definition 9.1 Ordinary differential equation

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Supports

• depends_on definition 32.3 Dynamical system, phase space, flow ¶
  • depends_on definition 32.8 Attractor and basin ¶
    • depends_on definition 32.29 Limit cycle ¶
      • depends_on phenomenon 32.35 Self-sustained oscillation ¶
        • depends_on phenomenon 32.37 Frequency demultiplication and the first heard chaos ¶
    • depends_on definition 32.67 Strange attractor ¶
      • depends_on example 32.70 The Hénon map ¶
      • depends_on phenomenon 32.97 Turbulence sets in after a few transitions ¶
      • depends_on phenomenon 32.68 Strange attractors ¶
    • depends_on theorem 32.74 Oseledets, quoted ¶
      • depends_on phenomenon 32.72 Sensitive dependence on initial conditions ¶
        • depends_on definition 32.67 Strange attractor ¶ ↺
  • depends_on definition 32.22 Bifurcation ¶
    • depends_on proposition 32.23 Saddle-node bifurcation ¶
      • depends_on proposition 32.95 Intermittency: the scaling of the laminar phase ¶
    • depends_on proposition 32.24 Transcritical and pitchfork bifurcations ¶
    • depends_on theorem 32.25 Hopf bifurcation, quoted ¶
      • depends_on proposition 32.26 The Hopf normal form ¶
  • depends_on definition 32.10 Fixed point ¶
    • depends_on definition 32.12 Linearization ¶
      • depends_on proposition 32.17 Classification of planar fixed points ¶
        • depends_on example 32.18 A linear centre that is really a stable focus ¶
        • depends_on theorem 32.25 Hopf bifurcation, 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 32.13 Linear stability ¶
        • depends_on proposition 32.65 Fixed points of the Lorenz system and their stability ¶
        • depends_on proposition 32.23 Saddle-node bifurcation ¶ ↺
        • depends_on proposition 32.24 Transcritical and pitchfork bifurcations ¶ ↺
        • depends_on proposition 28.69 Which branch is stable, and where it ends ¶
    • depends_on definition 32.19 Lyapunov function ¶
      • depends_on theorem 32.20 Lyapunov's direct method ¶
        • depends_on example 32.21 The damped pendulum ¶
    • depends_on definition 32.11 Lyapunov stability ¶
      • depends_on theorem 32.20 Lyapunov's direct method ¶ ↺
  • depends_on definition 32.41 The horseshoe map ¶
    • depends_on theorem 32.42 The horseshoe is a full shift, quoted ¶
      • depends_on example 32.80 The middle-thirds Cantor set ¶
      • depends_on proposition 32.43 What the shift does ¶
  • depends_on definition 32.83 The logistic map ¶
  • depends_on definition 32.38 Poincaré section and return map ¶
    • depends_on definition 32.91 The circle map ¶
      • depends_on proposition 32.93 The circle map loses invertibility at $K=1$ ¶
    • depends_on example 32.70 The Hénon map ¶ ↺
    • depends_on proposition 32.93 The circle map loses invertibility at $K=1$ ¶ ↺
    • depends_on proposition 32.95 Intermittency: the scaling of the laminar phase ¶ ↺
    • depends_on proposition 32.39 Stability of a periodic orbit ¶
      • depends_on phenomenon 32.87 Universal period doubling ¶
      • depends_on proposition 32.84 Fixed points and the first period doubling ¶
        • depends_on phenomenon 32.87 Universal period doubling ¶ ↺
  • depends_on proposition 32.4 The flow is a one-parameter group ¶
    • depends_on corollary 32.5 Trajectories do not cross ¶
      • depends_on definition 32.29 Limit cycle ¶ ↺
      • depends_on definition 32.38 Poincaré section and return map ¶ ↺
      • depends_on theorem 32.30 Poincaré–Bendixson, restated from Part II ¶
        • depends_on corollary 32.32 No chaos in a planar flow ¶
  • depends_on theorem 32.6 Evolution of phase volume ¶
    • depends_on definition 32.7 Conservative and dissipative flows ¶
      • depends_on definition 32.62 The Lorenz system ¶
        • depends_on theorem A.769 The Lorenz system ¶
    • depends_on definition 32.62 The Lorenz system ¶ ↺
    • depends_on proposition 32.64 The Lorenz flow contracts volume ¶
      • depends_on phenomenon 32.68 Strange attractors ¶ ↺
    • depends_on proposition 32.76 The exponents sum to the mean divergence ¶
• depends_on definition 9.86 Outer and inner expansions; matching ¶
• depends_on definition 9.2 Linear equation; homogeneity ¶
  • 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 theorem 9.39 Floquet ¶
      • 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 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 lemma A.567 The differential of the momentum map along the orbit ¶
        • depends_on proposition 13.131 Commuting fields have commuting flows ¶
        • 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 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 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 ¶
      • 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 ¶
  • depends_on proposition 30.57 Bending waves are dispersive ¶
    • depends_on example 30.58 A steel ruler ¶
    • depends_on remark 30.59 Where Euler–Bernoulli fails ¶
  • depends_on proposition 9.15 Abel's identity ¶
    • depends_on proposition 9.38 Liouville's formula ¶ ↺
    • depends_on proposition 9.16 Wronskian test for independence ¶ ↺
  • depends_on theorem 9.13 Dimension of the solution space ¶ ↺
• depends_on theorem 9.34 Poincaré–Bendixson; quoted ¶
  • depends_on theorem 32.30 Poincaré–Bendixson, restated from Part II ¶ ↺