Periodic Hamiltonian Systems Research Articles

Wave solutions of equations of Timoshenko type for the transverse vibrations of plates with parameters periodic on one coordinate

The hyperbolic system of equations that describes the vibrations of plates inhomogeneous along one rectangular coordinate in the context of the Timoshenko theory is presented in canonical hamiltonian form, assuming the solution is periodic on a second coordinate. In the case of periodic inhomogeneity we study the structure of the solutions of certain wave boundary-value problems for plates of this type using the general properties of periodic hamiltonian systems. Bibliography: 6 titles.

Construction of the solutions of the equations of vibration of the classical theory of plates with parameters periodic on one coordinate

The system of classical equations of transverse vibrations of plates that are inhomogeneous on one coordinate and have exponential dependence of the solution on the other coordinate and time are presented in the canonical Hamiltonian form with suitably chosen canonical variables. For periodically varying parameters we use the general properties of periodic Hamiltonian systems to study the structure of the solutions of boundary-value problems for stationary vibrations of plates.

Multibump homoclinic solutions for a class of second order, almost periodic Hamiltonian systems

We prove the existence of infinitely many homoclinic solutions for a class of second order Hamiltonian systems in R N of the form $-\ddot{q}=-q+\alpha(t)W'(q)$ where $\alpha$ is almost periodic and W is superquadratic.

Multibump solutions for perturbations of periodic second order Hamiltonian systems

Multibump solutions for perturbations of periodic second order Hamiltonian systems

Asymptotics beyond All Orders for a Certain Type of Nonlinear Oscillators

This paper shows that a special class of smooth nonlinear oscillators, called bisuperlinear, urn:x-wiley:00222526:media:sapm199696185:sapm199696185-math-0001 has a family of adiabatically symmetric solutions. This was motivated by a problem studied in sloshing water waves. A potential application of the work is to compute the nontrivial leading order term of the adiabatic invariants for a certain type of nonlinear nearly periodic Hamiltonian systems.

Existence and multiplicity of homoclinic orbits for a class of asymptotically periodic second order Hamiltonian systems

We consider the Hamiltonian system q=L(t)q−▽V(t, q) in R m,L and ▽V being asymptotic, as t→−∞, to certain periodic functions L_, ▽V_. Under suitable assumptions on the functions L, L_, V, V_, we prove for any k∈N, the existence of infinitely many k- bump homoclinic solutions of the Hamiltonian system.

On M. G. Krein's work in the theory of linear periodic Hamiltonian systems

On M. G. Krein's work in the theory of linear periodic Hamiltonian systems

The stability of linear periodic Hamiltonian systems under non-Hamiltonian perturbations

A definition of strong stability and strong instability is proposed for a linear periodic Hamiltonian system of differential equations under a given non-Hamiltonian perturbation. Such a system is subject to the action of periodic perturbations: an arbitrary Hamiltonian perturbation and a given non-Hamiltonian one. Sufficient conditions for strong stability and strong instability are established. Using the linear periodic Lagrange equations of the second kind, the effect of gyroscopic forces and specified dissipative and non-conservative perturbing forces on strong stability and strong instability is investigated on the assumption that the critical relations of combined resonances are satisfied.

Rotation type solutions for spatially periodic Hamiltonian systems

Along with solutions of (0.2) and in general for (O.l), the Hamiltonian H, also called the energy, is conserved. This one-dimensional situation can be studied in the phase space that is R x S’ since the sine function is periodic. When the energy His prescribed and bigger than 1, then system (0.2) has two periodic solutions. These periodic solutions, however, are not homotopic to a constant and they are usually called rotation type solutions. It is the purpose of this paper to generalize this situation to equation (0.1). Spatially periodic Hamiltonian systems have been studied by several authors. The nonautonomous case of (0.1) has been considered in [l-7], for the existence of periodic solutions (see also references therein). The papers of Liu [S] and the author [3] are specially related to this work. There the existence of periodic solutions is established through a generalized saddle point theorem. The hypotheses on H assumed in [3] will be considered here.

