# Instability of high dimensional Hamiltonian systems: Multiple resonances do not impede diffusion

```@article{Delshams2013InstabilityOH,
title={Instability of high dimensional Hamiltonian systems: Multiple resonances do not impede diffusion},
author={Amadeu Delshams and Rafael de la Llave and Tere M. Seara},
year={2013},
volume={294},
pages={689-755}
}```
• Published 19 June 2013
• Mathematics
Abstract We consider models given by Hamiltonians of the form H ( I , φ , p , q , t ; e ) = h ( I ) + ∑ j = 1 n ± ( 1 2 p j 2 + V j ( q j ) ) + e Q ( I , φ , p , q , t ; e ) where I ∈ I ⊂ R d , φ ∈ T d , p , q ∈ R n , t ∈ T 1 . These are higher dimensional analogues, both in the center and hyperbolic directions, of the models studied in [28] , [29] , [43] and are usually called “a-priori unstable Hamiltonian systems”. All these models present the large gap problem. We show that, for 0 e ≪ 1… Expand
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#### References

SHOWING 1-10 OF 122 REFERENCES
Arnold diffusion far from strong resonances in multidimensional a priori unstable Hamiltonian systems
We prove the existence of Arnold diffusion in a typical a priori unstable Hamiltonian system outside a small neighbourhood of strong resonances. More precisely, we consider a near-integrableExpand
Geography of resonances and Arnold diffusion in a priori unstable Hamiltonian systems
• Mathematics
• 2008
In the present paper we consider the case of a general \$\cont{r+2}\$ perturbation, for \$r\$ large enough, of an a priori unstable Hamiltonian system of \$2+1/2\$ degrees of freedom, and we provide Expand
A geometric mechanism for diffusion in Hamiltonian systems overcoming the large gap problem: announcement of results.
• Mathematics
• 2003
We present a geometric mechanism for diffusion in Hamiltonian systems. We also present tools that allow us to verify it in a concrete model. In particular, we verify it in a system which presents theExpand
Arnold diffusion in arbitrary degrees of freedom and crumpled 3-dimensional normally hyperbolic invariant cylinders
• Mathematics
• 2011
In the present paper we prove a form of Arnold diffusion. The main result says that for a ”generic” perturbation of a nearly integrable system of arbitrary degrees of freedom n > 2 H0(p) +Expand
Drift and diffusion in phase space
• Mathematics
• 1994
The problem of stability of action variables (i.e. of the adiabatic invariants) in perturbations of completely integrable (real analytic) hamiltonian systems with more than two degrees of freedom isExpand
TOPOLOGICAL METHODS IN THE INSTABILITY PROBLEM OF HAMILTONIAN SYSTEMS
• Mathematics
• 2005
We use topological methods to investigate some recently proposed mechanisms of instability (Arnol'd diffusion) in Hamiltonian systems. In these mechanisms, chains of heteroclinic connectionsExpand
Existence of Diffusion Orbits in a priori Unstable Hamiltonian Systems
• Mathematics
• 2004
Under open and dense conditions we show that Arnold diffusion orbits exist in a priori unstable and time-periodic Hamiltonian systems with two degrees of freedom. 1, Introduction and Results By theExpand
A strong form of Arnold diusion for two and a half degrees of freedom
• Mathematics
• 2013
In the present paper we prove a strong form of Arnold diusion. Let T 2 be the two torus and B 2 be the unit ball around the origin in R 2 . Fix > 0. Our main result says that for a \generic"Expand
Speed of Arnold diffusion for analytic Hamiltonian systems
For a convex, real analytic, ε-close to integrable Hamiltonian system with n≥5 degrees of freedom, we construct an orbit exhibiting Arnold diffusion with the diffusion time bounded byExpand
Drift in phase space: a new variational mechanism with optimal diffusion time
• Mathematics
• 2002
We consider nonisochronous, nearly integrable, a priori unstable Hamiltonian systems with a (trigonometric polynomial) O(µ)-perturbation which does not preserve the unperturbed tori. We prove theExpand