258.

[Cover] [Abbreviated Contents] [Contents] [Index]

Page xiii

		Summary

		1.6.1 Summary of Fractals

		108

		1.6.2 Where to Learn More about Fractals

		110

		Part II Chaos

		Introduction

		2.1.1 Two Sets of Data That Look Alike

		116

		2.1.2 The Difference between Randomness and Chaos

		118

		2.1.3 A Simple Equation Can Produce Complicated Output

		120

		2.1.4 How to Tell Randomness from Chaos

		122

		2.1.5 Definition of Chaos

		124

		2.1.6 The Properties of Chaos

		126

		Phase Space

		2.2.1 Phase Space

		130

		2.2.2 Attractors

		132

		2.2.3 The Fractal Dimension of the Attractor Tells Us the Number of Independent Variables Needed to Generate the Time Series

		134

		2.2.4 Random is High Dimensional; Chaos is Low Dimensional

		136

		2.2.5 Constructing the Phase Space Set from a Sequence of Data in Time

		138

		2.2.6 Example of Phase Space Sets Constructed from Direct Measurement: Motion of the Surface of Hair Cells in the Ear

		140

		2.2.7 Example of Phase Space Sets Constructed from the Measurement of One Variable

		142

		2.2.8 The Beating of Heart Cells

		144

		2.2.9 Example of Using the Phase Space Set to Differentiate Random and Deterministic Mechanisms: The Beating of Heart Cells

		146

		2.2.10 Overview of the Phase Space Analysis

		148

		2.2.11 The Fractal Dimension Is Not Equal to the Fractal Dimension

		150

		2.2.12 Biological Implications of the Phase Space Analysis

		152

		Sensitivity to Initial Conditions

		2.3.1 Lorenz System: Physical Description

		156

		2.3.2 Lorenz System: Phase Space Set

		158

		2.3.3 Lorenz System: Sensitivity to Initial Conditions

		160

		2.3.4 Chaos: Deterministic but Not Predictable

		162

		2.3.5 The Clockwork Universe and The Chaotic Universe

		164

		2.3.6 Lorenz System: A Strange Attractor

		166

		2.3.7 "Strange" and "Chaotic"

		168

		2.3.8 The Shadowing Theorem

		170

		2.3.9 Biological Implications of Sensitivity to Initial Conditions

		172

 

[Cover] [Abbreviated Contents] [Contents] [Index]