Related papers: A way to cross double resonance
For a nearly integrable Hamiltonian systems $H=h(p)+\epsilon P(p,q)$ with $(p,q)\in\mathbb{R}^3\times\mathbb{T}^3$, large normally hyperbolic invariant cylinders exist along the whole resonant path, except for the…
We consider a Hamiltonian system with 2 degrees of freedom, with a hyperbolic equilibrium point having a loop or homoclinic orbit (or, alternatively, two hyperbolic equilibrium points connected by a heteroclinic orbit), as a step towards…
We present a general mechanism to establish the existence of diffusing orbits in a large class of nearly integrable Hamiltonian systems. Our approach relies on successive applications of the `outer dynamics' along homoclinic orbits to a…
We study the continuation of periodic orbits from various compound of homoclinics in classical system. Together with the homoclinics, the periodic orbits make up a $C^1$-smooth, normally hyperbolic invariant cylinder with holes. It plays a…
We consider an integrable Hamiltonian system weakly coupled with a pendulum-type system. For each energy level within some range, the uncoupled system is assumed to possess a normally hyperbolic invariant manifold diffeomorphic to a…
In this paper, we study small perturbations of a class of non-convex integrable Hamiltonians with two degrees of freedom, and we prove a result of diffusion for an open and dense set of perturbations, with an optimal time of diffusion which…
We develop a geometric mechanism to prove the existence of orbits that drift along a prescribed sequence of cylinders, under some general conditions on the dynamics. This mechanism can be used to prove the existence of Arnold diffusion for…
In this paper we consider a representative a priori unstable Hamiltonian system with 2+1/2 degrees of freedom, to which we apply the geometric mechanism for diffusion introduced in the paper Delshams et al., Mem. Amer. Math. Soc. 2006, and…
We describe a topological mechanism for the existence of diffusing orbits in a dynamical system satisfying the following assumptions: (i) the phase space contains a normally hyperbolic invariant manifold diffeomorphic to a two-dimensional…
We consider models given by Hamiltonians of the form $$H(I,\phi,p,q,t;\epsilon) = h(I) + \sum_{j = 1}^n \pm(\frac{1}{2} p_j^2 + V_j(q_j)) + \epsilon Q(I,\phi,p,q,t;\epsilon)$$ where $I,\phi$ are d-dimensional actions and angles, $p,q$ are…
In circular accelerators, crossing the linear coupling resonance induces the exchange of the transverse emittances, provided the process is adiabatic. This has been considered in some previous works, where the description of the phenomenon…
We introduce the concept of a "transitory" dynamical system---one whose time-dependence is confined to a compact interval---and show how to quantify transport between two-dimensional Lagrangian coherent structures for the Hamiltonian case.…
We present a method that generalizes the periodic orbit dividing surface construction for Hamiltonian systems with three or more degrees of freedom. We construct a torus using as a basis a periodic orbit and we extend this to a $2n-2$…
This paper is the first of a series of three dedicated to a proof of the Arnold diffusion conjecture for perturbations of {convex} integrable Hamiltonian systems on $\mathbb{A}^3=\mathbb{T}^3\times \mathbb{R}^3$. We consider systems of the…
In the present paper we prove a strong form of Arnold diffusion. Let $\mathbb{T}^2$ be the two torus and $B^2$ be the unit ball around the origin in $\mathbb{R}^2$. Fix $\rho>0$. Our main result says that for a "generic" time-periodic…
Manipulation of wave propagation through open resonant systems has attracted tremendous interest. When accessible to the open system, the system under study is prone to tempering to out of equilibrium, and a lack of reciprocity is the rule…
An approximate Poincare map near equally strong multiple resonances is reduced by means the method of averaging. Near the resonance junction of three degrees of freedom, we find that some homoclinic orbits ``whiskers'' in single resonance…
In this paper, we analyze the adiabatic crossing of a resonance for Hamiltonian systems when a double-resonance condition is satisfied by the linear frequency at an elliptic fixed point. We discuss in detail the phase-space structure on a…
In this paper we consider two classes of resonant Hamiltonian PDEs on the circle with non-convex (respect to actions) first order resonant Hamiltonian. We show that, for appropriate choices of the nonlinearities we can find time-independent…
A continuous deformation of a Hamiltonian possessing at low energy two Dirac points of opposite chiralities can lead to a gap opening by merging of the two Dirac points. In two dimensions, the critical Hamiltonian possesses a semi-Dirac…