Differential equations, dynamical systems, and an introduction to chaos:
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Amsterdam [u.a.]
Elsevier
2007
|
| Edition: | 2. ed., [Nachdr.] |
| Series: | Pure and applied mathematics
60 |
| Subjects: | |
| Online Access: | Publisher description Table of contents Inhaltsverzeichnis |
| Item Description: | Früher u.d.T.: Differential equations, dynamical systems, and linear algebra |
| Physical Description: | XIII, 417 S. Ill., graph. Darst. |
| ISBN: | 0123497035 |
Staff View
MARC
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| 100 | 1 | |a Hirsch, Morris W. |d 1933- |e Verfasser |0 (DE-588)172139120 |4 aut | |
| 245 | 1 | 0 | |a Differential equations, dynamical systems, and an introduction to chaos |c Morris W. Hirsch, Stephen Smale, Robert L. Devaney |
| 250 | |a 2. ed., [Nachdr.] | ||
| 264 | 1 | |a Amsterdam [u.a.] |b Elsevier |c 2007 | |
| 300 | |a XIII, 417 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Pure and applied mathematics |v 60 | |
| 500 | |a Früher u.d.T.: Differential equations, dynamical systems, and linear algebra | ||
| 650 | 4 | |a Differential equations | |
| 650 | 4 | |a Algebras, Linear | |
| 650 | 4 | |a Chaotic behavior in systems | |
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Record in the Search Index
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| adam_text | Contents
Preface x
CHAPTER
1
First-Order Equations
1
1.1
The Simplest Example
1
1.2
The Logistic Population Model
4
1.3
Constant Harvesting and Bifurcations
7
1.4
Periodic Harvesting and Periodic
Solutions
9
1.5
Computing the
Poincaré Map
12
1.6
Exploration: A Two-Parameter Family
15
CHAPTER
2
Planar Linear Systems
21
2.1
Second-Order Differential Equations
23
2.2
Planar Systems
24
2.3
Preliminaries from Algebra
26
2.4
Planar Linear Systems
29
2.5
Eigenvalues and Eigenvectors
30
2.6
Solving Linear Systems
33
2.7
The Linearity Principle
36
CHAPTER
3
Phase Portraits for Planar Systems
39
3.1
Real Distinct Eigenvalues
39
3.2
Complex Eigenvalues
44
vi
Contents
3.3
Repeated Eigenvalues
47
3.4
Changing Coordinates
49
CHAPTER
4
Classification of Planar Systems
61
4.1
The Trace-Determinant Plane
61
4.2
Dynamical Classification
64
4.3
Exploration: A
3D
Parameter Space
71
CHAPTER
5
Higher Dimensional Linear Algebra
75
5.1
Preliminaries from Linear Algebra
75
5.2
Eigenvalues and Eigenvectors
83
5.3
Complex Eigenvalues
86
5.4
Bases and Subspaces
89
5.5
Repeated Eigenvalues
95
5.6
Genericity
101
CHAPTER
6
Higher Dimensional Linear Systems
107
6.1
Distinct Eigenvalues
107
6.2
Harmonic Oscillators
114
6.3
Repeated Eigenvalues
119
6.4
The Exponential of a Matrix
123
6.5
Nonautonomous Linear Systems
130
CHAPTER
7
Nonlinear Systems
139
7.1
Dynamical Systems
140
7.2
The Existence and Uniqueness
Theorem
142
7.3
Continuous Dependence of Solutions
147
7.4
The Variational Equation
149
7.5
Exploration: Numerical Methods
153
CHAPTER
8
Equilibria in Nonlinear Systems
159
8.1
Some Illustrative Examples
159
8.2
Nonlinear Sinks and Sources
165
8.3
Saddles
168
8.4
Stability
174
8.5
Bifurcations
176
8.6
Exploration: Complex Vector Fields
182
Contents
vii
CHAPTER
9
Global Nonlinear Techniques
189
9.1
Nullclines
189
9.2
Stability of Equilibria
194
9.3
Gradient Systems
203
9.4
Hamiltonian Systems
207
9.5
Exploration: The Pendulum with Constant
Forcing
210
CHAPTER
10
Closed Orbits and Limit Sets
215
10.1
Limit Sets
215
10.2
Local Sections and Flow Boxes
218
10.3
The
Poincaré Map
