Related papers: Homoclinic Orbits and Lagrangian Embeddings
We consider autonomous Lagrangian systems with two degrees of freedom, having an hyperbolic equilibrium of saddle-saddle type (that is the eingenvalues of the linearized system about the equilibrium are $\pm \lambda_1, \pm \lambda_2 $,…
We consider a Hamiltonian system which has an elliptic-hyperbolic equilibrium with a homoclinic loop. We identify the set of orbits which are homoclinic to the center manifold of the equilibrium via a Lyapunov- Schmidt reduction procedure.…
In this paper, we study the existence and multiplicity of homoclinic solutions for following Hamiltonian systems with asymptotically quadratic nonlinearities at infinity \begin{eqnarray*} \ddot{u}(t)-L(t)u+\nabla W(t,u)=0. {eqnarray*} We…
We consider a connected symplectic manifold $M$ acted on properly and in a Hamiltonian fashion by a connected Lie group $G$. Inspired to the recent paper \cite{gb2}, see also \cite{ch} and \cite{pacini}, we study Lagrangian orbits of…
In this article, we study the existence of homoclinic orbits for the first-order Hamiltonian system {equation*} J\dot{u}(t)+\nabla H(t,u(t))=0,\quad t\in\mathbb{R}. {equation*} Under the Ambrosetti-Rabinowitz's superquadraticy condition, or…
In this paper we study the dynamics near the equilibrium point of a family of Hamiltonian systems in the neighborhood of a $0^2 iw$ resonance. The existence of a family of periodic orbits surrounding the equilibrium is well-known and we…
For Hamiltonian flows we establish the existence of periodic orbits on a sequence of level sets approaching a Bott-nondegenerate symplectic extremum of the Hamiltonian. As a consequence, we show that a charge on a compact manifold with a…
We present a new existence mechanism, based on symplectic topology, for orbits of Hamiltonian flows connecting a pair of disjoint subsets in the phase space. The method involves function theory on symplectic manifolds combined with rigidity…
We consider a homoclinic orbit to a saddle fixed point of an arbitrary $C^\infty$ map $f$ on $\mathbb{R}^2$ and study the phenomenon that $f$ has an infinite family of asymptotically stable, single-round periodic solutions. From classical…
We study persistence of periodic and homoclinic orbits, first integrals and commutative vector fields in dynamical systems depending on a small parameter $\varepsilon>0$ and give several necessary conditions for their persistence. Here we…
The main result asserts the existence of noncontractible periodic orbits for compactly supported time dependent Hamiltonian systems on the unit cotangent bundle of the torus or of a negatively curved manifold whenever the generating…
We study bifurcations of homoclinic orbits to hyperbolic saddle equilibria in a class of four-dimensional systems which may be Hamiltonian or not. Only one parameter is enough to treat these types of bifurcations in Hamiltonian systems but…
An analytic reversible Hamiltonian system with two degrees of freedom is studied in a neighborhood of its symmetric heteroclinic connection made up of a symmetric saddle-center, a symmetric orientable saddle periodic orbit lying in the same…
In this paper, we prove the existence of infinitely many homoclinic orbits for the first order Hamiltonian systems $J\dot{x}-M(t)x+ R'(t,x)=0$, by the minimax methods in critical point theory, when $R(t,y)$ satisfies the superquadratic…
In this paper, we study the existence for the homoclinic orbits for the second order Hamiltonian systems. Under suitable conditions on the potential $V$, we apply the direct method of variations and the Fourier analysis to prove the…
In this work we consider a two-dimensional piecewise smooth system, defined in two domains separated by the switching manifold $x=0$. We assume that there exists a piecewise-defined continuous Hamiltonian that is a first integral of the…
Let M be a weakly monotone symplectic manifold, and H be a time-dependent Hamiltonian; we assume that the periodic orbits of the corresponding time-dependent Hamiltonian vector field are non-degenerate. We construct a refined version of the…
This article concerns arbitrary finite heteroclinic networks in any phase space dimension whose vertices can be a random mixture of equilibria and periodic orbits. In addition, tangencies in the intersection of un/stable manifolds are…
We will present a consistent description of Hamiltonian dynamics on the ``symplectic extended phase space'' that is analogous to that of a time-\underline{in}dependent Hamiltonian system on the conventional symplectic phase space. The…
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…