Chaos: an introduction to dynamical systems
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Buch |
| Sprache: | Undetermined |
| Veröffentlicht: |
New York, NY u.a.
New York, NY u.a.
1996
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XVII, 603 S. |
| ISBN: | 0387946772 |
Internformat
MARC
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| 100 | 1 | |a Alligood, Kathleen T. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Chaos |b an introduction to dynamical systems |c Kathleen T. Alligood ; Tim D. Sauer ; James A. Yorke |
| 264 | 1 | |a New York, NY u.a. |b New York, NY u.a. |c 1996 | |
| 300 | |a XVII, 603 S. | ||
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| 650 | 0 | 7 | |a Dynamisches System |0 (DE-588)4013396-5 |2 gnd |9 rswk-swf |
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| 700 | 1 | |a Yorke, James A. |e Verfasser |4 aut | |
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Datensatz im Suchindex
| _version_ | 1804128287989956608 |
|---|---|
| adam_text | Contents
INTRODUCTION
ONE-DIMENSIONAL MAPS I
1.1
One-Dimensional Maps
2
1.2
Cobweb Plot: Graphical Representation of an Orbit
5
1.3
Stability of Fixed Points
9
1.4
Periodic Points
13
1.5
The Family of Logistic Maps
17
1.6
The Logistic Map G(x)
=
4x(l
-
x)
22
1.7
Sensitive Dependence on Initial Conditions
25
1.8
Itineraries
27
Challenge
1 :
Period Three Implies Chaos
3 2
Exercises
36
Lab Visit
1 :
Boom, Bust, and Chaos in the Beetle Census
39
XIII
Contents
TWO-DIMENSIONAL MAPS
43
2.1
Mathematical Models
44
2.2
Sinks, Sources, and Saddles
58
2.3
Linear Maps
62
2.4
Coordinate Changes
67
2.5
Nonlinear Maps and the Jacobian Matrix
68
2.6
Stable and Unstable Manifolds
78
2.7
Matrix Times Circle Equals Ellipse
87
Challenge
2:
Counting the Periodic Orbits of
Linear Maps on a Torus
92
Exercises
98
Lab Visit
2:
Is the Solar System Stable?
99
CHAOS
105
3.1
Lyapunov Exponents
Ю6
3.2
Chaotic Orbits
109
3.3
Conjugacy and the Logistic Map
1
И
3.4
Transition Graphs and Fixed Points
124
3.5
Basins of Attraction
129
Challenge
3:
Sharkovskii s Theorem
135
Exercises
140
Lab Visit
3:
Periodicity and Chaos in a
Chemical Reaction
143
FRACTALS
149
4.1
Cantor Sets
150
4-2
Probabilistic Constructions of Fractals
156
4-3
Fractals from Deterministic Systems
161
4.4
Fractal Basin Boundaries
164
4-5
Fractal Dimension
172
4-6
Computing the Box-Count ing Dimension
177
4-7
Correlation Dimension
180
Challenge
4:
Fractal Basin Boundaries and the
Uncertainty Exponent
183
Exercises
186
Lab Visit
Ą-.
Fractal Dimension in Experiments
188
CHAOS IN TWO-DIMENSIONAL MAPS
193
5.1
Lyapunov Exponents
194
5.2
Numerical Calculation of Lyapunov Exponents
199
5.3
Lyapunov Dimension
203
5.4
A Two-Dimensional Fixed-Point Theorem
207
5.5
Markov Partitions
212
5.6
The Horseshoe Map
216
XIV
Contents
Challenge
5:
Computer Calculations and Shadowing
222
Exercises
226
Lab Visit
5·.
Chaos in Simple Mechanical Devices
228
CHAOTIC ATTRACTORS
231
6.1
Forward Limit Sets
233
6.2
Chaotic Attractors
238
6.3
Chaotic Attractors of Expanding Interval Maps
245
6.4
Measure
249
6.5
Natural Measure
253
6.6
Invariant Measure for One-Dimensional Maps
256
Challenge
6:
Invariant Measure for the Logistic Map
264
Exercises
266
Lab Visit
6:
Fractal Scum
267
DIFFERENTIAL EQUATIONS
273
7.1
One-Dimensional Linear Differential Equations
275
7.2
One-Dimensional Nonlinear Differential Equations
278
7.3
Linear Differential Equations in More than One Dimension
284
7.4
Nonlinear Systems
294
7.5
Motion in a Potential Field
300
7.6
Lyapunov Functions
304
7.7
Lotka-Volterra Models
309
Challenge
7:
A Limit Cycle in the Van
oer
Pol System
316
Exercises
321
Lar
Visit
Ъ
Fly vs. Fly
32 5
PERIODIC ORBITS AND LIMIT SETS
329
8.1
Limit Sets for Planar Differential Equations
331
8.2
Properties of
ω
-Limit
Sets
337
8.3
Proof of the Poincare-Bendixson Theorem
341
Challenge
8.·
Two Incommensurate Frequencies
Form a Torus
350
Exercises
353
Lab Visit
8:
Steady States and Periodicity in a
Squid Neuron
355
CHAOS IN DIFFERENTIAL EQUATIONS
359
9.1
The
Lorenz
Attractor
359
9.2
Stability in the Large, Instability in the Small
366
9.3
The
Rössler
Attractor
370
9.4
Chua s Circuit
375
9.5
Forced Oscillators
376
9.6
Lyapunov Exponents in Flows
379
xv
Contents
Challenge
9:
Synchronization of Chaotic; Orbits
387
Exercises
393
Lab Visit
9:
Lasers in Synchronization
394
10
STABLE MANIFOLDS AND CRISES
399
10.1
The Stable Manifold Theorem
401
10.2
Homoclinic and Heteroclinic Points
409
10.3
Crises
413
10.4
Proof of the
Stahle
Manifold Theorem
422
10.5
Stable and Unstable Manifolds for Higher Dimensional Maps
430
Challenge
10:
The Lakes of
Wada
432
Exercises
440
Lab Visit
10:
The Leaky Faucet Minor Irritation
or Crisis?
