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.

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.

In this work we develop a computer-assisted technique for proving existence of periodic solutions of nonlinear differential equations with non-polynomial nonlinearities. We exploit ideas from the theory of automatic differentiation in order to formulate an augmented polynomial system. We compute a numerical Fourier expansion of the periodic orbit for the augmented system, and prove the existence of a true solution nearby using an a-posteriori validation scheme (the radii polynomial approach). The problems considered here are given in terms of locally analytic vector fields (i.e. the field is analytic in a neighborhood of the periodic orbit) hence the computer-assisted proofs are formulated in a Banach space of sequences satisfying a geometric decay condition. In order to illustrate the use and utility of these ideas we implement a number of computer-assisted existence proofs for periodic orbits of the Planar Circular Restricted Three-Body Problem (PCRTBP).

We describe a method for computing an atlas for the stable or unstable manifold attached to an equilibrium point, and implement the method for the saddle-focus libration points of the planar equilateral restricted four body problem. We employ the method at the maximally symmetric case of equal masses, where we compute atlases for both the stable and unstable manifolds. The resulting atlases are comprised of thousands of individual chart maps, with each chart represented by a two variable Taylor polynomial. Post-processing the atlas data yields approximate intersections of the invariant manifolds, which we refine via a shooting method for an appropriate two point boundary value problem. Finally, we apply numerical continuation to some of the BVP problems. This breaks the symmetries and leads to connecting orbits for some non-equal values of the primary masses.Keywords Gravitational 4-body problem • invariant manifolds • high-order Taylor methods • automatic differentiation • numerical continuation

We prove the existence of chaotic motions in a planar restricted four body problem, establishing that the system is not integrable. The idea of the proof is to verify the hypotheses of a topological forcing theorem. The forcing theorem applies to two freedom Hamiltonian systems where the stable and unstable manifolds of a saddle-focus equilibrium intersect transversally in the energy level set of the equilibrium. We develop a mathematically rigorous computer assisted argument which verifies the hypotheses of the forcing theorem, and we implement our approach for the restricted four body problem. The method is constructive and works far from any perturbative regime and for non-equal masses. Being constructive, our argument results in useful byproducts like information about the locations of transverse connecting orbits, quantitative information about the invariant manifolds, and upper/lower bounds on transport times.

This work treats a functional analytic framework for computer assisted Fourier analysis which can be used to obtain mathematically rigorous error bounds on numerical approximations of solutions of differential equations. An abstract a-posteriori theorem is employed in order to obtain existence and regularity results for C k problems with 0 < k ≤ ∞ or k = ω. The main tools are certain infinite sequence spaces of rapidly decaying coefficients: we employ sequence spaces of algebraic and exponential decay rates in order to characterize the regularity our results. We illustrate the implementation and effectiveness of the method in a variety of regularity classes. We also examine the effectiveness of spaces of algebraic decays for studying solutions of problems near the breakdown of analyticity.

This work is concerned with high order polynomial approximation of stable and unstable manifolds for analytic discrete time dynamical systems. We develop a posteriori theorems for these polynomial approximations which allow us to obtain rigorous bounds on the truncation errors via a computer assisted argument. Moreover, we represent the truncation error as an analytic function, so that the derivatives of the truncation error can be bounded using classical estimates of complex analysis. As an application of these ideas we combine the approximate manifolds and rigorous bounds with a standard Newton-Kantorovich argument in order to obtain a kind of "analytic-shadowing" result for connecting orbits between fixed points of discrete time dynamical systems. A feature of this method is that we obtain the transversality of the connecting orbit automatically. Examples of the manifold computation are given for invariant manifolds which have dimension between two and ten. Examples of the a posteriori error bounds and the analytic-shadowing argument for connecting orbits are given for dynamical systems in dimension three and six.

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