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Foreword to the Classics Edition | |

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Preface to the First Edition | |

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Preface to the Second Edition | |

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Errata | |

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Preliminaries | |

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Preliminaries | |

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Basic theorems | |

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Smooth approximations | |

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Change of integration variables | |

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Notes | |

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Existence | |

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The Picard-Lindelof theorem | |

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Peano's existence theorem | |

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Extension theorem | |

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H. Kneser's theorem | |

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Example of nonuniqueness | |

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Notes | |

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Differential inequalities and uniqueness | |

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Gronwall's inequality | |

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Maximal and minimal solutions | |

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Right derivatives | |

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Differential inequalities | |

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A theorem of Wintner | |

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Uniqueness theorems | |

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van Kampen's uniqueness theorem | |

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Egress points and Lyapunov functions | |

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Successive approximations | |

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Notes | |

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Linear differential equations | |

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Linear systems | |

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Variation of constants | |

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Reductions to smaller systems | |

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Basic inequalities | |

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Constant coefficients | |

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Floquet theory | |

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Adjoint systems | |

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Higher order linear equations | |

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Remarks on changes of variables | |

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Analytic Linear Equations | |

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Fundamental matrices | |

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Simple singularities | |

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Higher order equations | |

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A nonsimple singularity | |

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Notes | |

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Dependence on initial conditions and parameters | |

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Preliminaries | |

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Continuity | |

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Differentiability | |

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Higher order differentiability | |

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Exterior derivatives | |

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Another differentiability theorem | |

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S- and L-Lipschitz continuity | |

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Uniqueness theorem | |

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A lemma | |

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Proof of Theorem 8.1 | |

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Proof of Theorem 6.1 | |

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First integrals | |

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Notes | |

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Total and partial differential equations | |

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A theorem of Frobenius | |

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Total differential equations | |

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Algebra of exterior forms | |

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A theorem of Frobenius | |

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Proof of Theorem 3.1 | |

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Proof of Lemma 3.1 | |

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The system (1.1) | |

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Cauchy's method of characteristics | |

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A nonlinear partial differential equation | |

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Characteristics | |

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Existence and uniqueness theorem | |

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Haar's lemma and uniqueness | |

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Notes | |

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The Poincare-Bendixson theory | |

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Autonomous systems | |

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Umlaufsatz | |

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Index of a stationary point | |

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The Poincare-Bendixson theorem | |

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Stability of periodic solutions | |

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Rotation points | |

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Foci, nodes, and saddle points | |

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Sectors | |

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The general stationary point | |

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A second order equation | |

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Poincare-Bendixson theory on 2-manifolds | |

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Preliminaries | |

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Analogue of the Poincare-Bendixson theorem | |

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Flow on a closed curve | |

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Flow on a torus | |

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Notes | |

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Plane stationary points | |

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Existence theorems | |

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Characteristic directions | |

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Perturbed linear systems | |

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More general stationary point | |

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Notes | |

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Invariant manifolds and linearlizations | |

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Invariant manifolds | |

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The maps T[superscript t] | |

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Modification of F([xi]) | |

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Normalizations | |

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Invariant manifolds of a map | |

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Existence of invariant manifolds | |

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Linearizations | |

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Linearization of a map | |

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Proof of Theorem 7.1 | |

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Periodic solution | |

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Limit cycles | |

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Smooth equivalence maps | |

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Smooth linearizations | |

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Proof of Lemma 12.1 | |

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Proof of Theorem 12.2 | |

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Smoothness of stable manifolds | |

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Notes | |

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Perturbed linear systems | |

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The case E = 0 | |

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A topological principle | |

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A theorem of Wazewski | |

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Preliminary lemmas | |

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Proof of Lemma 4.1 | |

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Proof of Lemma 4.2 | |

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Proof of Lemma 4.3 | |

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Asymptotic integrations. Logarithmic scale | |

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Proof of Theorem 8.2 | |

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Proof of Theorem 8.3 | |

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Logarithmic scale (continued) | |

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Proof of Theorem 11.2 | |

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Asymptotic integration | |

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Proof of Theorem 13.1 | |

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Proof of Theorem 13.2 | |

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Corollaries and refinements | |

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Linear higher order equations | |

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Notes | |

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Linear second order equations | |

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Preliminaries | |

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Basic facts | |

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Theorems of Sturm | |

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Sturm-Liouville boundary value problems | |

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Number of zeros | |

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Nonoscillatory equations and principal solutions | |

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Nonoscillation theorems | |

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Asymptotic integrations. Elliptic cases | |

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Asymptotic integrations. Nonelliptic cases | |

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Disconjugate systems | |

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Disconjugate systems | |

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Generalizations | |

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Notes | |

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Use of implicit function and fixed point theorems | |

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Periodic solutions | |

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Linear equations | |

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Nonlinear problems | |

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Second order boundary value problems | |

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Linear problems | |

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Nonlinear problems | |

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A priori bounds | |

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General theory | |

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Basic facts | |

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Green's functions | |

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Nonlinear equations | |

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Asymptotic integration | |

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Notes | |

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Dichotomies for solutions of linear equations | |

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General theory | |

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Notations and definitions | |

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Preliminary lemmas | |

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The operator T | |

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Slices of [double vertical line]Py(t)[double vertical line] | |

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Estimates for [double vertical line]y(t)[double vertical line] | |

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Applications to first order systems | |

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Applications to higher order systems | |

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P(B, D)-manifolds | |

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Adjoint equations | |

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Associate spaces | |

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The operator T' | |

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Individual dichotomies | |

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P'-admissible spaces for T' | |

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Applications to differential equations | |

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Existence of P D-solutions | |

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Notes | |

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Miscellany on monotony | |

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Monotone solutions | |

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Small and large solutions | |

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Monotone solutions | |

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Second order linear equations | |

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Second order linear equations (continuation) | |

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A problem in boundary layer theory | |

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The problem | |

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The case [lambda] > 0 | |

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The case [lambda] < 0 | |

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The case [lambda] = 0 | |

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Asymptotic behavior | |

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Global asymptotic stability | |

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Global asymptotic stability | |

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Lyapunov functions | |

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Nonconstant G | |

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On Corollary 11.2 | |

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On "J(y)x . x [less than or equal] 0 if x . f(y) = 0" | |

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Proof of Theorem 14.2 | |

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Proof of Theorem 14.1 | |

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Notes | |

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Hints for exercises | |

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References | |

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Index | |