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).
Author(s): Lena C. Zuchowski
Edition: 1st ed. 2017
Publisher: Palgrave Macmillan
Year: 2017
Language: English
Pages: 138
Tags: History of Ideas; Philosophy of Science;Science & Math;Chaos Theory;Physics;Science & Math;Science & Mathematics;Agriculture;Astronomy & Astrophysics;Biology
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
