Nonlinear ordinary differential equations: an introduction for scientists and engineers
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| Main Authors: | , |
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| Format: | Book |
| Language: | English |
| Published: |
Oxford
Oxford Univ. Press
2007
|
| Edition: | 4. ed. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | VIII, 531 S. Ill., graph. Darst. |
| ISBN: | 9780199208258 |
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| 245 | 1 | 0 | |a Nonlinear ordinary differential equations |b an introduction for scientists and engineers |c D. W. Jordan and P. Smith |
| 250 | |a 4. ed. | ||
| 264 | 1 | |a Oxford |b Oxford Univ. Press |c 2007 | |
| 300 | |a VIII, 531 S. |b Ill., graph. Darst. | ||
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-015918824 | ||
Record in the Search Index
| _version_ | 1804136934760513536 |
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| adam_text | Contents
Preface
to the fourth edition
vii
1
Second-order differential equations in the phase plane
1
1.1
Phase diagram for the pendulum equation
1
1.2
Autonomous equations in the phase plane
5
1.3
Mechanical analogy for the conservative system
χ
— ƒ
(χ)
14
1.4
The damped linear oscillator
21
1.5
Nonlinear damping: limit cycles
25
1.6
Some applications
32
1.7
Parameter-dependent conservative systems
37
1.8
Graphical representation of solutions
40
Problems
42
2
Plane autonomous systems and linearization
49
2.1
The general phase plane
49
2.2
Some population models
53
2.3
Linear approximation at equilibrium points
57
2.4
The general solution of linear autonomous plane systems
58
2.5
The phase paths of linear autonomous plane systems
63
2.6
Scaling in the phase diagram for a linear autonomous system
72
2.7
Constructing a phase diagram
73
2.8
Hamiltonian systems
75
Problems
79
3
Geometrical aspects of plane autonomous systems
89
3.1
The index of a point
89
3.2
The index at infinity
97
3.3
The phase diagram at infinity
100
3.4
Limit cycles and other closed paths
104
3.5
Computation of the phase diagram
107
3.6
Homoclinic and heteroclinic paths
111
Problems
113
Contents
4
Periodic
solutions;
averaging methods 125
4.1
An energy-balance method for limit cycles
125
4.2
Amplitude and frequency estimates: polar coordinates
130
4.3
An averaging method for spiral phase paths
134
4.4
Periodic solutions: harmonic balance
138
4.5
The equivalent linear equation by harmonic balance
140
Problems 143
5
Perturbation methods
149
5.1
Nonautonomous systems: forced oscillations
149
5.2
The direct perturbation method for the undamped Duffing s equation
153
5.3
Forced oscillations far from resonance
155
5.4
Forced oscillations near resonance with weak excitation
157
5.5
The amplitude equation for the undamped pendulum
159
5.6
The amplitude equation for a damped pendulum
163
5.7
Soft and hard springs
164
5.8
Amplitude-phase perturbation for the pendulum equation
167
5.9
Periodic solutions of autonomous equations (Lindstedt s method)
169
5.10
Forced oscillation of a self-excited equation
171
5.11
The perturbation method and Fourier series
173
5.12
Homoclinic bifurcation: an example
175
Problems
179
6
Singular perturbation methods
183
6.1
Non-uniform approximations to functions on an interval
183
6.2
Coordinate perturbation
185
6.3
Ughthill s method
190
6.4
Time-scaling for series solutions of autonomous equations
192
6.5
The multipie-scale technique applied to saddle points and nodes
199
6.6
Matching approximations on an interval
206
6.7
A matching technique for differential equations
211
Problems
217
7
Forced oscillations: harmonic and subharmonic response, stability,
and entrapment
223
7.1
General forced periodic solutions
223
7.2
Harmonic solutions, transients, and stability for Duffing s equation
225
7.3
The jump phenomenon
231
7.4
Harmonic oscillations, stability, and transients for the forced van
der
Pol equation
234
7.5
Frequency entrainment for the van
der Pol
equation
239
Contents
7.6 Subharmonics
of Duffing s equation by perturbation
242
7.7
Stability and transients for subharmonics of Duffing s equation
247
Problems
251
8
Stability
259
8.1
Poincaré
stability (stability of paths)
260
8.2
Paths and solution curves for general systems
265
8.3
Stability of time solutions: Liapunov stability