220
10.4
Monotone Sequences in Planar Dynamical
Systems
222
10.5
The
Poincaré-Bendixson
Theorem
225
10.6
Applications of
Poincaré-Bendixson
227
10.7
Exploration: Chemical Reactions That
Oscillate
230
CHAPTER
11
Applications in Biology
235
11.1
Infectious Diseases
235
11.2
Predator/Prey Systems
239
11.3
Competitive Species
246
11.4
Exploration: Competition and
Harvesting
252
CHAPTER
12
Applications in Circuit Theory
257
12.1
An RLC Circuit
257
12.2
The Lienard Equation
261
12.3
The van
der Pol
Equation
262
12.4
A
Hopf
Bifurcation
270
12.5
Exploration:
Neurodynamics
272
CHAPTER
13
Applications in Mechanics
277
13.1
Newton s Second Law
277
13.2
Conservative Systems
280
13.3
Centrai
Force Fields
281
13.4
The Newtonian Central Force
System
285
viii Contents
13.5
Kepler s
First
Law
289
13.6
The Two-Body Problem
292
13.7
Blowing Up the Singularity
293
13.8
Exploration: Other Central Force
Problems
297
13.9
Exploration: Classical Limits of Quantum
Mechanical Systems
298
CHAPTER
14
The
Lorenz
System
303
14.1
Introduction to the
Lorenz
System
304
14.2
Elementary Properties of the
Lorenz
System
306
14.3
The
Lorenz Attractor 310
14.4
A Model for the
Lorenz
Attractor
314
14.5
The Chaotic Attractor
319
14.6
Exploration: The
Rössler
Attractor
324
CHAPTER
15
Discrete Dynamical Systems
327
15.1
Introduction to Discrete Dynamical
Systems
327
15.2
Bifurcations
332
15.3
The Discrete Logistic Model
335
15.4
Chaos
337
15.5
Symbolic Dynamics
342
15.6
The Shift Map
347
15.7
The Cantor Middle-Thirds Set
349
15.8
Exploration: Cubic Chaos
352
15.9
Exploration: The Orbit Diagram
353
CHAPTER
16
Homoclinic Phenomena
359
16.1
The Shil nikov System
359
16.2
The Horseshoe Map
366
16.3
The Double Scroll Attractor
372
16.4
Homoclinic Bifurcations
375
16.5
Exploration: The Chua Circuit
379
CHAPTER
17
Existence and Uniqueness Revisited
383
17.1
The Existence and Uniqueness
Theorem
383
17.2
Proof of Existence and Uniqueness
385
Bibliography
407
Index
411
Contents
ix
17.3
Continuous Dependence on Initial
Conditions
392
17.4
Extending Solutions
395
17.5
Nonautonomous Systems
398
17.6
Differentiability of the Flow
400
|
| adam_txt |
Contents
Preface x
CHAPTER
1
First-Order Equations
1
1.1
The Simplest Example
1
1.2
The Logistic Population Model
4
1.3
Constant Harvesting and Bifurcations
7
1.4
Periodic Harvesting and Periodic
Solutions
9
1.5
Computing the
Poincaré Map
12
1.6
Exploration: A Two-Parameter Family
15
CHAPTER
2
Planar Linear Systems
21
2.1
Second-Order Differential Equations
23
2.2
Planar Systems
24
2.3
Preliminaries from Algebra
26
2.4
Planar Linear Systems
29
2.5
Eigenvalues and Eigenvectors
30
2.6
Solving Linear Systems
33
2.7
The Linearity Principle
36
CHAPTER
3
Phase Portraits for Planar Systems
39
3.1
Real Distinct Eigenvalues
39
3.2
Complex Eigenvalues
44
vi
Contents
3.3
Repeated Eigenvalues
47
3.4
Changing Coordinates
49
CHAPTER
4
Classification of Planar Systems
61
4.1
The Trace-Determinant Plane
61
4.2
Dynamical Classification
64
4.3
Exploration: A
3D
Parameter Space
71
CHAPTER
5
Higher Dimensional Linear Algebra
75
5.1
Preliminaries from Linear Algebra
75
5.2
Eigenvalues and Eigenvectors
83
5.3
Complex Eigenvalues
86
5.4
Bases and Subspaces
89
5.5
Repeated Eigenvalues
95
5.6
Genericity
101
CHAPTER
6
Higher Dimensional Linear Systems
107
6.1
Distinct Eigenvalues
107
6.2
Harmonic Oscillators
114
6.3
Repeated Eigenvalues
119
6.4
The Exponential of a Matrix
123
6.5
Nonautonomous Linear Systems
130
CHAPTER
7
Nonlinear Systems
139
7.1
