Nonlinear Dynamics in Complex Systems: Theory and Applications for the Life-, Neuro- and Natural Sciences
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Heidelberg u.a.
Springer
2013
|
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | XIII, 236 S. graph. Darst. |
| ISBN: | 9783642335518 9783642427817 |
Staff View
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Record in the Search Index
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CONTENTS
PART I: NONLINEAR DYNAMICAL SYSTEMS
1 INTRODUCTION 3
1.1 COMPLEX SYSTEMS 3
1.2 DIFFERENTIAL EQUATIONS - THE BASICS 6
1.3 FIRST STEPS 7
1.4 TERMINOLOGY - DYNAMICAL SYSTEMS JARGON 9
1.5 A FIRST ENCOUNTER WITH
PHASE
SPACE 11
1.6 LIMITATIONS OF LINEAR SYSTEMS 12
2 ONE-DIMENSIONAL SYSTEMS 15
2.1 LINEAR STABILITY ANALYSIS 16
2.1.1 A QUANTITATIVE MEASURE OF STABILITY 17
2.1.2 BISTABILITY, MULTISTABILITY AND QUALITATIVE CHANGE . 18
2.1.3 EXOTIC CASES: WHEN LINEAR STABILITY ANALYSIS
BREAKS DOWN 19
2.2 POTENTIAL FUNCTIONS 21
2.3 BIFURCATION TYPES 26
2.3.1 SADDLE-NODE BIFURCATION 27
2.3.2 TRANSCRITICAL BIFURCATION 27
2.3.3 SUPERCRITICAL PITCHFORK BIFURCATION 28
2.3.4 SUBCRITICAL PITCHFORK BIFURCATION 29
2.3.5 SYSTEMS WITH HYSTERESIS 30
2.4 PERIODIC ONE-DIMENSIONAL SYSTEMS 32
2.5 PROBLEMS FOR CHAPTER 2 34
3 TWO-DIMENSIONAL SYSTEMS 37
3.1 THE
HARMONIC OSCILLATOR 37
3.1.1 SECOND ORDER SYSTEMS - A FIRST LOOK 38
3.1.2 SECOND ORDER SYSTEMS - A SECOND LOOK 39
3.1.3 DAMPED HARMONIC OSCILLATOR 40
HTTP://D-NB.INFO/1025398025
XII
CONTENTS
3.2 CLASSIFICATION OF TWO-DIRRIENSIORIAL LINEAR SYSTEMS 42
3.3 NONLINEAR SYSTEMS: LINEAR STABILITY ANALYSIS IN TWO
DIMENSIONS 46
3.4 LIMIT CYCLES 52
3.5 HOPF BIFURCATION 53
3.6 POTENTIAL FUNCTIONS IN TWO-DIRRIENSIORIAL SYSTEMS 60
3.7 GENERALIZATION OF POTENTIALS: LYAPUNOV FUNCTIONS 64
3.8 NONLINEAR OSCILLATORS 66
3.8.1 VAN-DER-POL OSCILLATOR: N(X.
X) = X
2
X 67
3.8.2 RAYLEIGH OSCILLATOR: N(X,
X) = X
3
68
3.8.3 HYBRID OSCILLATOR: N(X,
X) = {X
2
X, X
3
} 69
3.9 POINCARE-BENDIXON THEOREM 71
3.10 PROBLEMS FOR CHAPTER 3 71
4 HIGHER-DIMENSIONAL SYSTEMS
AND CHAOS 73
4.1 PERIODICITY AND QUASI-PERIODICITY 73
4.2 LORENZ SYSTEM 74
4.3 ROSSLER SYSTEM 80
4.4 FRACTAL DIMENSION 82
4.5 LYAPUNOV EXPONENTS 84
4.6 TIME SERIES ANALYSIS 87
5 DISCRETE MAPS
AND ITERATIONS IN SPACE 89
5.1 LOGISTIC MAP 89
5.1.1 GEOMETRIC APPROACH 92
5.2 LYAPUNOV EXPONENTS 93
5.3 HENON MAP 95
5.4 FEIGENBAUM CONSTANTS 97
5.5 USING DISCRETE MAPS TO UNDERSTAND CONTINUOUS SYSTEMS . 98
5.6 ITERATED FUNCTION SYSTEMS 100
5.7 MANDELBROT SET 103
6 STOCHASTIC SYSTEMS 105
6.1 BROWNIAN MOTION 106
6.2 FEATURES OF NOISE 108
6.2.1 HOW TO DESCRIBE KICKS 108
6.2.2 AVERAGES 110
6.2.3 DISTRIBUTIONS ILL
6.2.4 PROPERTIES OF THE GAUSSIAN DISTRIBUTION 112
6.2.5 CORRELATIONS 113
6.2.6 COLORS OF NOISE 120
6.3 LANGEVIN AND FOKKER-PLANCK EQUATION 121
6.4 RECONSTRUCTION OF DRIFT AND DIFFUSION TERM 123
6.5 MULTIPLICATIVE NOISE 125
. 6.6 MEAN FIRST PASSAGE TIME 126
CONTENTS XIII
6.7 FINGERPRINTS OF STOCHASTICITY 127
6.8 PROBLEMS FOR CHAPTER 6 128
PART II: MODEL
SYSTEMS
7 HAKEN-KELSO-BUNZ (HKB) MODEL 133
7.1 BASIC LAW OF COORDINATION: RELATIVE PHASE 133
7.2 STABILITY: PERTURBATIONS AND FLUCTUATIONS 137
7.3 OSCILLATOR LEVEL 139
7.4 SYMMETRY BREAKING THROUGH THE COMPONENTS 141
8 SELF-ORGANIZATION AND SYNERGETICS 147
8.1 HAKEN-ZWANZIG SYSTEM 148
8.2 LORENZ REVISITED 150
8.3 FORMALISM OF SYNERGETICS 155
9 NEURONAL MODELS 159
