An exact trajectory of a dynamical system lying close to a numerical trajectory is called a shadow. We present a general-purpose method for proving the existence of finite-time shadows of numerical ODE integrations of arbitrary dimension in which some measure of hyperbolicity is present and there is either 0 or 1 expanding modes, or 0 or 1 contracting modes. Much of the rigor is provided automatically by interval arithmetic and validated ODE integration software that is freely available. The method is a generalization of a previously published containment process that was applicable only to two-dimensional maps. We extend it to handle maps of arbitrary dimension with the above restrictions, and finally to ODEs. The method involves building n-cubes around each point of the discrete numerical trajectory through which the shadow is guaranteed to pass at appropriate times. The proof consists of two steps: first, the rigorous computational verification of an inductive containment property; and second, a simple geometric argument showing that this property implies the existence of a shadow. The computational step is almost entirely automated and easily adaptable to any ODE problem. The method allows for the rescaling of time, which is a necessary ingredient for successfully shadowing ODEs. Finally, the method is local, in the sense that it builds the shadow inductively,

Introduction A gravitational N-body system consists of a set of N particles of mass m i , and position r i , i = 1; ::N , all moving under the force of their mutual gravitational attraction. The force particle j exerts on particle i is F ij = Gm i m j r 2 ij ¯ r ij = Gm i m j jr ij j 3 r ij = \GammaF ji (1) where G is Newton's gravitational constant, r ij = r j \Gamma r i , r ij = jr ij j, and ¯ r ij = r

A true trajectory of a chaotic dynamical system lying near an approximate trajectory is called a shadow. Finding shadows of numerical trajectories of ODE systems is very compute intensive, and until recently it has been infeasible to study shadows of higher-dimensional systems. We study the shadowing algorithm introduced by Grebogi, Hammel, Yorke and Sauer in 1990 and extended to arbitrary Hamiltonian systems by Quinlan and Tremaine in 1992, and introduce several major optimizations resulting in speedups of over 10 2 . This algorithm is used to shadow gravitational N-body systems with up to 150 phase space dimensions. 1 Introduction Computer simulation is a popular tool in the modern physical scientist's study of complex dynamical systems. Such systems often display sensitive dependence on initial conditions: small changes in initial conditions produce solutions that exponentially diverge from each other. Since a numerical solution introduces small perturbations like roundoff and tr...

This document outlines my proposed direction of research for my Ph.D., including the specific problems I'd like to solve and how I intend to attempt solving them. It will explain my ideas about rigorous shadowing, how shadowing relates to Boundary Value Problems, and how I intend to apply these ideas to shadowing the gravitational N-body problem. In summary, the main ideas are: ffl study the variational equation of the N-body problem to determine if/when it has exponential dichotomy, and if not, if there are ways to get around this (eg., using least-squares fits on the nonhyperbolic directions) ffl rigorous shadowing, using interval integration & my new containment algorithm, including accounting for short-time non-hyperbolicity. Some secondary ideas are: ffl if the variational equation has dichotomy, then study if particular integrators produce more or less shadowable trajectories. Clearly bounded energy error is necessary for "long" integrations, so symplectic or energy conserving...