267
8.4
Liapunov stability of plane autonomous linear systems
271
8.5
Structure of the solutions of
n-dimensional
linear systems
274
8.6
Structure of
n-dimensional
inhomogeneous linear systems
279
8.7
Stability and boundedness for linear systems
283
8.8
Stability of linear systems with constant coefficients
284
8.9
Linear approximation at equilibrium points for first-order systems in
η
variables
289
8.10
Stability of a class of non-autonomous linear systems in
η
dimensions
293
8.11
Stability of the zero solutions of nearly linear systems
298
Problems
300
9
Stability by solution perturbation: Mathieu s equation
305
9.1
The stability of forced oscillations by solution perturbation
305
9.2
Equations with periodic coefficients (Floquet theory)
308
9.3
Mathieu s equation arising from a Duffing equation
315
9.4
Transition curves for Mathieu s equation by perturbation
322
9.5
Mathieu s damped equation arising from a Duffing equation
325
Problems
330
10
Liapunov methods for determining stability of the zero solution
337
10.1
Introducing the Liapunov method
337
10.2
Topographic systems and the
Poincaré-Bendixson
theorem
338
10.3
Liapunov stability of the zero solution
342
10.4
Asymptotic stability of the zero solution
346
10.5
Extending weak Liapunov functions to asymptotic stability
349
10.6
A more general theory for autonomous systems
351
10.7
A test for instability of the zero solution:
η
dimensions
356
10.8
Stability and the linear approximation in two dimensions
357
10.9
Exponential function of a matrix
365
10.10
Stability and the linear approximation for nth order autonomous systems
367
10.11
Special systems
373
Problems
377
Contents
11
The existence of periodic solutions
383
11.1
The
Poincaré-Bendixson
theorem and periodic solutions
383
.1,2
A theorem on the existence of a centre
390
11.3
A theorem on the existence of a limit cycle
394
11.4
Van
der
Pol s equation with large parameter
400
Problems
403
12
Bifurcations and manifolds
405
12.1
Examples of simple bifurcations
405
12.2
The fold and the cusp
407
12.3
Further types of bifurcation
411
12.4 Hopf
bifurcations
419
12.5
Higher-order systems: manifolds
422
12.6
Linear approximation: centre manifolds
427
Problems
433
13
Poincaré
sequences, homoclinic bifurcation, and chaos
439
13.1
Poincaré
sequences
439
13.2
Poincaré
sections for nonautonomous systems
442
13.3
Subharmonics and period doubling
447
13.4
Homoclinic paths, strange attractors and chaos
450
13.5
The Duffing oscillator
453
13.6
A discrete system: the logistic difference equation
462
13.7
Liapunov exponents and difference equations
466
13.8
Homoclinic bifurcation for forced systems
469
13.9
The horseshoe map
476
13.10
Melnikov s method for detecting homoclinic bifurcation
477
13.11
Liapunov exponents and differential equations
483
13.12
Power spectra
491
13.13
Some further features of chaotic oscillations
492
Problems
494
Answers to the exercises
507
Appendices
511
A Existence and uniqueness theorems
511
В
Topographic systems
513
С
Norms for vectors and matrices
515
D
A contour integral
517
E
Useful results
518
References and further reading
521
Index
525
|
| adam_txt |
Contents
Preface
to the fourth edition
vii
1
Second-order differential equations in the phase plane
1
1.1
Phase diagram for the pendulum equation
1
1.2
Autonomous equations in the phase plane
5
1.3
Mechanical analogy for the conservative system
χ
— ƒ
(χ)
14
1.4
The damped linear oscillator
21
1.5
Nonlinear damping: limit cycles
25
1.6
Some applications
32
1.7
Parameter-dependent conservative systems
37
1.8
Graphical representation of solutions
40
Problems
42
2
Plane autonomous systems and linearization
49
2.1
The general phase plane
49
2.2
Some population models
53
2.3
Linear approximation at equilibrium points
57
2.4
The general solution of linear autonomous plane systems
58
2.5
The phase paths of linear autonomous plane systems
63
2.6
Scaling in the phase diagram for a linear autonomous system
72
2.7
Constructing a phase diagram
73
2.8
Hamiltonian systems
75
Problems
79
3
Geometrical aspects of plane autonomous systems
89
3.1
The index of a point
89
3.2
The index at infinity
97
3.3
The phase diagram at infinity
100
3.4
Limit cycles and other closed paths
104
3.5
Computation of the phase diagram
107
3.6
