Fractals and chaos simplified for the life sciences:
Gespeichert in:
| Beteilige Person: | |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
New York [u.a.]
Oxford Univ. Press
1998
|
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008038775&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | XV, 268 S. Ill., graph. Darst. |
| ISBN: | 0195120248 |
Internformat
MARC
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| 245 | 1 | 0 | |a Fractals and chaos simplified for the life sciences |c Larry S. Liebovitch |
| 264 | 1 | |a New York [u.a.] |b Oxford Univ. Press |c 1998 | |
| 300 | |a XV, 268 S. |b Ill., graph. Darst. | ||
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| 650 | 4 | |a Mathematik | |
| 650 | 4 | |a Medizin | |
| 650 | 4 | |a Biomathematics | |
| 650 | 4 | |a Chaotic behavior in systems | |
| 650 | 4 | |a Fractals | |
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Datensatz im Suchindex
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| adam_text |
Abbreviated Contents
Part I FRACTALS
1 Introduction 3
2 Self Similarity 11
3 Scaling 27
4 Dimension 45
5 Statistical Properties 73
6 Summary 107
Part II CHAOS
1 Introduction 115
2 Phase Space 129
3 Sensitivity to Initial Conditions 155
4 Bifurcations 175
5 Analyzing Data 191
6 Control of Chaos 225
7 Summary 237
Part III OTHER METHODS
1 The Big Picture 245
References 251
Illustration Credits 257
Index 259
iv
Contents
Part I FRACTALS
Introduction
1.1.1 The Difference between Non Fractal and Fractal Objects 4
1.1.2 The Sizes of the Features of Non Fractal and Fractal Objects 6
1.1.3 The Properties of Fractals 8
Self Similarity
1.2.1 Two Types of Self Similarity 12
1.2.2 Examples of Self Similarity in Space 14
1.2.3 Examples of Self Similarity in Time 16
1.2.4 The Currents through Ion Channels Are Self Similar in Time 18
1.2.5 The Open and Closed Times of Ion Channels Are Statistically
Self Similar 20
1.2.6 More Examples of Self Similarity 22
1.2.7 Biological Implications of Self Similarity 24
Scaling
1.3.1 Self Similarity Implies a Scaling Relationship 28
1.3.2 Scaling Relationships 30
1.3.3 Example of a Power Law Scaling of a Spatial Object:
The Length of the Coastline of Britain 32
1.3.4 Examples of Power Law Scalings of Spatial Objects:
The Surfaces of Cell Membranes 34
1.3.5 Example of a Power Law Scaling of a Process in Time:
Ion Channel Kinetics 36
1.3.6 The Physical Significance of the Scaling Relationship
of Ion Channel Kinetics 38
1.3.7 More Examples of Scaling Relationships 40
1.3.8 Biological Implications of Scaling Relationships 42
xi
Contents
Dimension
1.4.1 Dimension: A Quantitative Measure of Self Similarity and Scaling 46
1.4.2 The Simplest Fractal Dimension: The Self Similarity Dimension 48
1.4.3 More General Fractal Dimensions: The Capacity Dimension 50
1.4.4 More General Fractal Dimensions:
The Hausdorff Besicovitch Dimension 52
1.4.5 Example of Determining the Fractal Dimension:
Using the Self Similarity Dimension 54
1.4.6 Example of Determining the Fractal Dimension:
Using the Capacity Dimension and Box Counting 56
1.4.7 Example of Determining the Fractal Dimension:
Using the Scaling Relationship 58
1.4.8 The Topological Dimension 60
1.4.9 The Embedding Dimension 62
1.4.10 Definition of a Fractal 64
1.4.11 Example of the Fractal Dimension: Blood Vessels in the Lungs 66
1.4.12 More Examples of the Fractal Dimension 68
1.4.13 Biological Implications of the Fractal Dimension 70
Statistical Properties
1.5.1 The Statistical Properties of Fractals 74
1.5.2 Self Similarity Implies That the Moments Do Not Exist 76
1.5.3 Example Where the Average Does Not Exist:
The St. Petersburg Game 78
1.5.4 Example Where the Average Does Not Exist:
Diffusion Limited Aggregation (DLA) 80
1.5.5 Example Where the Variance Does Not Exist:
The Roughness of Rocks 82
1.5.6 Example Where the Variance Does Not Exist:
Electrical Activity in Nerve Cells 84
1.5.7 Statistical Analysis of the Electrical Activity in Nerve Cells 86
1.5.8 Example Where the Variance Does Not Exist:
Blood Flow in the Heart 88
1.5.9 Example Where the Variance Does Not Exist:
Volumes of Breaths 90
1.5.10 Example Where the Variance Does Not Exist: Evolution 92
1.5.11 The Distribution of Mutations Is the Same as in the
St. Petersburg Game 94
1.5.12 Statistical Properties and the Power Spectra 96
1.5.13 How to Measure the Properties of Fractal Data 98
1.5.14 More Examples of the Statistical Properties of Fractals 100
1.5.15 Biological Implications of the Statistical Properties of Fractals 102
1.5.16 Biological Implications of the Statistical Properties of Fractals
(continued) 104
