zbMATH — the first resource for mathematics

A “Birkhoff-Lewis” type result for a class of Hamiltonian systems. (English) Zbl 0616.58014
Consider the Hamiltonian system of differetial equations \[ (1)\quad - J\dot z=H_ z(t,z), \] where \(z=(p,q)\in {\mathbb{R}}^{2N}\), J denotes the symplectic structure in \({\mathbb{R}}^{2N}\) and \(H_ z\) denotes the gradient of the Hamiltonian function H with respect to z. Suppose that H is \(\tau\)-periodic in the t variable and \(H_ z(t,0)=0\). If H satisfies suitable assumptions it can be proved that (1) possesses a sequence \(z_ k\) of \(k\cdot \tau\) periodic solutions (k-prime) converging to zero uniformly in \(C^ 1({\mathbb{R}},{\mathbb{R}}^{2N})\).

37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
37G99 Local and nonlocal bifurcation theory for dynamical systems
58E30 Variational principles in infinite-dimensional spaces
Full Text: DOI EuDML
[1] - ARNOLD,V.I.: Mathematical Methods of Classical Mechanics. Berlin-Heidelberg-New York; Springer-Verlag 1978 · Zbl 0386.70001
[2] - BENCI,V.-CAPOZZI,A.-FORTUNATO,D.: Periodic solutions of Hamilt onian systems with superquadratic potential, Ann.Mat.Pura Appl.143, 1-46, (1986) · Zbl 0632.34036
[3] - BENCI,V. -RABINOWITZ,P.H.: Critical point theorems for indefinite functionals. Inv.Math.52, 336-352, (1979) · Zbl 0465.49006
[4] - BIRKHOFF,G.D.-LEWIS,D.C.: On the periodic motions near a given periodic motion of a dynamical system. Ann.Mat.Pura Appl.12, 117-133, (1933) · Zbl 0007.37104
[5] - GELFAND,I.-LIDSKY,V.: On the structure of regions of stability of linear canonical systems of differential equations with periodic coefficients, Uspekhi Mat.Naouk,10, (1955), 3-40 (A.M.S.Translation,8, 143-181, (1958))
[6] - HARRIS,T.C.: Periodic solutions of arbitrarily long periods in Hamiltonian systems. J.Diff.Eq.4, 131-141, (1968) · Zbl 0157.14504
[7] - J?RGENS, K. -WEIDMANN, J.: Spectral properties of Hamiltonian operators. Berlin-Heidelberg-New York; Springer-Verlag lecture notes in Math., (1973)
[8] - KREIN,M.: Generalization of certain investigations of A.H.Liapounov on linear differential equations with periodic coefficients. Doklady Akad.Naouk, USSR,73, 445-448, (1950)
[9] - MOSER, J.: Proof of a generalized form of a fixed point theorem due to G.D.Birkhoff. Berlin-Heidelberg-New York: Springer-Verlag lecture notes in Math.,597, 464-494, (1977)
[10] - RABINOWITZ,P.H.: On Subharmonic solutions of Hamiltonian systems. Comm.Pure Appl.Math.,33, 609-633, (1980) · Zbl 0437.34011
[11] - RABINOWITZ,P.H.: Minimax Methods in Critical point Theory with applications to differential equations. Conference series in Mathematics, A.M.S.,65, (1986) · Zbl 0609.58002
[12] - TARANTELLO,G.: Subharmonic solutions for Hamiltonian systems via aZ p pseudoindex theory, preprint · Zbl 0755.34035
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.