Annales Henri Poincaré

, Volume 15, Issue 1, pp 29–60 | Cite as

Localized Stable Manifolds for Whiskered Tori in Coupled Map Lattices with Decaying Interaction

  • Daniel BlazevskiEmail author
  • Rafael de la Llave


In this paper we consider lattice systems coupled by local interactions. We prove invariant manifold theorems for whiskered tori (we recall that whiskered tori are quasi-periodic solutions with exponentially contracting and expanding directions in the linearized system). The invariant manifolds we construct generalize the usual (strong) (un)stable manifolds and allow us to consider also non-resonant manifolds. We show that if the whiskered tori are localized near a collection of specific sites, then so are the invariant manifolds. We recall that the existence of localized whiskered tori has recently been proven for symplectic maps and flows in Fontich et al. (J Diff Equ, 2012), but our results do not need that the systems are symplectic. For simplicity we will present first the main results for maps, but we will show that the result for maps imply the results for flows. It is also true that the results for flows can be proved directly following the same ideas.


Manifold Invariant Manifold Unstable Manifold Stable Manifold Decay Function 
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© Springer Basel 2013

Authors and Affiliations

  1. 1.Department of Mechanical and Process EngineeringInstitute for Mechanical Systems, ETH ZurichZurichSwitzerland
  2. 2.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA

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