Introduction

In earlier chapters, we saw the means by which motion, bounded initially within a potential energy well, might spill over or escape either to infinity or to an adjacent energy well. In this chapter we take a closer look at global issues. We will see how basin boundaries and unstable fixed points have a considerable influence on behavior in the large. Dependence on initial conditions has been encountered earlier in this book in terms of multiple (point and periodic) attractors and the extreme sensitivity of chaos. We shall see that extreme sensitivity to initial conditions may also appear when the boundaries separating domains of attraction become fractal, causing transients to have arbitrarily long lengths (Eschenazi, Solari, and Gilmore, 1989; Grebogi, Ott, and Yorke, 1987; Gwinn and Westervelt, 1986). This is often a precursor of steady-state chaos. One specific aspect of interest is the appearance of indeterminate bifurcations. For the purposes of illustrating this behavior, we will revert back to the double-well Duffing oscillator of earlier chapters of this book. A chronological note here is that the experimental results to follow were obtained a couple of years after those described in Chapters 8 and 9 and, hence, some small adjustments appear in the basic system coefficients (Todd and Virgin, 1997b).

Dependence on Initial Conditions

One of the fundamental differences between a linear and a nonlinear system is that nonlinear systems often possess multiple stable solutions, and, hence, the final solution depends to an extent on the starting conditions. The standard theory of linear vibrations, even for high-order systems, obviates the need to consider this, with unique solutions capturing all possible initial conditions. We have seen that nonlinear systems (even unforced problems) typically have a variety of long-term solutions for a fixed set of parameter values. Although it can be argued that persistent (stable) solutions perhaps have the most practical importance (certainly in relation to their local region of phase space), it is the unstable solutions that have a profound influence on global behavior (Grebogi, Ott, and Yorke, 1986b).