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# Chaotic Dynamics and the Driven Bouncing Ball

## By Robert J. Moore, R.F. Martin, and B.K. Clark

Illinois State University Physics Department

## Introduction

- The purpose of this research was to study a real time chaotic system.
- Using standard methods for analyzing chaos, we can clearly show how chaotic processes differ from periodic and random processes.
- The system under study is an electronic analog of a ball bouncing on an oscillating table. This mechanical system was shown to be chaotic by Tufillaro and Albano (Am. J. Phys., Oct.,1986).
- The electronic circuit used in this project was developed by Zimmerman to simulate this mechanical system (Am. J. Phys., April,1992).

### What is chaos?

- Chaos is typified by behavior that appears to be random.
- Chaotic systems are not random, they are deterministic, i.e. their future behavior can be fully determined from the system's initial conditions.
- Strictly speaking, chaotic systems are deterministic systems for which nearby orbits diverge exponentially in time.

## The Experiment

### Experimental Setup

### Bouncing Ball Circuit

### Typical paths of the bouncing ball

## Conclusions

- Our results indicate that the principles of chaotic dynamics can be used to distinguish chaos from periodic and random behavior.
- The phase portraits provide a preliminary visual inspection.
- The power spectrum calculation discriminates between periodic and chaotic data sets.
- The fractal dimension of the chaotic data sets is higher than that of the periodic data sets, i.e. higher complexity yields higher dimensionality.
- The fractal dimension calculation further discriminates between chaotic and random data sets.

## Data Analysis

There are many methods available to analyze chaotic systems. We utilize the following:

- Examination of the system's parameter space.
- Examination of the system's phase portrait.
- Calculation of the power spectrum for the system.
- Calculation of the power spectrum for the system.

### Parameter Space

- There are three parameters for the driven bouncing ball. These are:
- the frequency, f
_{D},of the table's oscillations - the oscillation amplitude, A
_{D}, of the table - the ball's coefficient of restitution (a measure of how "stiff" the ball is)

- the frequency, f
- By mapping the system's behavior at several different values we can build a parameter map.
- A parameter map then allows us to see regions of periodic and chaotic behavior

### Phase Portraits and Phase Space

- A phase portrait is simply a picture of the system's trajectory in phase space.
- Phase space is the area described by the primary variables of the system, in this case the position and velocity of the ball.

### Power Spectrum

- The power spectrum is a measure of the power per unit frequency over a wide range of frequencies.
- Any discrete frequencies that are present in a function or time series of data, will then show up in a power spectrum as a sharp "spike".
- A power spectrum can therefore be used to distinguish between periodic data and chaotic data.

### Correlation Dimension

- There are many different methods used to calculate the fractal dimension of an object.
- The easiest to use with data in the form of a time series is the
*correlation dimension.* - To calculate the dimension we use the method of Procaccia and Grassberger (1983).
- By calculating the fractal dimension of the attractor we can distinguish between chaotic data and random data.

### Fractal Dimension

- When we discuss the concept of dimensionality we commonly think of only integer values of dimension.
- 0: point
- 1: line
- 2: surface etc. . .

- In 1960, Benoit Mandelbrot, a mathematician working for IBM came up with the concept that objects with a non-integer value of dimension can exist.
- Such objects came to be known as
*fractals*. - Orbits in dissipative mechanical systems will be attracted towards
an attractor. If the system's behavior within the attractor is chaotic, we
call it a
*strange attractor* - Strange attractors have a fractal structure, i.e. a non-integer dimension.
- Regular, or periodic, data will have an integer value of dimension.
- Random data will have a dimension of infinity.