Properties of numerical experiments in chaotic dynamical systems

This dissertation contains four projects that I have worked on during my graduate study at University of Maryland at College Park. These projects are all related to numerical simulations of chaotic dynamical systems. In particular, the two conjectures in Chapter 1 are inspired by the numerical discoveries in Hunt and Ott [1, 2]. In Chapter 2, statistical properties of scalar transport in chaotic flows are investigated by using numerical simulations. In Chapters 3 and 4, I take a different angle and discuss the limitations of numerical simulations; i.e. for certain "bad" systems numerical simulations will yield incorrect or at least unreliable results no matter how many digits of precision are used.;Chapter 1 discusses the properties of optimal orbits. Given a dynamical system and a function f from the state space to the real numbers, an optimal orbit for f is an orbit over which the average of f is maximal. In this chapter we discuss some basic mathematical aspects of optimal orbits: existence, sensitivity to perturbations of f, and approximability by periodic orbits with low period. For hyperbolic systems, we conjecture that (1) for (topologically) generic smooth functions, there exists an optimal periodic orbit, and (2) the optimal average can be approximated exponentially well by averages over certain periodic orbits with increasing period.;In Chapter 2 we theoretically study the power spectrum of passive scalars transported in two dimensional chaotic fluid flows. Using a wave-packet method introduced by Antonsen et al. [3] [4], we numerically investigate several model flows, and confirm that the power spectrum has the k --l-scaling predicted by Batchelor [5].;In Chapter 3 we consider a class of nonhyperbolic systems, for which there are two fixed points in an attractor having a dense trajectory; the unstable manifold of one fixed point has dimension one and the other's is two dimensional. Under the condition that there exists a direction which is more expanding than other directions, we show that such attractors are nonshadowable.;In Chapter 4, we discuss a numerical artifact called collapsing of chaos. In their numerical investigation of the family of one dimensional maps fℓ(x) = 1 -- 2|x| ℓ, where ℓ > 2, Diamond et al. have observed the surprising numerical phenomenon that a large fraction of initial conditions chosen at random eventually wind up at --1, a repelling fixed point. This is a numerical artifact because the continuous maps are chaotic and almost every (true) trajectory can be shown to be dense in [--1, 1]. The goal of this chapter is to explain and determine the extent of this paradox. We model the numerical simulation with a randomly selected map. (Abstract shortened by UMI.)...