proposition 32.4 The flow is a one-parameter group

open in the book · parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:122 · p. 1082

Rests on

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 13.71 Differentiable curve ¶
      - depends_on definition 13.48 Differentiable manifold ¶
    - depends_on definition 13.45 Differentiable map on a topological space ¶
      - depends_on definition 13.44 Atlas ¶
      - depends_on definition 13.43 Coordinate system ¶
    - depends_on definition 13.48 Differentiable manifold ¶ ↺
- 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 ¶
  - depends_on definition 7.20 Continuity at a point ¶
    - depends_on definition 7.16 Limit ¶
      - depends_on definition 7.2 Absolute value ¶
      - depends_on definition 7.9 Real function ¶
    - depends_on equation 7.7 eq:ana-limit-left ¶
    - depends_on equation 7.5 eq:ana-limit-right ¶
  - depends_on proposition 7.47 Comparison; absolute convergence ¶
    - depends_on corollary A.46 Monotone convergence ¶
      - depends_on theorem A.40 Least-upper-bound property ¶
      - proves proof app:A-long-proofs@proof-33 ¶
    - depends_on definition 7.45 Series ¶
      - depends_on definition 7.2 Absolute value ¶ ↺
      - depends_on definition 7.4 Convergence ¶
    - depends_on theorem 7.8 Cauchy criterion ¶
      - depends_on corollary A.47 Cauchy completeness ¶
      - depends_on definition 7.4 Convergence ¶ ↺
      - proves proof ch:05-real-analysis@proof-5 ¶
    - proves proof ch:05-real-analysis@proof-28 ¶
  - proves proof ch:07-odes-sturm-liouville@proof-1 ¶
- proves proof ch:07-odes-sturm-liouville@proof-2 ¶
proves proof ch:15-nonlinear-dynamics-chaos@proof-1 ¶

Supports

depends_on corollary 32.5 Trajectories do not cross ¶
- depends_on definition 32.29 Limit cycle ¶
  - depends_on phenomenon 32.35 Self-sustained oscillation ¶
    - depends_on phenomenon 32.37 Frequency demultiplication and the first heard chaos ¶
      - depends_on definition 32.91 The circle map ¶
- depends_on definition 32.38 Poincaré section and return map ¶
  - depends_on definition 32.91 The circle map ¶ ↺
  - depends_on example 32.70 The Hénon map ¶
  - depends_on proposition 32.93 The circle map loses invertibility at $K=1$ ¶
  - depends_on proposition 32.95 Intermittency: the scaling of the laminar phase ¶
  - depends_on proposition 32.39 Stability of a periodic orbit ¶
    - 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 ¶ ↺
- depends_on theorem 32.30 Poincaré–Bendixson, restated from Part II ¶
  - depends_on corollary 32.32 No chaos in a planar flow ¶

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`	→	Dynamical system, phase space, flow	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:131
`depends_on`	→	Picard–Lindelöf	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:131
`depends_on`	←	Trajectories do not cross	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:154
`proves`	←	ch:15-nonlinear-dynamics-chaos@proof-1	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:134