Periodic Solutions Of Hamiltonian Systems Research Articles

Periodic solutions of Hamiltonian systems coupling twist with an isochronous center

Periodic solutions of Hamiltonian systems coupling twist with an isochronous center

Periodic solutions of Hamiltonian systems coupling twist with generalized lower/upper solutions

The Hamiltonian systems considered in this paper are obtained by weakly coupling two systems having completely different behaviors. The first one satisfies the twist assumptions usually considered for the application of the Poincaré–Birkhoff Theorem, while the second one presents the existence of some well-ordered lower and upper solutions. In the higher dimensional case, we also treat a coupling situation where the classical Hartman condition is assumed.

Existence and multiplicity of rotating periodic solutions for Hamiltonian systems with a general twist condition

In this paper, we consider rotating periodic solutions of the Hamiltonian systemx˙=JH′(t,x),x∈R2N, having the form x(t+T)=Qx(t),∀t∈R, for some T>0 and a symplectic orthogonal matrix Q. We study the system under a general twist condition: the nonlinear term H′(t,x) is required to be of linear growth but not necessarily to be asymptotically linear at infinity. The twist is reflected in the difference of the generalized Morse index at the origin and at infinity. By combining a finite dimensional reduction method, Morse theory and minimax principle, we establish the existence and multiplicity of nontrivial rotating periodic solutions.

The equivariant spectral flow and bifurcation of periodic solutions of Hamiltonian systems

We define a spectral flow for paths of selfadjoint Fredholm operators that are equivariant under the orthogonal action of a compact Lie group as an element of the representation ring of the latter. This G-equivariant spectral flow shares all common properties of the integer valued classical spectral flow, and it can be non-trivial even if the classical spectral flow vanishes. Our main theorem uses the G-equivariant spectral flow to study bifurcation of periodic solutions for autonomous Hamiltonian systems with symmetries.

Multiple minimal periodic solutions for subquadratic second-order Hamiltonian systems

By using a generalized version of the Weierstrass theorem and a new space decomposition, we obtain the existence of two minimal periodic solutions for a class of subquadratic even Hamiltonian systems. To our best knowledge, this is the first result of the existence of multiple minimal periodic solutions for Hamiltonian systems with subquadratic potentials.

A generalized mountain pass lemma with a closed subset for locally Lipschitz functionals

ABSTRACT The classical Mountain Pass Lemma of Ambrosetti-Rabinowitz has been studied, extended and modified in several directions. Notable examples would certainly include the generalization to locally Lipschitz functionals by K. C. Chang, analyzing the structure of the critical set in the mountain pass theorem in the works of Hofer, Pucci-Serrin and Tian, and the extension by Ghoussoub-Preiss to closed subsets in a Banach space with recent variations. In this paper, we utilize the generalized gradient of Clarke and Ekeland's variatonal principle to generalize the Ghoussoub-Preiss's Theorem in the setting of locally Lipschitz functionals. We give an application to periodic solutions of Hamiltonian systems.

A dihedral Bott-type iteration formula and stability of symmetric periodic orbits

Motivated by the recent discoveries on the stability properties of symmetric periodic solutions of Hamiltonian systems, we establish a Bott-type iteration formula for dihedraly equivariant Hamiltonian systems. We apply the abstract theory for computing the Morse indices of the celebrated Chenciner and Montgomery figure-eight orbit for the planar three body problem in different equivariant spaces. Finally we provide a hyperbolicity criterion for reversible Lagrangian systems.

Periodic solutions of perturbed central Hamiltonian systems

We search for periodic solutions of Hamiltonian systems having two degrees of freedom, which are periodic perturbations of a central system. Assuming the perturbing term to be symmetric under a given rotation, we prove the persistence of circular orbits of the unperturbed problem satisfying some non-degeneracy assumptions. This work is a generalization of the perturbation results obtained in [1] Ambrosetti and Zelati (Math Z 201:227–242, 1989) and [5] Fonda and Gallo (J Differ Equ 264:7055–7068, 2018) in the framework of classical central force systems. Applications are given also for particles in relativistic regimes.

Hill-type formula for Hamiltonian system with Lagrangian boundary conditions

In this paper, we build up Hill-type formula for linear Hamiltonian systems with Lagrangian boundary conditions, which include standard Neumann, Dirichlet boundary conditions. Such a kind of boundary conditions comes from the N-reversible symmetry periodic orbits in n-body problem naturally, where N is an anti-symplectic orthogonal matrix with N2=I. The Hill-type formula connects the infinite determinant of the Hessian of the action functional with the determinant of matrices which depend on the monodromy matrix and boundary conditions. Consequently, we derive the Krein-type trace formula and give nontrivial estimation for the eigenvalue problem. Combined with the Maslov-type index theory, we give some new stability criteria for the N-reversible symmetry periodic solutions of Hamiltonian systems. As an application, we study the linear stability of elliptic relative equilibria in planar 3-body problem.

Periodic solutions of Euler-Lagrange equations with sublinear potentials in an Orlicz-Sobolev space setting

In this paper, we obtain existence results of periodic solutions of hamiltonian systems in the Orlicz-Sobolev space \(W^1L^\Phi([0,T])\). We employ the direct method of calculus of variations and we consider a potential function \(F\) satisfying the inequality \(|\nabla F(t,x)|\leq b_1(t) \Phi_0'(|x|)+b_2(t)\), with \(b_1, b_2\in L^1\) and certain \(N\)-functions \(\Phi_0\).