Dynamical Systems
140
7.2
The Existence and Uniqueness
Theorem
142
7.3
Continuous Dependence of Solutions
147
7.4
The Variational Equation
149
7.5
Exploration: Numerical Methods
153
CHAPTER
8
Equilibria in Nonlinear Systems
159
8.1
Some Illustrative Examples
159
8.2
Nonlinear Sinks and Sources
165
8.3
Saddles
168
8.4
Stability
174
8.5
Bifurcations
176
8.6
Exploration: Complex Vector Fields
182
Contents
vii
CHAPTER
9
Global Nonlinear Techniques
189
9.1
Nullclines
189
9.2
Stability of Equilibria
194
9.3
Gradient Systems
203
9.4
Hamiltonian Systems
207
9.5
Exploration: The Pendulum with Constant
Forcing
210
CHAPTER
10
Closed Orbits and Limit Sets
215
10.1
Limit Sets
215
10.2
Local Sections and Flow Boxes
218
10.3
The
Poincaré Map
220
10.4
Monotone Sequences in Planar Dynamical
Systems
222
10.5
The
Poincaré-Bendixson
Theorem
225
10.6
Applications of
Poincaré-Bendixson
227
10.7
Exploration: Chemical Reactions That
Oscillate
230
CHAPTER
11
Applications in Biology
235
11.1
Infectious Diseases
235
11.2
Predator/Prey Systems
239
11.3
Competitive Species
246
11.4
Exploration: Competition and
Harvesting
252
CHAPTER
12
Applications in Circuit Theory
257
12.1
An RLC Circuit
257
12.2
The Lienard Equation
261
12.3
The van
der Pol
Equation
262
12.4
A
Hopf
Bifurcation
270
12.5
Exploration:
Neurodynamics
272
CHAPTER
13
Applications in Mechanics
277
13.1
Newton's Second Law
277
13.2
Conservative Systems
280
13.3
Centrai
Force Fields
281
13.4
The Newtonian Central Force
System
285
viii Contents
13.5
Kepler's
First
Law
289
13.6
The Two-Body Problem
292
13.7
Blowing Up the Singularity
293
13.8
Exploration: Other Central Force
Problems
297
13.9
Exploration: Classical Limits of Quantum
Mechanical Systems
298
CHAPTER
14
The
Lorenz
System
303
14.1
Introduction to the
Lorenz
System
304
14.2
Elementary Properties of the
Lorenz
System
306
14.3
The
Lorenz Attractor 310
14.4
A Model for the
Lorenz
Attractor
314
14.5
The Chaotic Attractor
319
14.6
Exploration: The
Rössler
Attractor
324
CHAPTER
15
Discrete Dynamical Systems
327
15.1
Introduction to Discrete Dynamical
Systems
327
15.2
Bifurcations
332
15.3
The Discrete Logistic Model
335
15.4
Chaos
337
15.5
Symbolic Dynamics
342
15.6
The Shift Map
347
15.7
The Cantor Middle-Thirds Set
349
15.8
Exploration: Cubic Chaos
352
15.9
Exploration: The Orbit Diagram
353
CHAPTER
16
Homoclinic Phenomena
359
16.1
The Shil'nikov System
359
16.2
The Horseshoe Map
366
16.3
The Double Scroll Attractor
372
16.4
Homoclinic Bifurcations
375
16.5
Exploration: The Chua Circuit
379
CHAPTER
17
Existence and Uniqueness Revisited
383
17.1
The Existence and Uniqueness
Theorem
383
17.2
Proof of Existence and Uniqueness
385
Bibliography
407
Index
411
Contents
ix
17.3
Continuous Dependence on Initial
Conditions
392
17.4
Extending Solutions
395
17.5
Nonautonomous Systems
398
17.6
Differentiability of the Flow
400 |
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| id | DE-604.BV035164591 |
| illustrated | Illustrated |
| index_date | 2024-07-02T22:52:02Z |
| indexdate | 2024-07-09T21:26:28Z |
| institution | BVB |
| isbn | 0123497035 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016971639 |
| oclc_num | 633816945 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR |
| owner_facet | DE-355 DE-BY-UBR |
| physical | XIII, 417 S. Ill., graph. Darst. |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | Elsevier |
| record_format | marc |
| series | Pure and applied mathematics |
| series2 | Pure and applied mathematics |
