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Periodic and heteroclinic orbits for a periodic Hamiltonian system. (English) Zbl 0701.58023
Der Autor betrachtet Hamiltonsche Systeme der Form \((1)\quad \ddot q+V'(q)=0,\) wobei \(q=(q_ 1,...,q_ n)\) und V periodisch in \(q_ i\), \(1\leq i\leq n\), ist. Es werden Kriterien angegeben, unter denen das System (1) nicht konstante periodische Lösungen besitzt, und die Form der Trajektorien in der Nähe von Maxima der Funktion V werden untersucht.
Reviewer: W.Wendt

MSC:
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
37G99 Local and nonlocal bifurcation theory for dynamical systems
34C25 Periodic solutions to ordinary differential equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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References:
[1] K. C. Chang, On the Periodic Nonlinearity and Multiplicity of Solutions, Nonlinear Analysis, T.M.A. (to appear). · Zbl 0681.58036
[2] A. Fonda and J. Mawhin, Multiple Periodic Solutions of Conservative Systems with Periodic Nonlinearity, preprint. · Zbl 0718.34054
[3] J. Franks, Generalizations of the Poincaré-Birkhoff Theorem, preprint. · Zbl 0676.58037
[4] Li Shujie, Multiple Critical Points of Periodic Functional and Some Applications, International Center for Theoretical Physics Tech. Rep. IC-86-191.
[5] J. Mawhin, Forced Second Order Conservative Systems with Periodic Nonlinearity, Analyse Nonlineaire (to appear). · Zbl 0678.34050
[6] Mawhin, J.; Willem, M., Multiple solutions of the periodic boundary value problem for some forced pendulum-type equations, J. Diff. Eq., Vol. 52, 264-287, (1984) · Zbl 0557.34036
[7] Pucci, P.; Serrin, J., A mountain pass theorem, J. Diff. Eq., Vol. 60, 142-149, (1985) · Zbl 0585.58006
[8] P. Pucci and J. Serrin, Extensions of the Mountain Pass Theorem, Univ. of Minnesota Math. Rep. 83-150. · Zbl 0564.58012
[9] P. H. Rabinowitz, On a Class of Functionals Invariant Under a Z^n Action, Trans. A.M.S. (to appear). · Zbl 0718.34057
[10] P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, C.B.M.S, Reg. Conf. Ser. No. 56, Amer. Math. Soc., Providence, RI, 1986. · Zbl 0609.58002
[11] A. M. Lyapunov, The General Problem of Instability of a Motion, ONTI, Moscow-Leningrad, 1935.
[12] Kozlov, V. V., Instability of equilibrium in a potential field, Russian Math. Surveys, Vol. 36, 238-239, (1981) · Zbl 0478.70004
[13] Kozlov, V. V., On the instability of equilibrium in a potential field, Russian Math. Surveys, Vol. 36, 257-258, (1981) · Zbl 0556.70003
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