Multiple periodic solutions of Hamiltonian systems confined in a box

We consider a nonautonomous Hamiltonian system, $T$-periodic in time, possibly defined on a bounded space region, the boundary of which consists of singularity points which can never be attained. Assuming that the system has an interior equilibrium point, we prove the existence of infinitely many $T$-periodic solutions, by the use of a generalized version of the Poincaré-Birkhoff theorem.

Periodic perturbations of Hamiltonian systems

Abstract We prove existence and multiplicity results for periodic solutions of Hamiltonian systems, by the use of a higher dimensional version of the Poincaré–Birkhoff fixed point theorem. The first part of the paper deals with periodic perturbations of a completely integrable system, while in the second part we focus on some suitable global conditions, so to deal with weakly coupled systems.

The Elusive Liapunov Periodic Solutions

We illustrate the use of regular and singular reduction methods in conjunction with classical perturbation theory to find periodic solutions of Hamiltonian systems. In particular, we use these methods to find families of periodic solutions when the classical Liapunov center theorem fails due to a resonance.

Характеристические показатели периодических решений гамильтоновых систем и необходимые условия устойчивости

Характеристические показатели периодических решений гамильтоновых систем и необходимые условия устойчивости

Continuity of the Peierls barrier and robustness of laminations

AbstractWe study the Peierls barrier$P_{\omega }(\xi )$for a broad class of monotone variational problems. These problems arise naturally in solid state physics and from Hamiltonian twist maps. We start by deriving an estimate for the difference$\vert P_{\omega }(\xi ) - P_{q/p}(\xi ) \vert $of the Peierls barriers of rotation numbers$\omega \in {{\mathbb{R}}}$and$q/p\in {\mathbb{Q}}$. A similar estimate was obtained by Mather [Modulus of continuity for Peierls’s barrier.Proc. NATO Advanced Research Workshop on Periodic Solutions of Hamiltonian Systems and Related Topics (Il Ciocco, Italy, 13–18 October 1986) (NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 209).Eds. P. H. Rabinowitz, A. Ambrosetti and I. Eckeland. D. Reidel, Dordrecht, 1987, pp. 177–202] in the context of twist maps, but our proof is different and applies more generally. It follows from the estimate that$\omega \mapsto P_{\omega }(\xi )$is continuous at irrational points. Moreover, we show that the Peierls barrier depends continuously on parameters and hence that the property that a monotone variational problem admits a lamination of minimizers of rotation number$\omega \in {{\mathbb{R}}}\delimiter "026E30F {\mathbb{Q}}$is open in the$C^1$-topology.

On the influence of numerical preservation of invariants when simulating Hamiltonian relative periodic orbits

We study the structure of the error when simulating relative periodic solutions of Hamiltonian systems with symmetries. We identify the mechanisms for which the preservation, in the numerical integration, of the Hamiltonian and the invariants associated to the symmetry group, implies a better time behavior of the error. A second consequence is a more correct simulation of the parameters that characterize the relative periodic orbit.

Periodic solutions for Hamiltonian systems without Ambrosetti–Rabinowitz condition and spectrum 0

In this paper, we consider the superquadratic second order Hamiltonian system u ″ ( t ) + A ( t ) u ( t ) + ∇ H ( t , u ( t ) ) = 0 , t ∈ R . Our main results here allow the classical Ambrosetti–Rabinowitz superlinear condition to be replaced by a general superquadratic condition, and 0 lies in a gap of σ ( B ) , where B : = − d 2 d t 2 − A ( t ) . We will study the ground state periodic solutions for this problem. The main idea here lies in an application of a variant generalized weak linking theorem for strongly indefinite problem developed by Schechter and Zou.

Periodic solutions of Hamiltonian systems with anisotropic growth

In this paper we obtain some existence and multiplicity results for periodic solutions of nonautonomous Hamiltonian systems $\dot z(t)=J\nabla H(z(t),t)$ whose Hamiltonian functions may have simultaneously, in different components, superquadratic, subquadratic and quadratic behaviors. Our results generalize some earlier work [3] of P. Felmer and [5] of P. Felmer and Z.-Q. Wang.

Multiple periodic solutions for Hamiltonian systems with not coercive potential

Under an appropriate oscillating behavior of the nonlinear term, the existence of infinitely many periodic solutions for a class of second order Hamiltonian systems is established. Moreover, the existence of two non-trivial periodic solutions for Hamiltonian systems with not coercive potential is obtained, and the existence of three periodic solutions for Hamiltonian systems with coercive potential is pointed out. The approach is based on critical point theorems.

Morse index and stability of elliptic Lagrangian solutions in the planar three-body problem

We illustrate a new way to study the stability problem in celestial mechanics. In this paper, using the variational nature of elliptic Lagrangian solutions in the planar three-body problem, we study the relation between Morse index and its stability via Maslov-type index theory of periodic solutions of Hamiltonian system. For elliptic Lagrangian solutions we get an estimate of the algebraic multiplicity of unit eigenvalues of its monodromy matrix in terms of the Morse index, which is the key to understand the stability problem. As a special case, we provide a criterion to spectral stability of relative equilibrium.