phenomenon 32.68 Strange attractors

Rests on

• depends_on definition 32.67 Strange attractor ¶
  • depends_on definition 32.8 Attractor and basin ¶
    • depends_on definition 32.3 Dynamical system, phase space, flow ¶
      • depends_on definition 13.82 Vector field ¶
        • depends_on definition 13.81 Vector on a manifold ¶
      • depends_on definition 9.1 Ordinary differential equation ¶
    • depends_on definition 6.9 Compact set ¶
      • depends_on definition 6.5 Open cover ¶
        • depends_on definition 6.2 Open set ¶
        • depends_on equation 3.51 eq:set-indexed ¶
  • depends_on phenomenon 32.72 Sensitive dependence on initial conditions ¶
    • depends_on equation 32.4 eq:chaos-linearization ¶
    • depends_on theorem 32.74 Oseledets, quoted ¶
      • depends_on definition 32.8 Attractor and basin ¶ ↺
      • depends_on definition 32.73 Lyapunov spectrum ¶
        • depends_on equation 32.45 eq:chaos-variational ¶
    • proves proof ch:15-nonlinear-dynamics-chaos@proof-25 ¶
• depends_on proposition 32.64 The Lorenz flow contracts volume ¶
  • depends_on equation 32.36 eq:chaos-lorenz ¶
  • depends_on theorem 32.6 Evolution of phase volume ¶
    • depends_on definition 32.3 Dynamical system, phase space, flow ¶ ↺
    • depends_on theorem 7.133 Gauss ¶
      • depends_on definition 7.127 Simple regions ¶
        • depends_on definition 7.98 Functions of class $C^{1}$ ¶
        • depends_on definition 6.9 Compact set ¶ ↺
      • depends_on remark 7.128 What the derivations below take as given ¶
        • depends_on definition 7.125 Multiple integral ¶
        • depends_on theorem 7.40 Continuous functions are integrable ¶
        • depends_on theorem 7.25 Heine–Cantor: uniform continuity ¶
      • depends_on theorem 7.43 Fundamental theorem of calculus, II ¶
        • depends_on definition 7.41 Antiderivative ¶
        • depends_on theorem 7.42 Fundamental theorem of calculus, I ¶
        • proves proof ch:05-real-analysis@proof-25 ¶
      • proves proof ch:05-real-analysis@proof-79 ¶
    • proves proof ch:15-nonlinear-dynamics-chaos@proof-3 ¶
  • proves proof ch:15-nonlinear-dynamics-chaos@proof-22 ¶
• proves proof ch:15-nonlinear-dynamics-chaos@proof-24 ¶

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