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Tables of Contents for Ordinary Differential Equations
Foreword to the Classics Edition
xv
Preface to the First Edition
xvii
Preface to the Second Edition
xviii
Change of integration variables
7
1
Notes7
1
The Picard-Lindelof theorem
8
2
Peano's existence theorem
10
2
Example of nonuniqueness
18
6
Notes23
1
Differential inequalities and uniqueness
24
21
Maximal and minimal solutions
25
1
Differential inequalities
26
3
van Kampen's uniqueness theorem
35
2
Egress points and Lyapunov functions
37
3
Successive approximations
40
5
Notes44
1
Linear differential equations
45
48
Variation of constants
48
1
Reductions to smaller systems
49
5
Higher order linear equations
63
5
Remarks on changes of variables
68
2
Appendix. Analytic Linear Equations70
1
Higher order equations
84
3
A nonsimple singularity
87
6
Notes91
2
Dependence on initial conditions and parameters
93
24
Higher order differentiability
100
1
Another differentiability theorem
104
3
S-and L-Lipschitz continuity
107
2
Notes116
1
Total and partial differential equations
117
27
A Theorem of Frobenius117
1
Total differential equations
117
3
Algebra of exterior forms
120
2
A theorem of Frobenius
122
2
Cauchy's Method of Characteristics131
1
A nonlinear partial differential equation
131
4
Existence and uniqueness theorem
137
2
Haar's lemma and uniqueness
139
5
Notes142
2
The Poincare-Bendixson theory
144
58
Index of a stationary point
149
2
The Poincare-Bendixson theorem
151
5
Stability of periodic solutions
156
2
Foci, nodes, and saddle points
160
1
The general stationary point
166
8
A second order equation
174
8
Appendix. Poincare-Bendixson Theory on 2-Manifolds182
1
Analogue of the Poincare-Bendixson theorem
185
5
Flow on a closed curve
190
5
Notes201
1
Plane stationary points
202
26
Characteristic directions
209
3
Perturbed linear systems
212
8
More general stationary point
220
8
Notes227
1
Invariant manifolds and linearizations
228
45
Invariant manifolds of a map
234
8
Existence of invariant manifolds
242
2
Linearization of a map
245
5
Appendix. Smooth Equivalence Maps256
1
Smooth linearizations
256
3
Proof of Theorem 12.2
261
12
Appendix. Smoothness of Stable Manifolds271
1
Notes271
2
Perturbed linear systems
273
49
A topological principle
278
2
A theorem of Wazewski
280
3
Asymptotic integrations. Logarithmic scale
294
3
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
Notes320
2
Linear second order equations
322
82
Sturm-Liouville boundary value problems
337
7
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 Systems384
1
Disconjugate systems
384
12
Notes401
3
Use of implicit function and fixed point theorems
404
46
Periodic Solutions407
1
Second Order Boundary Value Problems418
1
General Theory435
1
Asymptotic integration
445
5
Notes447
3
Dichotomies for solutions of linear equations
450
50
General Theory451
1
Notations and definitions
451
4
Estimates for ||y(t)||
470
4
Applications to first order systems
474
4
Applications to higher order systems
478
5
Adjoint Equations484
1
Individual dichotomies
486
4
P'-admissible spaces for T'
490
3
Applications to differential equations
493
4
Existence of PD-solutions
497
3
Notes498
2
Miscellany on monotony
500
57
Monotone Solutions500
1
Small and large solutions
500
6
Second order linear equations
510
5
Second order linear equations (continuation)
515
4
A Problem in Boundary Layer Theory519
1
Global Asymptotic Stability537
1
Global asymptotic stability
537
2
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
Notes554
3
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