example 32.70 The Hénon map

open in the book · parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:2112 · p. 1105

example -- no derivation owed by its kind

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 definition 32.67 Strange attractor ¶
- depends_on definition 32.8 Attractor and basin ¶
  - depends_on definition 32.3 Dynamical system, phase space, flow ¶ ↺
  - depends_on definition 6.9 Compact set ¶
    - depends_on definition 6.5 Open cover ¶
      - depends_on definition 6.2 Open set ¶
      - depends_on equation 3.51 eq:set-indexed ¶
- depends_on phenomenon 32.72 Sensitive dependence on initial conditions ¶
  - depends_on equation 32.4 eq:chaos-linearization ¶
  - depends_on theorem 32.74 Oseledets, quoted ¶
    - depends_on definition 32.8 Attractor and basin ¶ ↺
    - depends_on definition 32.73 Lyapunov spectrum ¶
      - depends_on equation 32.45 eq:chaos-variational ¶
  - proves proof ch:15-nonlinear-dynamics-chaos@proof-25 ¶

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type	direction	node	provenance	where
`cites`	→	Characterization of Strange Attractors	derived	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:2128
`cites`	→	A two-dimensional mapping with a strange attractor	derived	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:2118
`depends_on`	→	Poincaré section and return map	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:2133
`depends_on`	→	Strange attractor	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:2133