Periodic solutions of spatially periodic hamiltonian systems

Abstract : This work is concerned with the study of existence and multiplicity of periodic solutions of Hamiltonian systems of ordinary differential equations z=J(Hz(z,t) + f(t)) when the Hamiltonian H(z,t) = H(p,q,t) is periodic in the variable q and superlinear in the variable p. By imposing a growth condition on the derivative of H, we obtain the existence of at least n + 1 periodic solutions, where n is the dimension of the system. The existence of periodic solutions is obtained by using a Saddle Point Theorem recently proved by Lui. We consider a functional over E X M, where E is a Hilbert space and M is a compact manifold, satisfying a saddle point condition on E, uniformly on M. We present a proof of this Saddle Point Theorem using standard minimax techniques based on the cup length of M.

Heteroclinic orbits for spatially periodic Hamiltonian systems

We study the existence of heteroclinic orbits for a Hamiltonian system p˙=−Hq(p,q)q˙=Hp(p,q)where the Hamiltonian is periodic in the space variable q and superlinear in p. We use the Saddle Point Theorem to obtain existence of solutions for a finite time interval, and then we obtain heteroclinic orbits as limit of them. Our hypothesis on H are motivated by the second order Lagrangean systems on the torus.

Espaces de Krein et index des systèmes hamiltoniens

Some concepts and some results of the Krein-Spaces theory are introduced. We define the concept of “Krein trace”. This is then applied to a new, general definition of the index of a linear periodic hamiltonian system, which is shown to generalize the one of Morse-Ekeland and the one of Conley and Zehnder, which are then equivalent.

The construction of phase paths of a hamiltonian system in the neighbourhood of an equilibrium

A non-autonomous periodic Hamiltonian system with one degree of freedom is studied in the neighbourhood of an elliptical equilibrium point. A uniform approximation of the solution in a finite time interval in the resonance case is determined using the projection method, instead of the traditional perturbation theoretical method. The Cauchy problem is reduced to a functional equation in the space of the derivatives, and a Galerkin scheme is constructed for this equation. A theorem is proved on convergence of the sequence of approximations to the exact solution. Every finite-dimensional approximation of sufficiently high order may be found by explicit iterations. The results can be generalized to dynamical systems of higher dimensions.

The stability of multidimensional hamiltonian systems

The Lyapunov stability of the zero solution of autonomous periodic Hamiltonian systems is investigated. It is assumed that the linearized system is stable, its matrix can be reduced to diagonal form, and all characteristic indices (or roots of characteristic equations) are purely imaginary. The problem is completely solved for periodic systems with one degree of freedom, and for autonomous systems with two degrees of freedom /1–7/. The conditions of stability for systems with an arbitrary number of degrees of freedom are obtained below. The results are used to solve the problem of the Lyapunov stability of triangular libration points of the restricted three-body problem (plane elliptic and three-dimensional circular).

Classical and quantum-mechanical systems of Toda lattice type. I

Solutions to the classical periodic and non-periodic Toda lattice type Hamiltonian systems are expressed in terms of an Iwasawa-type factorization of a “large” Lie group. The scattering of these systems is determined in the non-periodic case. For the generalized periodic Toda lattices a generalization of Kostant's formula is obtained using standard representations of affine Lie groups.

Adiabatic Invariance and Passage Through Resonance for Nearly Periodic Hamiltonian Systems

This is a continuation of previous work on passage through resonance to nearly periodic Hamiltonian systems. We review the classical technique for calculating adiabatic invariants and exhibit the occurrence of zero divisors in the results as a certain critical term evolves slowly through a resonance condition. We then isolate the coordinate associated with the singularity and remove the remaining coordinate from the Hamiltonian to any desired order by successive canonical transformations. This is a variant of the von Zeipel procedure used extensively in celestial mechanics. The momentum conjugate to the cyclic coordinate is an adiabatic invariant, and the reduced Hamiltonian is then solved by constructing and matching three multiple variable expansions which describe the solution before, during, and after resonance passage.

Asymptotic Stability of Randomly Perturbed Linear Periodic Systems

For a class of linear periodic systems with randomly perturbed coefficients, an explicit criterion is obtained for the system to be asymptotically stable in the mean-square or an almost sure sense. The criterion relates the moduli of the characteristic multipliers for the reduced periodic system to the size and the correlation length of the random perturbation. The result is applied to a randomly perturbed, periodic Hamiltonian system and to the stability of periodic solutions for nonlinear systems with state-dependent noises.

The behavior of the index of a periodic linear Hamiltonian system under iteration

The behavior of the index of a periodic linear Hamiltonian system under iteration

Numerical study of periodic Hamiltonian systems by means of associated point mappings

Numerical study of periodic Hamiltonian systems by means of associated point mappings