Periodic Solutions Of Hamiltonian Systems Research Articles

Subharmonic and multiple periodic solutions for Hamiltonian systems with local partial sublinear nonlinearity

AbstractThe existence of subharmonic and multiple periodic solutions as well as the minimality of periods are obtained for the nonautonomous Hamiltonian systems

A twist condition and periodic solutions of Hamiltonian systems

In this paper, we investigate existence of nontrivial periodic solutions to the Hamiltonian system (HS) − J z ˙ = H ′ ( t , z ) , z ∈ R 2 N . Under a general twist condition for the Hamiltonian function in terms of the difference of the Conley–Zehnder index at the origin and at infinity we establish existence of nontrivial periodic solutions. Compared with the existing work in the literature, our results do not require the Hamiltonian function to have linearization at infinity. Our results allow interactions at infinity between the Hamiltonian and the linear spectra. The general twist condition raised here seems to resemble more the spirit of Poincaré's last geometric theorem.

Hamiltonian systems as selfdual equations

Hamiltonian systems with various time boundary conditions are formulated as absolute minima of newly devised non-negative action functionals obtained by a generalization of Bogomolnyi’s trick of ‘dcompleting squares’. Reminiscent of the selfdual Yang-Mills equations, they are not derived from the fact that they are critical points (i.e., from the corresponding Euler-Lagrange equations) but from being zeroes of the corresponding non-negative Lagrangians. A general method for resolving such variational problems is also described and applied to the construction of periodic solutions for Hamiltonian systems, but also to study certain Lagrangian intersections.

Periodic solutions of some resonant Hamiltonian systems

Based on the Galerkin approximation and the Morse inequality, an existence theorem of periodic solutions of the Hamiltonian system J z ′ = H ′ ( t , z ) is given, where H ′ ( t , z ) is asymptotically linear and resonant at infinity. Existence of periodic solutions of the resonant asymmetric p-Laplacian equation is also discussed.

On a product formula for the Conley–Zehnder index of symplectic paths and its applications

Using invariance by fixed-endpoints homotopies and a generalized notion of symplectic Cayley transform, we prove a product formula for the Conley–Zehnder index of continuous paths with arbitrary endpoints in the symplectic group. We discuss two applications of the formula, to the metaplectic group and to periodic solutions of Hamiltonian systems.

Multiple periodic solutions of Hamiltonian systems with prescribed energy

Consider the periodic solutions of autonomous Hamiltonian systems x ˙ = J ∇ H ( x ) on the given compact energy hypersurface Σ = H −1 ( 1 ) . If Σ is convex or star-shaped, there have been many remarkable contributions for existence and multiplicity of periodic solutions. It is a hard problem to discuss the multiplicity on general hypersurfaces of contact type. In this paper we prove a multiplicity result for periodic solutions on a special class of hypersurfaces of contact type more general than star-shaped ones.

Computations of cohomology groups and nontrivial periodic solutions of Hamiltonian systems

By computing the E -critical groups at θ and infinity of the corresponding functional of Hamiltonian systems, we proved the existence of nontrivial periodic solutions for the systems which may be resonant at θ and infinity under some new conditions. Some results in the literature are extended and some new type of theorems are proved. The main tool is the E -Morse theory developed by Kryszewski and Szulkin.

On the minimal periodic solutions of nonconvex superlinear Hamiltonian systems

This paper proves a multiplicity result for the minimal periodic solutions of Hamiltonian system x ˙ ( t ) = J ∇ H ( x ( t ) ) under a superquadratic hypothesis on H weaker than the Ambrosetti–Rabinowitz-type condition. We also define a class of homogeneous functions, that are more general than the classical ones, and prove the Rabinowitz's conjecture in the case of H satisfying such new homogeneity.

Periodic solutions for Hamiltonian systems under strong constraining forces

We consider periodic solutions of Hamiltonian systems in Euclidean spaces whose motion is constrained to a submanifold M. We prove that under some nondegeneracy assumptions, periodic solutions persist when the constraint is replaced by a strong restoring potential.

Proof and generalization of Kaplan-Yorke’s conjecture under the condition f ' (0) > 0 on periodic solution of differential delay equations

Using the theory of existence of periodic solutions of Hamiltonian systems, it is shown that many periodic solutions of differential delay equations can be yielded from many families of periodic solutions of the coupled generalized Hamiltonian systems. Some sufficient conditions on the existence of periodic solutions of differential delay equations are obtained. As a corollary of our results, the conjecture of Kaplan-Yorke on the search for periodic solutions for certain special classes of scalar differential delay equations is shown to be true when\(\mathbb{W}^t \).

