A Philosophical Analysis of Chaos Theory (New Directions in the Philosophy of Science)

Over the last 40 years, chaos theory has had a huge impact on science and philosophy. This is evidenced by the astonishing volume of chaos-related publications; even a cursory survey shows that chaos has been detected virtually everywhere, from cardiac rhythms to Joyce’s Ulysses (e.g. Kellert, 2008). Given the high export appeal of chaos theory, it is surprising that there are fundamental aspects of the field that still remain poorly understood and, in some cases, permanently debated. In particular: (i) it is still not clear how chaos should be defined and how the large number of coexisting chaos definitions relate to each other (e.g. Smith, 1998; Werndl, 2009c); (ii) there are still (largely unarticulated) questions about the faithfulness and predictiveness of the numerical and theoretical models on which chaos theory is based; and, finally, (iii) it has not been unequivocally resolved whether there is chaos in nature (e.g. Kellert, 1993) and how it should be diagnosed (e.g. Pool, 1989; Hastings et al., 1993). The three aspects are not independent of each other and it is evident that difficulties (i) and (ii) contribute to difficulty (iii). Together, they have made it very difficult to judge the diverse contributions to chaos theory comparatively and to enforce universal standards of quality and rigour. This book aims to clarify aspects (i)–(iii) by providing a structured survey of the construction, diagnosis and evaluation of chaotic models. Although the book follows a survey approach in that it aims to achieve a certain degree of comprehensiveness and to comparatively cover different aspects of chaos theory, it is not a mere review. I will pursue a modelling-centred strategy and thereby aim to provide the first in-depth analysis of all three stages, i.e. construction, diagnosis and evaluation, of modelling in chaos theory. This allows me to draw on a large amount of recently developed work on the use of models in science, which has so far not been applied to the field of chaos theory. In particular, the book uses, and develops further, several results of both the fictionalist approach to modelling (e.g. Frigg, 2010; Toon, 2012; Suarez, 2013) as well as the work on horizontal modelling by Bokulich (2003), which were not available to authors of earlier philosophical analyses of chaos theory (e.g. Kellert, 1993; Smith, 1998). A Philosophical Analysis of Chaos Theory 4 Acknowledgements 6 Contents 7 List of Figures 10 List of Tables 11 1 Introduction 12 2 Vertical and Horizontal Models in Chaos Theory 16 2.1 Introduction 16 2.2 Vertical and Horizontal Models 17 2.2.1 Vertical Models 18 Construction of Vertical Models 19 Evaluation of Vertical Models 20 Model Faithfulness and the Transference of Conditionals 22 2.2.2 Horizontal Models 25 Construction of Horizontal Models 25 Evaluation of Horizontal Models 26 2.3 A Lineage of Logistic Models 27 2.3.1 Continuous Logistic Model 27 Construction of the Continuous Logistic Model 28 Behaviour of the Continuous Logistic Model 28 2.3.2 Discrete Logistic Model 28 Construction of the Discrete Logistic Model 29 Behaviour of the Discrete Logistic Model 30 2.3.3 Iterated Logistic Model 31 Construction of the Iterated Logistic Model 31 Behaviour of the Iterated Logistic Model 34 2.4 A Lineage of Lorenz Models 38 2.4.1 Discrete Lorenz Model 38 Construction of the Discrete Lorenz Model 39 Behaviour of the Discrete Lorenz Model 40 2.4.2 Iterated Lorenz Model 42 Construction of the Iterated Lorenz Model 43 Behaviour of the Iterated Lorenz Model 44 2.5 Conclusion 45 3 Chaos Criteria and Definitions 48 3.1 Introduction 48 3.1.1 Sufficient Conditions and Criteria for Chaos 50 3.1.2 Outline of the Chapter’s Content 51 3.2 Diagnosis of Chaos in the Logistic Lineage 53 3.2.1 No Chaos in the Continuous Logistic Model 54 3.2.2 Stochastic Chaos in the Discrete Logistic Model 54 Determinism 54 Aperiodicity 55 Sensitivity to Initial Conditions 56 3.2.3 Devaney Chaos in the Iterated