definition 32.10 Fixed point

open in the book · parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:246 · p. 1083

ground object -- no derivation owed

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 ¶

Supports

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 proposition 32.26 The Hopf normal form ¶
- depends_on theorem 32.15 Hartman–Grobman, restated from Part II ¶
  - 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 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 ¶ ↺

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:251
`depends_on`	←	Linearization	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:283
`depends_on`	←	Lyapunov function	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:454
`depends_on`	←	Lyapunov stability	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:262