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Tables of Contents for Ordinary Differential Equations
Chapter/Section Title
Page #
Page Count
Foreword to the Classics Edition
xv
Preface to the First Edition
xvii
Preface to the Second Edition
xviii
Errata
xix
Preliminaries
1
7
Preliminaries
1
1
Basic theorems
2
4
Smooth approximations
6
1
Change of integration variables
7
1
Notes
7
1
Existence
8
16
The Picard-Lindelof theorem
8
2
Peano's existence theorem
10
2
Extension theorem
12
3
H. Kneser's theorem
15
3
Example of nonuniqueness
18
6
Notes
23
1
Differential inequalities and uniqueness
24
21
Gronwall's inequality
24
1
Maximal and minimal solutions
25
1
Right derivatives
26
1
Differential inequalities
26
3
A theorem of Wintner
29
2
Uniqueness theorems
31
4
van Kampen's uniqueness theorem
35
2
Egress points and Lyapunov functions
37
3
Successive approximations
40
5
Notes
44
1
Linear differential equations
45
48
Linear systems
45
3
Variation of constants
48
1
Reductions to smaller systems
49
5
Basic inequalities
54
3
Constant coefficients
57
3
Floquet theory
60
2
62
1
Higher order linear equations
63
5
Remarks on changes of variables
68
2
Appendix. Analytic Linear Equations
70
1
Fundamental matrices
70
3
Simple singularities
73
11
Higher order equations
84
3
A nonsimple singularity
87
6
Notes
91
2
Dependence on initial conditions and parameters
93
24
Preliminaries
93
1
Continuity
94
1
Differentiability
95
5
Higher order differentiability
100
1
Exterior derivatives
101
3
Another differentiability theorem
104
3
S-and L-Lipschitz continuity
107
2
Uniqueness theorem
109
1
A lemma
110
1
Proof of Theorem 8.1
111
2
Proof of Theorem 6.1
113
1
First integrals
114
3
Notes
116
1
Total and partial differential equations
117
27
A Theorem of Frobenius
117
1
Total differential equations
117
3
Algebra of exterior forms
120
2
A theorem of Frobenius
122
2
Proof of Theorem 3.1
124
3
Proof of Lemma 3.1
127
1
The system (1.1)
128
3
Cauchy's Method of Characteristics
131
1
A nonlinear partial differential equation
131
4
Characteristics
135
2
Existence and uniqueness theorem
137
2
Haar's lemma and uniqueness
139
5
Notes
142
2
The Poincare-Bendixson theory
144
58
Autonomous systems
144
2
Umlaufsatz
146
3
Index of a stationary point
149
2
The Poincare-Bendixson theorem
151
5
Stability of periodic solutions
156
2
Rotation points
158
2
160
1
Sectors
161
5
The general stationary point
166
8
A second order equation
174
8
Appendix. Poincare-Bendixson Theory on 2-Manifolds
182
1
Preliminaries
182
3
Analogue of the Poincare-Bendixson theorem
185
5
Flow on a closed curve
190
5
Flow on a torus
195
7
Notes
201
1
Plane stationary points
202
26
Existence theorems
202
7
Characteristic directions
209
3
Perturbed linear systems
212
8
More general stationary point
220
8
Notes
227
1
Invariant manifolds and linearizations
228
45
Invariant manifolds
228
3
The maps Tt
231
1
Modification of F(ξ)
232
1
Normalizations
233
1
Invariant manifolds of a map
234
8
Existence of invariant manifolds
242
2
Linearizations
244
1
Linearization of a map
245
5
Proof of Theorem 7.1
250
1
Periodic solution
251
2
Limit cycles
253
3
Appendix. Smooth Equivalence Maps
256
1
Smooth linearizations
256
3
Proof of Lemma 12.1
259
2
Proof of Theorem 12.2
261
12
Appendix. Smoothness of Stable Manifolds
271
1
Notes
271
2
Perturbed linear systems
273
49
The case E = 0
273
5
A topological principle
278
2
A theorem of Wazewski
280
3
Preliminary lemmas
283
7
Proof of Lemma 4.1
290
1
Proof of Lemma 4.2
291
1
Proof of Lemma 4.3
292
2
Asymptotic integrations. Logarithmic scale
294
3
Proof of Theorem 8.2
297
2
Proof of Theorem 8.3
299
1
Logarithmic scale (continued)
300
3
Proof of Theorem 11.2
303
1
Asymptotic integration
304
3
Proof of Theorem 13.1
307
3
Proof of Theorem 13.2
310
1
Corollaries and refinements
311
3
Linear higher order equations
314
8
Notes
320
2
Linear second order equations
322
82
Preliminaries
322
3
Basic facts
325
8
Theorems of Sturm
333
4
Sturm-Liouville boundary value problems
337
7
Number of zeros
344
6
Nonoscillatory equations and principal solutions
350
12
Nonoscillation theorems
362
7
Asymptotic integrations. Elliptic cases
369
6
Asymptotic integrations. Nonelliptic cases
375
9
Appendix. Disconjugate Systems
384
1
Disconjugate systems
384
12
Generalizations
396
8
Notes
401
3
Use of implicit function and fixed point theorems
404
46
Periodic Solutions
407
1
Linear equations
407
5
Nonlinear problems
412
6
Second Order Boundary Value Problems
418
1
Linear problems
418
4
Nonlinear problems
422
6
A priori bounds
428
7
General Theory
435
1
Basic facts
435
4
Green's functions
439
2
Nonlinear equations
441
4
Asymptotic integration
445
5
Notes
447
3
Dichotomies for solutions of linear equations
450
50
General Theory
451
1
Notations and definitions
451
4
Preliminary lemmas
455
6
The operator T
461
4
Slices of ||Py(t)||
465
5
Estimates for ||y(t)||
470
4
Applications to first order systems
474
4
Applications to higher order systems
478
5
P(B, D)-manifolds
483
1
484
1
Associate spaces
484
2
The operator T'
486
1
Individual dichotomies
486
4
490
3
Applications to differential equations
493
4
Existence of PD-solutions
497
3
Notes
498
2
Miscellany on monotony
500
57
Monotone Solutions
500
1
Small and large solutions
500
6
Monotone solutions
506
4
Second order linear equations
510
5
Second order linear equations (continuation)
515
4
A Problem in Boundary Layer Theory
519
1
The problem
519
1
The Case λ > 0
520
5
The case λ < 0
525
6
The case λ = 0
531
3
Asymptotic behavior
534
3
Global Asymptotic Stability
537
1
Global asymptotic stability
537
2
Lyapunov functions
539
1
Nonconstant G
540
5
On Corollary 11.2
545
3
On ``J(y)x x 0 if x f(y) = 0''
548
2
Proof of Theorem 14.2
550
4
Proof of Theorem 14.1
554
3
Notes
554
3
Hints For Exercises
557
24
References
581
26
Index
607
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