Large Torsional Oscillations in Suspension Bridges Revisited: Fixing an Old Approximation

Click to increase image sizeClick to decrease image size Additional informationNotes on contributorsP. J. McKennaP. J. MCKENNA did his undergraduate work in Dublin (at U.C.D.) and his graduate work in Ann Arbor. The central theme of his research is nonlinear analysis, in particular, the existence, multiplicity, and numerical approximation of solutions of nonlinear boundary value problems. The work arises naturally from his research in multiple periodic solutions of Hamiltonian systems. Kristen Moore, his doctoral student, is extending this analysis to the partial differential equations that describe the spatial behaviour along the length of the bridge. Movies showing computed solutions can be seen at http:\\www.math.uconn.edu\~kmoore\

Bott formula of the Maslov-type index theory

In this paper the integer valued !-index theory parameterized by all ! on the unit circle for paths in the symplectic group Sp(2n) is established. Based on this index theory, the Bott formula of the Maslov-type index theory for iterated paths in Sp(2n) is estalished, the mean index for periodic solutions of Hamiltonian systems is dened, and the increasing estimate of the iterated Maslov-type index in terms of the mean index is proved.

Multiplicity for symmetric indefinite functionals: applications to Hamiltonian and elliptic systems

In this article we study the existence of critical points for certain superquadratic strongly indefinite even functionals appearing in the study of periodic solutions of Hamiltonian systems and solutions of certain class of Elliptic Systems. We first present two abstract critical point theorems for even functionals. These results are well suited for our applications, but they are interesting by their own. These theorems are then applied to the specific problems we mentioned, when the corresponding Hamiltonians are superquadratic. Critical points theory for even indefinite functionals was studied by Benci in [2], where he developed a general pseudo index theory, a variant of the classical genus theory (c.f. [10]). We want to point out that our approach is totally different from [2]. Taking advantages of the even nature of the functional we shall construct a geometric linking structure which does not require conditions near the origin (see [10] for examples of some standard linking and compare the set up of our linking in Theorem 1.1).

Symplectic Transformations and Periodic Solutions of Hamiltonian Systems

Symplectic transformations with a kind of homogeneity are introduced, which enable us to give a unified approach to some existence problems on the periodic solutions for the first- and second-order Hamiltonian systems.

The space of loops on configuration spaces and the Majer-Terracini index

The topology of configuration spaces and their free and based loop spaces plays an important role in the study of the existence of periodic solutions of Hamiltonian systems of n-body type (see e.g. [1], [6], [7]). In particular, if Fn(R) is the configuration space of n-particles u = (u1, . . . , un) in R, then the free loop space ΛFn(R) is, up to homotopy type, the domain of a functional f whose critical points are solutions of a corresponding n-body problem. Thus, the homotopy-type invariants of ΛFn(R) such as Lusternik–Schnirelmann (LS) category (and its generalizations — commonly called “index theories”), homotopy and homology play an important role in the subject. In some recent work ([15], [16], [17]), P. Majer and S. Terracini introduced an interesting “collision index” in the space ΛFn(R) which allowed contractions of subsets of ΛFn(R) to move through subspaces intermediate to ΛFn(R) and Λ(R), thus allowing a limited number of collisions during the contraction. However, their index is based upon an equivalence relation which results in a generalized notion of collision. Namely, that ui “collides” with uj during the deformation h if there is a chain of indices i1, . . . , is, such that ui = ui1 , uj = uis and uir collides with uir+1 , 1 ≤ r ≤ s.

Existence of periodic solutions of Hamiltonian systems with potential indefinite in sign

Existence of periodic solutions of Hamiltonian systems with potential indefinite in sign

Multiple periodic solutions of Hamiltonian systems with strong resonance at infinity

An asymptotically linear Hamiltonian system with strong resonance at infinity is considered. The existence of multiple periodic solutions is proved via variational methods in an equivariant setting.

Existence of periodic solutions for Hamiltonian systems at resonance

Existence of periodic solutions for Hamiltonian systems at resonance

Computation of periodic solutions of Hamiltonian systems

This paper attempts to give a practical method to compute global periodic solutions of autonomous Hamiltonian systems of arbitrary finite order. The proposed numerical method is based on continuation of solutions branching from equlibrium points and requires no iterations. Moreover, during computation of one-parameter families of periodic orbits, their possible bifurcations are determined as well.

Periodic solutions of Hamiltonian systems on hypersurfaces in a torus

The existence of periodic solutions to Hamiltonian systems on the symplectic manifold (T 2n, ω) is studied. We show that on a class of hypersurfaces in the torusT 2n there is a periodic solution, which generalizes the results due to Long and Zehnder.