A Philosophical Analysis of Chaos Theory

1 Introduction 1
2 Vertical and Horizontal Models in Chaos Theory 5
2.1 Introduction 5
2.2 Vertical and Horizontal Models 6
2.2.1 Vertical Models 7
2.2.2 Horizontal Models 14
2.3 A Lineage of Logistic Models 16
2.3.1 Continuous Logistic Model 16
2.3.2 Discrete Logistic Model 17
2.3.3 Iterated Logistic Model 20
2.4 A Lineage of Lorenz Models 27
2.4.1 Discrete Lorenz Model 27
2.4.2 Iterated Lorenz Model 31
2.5 Conclusion 34
3 Chaos Criteria and Definitions 37
3.1 Introduction 37
3.1.1 Sufficient Conditions and Criteria for Chaos 39
3.1.2 Outline of the Chapter’s Content 40
3.2 Diagnosis of Chaos in the Logistic Lineage 42
3.2.1 No Chaos in the Continuous Logistic Model 43
3.2.2 Stochastic Chaos in the Discrete Logistic Model 43
3.2.3 Devaney Chaos in the Iterated Logistic Model 46
3.3 Dynamical and Phenomenological Criteria for Chaos 48
3.3.1 Determinism: A Dynamical (Pre-)criterion 50
3.3.2 Transitivity 54
3.3.3 Periodicity 56
3.3.4 Aperiodicity 57
3.3.5 Sensitive Dependence on Initial Conditions 60
3.4 Analysis and Comparison of Chaos Definitions 66
3.4.1 Devaney Chaos 67
3.4.2 Mixing 69
3.4.3 Positive Lyapunov Exponents 71
3.4.4 Stochastic Chaos 72
3.4.5 Strange Attractors 74
3.5 Conclusion 77
4 Evaluation of Chaotic Models 81
4.1 Introduction 81
4.1.1 A Three-Step Framework for the Evaluation of
Vertical Chaotic Models 81
4.1.2 Outline of the Chapter’s Content 83
4.2 Evaluation of the Lineage of Logistic Models 86
4.2.1 Evaluation of the Discrete Logistic Model 87
4.2.2 Evaluation of the Iterated Logistic Model 93
4.3 Evaluation of Vertical Chaotic Models 98
4.3.1 Determining the Conditional to be Evaluated 98
4.3.2 Determining the Existence of Chaos 107
4.3.3 Determining Model Faithfulness 112
4.4 Evaluation of Horizontal Chaotic Models 116
4.4.1 Investigation of Mathematical Properties 116
4.4.2 Investigative Use 117
4.5 Smale’s 14th Problem 118
4.5.1 Chaotic Conditionals in the Lorenz Models 119
4.5.2 Construction of the Rigorous Lorenz Model 121
4.5.3 Behaviour of the Rigorous Lorenz Model 122
4.5.4 Evaluation of the Rigorous Lorenz Model 123
4.6 Conclusion 124
5 Conclusion 127
5.1 Construction of Models in Chaos Theory 128
5.2 Diagnosis of Models as Being Chaotic 129
5.3 Evaluation of Chaotic Models 130
5.4 Interplay of Models as a Characteristic Feature
of Chaos Theory 131
Bibliography 133
Index 137