Homoclinic and heteroclinic paths
111
Problems
113
Contents
4
Periodic
solutions;
averaging methods 125
4.1
An energy-balance method for limit cycles
125
4.2
Amplitude and frequency estimates: polar coordinates
130
4.3
An averaging method for spiral phase paths
134
4.4
Periodic solutions: harmonic balance
138
4.5
The equivalent linear equation by harmonic balance
140
Problems 143
5
Perturbation methods
149
5.1
Nonautonomous systems: forced oscillations
149
5.2
The direct perturbation method for the undamped Duffing's equation
153
5.3
Forced oscillations far from resonance
155
5.4
Forced oscillations near resonance with weak excitation
157
5.5
The amplitude equation for the undamped pendulum
159
5.6
The amplitude equation for a damped pendulum
163
5.7
Soft and hard springs
164
5.8
Amplitude-phase perturbation for the pendulum equation
167
5.9
Periodic solutions of autonomous equations (Lindstedt's method)
169
5.10
Forced oscillation of a self-excited equation
171
5.11
The perturbation method and Fourier series
173
5.12
Homoclinic bifurcation: an example
175
Problems
179
6
Singular perturbation methods
183
6.1
Non-uniform approximations to functions on an interval
183
6.2
Coordinate perturbation
185
6.3
Ughthill's method
190
6.4
Time-scaling for series solutions of autonomous equations
192
6.5
The multipie-scale technique applied to saddle points and nodes
199
6.6
Matching approximations on an interval
206
6.7
A matching technique for differential equations
211
Problems
217
7
Forced oscillations: harmonic and subharmonic response, stability,
and entrapment
' 223
7.1
General forced periodic solutions
223
7.2
Harmonic solutions, transients, and stability for Duffing's equation
225
7.3
The jump phenomenon
231
7.4
Harmonic oscillations, stability, and transients for the forced van
der
Pol equation
234
7.5
Frequency entrainment for the van
der Pol
equation
239
Contents
7.6 Subharmonics
of Duffing's equation by perturbation
242
7.7
Stability and transients for subharmonics of Duffing's equation
247
Problems
251
8
Stability
259
8.1
Poincaré
stability (stability of paths)
260
8.2
Paths and solution curves for general systems
265
8.3
Stability of time solutions: Liapunov stability
267
8.4
Liapunov stability of plane autonomous linear systems
271
8.5
Structure of the solutions of
n-dimensional
linear systems
274
8.6
Structure of
n-dimensional
inhomogeneous linear systems
279
8.7
Stability and boundedness for linear systems
283
8.8
Stability of linear systems with constant coefficients
284
8.9
Linear approximation at equilibrium points for first-order systems in
η
variables
289
8.10
Stability of a class of non-autonomous linear systems in
η
dimensions
293
8.11
Stability of the zero solutions of nearly linear systems
298
Problems
300
9
Stability by solution perturbation: Mathieu's equation
305
9.1
The stability of forced oscillations by solution perturbation
305
9.2
Equations with periodic coefficients (Floquet theory)
308
9.3
Mathieu's equation arising from a Duffing equation
315
9.4
Transition curves for Mathieu's equation by perturbation
322
9.5
Mathieu's damped equation arising from a Duffing equation
325
Problems
330
10
Liapunov methods for determining stability of the zero solution
337
10.1
Introducing the Liapunov method
337
10.2
Topographic systems and the
Poincaré-Bendixson
theorem
338
10.3
Liapunov stability of the zero solution
342
10.4
Asymptotic stability of the zero solution
346
10.5
Extending weak Liapunov functions to asymptotic stability
349
10.6
A more general theory for autonomous systems
351
10.7
A test for instability of the zero solution:
η
dimensions
356
10.8
Stability and the linear approximation in two dimensions
357
10.9
Exponential function of a matrix
365
10.10
Stability and the linear approximation for nth order autonomous systems
367
10.11
Special systems
373
Problems
377
Contents
11
The existence of periodic solutions
383
11.1
The
Poincaré-Bendixson
theorem and periodic solutions
383
.1,2
A theorem on the existence of a centre
390
11.3
A theorem on the existence of a limit cycle
394
11.4
Van
der
Pol's equation with large parameter
400
Problems
403
12
Bifurcations and manifolds
405
12.1
Examples of simple bifurcations
405
12.2
The fold and the cusp
407
12.3
Further types of bifurcation
