remark 9.3 Normal form and first-order systems

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• depends_on corollary 9.9 Linear equations: existence on the whole interval ¶
  • depends_on definition 9.37 Fundamental matrix ¶
    • depends_on proposition 9.38 Liouville's formula ¶
      • depends_on corollary 9.41 Hill's equation ¶
        • depends_on proposition 9.43 The unmodulated line, and the feet of the tongues ¶
      • depends_on theorem 9.39 Floquet ¶
        • depends_on corollary 9.41 Hill's equation ¶ ↺
    • depends_on theorem 9.39 Floquet ¶ ↺
  • depends_on lemma A.439 Existence, uniqueness, and the structure of the zeros ¶
    • depends_on lemma A.442 The Wronskian of the accessory equation is constant ¶
      • depends_on definition A.467 Green function of the shifted problem ¶
        • depends_on lemma A.468 The kernel is well defined, symmetric and Lipschitz ¶
        • depends_on theorem A.469 The Green operator inverts $L_{K}$ ¶
      • depends_on theorem A.443 Sturm separation theorem ¶
    • depends_on theorem A.441 Continuous dependence on the data and on the initial point ¶
      • depends_on corollary A.445 Displacing the initial point ¶
  • depends_on proposition 20.9 Linear resistance: exact motion, and the lost range ¶
    • depends_on remark 20.11 What has no closed form, and what to do about it ¶
  • depends_on proposition 9.103 The positive zeros and their spacing ¶
  • depends_on proposition 9.18 The linear equation: integrating factor ¶
    • depends_on proposition 9.87 The distinguished limit, and which end carries the layer ¶
      • depends_on proposition 9.88 The model problem, and the error of the composite ¶
  • depends_on theorem A.74 Flow of a time-dependent vector field ¶
    • depends_on theorem 13.125 Existence, uniqueness and smoothness of the flow ¶
      • depends_on lemma A.548 The joint flow is a translation action on the level set ¶
        • depends_on corollary A.553 Quasi-periodic motion ¶
        • depends_on lemma A.549 The action is transitive ¶
        • depends_on proposition A.552 The component is a torus ¶
      • depends_on lemma A.567 The differential of the momentum map along the orbit ¶
        • depends_on theorem A.568 The radical is the isotropy orbit ¶
      • depends_on proposition 13.131 Commuting fields have commuting flows ¶
        • depends_on lemma A.548 The joint flow is a translation action on the level set ¶ ↺
        • depends_on proposition 13.132 Simultaneous straightening of commuting fields ¶
        • depends_on theorem 13.137 Commuting complete fields on a compact manifold ¶
      • depends_on proposition 13.132 Simultaneous straightening of commuting fields ¶ ↺
      • depends_on theorem 13.137 Commuting complete fields on a compact manifold ¶ ↺
  • depends_on theorem 9.23 Solution of a constant-coefficient system ¶
    • depends_on corollary 9.25 Normal modes of a diagonalizable system ¶
      • depends_on corollary 9.27 Linear stability ¶
        • depends_on corollary 9.41 Hill's equation ¶ ↺
        • depends_on proposition 9.31 Lyapunov's first method: asymptotic stability ¶
        • depends_on proposition 9.29 The planar classification ¶
        • depends_on theorem 9.39 Floquet ¶ ↺
      • depends_on proposition 9.29 The planar classification ¶ ↺
    • depends_on proposition 9.87 The distinguished limit, and which end carries the layer ¶ ↺
    • depends_on proposition 9.31 Lyapunov's first method: asymptotic stability ¶ ↺
    • depends_on proposition 9.43 The unmodulated line, and the feet of the tongues ¶ ↺
    • depends_on theorem 9.32 Hartman–Grobman; quoted ¶
      • depends_on theorem 32.15 Hartman–Grobman, restated from Part II ¶
        • depends_on definition 32.22 Bifurcation ¶
        • depends_on example 32.18 A linear centre that is really a stable focus ¶
    • depends_on theorem 9.26 Structure of the solutions ¶
      • depends_on corollary 9.27 Linear stability ¶ ↺
  • depends_on theorem 9.13 Dimension of the solution space ¶
    • depends_on definition 9.37 Fundamental matrix ¶ ↺
    • depends_on proposition 28.18 The three regimes ¶
    • depends_on proposition 28.4 General solution and isochronism ¶
      • depends_on phenomenon 28.9 Isochronism of the small-amplitude pendulum ¶
        • assumes experiment Kater: the reversible pendulum ¶
        • tests experiment Galileo and Huygens: isochronism ¶
    • depends_on proposition 9.106 The modified series and what it does not do ¶
    • depends_on proposition 9.28 The scalar constant-coefficient equation ¶
      • depends_on proposition 9.88 The model problem, and the error of the composite ¶ ↺
    • depends_on proposition 9.16 Wronskian test for independence ¶