Construction of invariant whiskered tori by a parameterization method. Part II: Quasi-periodic and almost periodic breathers in coupled map lattices

Abstract

Abstract. We construct quasi-periodic and almost periodic solutions for coupled Hamiltonian systems on an infinite lattice which is translation invariant. The couplings can be long range, provided that they decay moderately fast with respect to the distance.For the solutions we construct, most of the sites are moving in a neighborhood of a hyperbolic fixed point, but there are oscillating sites clustered around a sequence of nodes. The amplitude of these oscillations does not need to tend to zero. In particular, the almost periodic solutions do not decay at infinity.The main result is an a-posteriori theorem. We formulate an invariance equation. Solutions of this equation are embeddings of an invariant torus on which the motion is conjugate to a rotation. We show that, if there is an approximate solution of the invariance equation that satisfies some nondegeneracy conditions, there is a true solution close by.This does not require that the system is close to integrable, hence it can be used to validate numerical calculations or formal expansions.The proof of this a-posteriori theorem is based on a Nash-Moser iteration, which does not use transformation theory. Simpler versions of the scheme were developed in E. Fontich, R. de la Llave,Y. Sire J. Differential. Equations. 246, 3136 (2009).One technical tool, important for our purposes, is the use of weighted spaces that capture the idea that the maps under consideration are local interactions. Using these weighted spaces, the estimates of iterative steps are similar to those in finite dimensional spaces. In particular, the estimates are independent of the number of nodes that get excited. Using these techniques, given two breathers, we can place them apart and obtain an approximate solution, which leads to a true solution nearby. By repeating the process infinitely often, we can get solutions with infinitely many frequencies which do not tend to zero at infinity.

Fourier–Taylor parameterization of unstable manifolds for parabolic partial differential equations: Formalism, implementation and rigorous validation

“…• Map Lattices: stable/unstable manifolds for invariant tori [39], almost-periodic breathers and their stable/unstable manifolds [9].…”

“…Equating like powers of θ in Equation (38) and Equation (39) gives that for each m ∈ N d with |m| ≥ 2 the coefficient p m ∈ X is a solution of the equation…”

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

“…In this section we introduce several technical definitions that follow the setup in [FdlLS12,FdlLM11a,FdlLM11b]. This section can be used as a reference.…”

“…The central idea is to make precise the notion that objects are localized by imposing that the derivatives of a component with respect to a variable are small if the distance between index of the component and the variable is large. To avoid unnecessary repetitions, but to maintain some readability, we note that the definitions of Sections 2.1,2.2, 2.2.1 are the same as in [FdlLS12,FdlLM11a,FdlLM11b] (even if we suppress the references to symplectic forms,etc. in [FdlLS12]).…”

“…We say that the whiskered tori are "localized" when the oscillations are concentrated near a specific collection of sites (which we allow to be finite or infinite, see Definition 4). The existence of localized whiskered tori has been proved (under some hypotheses) for symplectic maps and flows in [FdlLS12] (e.g. for coupled pendula or Fermi-Pasta-Ulam systems [FPU55] ).…”