proposition 32.84 Fixed points and the first period doubling

open in the book · parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:2480 · p. 1109

Rests on

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 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 theorem 5.77 Criterion for diagonalizability ¶
  - depends_on definition 5.76 Diagonalizable operator ¶
    - depends_on definition 5.56 Endomorphism ¶
      - depends_on definition 5.37 Linear transformation ¶
    - 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 definition 5.37 Linear transformation ¶ ↺
      - depends_on definition 5.7 Vector subspace ¶
      - proves proof ch:03-linear-algebra-representations@proof-12 ¶
  - 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 ¶
      - depends_on definition 5.5 Linear combination ¶
    - 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 ¶

Supports

depends_on phenomenon 32.87 Universal period doubling ¶

Neighborhood

Every logical edge within two steps of this node.

declared and complete
partly declared
a check failed
not graded
declared in the source
inferred from structure

Edges

type	direction	node	provenance	where
`depends_on`	→	eq:chaos-logistic	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:2490
`depends_on`	→	Stability of a periodic orbit	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:2490
`depends_on`	←	Universal period doubling	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:2605
`proves`	←	ch:15-nonlinear-dynamics-chaos@proof-28	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:2493