Related papers: Normal modes in symplectic stratified spaces

We generalize the Weinstein-Moser theorem on the existence of nonlinear normal modes near an equilibrium in a Hamiltonian system to a theorem on the existence of relative perodic orbits near a relative equilibrium in a Hamiltonian system…

An estimate on the number of distinct relative periodic orbits around a stable relative equilibrium in a Hamiltonian system with continuous symmetry is given. This result constitutes a generalization to the Hamiltonian symmetric framework…

We show existence of relative periodic orbits (a.k.a. relative nonlinear normal modes) near relative equilibria of a symmetric Hamiltonian system under an appropriate assumption on the Hessian of the Hamiltonian. This gives a relative…

In this paper, we establish the existence of periodic orbits of a twisted geodesic flow on all low energy levels and in all dimensions whenever the magnetic field form is symplectic and spherically rational. This is a consequence of a more…

The classical theorem of Moser, on the existence of a normal form in the neighbourhood of a hyperbolic equilibrium, is extended to a class of real-analytic Hamiltonians with aperiodically time-dependent perturbations. A stronger result is…

In this article we formulate and prove sufficient conditions for the existence of trajectories of nonstationary periodic solutions of autonomous Hamiltonian systems in a neighbourhood of equilibria. It is worth pointing out that assumptions…

The Hamiltonian flow of the standard metric Hamiltonian with respect to the twisted symplectic structure on the cotangent bundle describes the motion of a charged particle on the base. We prove that under certain natural hypotheses the…

We prove a normal form theorem for Poisson structures around Poisson transversals (also called cosymplectic submanifolds), which simultaneously generalizes Weinstein's symplectic neighborhood theorem from symplectic geometry and Weinstein's…

We show that the presence of one non-degenerate, non-contractible periodic orbit of a Hamiltonian on the standard symplectic torus implies the existence of infinitely many simple non-contractible periodic orbits.

In this paper we give an extension of the Birkhoff--Lewis theorem to some semilinear PDEs. Accordingly we prove existence of infinitely many periodic orbits with large period accumulating at the origin. Such periodic orbits bifurcate from…

We consider Hamiltonian diffeomorphisms of the Euclidean space, generated by compactly supported time-dependent perturbations of hyperbolic quadratic forms. We prove that, under some natural assumptions, such a diffeomorphism must have…

Hamiltonian dynamical systems tend to have infinitely many periodic orbits. For example, for a broad class of symplectic manifolds almost all levels of a proper smooth Hamiltonian carry periodic orbits. The Hamiltonian Seifert conjecture is…

We prove several generic existence results for infinitely many periodic orbits of Hamiltonian diffeomorphisms or Reeb flows. For instance, we show that a Hamiltonian diffeomorphism of a complex projective space or Grassmannian generically…

We develop a general stability theory for equilibrium points of Poisson dynamical systems and relative equilibria of Hamiltonian systems with symmetries, including several generalisations of the Energy-Casimir and Energy-Momentum methods.…

In 1967 Moser proved the existence of a normal form for real analytic perturbations of vector fields possessing a reducible Diophantine invariant quasi-periodic torus. In this paper we present a proof of existence of this normal form based…

We devise a generalisation of the energy momentum-method for studying the stability of non-autonomous Hamiltonian systems with a Lie group of Hamiltonian symmetries. A generalisation of the relative equilibrium point notion to a…

Recent well-posedness results have identified the Hardy space $L^2_+$ as the natural phase space for continuum Calogero-Moser models, both focusing and defocusing, on the line and on the torus. In this paper, we introduce a symplectic form…

The aim of this paper is to prove the existence of periodic solutions to symmetric Newtonian systems in any neighborhood of an isolated orbit of equilibria. Applying equivariant bifurcation techniques we obtain a generalization of the…

We revisit the linearization theorems for proper Lie groupoids around general orbits (statements and proofs). In the the fixed point case (known as Zung's theorem) we give a shorter and more geometric proof, based on a Moser deformation…

In this article, we consider solutions starting close to some linearly stable invariant tori in an analytic Hamiltonian system and we prove results of stability for a super-exponentially long interval of time, under generic conditions. The…