| spelling | Hirsch, Morris W. 1933- Verfasser (DE-588)172139120 aut Differential equations, dynamical systems, and an introduction to chaos Morris W. Hirsch, Stephen Smale, Robert L. Devaney 2. ed., [Nachdr.] Amsterdam [u.a.] Elsevier 2007 XIII, 417 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Pure and applied mathematics 60 Früher u.d.T.: Differential equations, dynamical systems, and linear algebra Differential equations Algebras, Linear Chaotic behavior in systems Differentialgleichung (DE-588)4012249-9 gnd rswk-swf Lineare Algebra (DE-588)4035811-2 gnd rswk-swf Dynamisches System (DE-588)4013396-5 gnd rswk-swf Differentialgleichung (DE-588)4012249-9 s Dynamisches System (DE-588)4013396-5 s Lineare Algebra (DE-588)4035811-2 s DE-604 Smale, Stephen 1930- Verfasser (DE-588)119122472 aut Devaney, Robert L. 1948- Verfasser (DE-588)171063988 aut Pure and applied mathematics 60 (DE-604)BV010177228 60 http://www.loc.gov/catdir/description/els031/2003058255.html Publisher description http://www.loc.gov/catdir/toc/els031/2003058255.html Table of contents Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016971639&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Hirsch, Morris W. 1933- Smale, Stephen 1930- Devaney, Robert L. 1948- Differential equations, dynamical systems, and an introduction to chaos Pure and applied mathematics Differential equations Algebras, Linear Chaotic behavior in systems Differentialgleichung (DE-588)4012249-9 gnd Lineare Algebra (DE-588)4035811-2 gnd Dynamisches System (DE-588)4013396-5 gnd |
| subject_GND | (DE-588)4012249-9 (DE-588)4035811-2 (DE-588)4013396-5 |
| title | Differential equations, dynamical systems, and an introduction to chaos |
| title_auth | Differential equations, dynamical systems, and an introduction to chaos |
| title_exact_search | Differential equations, dynamical systems, and an introduction to chaos |
| title_exact_search_txtP | Differential equations, dynamical systems, and an introduction to chaos |
| title_full | Differential equations, dynamical systems, and an introduction to chaos Morris W. Hirsch, Stephen Smale, Robert L. Devaney |
| title_fullStr | Differential equations, dynamical systems, and an introduction to chaos Morris W. Hirsch, Stephen Smale, Robert L. Devaney |
| title_full_unstemmed | Differential equations, dynamical systems, and an introduction to chaos Morris W. Hirsch, Stephen Smale, Robert L. Devaney |
| title_short | Differential equations, dynamical systems, and an introduction to chaos |
| title_sort | differential equations dynamical systems and an introduction to chaos |
| topic | Differential equations Algebras, Linear Chaotic behavior in systems Differentialgleichung (DE-588)4012249-9 gnd Lineare Algebra (DE-588)4035811-2 gnd Dynamisches System (DE-588)4013396-5 gnd |
| topic_facet | Differential equations Algebras, Linear Chaotic behavior in systems Differentialgleichung Lineare Algebra Dynamisches System |
| url | http://www.loc.gov/catdir/description/els031/2003058255.html http://www.loc.gov/catdir/toc/els031/2003058255.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016971639&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV010177228 |
| work_keys_str_mv | AT hirschmorrisw differentialequationsdynamicalsystemsandanintroductiontochaos AT smalestephen differentialequationsdynamicalsystemsandanintroductiontochaos AT devaneyrobertl differentialequationsdynamicalsystemsandanintroductiontochaos |