Fractals and Chaos Simplified for the Life Sciences
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
Contents
Bifurcations
2.4.1 Bifurcation: An Abrupt Shift in Behavior 176
2.4.2 Bifurcation Diagrams 178
2.4.3 Example of Bifurcations: Theory of Glycolysis 180
2.4.4 Example of Bifurcations: Experiments of Glycolysis 182
2.4.5 Example of Bifurcation Diagrams: Theory and Experiments of
Glycolysis 184
2.4.6 Example of Bifurcations: Motor and Sensory Phase Transitions 186
2.4.7 Biological Implications of Bifurcations 188
Analyzing Data
2.5.1 Analyzing Data: The Good News 192
2.5.2 Example of the Fractal Dimension of the Phase Space Set:
Epidemics 194
2.5.3 Example of the Fractal Dimension of the Phase Space Set:
The Heart 196
2.5.4 Example of the Fractal Dimension of the Phase Space Set:
The Brain 198
2.5.5 Ion Channel Kinetics: Random or Deterministic? 200
2.5.6 Ion Channel Kinetics: An Interpretation of the Deterministic Model 202
2.5.7 Ion Channel Kinetics: Physical Meaning of the Random and
Deterministic Models 204
2.5.8 Analyzing Data: The Bad News 206
2.5.9 Some of the Problems of the Phase Space Analysis 208
2.5.10 Problems: How Much Data? 210
2.5.11 Problems: The Lag At 212
2.5.12 Problems: The Embedding Theorems 214
2.5.13 Problems: The Meaning of a Low Dimensional Attractor 216
2.5.14 New Methods: Average Direction of the Trajectories 218
2.5.15 New Methods: Surrogate Data Set 220
2.5.16 Biological Implications of the Strengths and Weakness in
Analyzing Data 222
Control of Chaos
2.6.1 The Advantages of Chaos 226
2.6.2 Example of Chaotic Control: The Light Output from a Laser 228
2.6.3 Example of Chaotic Control: The Motion of a Magnetoelastic
Ribbon 230
2.6.4 Biological Implications of the Control of Chaos 232
2.6.5 Biological Implications of the Control of Chaos (continued) 234
Summary
2.7.1 Summary of Chaos 238
2.7.2 Where to Learn More about Chaos 240
Fractals and Chaos Simplified for the Life Sciences
Part III OTHER METHODS
The Big Picture
3.1.1 The Big Picture 246
3.1.2 Some Other Nonlinear Tools 248 |
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| indexdate | 2025-01-28T19:15:00Z |
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| isbn | 0195120248 |
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| spelling | Liebovitch, Larry S. Verfasser aut Fractals and chaos simplified for the life sciences Larry S. Liebovitch New York [u.a.] Oxford Univ. Press 1998 XV, 268 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Chaos gtt Fractals gtt Mathematik Medizin Biomathematics Chaotic behavior in systems Fractals Medicine Mathematics Fraktal (DE-588)4123220-3 gnd rswk-swf Chaostheorie (DE-588)4009754-7 gnd rswk-swf Medizin (DE-588)4038243-6 gnd rswk-swf Biologie (DE-588)4006851-1 gnd rswk-swf (DE-588)4151278-9 Einführung gnd-content Medizin (DE-588)4038243-6 s Chaostheorie (DE-588)4009754-7 s DE-604 Biologie (DE-588)4006851-1 s Fraktal (DE-588)4123220-3 s DE-188 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008038775&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Liebovitch, Larry S. Fractals and chaos simplified for the life sciences Chaos gtt Fractals gtt Mathematik Medizin Biomathematics Chaotic behavior in systems Fractals Medicine Mathematics Fraktal (DE-588)4123220-3 gnd Chaostheorie (DE-588)4009754-7 gnd Medizin (DE-588)4038243-6 gnd Biologie (DE-588)4006851-1 gnd |
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| title | Fractals and chaos simplified for the life sciences |
| title_auth | Fractals and chaos simplified for the life sciences |
| title_exact_search | Fractals and chaos simplified for the life sciences |
| title_full | Fractals and chaos simplified for the life sciences Larry S. Liebovitch |
| title_fullStr | Fractals and chaos simplified for the life sciences Larry S. Liebovitch |
| title_full_unstemmed | Fractals and chaos simplified for the life sciences Larry S. Liebovitch |
| title_short | Fractals and chaos simplified for the life sciences |
| title_sort | fractals and chaos simplified for the life sciences |
| topic | Chaos gtt Fractals gtt Mathematik Medizin Biomathematics Chaotic behavior in systems Fractals Medicine Mathematics Fraktal (DE-588)4123220-3 gnd Chaostheorie (DE-588)4009754-7 gnd Medizin (DE-588)4038243-6 gnd Biologie (DE-588)4006851-1 gnd |
| topic_facet | Chaos Fractals Mathematik Medizin Biomathematics Chaotic behavior in systems Medicine Mathematics Fraktal Chaostheorie Biologie Einführung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008038775&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT liebovitchlarrys fractalsandchaossimplifiedforthelifesciences |