## Homoclinic orbits in a first order superquadratic Hamiltonian system: Convergence of subharmonic orbits.(English)Zbl 0787.34041

Homoclinic orbits for a time-periodic Hamiltonian system $$(*)$$ $$\dot z =JH_ z(t,z)$$, $$H={1 \over 2}\langle Az,z \rangle+W(T,z)$$ are found, assuming that $$z=0$$ is a hyperbolic equilibrium point and that $$W$$ has global superquadratic growth in $$z$$. They are obtained as local $$C^ 1$$- limits of certain nontrivial $$T$$-periodic solutions of $$(*)$$ as $$T \to \infty$$, where the hyperbolicity prevents them from shrinking to zero. This approach extends results by Rabinowitz for second order Hamiltonian systems, and it differs from corresponding results by Coti-Zelati, Ekeland & Séré and Hofer & Wysocki. The references are given in the paper.

### MSC:

 34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
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### References:

 [1] Benci, V; Giannoni, F, Homoclinic orbits on compact manifolds, J. math. anal. appl., 157, 568-576, (1991) · Zbl 0737.58052 [2] Coti-Zalati, V; Ekeland, I; Séré, E, A variational approach to homoclinic orbits in Hamiltonian systems, Math. ann., 288, 133-160, (1990) · Zbl 0731.34050 [3] Edwards, R.E; Gaudry, G.I, Littlewood-Paley and multiplier theory, () · Zbl 0464.42013 [4] Felmer, P.L, Subharmonic solutions near an equilibrium point for Hamiltonian systems, Manuscripta math., 66, 359-396, (1990) · Zbl 0688.34027 [5] Felmer, P.L, Heteroclinic orbits for spatially periodic Hamiltonian systems, (1989), preprint · Zbl 0749.58021 [6] Hofer, H; Wysocki, K, First order elliptic system and the existence of homoclinic orbits in Hamiltonian system, Math. ann., 288, 483-503, (1990) · Zbl 0702.34039 [7] Rabinowitz, P.H, On subharmonic solutions of Hamiltonian systems, Comm. pure appl. math., 33, 609-633, (1980) · Zbl 0425.34024 [8] Rabinowitz, P.H, Minimax methods in critical point theory with applications to differential equations, (), 21 [9] Rabinowitz, P.H, Periodic and heteroclinic orbits for a periodic Hamiltonian system, Ann. inst. H. Poincaré anal. non linéaire, 6, 331-346, (1989) · Zbl 0701.58023 [10] Rabinowitz, P.H, Homoclinic orbits for a class of Hamiltonian systems, (), 33-38 · Zbl 0705.34054 [11] Rabinowitz, P.H; Tanaka, K, Some results on connecting orbits for a class of Hamiltonian system, Math. zeit., 206, 473-499, (1991) · Zbl 0707.58022 [12] Tanaka, K, Homoclinic orbits for a singular second order Hamiltonian system, Ann. inst. H. Poincaré anal. non linéaire, 7, 427-438, (1990) · Zbl 0712.58026
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