411
12.4 Hopf
bifurcations
419
12.5
Higher-order systems: manifolds
422
12.6
Linear approximation: centre manifolds
427
Problems
433
13
Poincaré
sequences, homoclinic bifurcation, and chaos
439
13.1
Poincaré
sequences
439
13.2
Poincaré
sections for nonautonomous systems
442
13.3
Subharmonics and period doubling
447
13.4
Homoclinic paths, strange attractors and chaos
450
13.5
The Duffing oscillator
453
13.6
A discrete system: the logistic difference equation
462
13.7
Liapunov exponents and difference equations
466
13.8
Homoclinic bifurcation for forced systems
469
13.9
The horseshoe map
476
13.10
Melnikov's method for detecting homoclinic bifurcation
477
13.11
Liapunov exponents and differential equations
483
13.12
Power spectra
491
13.13
Some further features of chaotic oscillations
492
Problems
494
Answers to the exercises
507
Appendices
511
A Existence and uniqueness theorems
511
В
Topographic systems
513
С
Norms for vectors and matrices
515
D
A contour integral
517
E
Useful results
518
References and further reading
521
Index
525 |
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| author | Jordan, Dominic W. Smith, Peter 1935- |
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| classification_tum | MAT 340f |
| ctrlnum | (OCoLC)137312934 (DE-599)BVBBV022712991 |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
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| dewey-search | 515/.352 |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| edition | 4. ed. |
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| id | DE-604.BV022712991 |
| illustrated | Illustrated |
| index_date | 2024-07-02T18:27:03Z |
| indexdate | 2024-07-09T21:04:17Z |
| institution | BVB |
| isbn | 9780199208258 |
| language | English |
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| spelling | Jordan, Dominic W. Verfasser (DE-588)115172602 aut Nonlinear ordinary differential equations an introduction for scientists and engineers D. W. Jordan and P. Smith 4. ed. Oxford Oxford Univ. Press 2007 VIII, 531 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Differential equations, Nonlinear Nichtlineare gewöhnliche Differentialgleichung (DE-588)4478411-9 gnd rswk-swf Nichtlineare gewöhnliche Differentialgleichung (DE-588)4478411-9 s DE-604 Smith, Peter 1935- Verfasser (DE-588)141762349 aut Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015918824&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Jordan, Dominic W. Smith, Peter 1935- Nonlinear ordinary differential equations an introduction for scientists and engineers Differential equations, Nonlinear Nichtlineare gewöhnliche Differentialgleichung (DE-588)4478411-9 gnd |
| subject_GND | (DE-588)4478411-9 |
| title | Nonlinear ordinary differential equations an introduction for scientists and engineers |
| title_auth | Nonlinear ordinary differential equations an introduction for scientists and engineers |
| title_exact_search | Nonlinear ordinary differential equations an introduction for scientists and engineers |
| title_exact_search_txtP | Nonlinear ordinary differential equations an introduction for scientists and engineers |
| title_full | Nonlinear ordinary differential equations an introduction for scientists and engineers D. W. Jordan and P. Smith |
| title_fullStr | Nonlinear ordinary differential equations an introduction for scientists and engineers D. W. Jordan and P. Smith |
| title_full_unstemmed | Nonlinear ordinary differential equations an introduction for scientists and engineers D. W. Jordan and P. Smith |
| title_short | Nonlinear ordinary differential equations |
| title_sort | nonlinear ordinary differential equations an introduction for scientists and engineers |
| title_sub | an introduction for scientists and engineers |
| topic | Differential equations, Nonlinear Nichtlineare gewöhnliche Differentialgleichung (DE-588)4478411-9 gnd |
| topic_facet | Differential equations, Nonlinear Nichtlineare gewöhnliche Differentialgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015918824&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT jordandominicw nonlinearordinarydifferentialequationsanintroductionforscientistsandengineers AT smithpeter nonlinearordinarydifferentialequationsanintroductionforscientistsandengineers |