Order and Chaos on Your Desk
Susan Bassein · American Mathematical Monthly · 1995
This paper describes a physical system which is easy to build and can fit in a small clearing on your desk and whose dynamics can be easily varied to demonstrate some fundamental concepts of dynamical systems. And, because its dynamics take place in one dimension, they are simple to analyze: although the restriction to one dimension excludes some of the phenomena which are responsible for much of the current interest in dynamical systems [1, 4], it allows one to draw pictures which illuminate some of the central ideas of the subject [1, 2, 3]. On the other hand, examples of chaotic systems (e.g., [1, 4]) are typically either difficult to realize in practice or result in dynamics in dimension 2 or higher. For example, the complicated (albeit fascinating!) dynamics of a periodically, externally forced, damped pendulum in S1 x R are described in [1, 4]. To understand what the challenge of designing a system with one-dimensional dynamics entails, let us see why the dynamics of the forced pendulum take place in S1 x R. As shown in Figure 1, the state of the pendulum system can be described by three parameters: the position 0 of the pendulum (in S1), the (angular) velocity d0/dt of the pendulum (in R), and the phase of the oscillation of the external force applied at that time (in S1). If we take periodic snapshots of the system at the moments that the oscillation of the external force passes through some given, fixed phase, then the state of the system at those moments can be described by the remaining two parameters. Since the state in the next snapshot is a function of the state in the current snapshot, we obtain a function from S1 x R to S1 x R whose iteration describes the sequence of states observable in the sequence of snapshots. For a system's dynamics to take place in R instead, its state in each snapshot must be described completely by a single parameter.