proposition 32.93 The circle map loses invertibility at $K=1$

Rests on

• depends_on definition 32.91 The circle map ¶
  • depends_on definition 32.38 Poincaré section and return map ¶
    • depends_on corollary 32.5 Trajectories do not cross ¶
      • depends_on proposition 32.4 The flow is a one-parameter group ¶
        • depends_on definition 32.3 Dynamical system, phase space, flow ¶
        • depends_on theorem 9.8 Picard–Lindelöf ¶
        • proves proof ch:15-nonlinear-dynamics-chaos@proof-1 ¶
      • depends_on theorem 9.8 Picard–Lindelöf ¶ ↺
      • proves proof ch:15-nonlinear-dynamics-chaos@proof-2 ¶
    • depends_on definition 32.3 Dynamical system, phase space, flow ¶ ↺
  • depends_on phenomenon 32.37 Frequency demultiplication and the first heard chaos ¶
    • depends_on phenomenon 32.35 Self-sustained oscillation ¶
      • depends_on definition 32.29 Limit cycle ¶
        • depends_on corollary 32.5 Trajectories do not cross ¶ ↺
        • depends_on definition 32.8 Attractor and basin ¶
      • depends_on equation 32.10 eq:chaos-vdp-si ¶
      • proves proof ch:15-nonlinear-dynamics-chaos@proof-12 ¶
    • proves proof ch:15-nonlinear-dynamics-chaos@proof-13 ¶
• depends_on definition 32.38 Poincaré section and return map ¶ ↺
• proves proof ch:15-nonlinear-dynamics-chaos@proof-31 ¶

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