proposition 32.39 Stability of a periodic orbit

open in the book · parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:1007 · p. 1092

Rests on

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 definition 13.82 Vector field ¶
      - depends_on definition 9.1 Ordinary differential equation ¶
    - depends_on theorem 9.8 Picard–Lindelöf ¶
      - depends_on definition 9.4 Lipschitz condition ¶
      - depends_on equation 9.3 eq:slt-ode-system ¶
      - depends_on lemma 9.6 Weierstrass $M$-test; uniform limits are continuous ¶
      - proves proof ch:07-odes-sturm-liouville@proof-2 ¶
    - 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 definition 4.33 Vector space ¶
  - 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 ¶
      - depends_on definition 5.5 Linear combination ¶
    - 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 ¶

Supports

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 ¶ ↺

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	Poincaré section and return map	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:1015
`depends_on`	→	Criterion for diagonalizability	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:1015
`depends_on`	←	Universal period doubling	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:2605
`depends_on`	←	Fixed points and the first period doubling	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:2490
`proves`	←	ch:15-nonlinear-dynamics-chaos@proof-14	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:1018