Gevrey estimates for one dimensional parabolic invariant manifolds of non-hyperbolic fixed points

We study the Gevrey character of a natural parameterization of one dimensional invariant manifolds associated to a parabolic direction of fixed points of analytic maps, that is, a direction associated with an eigenvalue equal to $1$. We show that, under general hypotheses, these invariant manifolds are Gevrey with type related to some explicit constants. We provide examples of the optimality of our results as well as some applications to celestial mechanics, namely, the Sitnikov problem and the restricted planar three body problem

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

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. 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 non-degeneracy conditions, there is a true solution close by. The proof of this \emph{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 \emph{J. Differential. Equations.} {\bf 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.Comment: This is a revised version of the paper located at http://www.ma.utexas.edu/mp_arc-bin/mpa?yn=12-2

Construction of invariant whiskered tori by a parameterization method. Part I: Maps and flows in finite dimensions

We present theorems which provide the existence of invariant whiskered tori in finite-dimensional exact symplectic maps and flows. The method is based on the study of a functional equation expressing that there is an invariant torus. We show that, given an approximate solution of the invariance equation which satisfies some non-degeneracy conditions, there is a true solution nearby. We call this an {\sl a posteriori} approach. The proof of the main theorems is based on an iterative method to solve the functional equation. The theorems do not assume that the system is close to integrable nor that it is written in action-angle variables (hence we can deal in a unified way with primary and secondary tori). It also does not assume that the hyperbolic bundles are trivial and much less that the hyperbolic motion can be reduced to constant. The a posteriori formulation allows us to justify approximate solutions produced by many non-rigorous methods (e.g. formal series expansions, numerical methods). The iterative method is not based on transformation theory, but rather on succesive corrections. This makes it possible to adapt the method almost verbatim to several infinite-dimensional situations, which we will discuss in a forthcoming paper. We also note that the method leads to fast and efficient algorithms. We plan to develop these improvements in forthcoming papers.Comment: To appear in JD

Exponentially small splitting of separatrices beyond Melnikov analysis: rigorous results

We study the problem of exponentially small splitting of separatrices of one degree of freedom classical Hamiltonian systems with a non-autonomous perturbation fast and periodic in time. We provide a result valid for general systems which are algebraic or trigonometric polynomials in the state variables. It consists on obtaining a rigorous proof of the asymptotic formula for the measure of the splitting. We obtain that the splitting has the asymptotic behavior $K \varepsilon^{\beta} \text{e}^{-a/\varepsilon}$, identifying the constants $K,\beta,a$ in terms of the system features. We consider several cases. In some cases, assuming the perturbation is small enough, the values of $K,\beta$ coincide with the classical Melnikov approach. We identify the limit size of the perturbation for which this theory holds true. However for the limit cases, which appear naturally both in averaging and bifurcation theories, we encounter that, generically, $K$ and $\beta$ are not well predicted by Melnikov theory

Differentiable invariant manifolds of nilpotent parabolic points

We consider a map $F$ of class $C^r$ with a fixed point of parabolic type whose differential is not diagonalizable and we study the existence and regularity of the invariant manifolds associated with the fixed point using the parameterization method. Concretely, we show that under suitable conditions on the coefficients of $F$, there exist invariant curves of class $C^r$ away from the fixed point, and that they are analytic when $F$ is analytic. The differentiability result is obtained as an application of the fiber contraction theorem. We also provide an algorithm to compute an approximation of a parameterization of the invariant curves and a normal form of the restricted dynamics of $F$ on them.Comment: 43 page

Exponentially small splitting of invariant manifolds of parabolic points

We consider families of one and a half degrees of freedom Hamiltonians with high frequency periodic dependence on time, which are perturbations of an autonomous system. We suppose that the origin is a parabolic xed point with non-diagonalizable linear part and that the unperturbed system has a homoclinic connexion associated to it. We provide a set of hypotheses under which the splitting is exponentially small and is given by the Poincaré-Melnikov function