phenomenon 32.87 Universal period doubling

Rests on

• depends_on proposition 32.84 Fixed points and the first period doubling ¶
  • depends_on equation 32.50 eq:chaos-logistic ¶
  • depends_on proposition 32.39 Stability of a periodic orbit ¶
    • 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 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 definition 13.82 Vector field ¶
        • depends_on definition 9.1 Ordinary differential equation ¶
    • depends_on theorem 5.77 Criterion for diagonalizability ¶
      • depends_on definition 5.76 Diagonalizable operator ¶
        • depends_on definition 5.56 Endomorphism ¶
        • depends_on equation 5.74 eq:lin-matrix-rep ¶
      • depends_on definition 5.69 Eigenvector, eigenvalue, eigenspace ¶
        • depends_on definition 5.56 Endomorphism ¶ ↺
        • depends_on definition 5.39 Kernel, image, nullity, rank ¶
      • depends_on proposition 5.75 Eigenvectors for distinct eigenvalues are independent ¶
        • depends_on definition 5.69 Eigenvector, eigenvalue, eigenspace ¶ ↺
        • depends_on definition 5.14 Linear independence ¶
        • proves proof ch:03-linear-algebra-representations@proof-29 ¶
      • proves proof ch:03-linear-algebra-representations@proof-30 ¶
    • proves proof ch:15-nonlinear-dynamics-chaos@proof-14 ¶
  • proves proof ch:15-nonlinear-dynamics-chaos@proof-28 ¶
• depends_on proposition 32.39 Stability of a periodic orbit ¶ ↺
• proves proof ch:15-nonlinear-dynamics-chaos@proof-30 ¶

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