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.
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The Experiment

Experimental Setup

Design

Bouncing Ball Circuit

Design

Typical paths of the bouncing ball

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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:

  1. Examination of the system's parameter space.
  2. Examination of the system's phase portrait.
  3. Calculation of the power spectrum for the system.
  4. Calculation of the power spectrum for the system.

Parameter Space

  • There are three parameters for the driven bouncing ball. These are:
    1. the frequency, fD,of the table's oscillations
    2. the oscillation amplitude, AD, of the table
    3. the ball's coefficient of restitution (a measure of how "stiff" the ball is)
  • 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

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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.

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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.

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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.

graph grpah graph graph

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.