441
11 BIFURCATIONS
447
11.1
Saddle-Node and Period-Doubling Bifurcations
448
11.2
Bifurcation Diagrams
453
11.3
Continuability
460
11.4
Bifurcations of One-Dimensional Maps
464
11.5
Bifurcations in Plane Maps: Area-Contracting Case
468
11.6
Bifurcations in Plane Maps: Area-Preserving Case
471
11.7
Bifurcations in Differential Equations
478
11.8 Hopf
Bifurcations
483
Challenge
11:
Hamiltonian Systems and the
Lyafunov Center Theorem
491
Exercises
494
Lab Visit
11:
Iron
+
Sulfuric
Acid
—> Hopf
Bifurcation
496
12
CASCADES
499
12.1
Cascades and
4.669201609. .. 500
12.2
Schematic Bifurcation Diagrams
504
12.3
Generic Bifurcations
510
12.4
The Cascade Theorem
518
challenge
12:
universality in bifurcation diagrams
525
Exercises
531
Lab Visit
12:
Experimental Cascades
532
13
STATE RECONSTRUCTION FROM DATA
537
1.3.1
Delay Plots from Time Series
537
13.2
Delay Coordinates
541
13.3
Emhedology
545
challenge
13:
box-counting dimension
and Intersection
553
XVI
Contents
A MATRIX
ALGEBRA
557
АЛ
Eigenvalues and Eigenvectors
557
A.2 Coordinate Changes
561
A.3 Matrix Times Circle Equals Ellipse
563
В
COMPUTER SOLUTION OF ODES
567
B.I ODE Solvers
568
B.2 Error in Numerical Integration
570
B.3 Adaptive Step-Size Methods
574
ANSWERS AND HINTS TO SELECTED EXERCISES
577
BIBLIOGRAPHY
587
INDEX
595
XVII
|
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| author | Alligood, Kathleen T. Sauer, Tim D. Yorke, James A. |
| author_facet | Alligood, Kathleen T. Sauer, Tim D. Yorke, James A. |
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| building | Verbundindex |
| bvnumber | BV013497461 |
| ctrlnum | (OCoLC)635001650 (DE-599)BVBBV013497461 |
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| id | DE-604.BV013497461 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T18:46:51Z |
| institution | BVB |
| isbn | 0387946772 |
| language | Undetermined |
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| physical | XVII, 603 S. |
| publishDate | 1996 |
| publishDateSearch | 1996 |
| publishDateSort | 1996 |
| publisher | New York, NY u.a. |
| record_format | marc |
| spelling | Alligood, Kathleen T. Verfasser aut Chaos an introduction to dynamical systems Kathleen T. Alligood ; Tim D. Sauer ; James A. Yorke New York, NY u.a. New York, NY u.a. 1996 XVII, 603 S. txt rdacontent n rdamedia nc rdacarrier Dynamisches System (DE-588)4013396-5 gnd rswk-swf Chaos (DE-588)4191419-3 gnd rswk-swf Chaostheorie (DE-588)4009754-7 gnd rswk-swf Chaostheorie (DE-588)4009754-7 s 1\p DE-604 Chaos (DE-588)4191419-3 s 2\p DE-604 Dynamisches System (DE-588)4013396-5 s 3\p DE-604 Sauer, Tim D. Verfasser aut Yorke, James A. Verfasser aut Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009213083&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 3\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Alligood, Kathleen T. Sauer, Tim D. Yorke, James A. Chaos an introduction to dynamical systems Dynamisches System (DE-588)4013396-5 gnd Chaos (DE-588)4191419-3 gnd Chaostheorie (DE-588)4009754-7 gnd |
| subject_GND | (DE-588)4013396-5 (DE-588)4191419-3 (DE-588)4009754-7 |
| title | Chaos an introduction to dynamical systems |
| title_auth | Chaos an introduction to dynamical systems |
| title_exact_search | Chaos an introduction to dynamical systems |
| title_full | Chaos an introduction to dynamical systems Kathleen T. Alligood ; Tim D. Sauer ; James A. Yorke |
| title_fullStr | Chaos an introduction to dynamical systems Kathleen T. Alligood ; Tim D. Sauer ; James A. Yorke |
| title_full_unstemmed | Chaos an introduction to dynamical systems Kathleen T. Alligood ; Tim D. Sauer ; James A. Yorke |
| title_short | Chaos |
| title_sort | chaos an introduction to dynamical systems |
| title_sub | an introduction to dynamical systems |
| topic | Dynamisches System (DE-588)4013396-5 gnd Chaos (DE-588)4191419-3 gnd Chaostheorie (DE-588)4009754-7 gnd |
| topic_facet | Dynamisches System Chaos Chaostheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009213083&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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