# Introduction

## Definitions

Throughout this thesis, when referring to mathematical variables, boldface will refer to vectors, and italic will refer to scalars, matrices and functions. Scalars are written in small letters and matrices in CAPITALS.

Some of the following definitions are taken, with minor modifications, from Grebogi, Hammel, Yorke, and Sauer [11], hereinafter referred to as GHYS. The terms trajectory, orbit, and solution are used interchangably throughout this thesis.

Definition. A true trajectory of f satisfies for . We are interested in the case where a and b are finite integers. For a chaotic map, f may be a simple equation, such as the logistic equation , which always maps the interval [-1,1] onto itself. For an ODE system like the N-body problem, represents the true solution of integrating the phase-space co-ordinates for one timestep.

Definition. is a -pseudo-trajectory, also called a noisy orbit, for f if for , where is the noise amplitude.

Definition. For , the 1-step error made between step i and step i+1 of the pseudo-trajectory is the vector . Thus, a true trajectory is one whose 1-step errors are identically zero.

Definition of shadowing. The true trajectory -shadows on if for . The two stages, of proving that a shadow exists, are refinement and containment, defined below.

Definition. A shadow step is an interval, that can be larger than the internal timestep of a numerical integrator, across which a 1-step error is computed.

Definition. The pseudo-trajectory has a glitch at point if for some relevant there exists a true trajectory that -shadows for , but no true trajectory that -shadows it for , for .

The first group of chaotic systems for which it was proven that shadow orbits exist was hyperbolic systems [2, 7]. In a 2-dimensional hyperbolic system, there are two special directions called the unstable (or expanding) and the stable (or contracting) directions, which are generally not orthogonal. Small perturbations along the stable direction decay exponentially in forward time, while small perturbations in the unstable direction grow exponentially in forward time. The two directions reverse roles in backwards time. A ``trajectory'' for such a system can be imagined as a point moving in a 2D plane, evolving through a third dimension, representing time.

For such a system it was shown that, if the angle between the stable and unstable directions is uniformly bounded away from 0, then a noisy trajectory can be shadowed for all time. For non-hyperbolic systems, it appears that shadows may exist only for finite time. The most important question in this regard is, how long can a noisy orbit be shadowed? If the time is at least as long as most typical simulations of chaotic non-hyperbolic systems, then simulations have great validity; if the shadowing time turns out to be too short, then a less stringent error measure must be resorted to, such as one listed previously.

Definition.   Refinement is an iterative process that perturbs each point of a noisy orbit in an attempt to produce a nearby orbit with less noise. A refinement iteration is successful if the trajectory before the iteration has noise and the trajectory after the iteration has noise , and , for some reasonable . Otherwise the iteration is unsuccessful.

Definition. Containment is a rigorous method to prove the existence of a shadow orbit. See GHYS for details. Although this is a good area for further work, containment is beyond the scope of this thesis.

The refinement algorithm that concerns us in this thesis is the one first presented in two dimensions by GHYS, and generalized to handle arbitrary Hamiltonian systems by Quinlan and Tremaine [29], hereinafter referred to as QT.

QT make the distinction between dynamical noise and observational noise. Observational noise does not effect the future evolution of the system. Laboratory measurements of a macroscopic system are usually of this type; another example is computer output that prints fewer digits than are represented internally. In contrast, dynamical noise does effect the future evolution of the system. The noise introduced by numerical solution of a system of ODE's is dynamical. QT studied some existing noise-reduction algorithms in an attempt to refine noisy trajectories of chaotic systems, but none worked as well as that presented by GHYS. QT postulates that this may be because the other noise reduction algorithms were designed to reduce observational noise, whereas the GHYS procedure was designed to reduce dynamical noise in a chaotic system.