definition 32.91 The circle map

open in the book · parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:2793 · p. 1113

ground object -- no derivation owed

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 phenomenon 32.37 Frequency demultiplication and the first heard chaos ¶
- depends_on phenomenon 32.35 Self-sustained oscillation ¶
  - depends_on definition 32.29 Limit cycle ¶
    - depends_on corollary 32.5 Trajectories do not cross ¶ ↺
    - 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 equation 32.10 eq:chaos-vdp-si ¶
  - proves proof ch:15-nonlinear-dynamics-chaos@proof-12 ¶
- proves proof ch:15-nonlinear-dynamics-chaos@proof-13 ¶

Supports

depends_on proposition 32.93 The circle map loses invertibility at $K=1$ ¶

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`	→	Poincaré section and return map	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:2806
`depends_on`	→	Frequency demultiplication and the first heard chaos	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:2806
`depends_on`	←	The circle map loses invertibility at $K=1$	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:2845