definition 32.12 Linearization

open in the book · parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:272 · p. 1084

ground object -- no derivation owed

Rests on

depends_on definition 32.10 Fixed point ¶
- 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 13.45 Differentiable map on a topological space ¶
      - depends_on definition 13.48 Differentiable manifold ¶
  - depends_on definition 9.1 Ordinary differential equation ¶
depends_on theorem 7.38 Taylor's theorem with Lagrange remainder ¶
- depends_on proposition 7.30 Leibniz rule ¶
  - depends_on definition 7.12 Product of functions ¶
    - depends_on definition 7.9 Real function ¶
      - depends_on definition 3.43 Map ¶
  - depends_on equation 7.14 eq:ana-derivh ¶
  - depends_on proposition 7.27 Differentiable implies continuous ¶
    - depends_on definition 7.20 Continuity at a point ¶
      - depends_on definition 7.16 Limit ¶
      - depends_on equation 7.7 eq:ana-limit-left ¶
      - depends_on equation 7.5 eq:ana-limit-right ¶
    - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on definition 7.16 Limit ¶ ↺
      - depends_on definition 7.9 Real function ¶ ↺
    - depends_on proposition 7.6 Algebra of limits ¶
      - depends_on definition 7.4 Convergence ¶
      - depends_on proposition 7.3 Triangle inequality ¶
      - proves proof ch:05-real-analysis@proof-3 ¶
    - proves proof ch:05-real-analysis@proof-11 ¶
  - depends_on proposition 7.6 Algebra of limits ¶ ↺
  - proves proof ch:05-real-analysis@proof-14 ¶
- depends_on theorem 7.34 Rolle ¶
  - depends_on lemma 7.33 Fermat: interior extremum ¶
    - depends_on definition 7.26 Derivative of a function at a point ¶ ↺
    - depends_on proposition 7.18 Two-sided limit from one-sided limits ¶
      - depends_on definition 7.16 Limit ¶ ↺
      - depends_on equation 7.7 eq:ana-limit-left ¶ ↺
      - depends_on equation 7.5 eq:ana-limit-right ¶ ↺
      - proves proof ch:05-real-analysis@proof-6 ¶
    - proves proof ch:05-real-analysis@proof-17 ¶
  - depends_on theorem 7.24 Extreme value theorem ¶
    - depends_on axiom 7.1 Completeness of $\R$ ¶
    - depends_on proposition 7.22 Sequential characterization ¶
      - depends_on definition 7.20 Continuity at a point ¶ ↺
      - depends_on definition 7.4 Convergence ¶ ↺
      - proves proof ch:05-real-analysis@proof-7 ¶
    - depends_on theorem 7.7 Bolzano–Weierstrass ¶
      - depends_on corollary A.45 Archimedean property and density of $\Q$ ¶
      - depends_on corollary A.46 Monotone convergence ¶
      - proves proof ch:05-real-analysis@proof-4 ¶
    - proves proof ch:05-real-analysis@proof-9 ¶
  - proves proof ch:05-real-analysis@proof-18 ¶
- proves proof ch:05-real-analysis@proof-22 ¶

Supports

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 ¶

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`	→	Fixed point	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:283
`depends_on`	→	Taylor's theorem with Lagrange remainder	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:283
`depends_on`	←	Classification of planar fixed points	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:378
`depends_on`	←	Hartman–Grobman, restated from Part II	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:346
`depends_on`	←	Linear stability	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:291