Parameterization of Invariant Manifolds for Lagrangian Systems with Long-range Interactions

Series: 
CDSNS Colloquium
Monday, February 13, 2012 - 11:05
1 hour (actually 50 minutes)
Location: 
Skiles 006
,  
Univ. of Texas at Austin/ITAM
Organizer: 
We generalize some notions that have played an important role in dynamics, namely invariant manifolds, to the more general context of difference equations. In particular, we study Lagrangian systems in discrete time. We define invariant manifolds, even if the corresponding difference equations can not be transformed in a dynamical system. The results apply to several examples in the Physics literature: the Frenkel-Kontorova model with long-range interactions and the Heisenberg model of spin chains with a perturbation. We use a modification of the parametrization method to show the existence of Lagrangian stable manifolds. This method also leads to efficient algorithms that we present with their implementations. (Joint work with Rafael de la Llave.)