9.1 HODGKIN-HUXLEY MODEL 160
9.2 FITZHUGH-NAGUMO MODEL 166
9.3 HINDMARSH-ROSE MODEL 171
PART III: MATHEMATICAL BASICS
10 MATHEMATICAL BASICS 181
10.1 COMPLEX NUMBERS 181
10.2 LINEAR SYSTEMS OF EQUATIONS 185
10.3 EIGENVALUES AND EIGENVECTORS 187
10.4 TAYLOR SERIES 193
10.5 DELTA AND SIGMOID FUNCTIONS 195
A THE
COUPLED HKB
SYSTEM 199
A.L THE OSCILLATORS 199
A.2 THE COUPLING 200
A.3 THE
COUPLED SYSTEM 202
B NUMERICAL PROCEDURES AND
COMPUTER SIMULATIONS 205
B.L INTEGRATION SCHEMES FOR ORDINARY DIFFERENTIAL EQUATIONS . 205
C SOLUTIONS 213
C.L CHAPTER 2 213
C.2 CHAPTER 3 218
C.3 CHAPTER 6 223
REFERENCES 229
INDEX 231 |
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| discipline | Maschinenbau / Maschinenwesen Informatik Mathematik |
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| id | DE-604.BV040502984 |
| illustrated | Illustrated |
| indexdate | 2024-09-06T00:24:28Z |
| institution | BVB |
| isbn | 9783642335518 9783642427817 |
| language | English |
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| owner_facet | DE-384 DE-1102 |
| physical | XIII, 236 S. graph. Darst. |
| publishDate | 2013 |
| publishDateSearch | 2013 |
| publishDateSort | 2013 |
| publisher | Springer |
| record_format | marc |
| spelling | Fuchs, Armin Verfasser aut Nonlinear Dynamics in Complex Systems Theory and Applications for the Life-, Neuro- and Natural Sciences Armin Fuchs Heidelberg u.a. Springer 2013 XIII, 236 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Komplexes System (DE-588)4114261-5 gnd rswk-swf Nichtlineares dynamisches System (DE-588)4126142-2 gnd rswk-swf Nichtlineares dynamisches System (DE-588)4126142-2 s Komplexes System (DE-588)4114261-5 s DE-604 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=4107170&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025349704&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Fuchs, Armin Nonlinear Dynamics in Complex Systems Theory and Applications for the Life-, Neuro- and Natural Sciences Komplexes System (DE-588)4114261-5 gnd Nichtlineares dynamisches System (DE-588)4126142-2 gnd |
| subject_GND | (DE-588)4114261-5 (DE-588)4126142-2 |
| title | Nonlinear Dynamics in Complex Systems Theory and Applications for the Life-, Neuro- and Natural Sciences |
| title_auth | Nonlinear Dynamics in Complex Systems Theory and Applications for the Life-, Neuro- and Natural Sciences |
| title_exact_search | Nonlinear Dynamics in Complex Systems Theory and Applications for the Life-, Neuro- and Natural Sciences |
| title_full | Nonlinear Dynamics in Complex Systems Theory and Applications for the Life-, Neuro- and Natural Sciences Armin Fuchs |
| title_fullStr | Nonlinear Dynamics in Complex Systems Theory and Applications for the Life-, Neuro- and Natural Sciences Armin Fuchs |
| title_full_unstemmed | Nonlinear Dynamics in Complex Systems Theory and Applications for the Life-, Neuro- and Natural Sciences Armin Fuchs |
| title_short | Nonlinear Dynamics in Complex Systems |
| title_sort | nonlinear dynamics in complex systems theory and applications for the life neuro and natural sciences |
| title_sub | Theory and Applications for the Life-, Neuro- and Natural Sciences |
| topic | Komplexes System (DE-588)4114261-5 gnd Nichtlineares dynamisches System (DE-588)4126142-2 gnd |
| topic_facet | Komplexes System Nichtlineares dynamisches System |
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