We introduce a method to prove existence of invariant manifolds and, at the same time to find simple polynomial maps which are conjugated to the dynamics on them. As a first application, we consider the dynamical system given by a C r map F in a Banach space X close to a fixed point: F (x) = Ax + N (x), A linear, N (0) = 0, DN (0) = 0. We show that if X 1 is an invariant subspace of A and A satisfies certain spectral properties, then there exists a unique C r manifold which is invariant under F and tangent to X 1. When X 1 corresponds to spectral subspaces associated to sets of the spectrum contained in disks around the origin or their complement, we recover the classical (strong) (un)stable manifold theorems. Our theorems, however, apply to other invariant spaces. Indeed, we do not require X 1 to be an spectral subspace or even to have a complement invariant under A.

We describe a method to establish existence and regularity of invariant manifolds and, at the same time to find simple maps which are conjugated to the dynamics on them. The method establishes several invariant manifold theorems. For instance, it reduces the proof of the usual stable manifold theorem near hyperbolic points to an application of the implicit function theorem in Banach spaces. We also present several other applications of the method.

Abstract. We study the regularity with respect to parameters of the invariant manifolds associated to non-resonant subspaces obtained in the previous article [CFdlL00].

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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 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 linear map. 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 successive 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.

Abstract. We study families of diffeomorphisms close to the identity, which tend to it when the parameter goes to zero, and having homoclinic points. We consider the analytical case and we find that the maximum separation between the invariant manifolds, in a given region, is exponentially small with respect to the parameter. The exponent is related to the complex singularities of a flow which is taken as an unperturbed problem. Finally several examples are given. IntroductionIn a previous paper [5] we considered families of differentiable diffeomorphisms with hyperbolic points which reduce to the identity for a certain value of the parameter. We studied the separation between the stable and unstable manifolds near homo-heteroclinic points associated with the hyperbolic ones. Now we shall study the analytic and conservative case. We refer the reader to [5] for the motivations of this problem. The results were inspired by the previous work by Lazutkin [10] for the standard map.We study the complex invariant manifolds through the Birkhoff normal form of a diffeomorphism in a neighbourhood of a hyperbolic point. For that we need uniform behaviour of the normal form with respect to the parameter of the family. As in the differentiable case we use an auxiliary family of diffeomorphisms defined through the flow of an autonomous vector field with a homoclinic orbit. We compare the invariant manifolds of the two families, first locally and then globally. From the Birkhoff normal form we get a local first integral for the diffeomorphisms, which can be extended in a neighbourhood of the invariant manifolds as a multivalued function. With a suitable parametrization of the invariant manifolds, the evaluation of the first integral extended along one manifold on the other manifold gives a periodic function. The Fourier coefficients of that function can be bounded by using integration over complex paths. From that we get the bound of the separation between the invariant manifolds. Now we state the main result in a precise way.

We consider weakly coupled map lattices with a decaying interaction. That is, we consider systems which consist of a phase space at every site such that the dynamics at a site is little affected by the dynamics at far away sites. We develop a functional analysis framework which formulates quantitatively the decay of the interaction and is able to deal with lattices such that the sites are manifolds. This framework is very well suited to study systematically invariant objects. One obtains that the invariant objects are essentially local. We use this framework to prove a stable manifold theorem and show that the manifolds are as smooth as the maps and have decay properties (i.e. the derivatives of one of the coordinates of the manifold with respect to the coordinates at far away sites are small). Other applications of the framework are the study of the structural stability of maps with decay close to uncoupled possessing hyperbolic sets and the decay properties of the invariant manifolds of their hyperbolic sets, in the companion paper by Fontich et al. (2011) [10].

Saddle-node bifurcations have been described in a multitude of nonlinear dynamical systems modeling physical, chemical, as well as biological systems. Typically, this type of bifurcation involves the transition of a given set of fixed points from the real to the complex phase space. After the bifurcation, a saddle remnant can continue influencing the flows and generically, for non-degenerate saddle-node bifurcations, the time the flows spend in the bottleneck region of the ghost follows the inverse square root scaling law. Here we analytically derive this scaling law for a general one-dimensional, analytical, autonomous dynamical system undergoing a not necessarily non-degenerate saddle-node bifurcation, in terms of the degree of degeneracy by using complex variable techniques. We then compare the analytic calculations with a one-dimensional equation modeling the dynamics of an autocatalytic replicator. The numerical results are in agreement with the analytical solution.