In this work we develop some automatic procedures for computing high order polynomial expansions of local (un)stable manifolds for equilibria of differential equations. Our method incorporates validated truncation error bounds, and maximizes the size of the image of the polynomial approximation relative to some specified constraints. More precisely we use that the manifold computations depend heavily on the scalings of the eigenvectors: indeed we study the precise effects of these scalings on the estimates which determine the validated error bounds. This relationship between the eigenvector scalings and the error estimates plays a central role in our automatic procedures. In order to illustrate the utility of these methods we present several applications, including visualization of invariant manifolds in the Lorenz and FitzHugh-Nagumo systems and an automatic continuation scheme for (un)stable manifolds in a suspension bridge problem. In the present work we treat explicitly the case where the eigenvalues satisfy a certain non-resonance condition. Remark 1.2. We fix the domain of our approximate parameterization to be the unit ball in C m (where m is the number of (un)stable eigenvalues, i.e. the dimension of the manifold) and vary the scalings of the eigenvectors in order to optimize with respect to the constraints. Another (theoretically equivalent approach) would be to fix the scalings of the eigenvectors and vary the size of the domain. However the scalings of the eigenvectors determine the decay rates of the power series coefficients, and working with analytic functions of fast decay seems to stabilize the problem numerically. Remark 1.3. In many previous applications of the parameterization method the free constants were selected by some "numerical experimentation." See for example the introduction and discussion in Section 5 of [8], Remark 3.6 of [9], Remark 2.18 and 2.20 of [10], the discussion of Example 5.2 in [11], Remark 2.4 of [12], and the discussion in Sections 4.2 and 6 of [12]. This motivates the need for systematic procedures developed here.

Judicious use of interval arithmetic, combined with careful pen and paper estimates, leads to effective strategies for computer assisted analysis of nonlinear operator equations. The method of radii polynomials is an efficient tool for bounding the smallest and largest neighborhoods on which a Newton-like operator associated with a nonlinear equation is a contraction mapping. The method has been used to study solutions of ordinary, partial, and delay differential equations such as equilibria, periodic orbits, solutions of initial value problems, heteroclinic and homoclinic connecting orbits in the C k category of functions. In the present work we adapt the method of radii polynomials to the analytic category. For ease of exposition we focus on studying periodic solutions in Cartesian products of infinite sequence spaces. We derive the radii polynomials for some specific application problems, and give a number of computer assisted proofs in the analytic framework.

In this paper we propose a rigorous numerical technique for the computation of symmetric connecting orbits for ordinary differential equations. The idea is to solve a projected boundary value problem (BVP) in a function space via a fixed point argument. The formulation of the projected BVP involves a high order parameterization of the invariant manifolds at the steady states. Using this parameterization, one can obtain explicit exponential asymptotic bounds for the coefficients of the expansion of the manifolds. Combining these bounds with piecewise linear approximations, one can construct a contraction in a function space whose unique fixed point corresponds to the wanted connecting orbit. We have implemented the method to demonstrate its effectiveness, and we have used it to prove the existence of a family of even homoclinic orbits for the Gray-Scott equation.1. Introduction. Equilibria, periodic orbits, connecting orbits, and, more generally, invariant manifolds are the fundamental components through which much of the structure of the dynamics of nonlinear differential equations is explained. Thus it is not surprising that there is a vast literature on numerical techniques for approximating these objects. In particular, the last 30 years have witnessed a strong interest in developing computational methods for connecting orbits [5,10,12,14,15,23]. As mentioned in [13], most algorithms for computing heteroclinic or homoclinic orbits reduce the question to solving a boundary value problem (BVP) on a finite interval where the boundary conditions are given in terms of linear or higher order approximations of invariant manifolds near the steady states. We adopt the same philosophy in this paper. The novelty of our approach is that our computational techniques provide existence results and bounds on approximations that are mathematically rigorous. We hasten to add that a variety of authors have already developed methods that involve a combination of interval arithmetic with analytical and topological tools and provide proofs for the existence of homoclinic and heteroclinic solutions to differential equations [28,22,31,6,32]. However, the combination of techniques we propose appears to be unique, perhaps because our approach is being developed with additional goals in mind. We return to this point later.

In this paper we introduce a computational method for proving the existence of generic saddle-to-saddle connections between equilibria of first order vector fields. The first step consists of rigorously computing high order parametrizations of the local stable and unstable manifolds. If the local manifolds intersect, the NewtonKantorovich theorem is applied to validate the existence of a so-called short connecting orbit. If the local manifolds do not intersect, a boundary value problem with boundary values in the local manifolds is rigorously solved by a contraction mapping argument on a ball centered at the numerical solution, yielding the existence of a so-called long connecting orbit. In both cases our argument yields transversality of the corresponding intersection of the manifolds. The method is applied to the Lorenz equations, where a study of a pitchfork bifurcation with saddle-to-saddle stability is done and where several proofs of existence of short and long connections are obtained.

We present an efficient numerical method for computing Fourier-Taylor expansions of stable/unstable manifolds associated with hyperbolic periodic orbits. Three features of the method are (1) that we obtain accurate representation of the invariant manifold as well as the dynamics on the manifold, (2) that the method admits natural a-posteriori error analysis, and (3) that the method does not require numerical integrating the vector field. Our method is based on the Parameterization Method for invariant manifolds, and studies a certain partial differential equation which characterizes a chart map of the (un)stable manifold. The method requires only that some mild non-resonance conditions hold between the Floquet multipliers of the periodic orbit. The novelty of the the present work is that we exploit the Floquet normal form in order to efficiently compute the Fourier-Taylor expansion. We present a number of example computations, including stable/unstable manifolds in phase space dimension as hight as ten, computation of manifolds which are two and three dimensional, and computation of some homoclinic connecting orbits.