Introduction

 

A ball bouncing on an oscillating table gives rise to complicated phenomena which appear to defy our comprehension and analysis. The motions in the bouncing ball system are truly complex. However, part of the problem is that we do not, as yet, have the right language with which to discuss nonlinear phenomena. We thus need to develop a vocabulary for nonlinear dynamics.

A good first step in developing any scientific vocabulary is the detailed analysis of some simple examples . In this chapter we will begin by exploring the quadratic map. In linear dynamics, the corresponding example used for building a scientific vocabulary is the simple harmonic oscillator  (see Figure 2.1).

  
Figure 2.1: Simple harmonic oscillator.

As its name implies, the harmonic oscillator is a simple model which illustrates many key notions useful in the study of linear systems. The image of a mass on a spring is usually not far from one's mind even when dealing with the most abstract problems in linear physics.

The similarities among linear systems are easy to identify because of the extensive development of linear theory over the past century. Casual inspection of nonlinear systems suggests little similarity. Careful inspection, though, reveals many common features. Our original intuition is misleading because it is steeped in linear theory. Nonlinear systems possess as many similarities as differences. However, the vocabulary of linear dynamics is inadequate to name these common structures. Thus, our task is to discover the common elements of nonlinear systems and to analyze their structure.

A simple model of a nonlinear system is given by the difference equation known as a quadratic map ,

 

For instance, if we set the value and initial condition in the quadratic map we find that

and in this case the value appears to be approaching 1/2.

Phenomena illustrated in the quadratic map arise in a wide variety of nonlinear systems. The quadratic map is also known as the logistic map , and it was studied as early as 1845 by P. F. Verhulst  as a model for population growth. Verhulst was led to this difference equation by the following reasoning. Suppose in any given year, indexed by the subscript n, that the (normalized) population is . Then to find the population in the next year ( ) it seems reasonable to assume that the number of new births will be proportional to the current population, , and the remaining inhabitable space, . The product of these two factors and gives the quadratic map, where is some parameter that depends on the fertility rate, the initial living area, the average disease rate, and so on.

Given the quadratic map as our model for population dynamics, it would now seem like an easy problem to predict the future population. Will it grow, decline, or vary in a cyclic pattern? As we will see, the answer to this question is easy to discover for some values of , but not for others. The dynamics are difficult to predict because, in addition to exhibiting cyclic behavior, it is also possible for the population to vary in a chaotic manner.

In the context of physical systems, the study of the quadratic map was first advocated by E. N. Lorenz  in 1964 [1]. At the time, Lorenz was looking at the convection of air in certain models of weather prediction. Lorenz was led to the quadratic map by the following reasoning, which also applies to the bouncing ball system as well as to Lorenz's original model (or, for that matter, to any highly dissipative system). Consider a time series  that comes from the measurement of a variable in some physical system,

 

For instance, in the bouncing ball system this time series could consist of the sequence of impact phases, so that , and so on. We require that this time series arise from motion on an attractor. To meet this requirement, we throw out any measurements that are part of the initial transient motion. In addition, we assume no foreknowledge of how to model the process giving rise to this time series. Given our ignorance, it then seems natural to try to predict the n+1st element of the time series from the previous nth value. Formally, we are seeking a function, f, such that

 

In the bouncing ball example this idea suggests that the next impact phase would be a function of the previous impact phase; that is, .

If such a simple relation exists, then it should be easy to see by plotting on the vertical axis and on the horizontal axis. Formally, we are taking our original time series, equation (2.2), and creating an embedded time series  consisting of the ordered pairs, ,

 

The idea of embedding  a time series will be central to the experimental study of nonlinear systems discussed in section 3.8.2. In Figure 2.2 we show an embedded time series of the impact phases for chaotic motions in the bouncing ball system.

  
Figure 2.2: Embedded time series of chaotic motion in the bouncing ball system. (Generated by the Bouncing Ball program.)

The points for this embedded time series appear to lie close to a region that resembles an upside-down parabola. The exact details of the curve depend, of course, on the specific parameter values, but as a first approximation the quadratic map provides a reasonable fit to this curve (see Figure 2.3).

  
Figure 2.3: The quadratic function. (Generated by the Quadratic Map program.)

Note that the curve's maximum amplitude (located at the point x = 1/2) rises as the parameter increases. We think of the parameter in the quadratic map as representing some parameter in our process; could be analogous to the table's forcing amplitude in the bouncing ball system. Such single-humped maps often arise when studying highly dissipative nonlinear systems. Of course, more complicated many-humped maps can and do occur; however, the single-humped map is the simplest, and is therefore a good place to start in developing our new vocabulary.