. Poincaré, 10, 561-590, (1993) · Zbl 0803.58013 [20] Stuart, C.A., Bifurcation into spectral gaps, Bull. belg. math. soc., (1995) · Zbl 0864.47037 [21] Tanaka, K., Homoclinic orbits in a first order superquadratic Hamiltonian system: convergence of subharmonic orbits, J. differential equations, 94, 315-339, (1991) · Zbl 0787.34041 [22] Willem, M., Minimax theorems, (1996), Birkhäuser Basel · Zbl 0856.49001
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.