Instability for a priori unstable Hamiltonian systems: A dynamical approach

Abed Bounemoura; Edouard Pennamen

doi:10.3934/dcds.2012.32.753

Article Contents

2012, Volume 32, Issue 3: 753-793. Doi: 10.3934/dcds.2012.32.753

This issue Previous Article Localized asymptotic behavior for almost additive potentials Next Article Existence of multiple solutions for a nonhomogeneous semilinear elliptic equation involving critical exponent

Instability for a priori unstable Hamiltonian systems: A dynamical approach

Abed Bounemoura^1, and
Edouard Pennamen^2,

1.
IMPA, Estrada Dona Castorina 110, Rio de Janeiro, Brasil, 22460-320
2.
IMJ, 4 place Jussieu, 75252 Paris Cedex

Received: September 30, 2010

Revised: April 30, 2011

Published: February 2012

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Abstract

In this article, we consider an a priori unstable Hamiltonian system with three degrees of freedom, for which we construct a drifting solution with an optimal time of instability. Such a result has been already proved by Berti, Bolle and Biasco using variational arguments, and by Treschev using his separatrix map theory. Our approach is new: it is based on a special type of symbolic dynamics corresponding to the random iteration of a family of twist maps of the annulus, and it gives the first concrete application of this idea introduced by Moeckel in an abstract setting and further studied by Marco. Our method should also be useful in obtaining the optimal time of instability in the more difficult context of a priori stable Hamiltonian systems.

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Mathematics Subject Classification: Primary: 37J40; Secondary: 37D05.

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