definition 32.3 Dynamical system, phase space, flow

open in the book · parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:104 · p. 1082

ground object -- no derivation owed

Rests on

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 13.47 Differentiable structure ¶
  - 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 13.43 Coordinate system ¶ ↺
  - depends_on definition 13.48 Differentiable manifold ¶ ↺
depends_on definition 9.1 Ordinary differential equation ¶

Supports

depends_on definition 32.8 Attractor and basin ¶
- 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 32.67 Strange attractor ¶
  - depends_on example 32.70 The Hénon map ¶
  - depends_on phenomenon 32.97 Turbulence sets in after a few transitions ¶
  - depends_on phenomenon 32.68 Strange attractors ¶
- depends_on theorem 32.74 Oseledets, quoted ¶
  - depends_on phenomenon 32.72 Sensitive dependence on initial conditions ¶
    - depends_on definition 32.67 Strange attractor ¶ ↺
depends_on definition 32.22 Bifurcation ¶
- depends_on proposition 32.23 Saddle-node bifurcation ¶
  - depends_on proposition 32.95 Intermittency: the scaling of the laminar phase ¶
- depends_on proposition 32.24 Transcritical and pitchfork bifurcations ¶
- depends_on theorem 32.25 Hopf bifurcation, quoted ¶
  - depends_on proposition 32.26 The Hopf normal form ¶
depends_on definition 32.10 Fixed point ¶
- depends_on definition 32.12 Linearization ¶
  - depends_on proposition 32.17 Classification of planar fixed points ¶
    - depends_on example 32.18 A linear centre that is really a stable focus ¶
    - depends_on theorem 32.25 Hopf bifurcation, quoted ¶ ↺
  - depends_on theorem 32.15 Hartman–Grobman, restated from Part II ¶
    - depends_on definition 32.22 Bifurcation ¶ ↺
    - depends_on example 32.18 A linear centre that is really a stable focus ¶ ↺
  - depends_on theorem 32.13 Linear stability ¶
    - depends_on proposition 32.65 Fixed points of the Lorenz system and their stability ¶
    - depends_on proposition 32.23 Saddle-node bifurcation ¶ ↺
    - depends_on proposition 32.24 Transcritical and pitchfork bifurcations ¶ ↺
    - depends_on proposition 28.69 Which branch is stable, and where it ends ¶
- depends_on definition 32.19 Lyapunov function ¶
  - depends_on theorem 32.20 Lyapunov's direct method ¶
    - depends_on example 32.21 The damped pendulum ¶
- depends_on definition 32.11 Lyapunov stability ¶
  - depends_on theorem 32.20 Lyapunov's direct method ¶ ↺
depends_on definition 32.41 The horseshoe map ¶
- depends_on theorem 32.42 The horseshoe is a full shift, quoted ¶
  - depends_on example 32.80 The middle-thirds Cantor set ¶
  - depends_on proposition 32.43 What the shift does ¶
depends_on definition 32.83 The logistic 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 proposition 32.4 The flow is a one-parameter group ¶
- depends_on corollary 32.5 Trajectories do not cross ¶
  - depends_on definition 32.29 Limit cycle ¶ ↺
  - depends_on definition 32.38 Poincaré section and return map ¶ ↺
  - depends_on theorem 32.30 Poincaré–Bendixson, restated from Part II ¶
    - depends_on corollary 32.32 No chaos in a planar flow ¶
depends_on theorem 32.6 Evolution of phase volume ¶
- depends_on definition 32.7 Conservative and dissipative flows ¶
  - depends_on definition 32.62 The Lorenz system ¶
    - depends_on theorem A.769 The Lorenz system ¶
      - depends_on remark A.771 What was assumed, and where it fails ¶
      - depends_on remark A.770 What the three variables and three parameters are ¶
- depends_on definition 32.62 The Lorenz system ¶ ↺
- depends_on proposition 32.64 The Lorenz flow contracts volume ¶
  - depends_on phenomenon 32.68 Strange attractors ¶ ↺
- depends_on proposition 32.76 The exponents sum to the mean divergence ¶

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`	→	Vector field	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:119
`depends_on`	→	Ordinary differential equation	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:119
`depends_on`	←	Attractor and basin	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:219
`depends_on`	←	Bifurcation	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:521
`depends_on`	←	Fixed point	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:251
`depends_on`	←	The horseshoe map	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:1086
`depends_on`	←	The logistic map	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:2477
`depends_on`	←	Poincaré section and return map	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:998
`depends_on`	←	The flow is a one-parameter group	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:131
`depends_on`	←	Evolution of phase volume	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:181