Logistic Model 57 Determinism 57 SDIC, Transitivity and Periodicity 57 3.3 Dynamical and Phenomenological Criteria for Chaos 59 3.3.1 Determinism: A Dynamical (Pre-)criterion 61 Indeterministic Models 62 Deterministic Models 64 Determinism as Demarcation Criterion 64 3.3.2 Transitivity 65 Topological Embodiments 65 Statistical Embodiments 66 3.3.3 Periodicity 67 Strong Embodiments 67 Weak Embodiments 68 3.3.4 Aperiodicity 68 Absence of Periodicity 68 Apparent Stochasticity 69 Local Unpredictability 70 3.3.5 Sensitive Dependence on Initial Conditions 71 Exponential Divergence 73 SDIC and Other Chaos Criteria 73 Global Unpredictability 74 Global Unpredictability as a Defining Criterion of Chaos 75 3.4 Analysis and Comparison of Chaos Definitions 77 3.4.1 Devaney Chaos 78 Extension of Devaney Chaos 79 3.4.2 Mixing 80 Extension of Mixing 81 3.4.3 Positive Lyapunov Exponents 82 Extension of the Definition of Chaos in Terms of Positive Lyapunov Exponents 82 3.4.4 Stochastic Chaos 83 Extension of Stochastic Chaos 84 3.4.5 Strange Attractors 85 Requirement of Fractal Geometry 87 Requirement of Other Criteria 87 3.5 Conclusion 88 4 Evaluation of Chaotic Models 92 4.1 Introduction 92 4.1.1 A Three-Step Framework for the Evaluation of Vertical Chaotic Models 92 4.1.2 Outline of the Chapter’s Content 94 4.2 Evaluation of the Lineage of Logistic Models 97 4.2.1 Evaluation of the Discrete Logistic Model 98 Determining the Conditional to Be Transferred 98 Determining the Existence of Chaos 100 Determining Model Faithfulness 101 Further Evaluation of Chaotic Models in Population Dynamics 103 4.2.2 Evaluation of the Iterated Logistic Model 104 Determination of the Chaotic Conditional 104 Investigative Use of the Iterated Logistic Model 106 4.3 Evaluation of Vertical Chaotic Models 109 4.3.1 Determining the Conditional to be Evaluated 109 Variability in the Chaotic Behaviour B 110 Two Types of Sufficient Conditions C 111 Type 1 Conditionals 111 Type 2 Conditionals 112 Correlations Between Types of Conditional and Kinds of Chaos 113 Determining Conditionals in Numerical Models 114 4.3.2 Determining the Existence of Chaos 118 Phase-Space Reconstruction as a Diagnostic Tool 119 Diagnosing Determinism: The Route-to-Chaos Approach 121 4.3.3 Determining Model Faithfulness 123 Model Faithfulness and Type 1 Conditionals 125 Model Faithfulness and Type 2 Conditionals 126 4.4 Evaluation of Horizontal Chaotic Models 127 4.4.1 Investigation of Mathematical Properties 127 4.4.2 Investigative Use 128 4.5 Smale’s 14th Problem 129 4.5.1 Chaotic Conditionals in the Lorenz Models 130 4.5.2 Construction of the Rigorous Lorenz Model 132 4.5.3 Behaviour of the Rigorous Lorenz Model 133 4.5.4 Evaluation of the Rigorous Lorenz Model 134 4.6 Conclusion 135 5 Conclusion 138 5.1 Construction of Models in Chaos Theory 139 5.2 Diagnosis of Models as Being Chaotic 140 5.3 Evaluation of Chaotic Models 141 5.4 Interplay of Models as a Characteristic Feature of Chaos Theory 142 Bibliography 144 Index 148 Annotation This text provides an analysis of the construction, diagnosis (as chaotic) and evaluation of models in chaos theory. It contains a detailed look at the interaction of the different models used in chaos theory and analyses how these models influence the way chaos is defined. Furthermore, the book discusses the conditions for the occurrence of chaos and the detection of chaos in nature This book provides an analysis of the construction, diagnosis (as chaotic) and evaluation of models in chaos theory. It contains a detailed look at the interaction of the different models used in chaos theory and analyses how these models influence the way chaos is defined. Furthermore, the book discusses the conditions for the occurrence of chaos and the detection of chaos in nature. Front Matter....Pages i-xiii Introduction....Pages 1-4 Vertical and Horizontal Models in Chaos Theory....Pages 5-36 Chaos Criteria and Definitions....Pages 37-80 Evaluation of Chaotic Models....Pages 81-126 Conclusion....Pages 127-132 Back Matter....Pages 133-138