proposition 32.17 Classification of planar fixed points

open in the book · parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:369 · p. 1085

Rests on

depends_on definition 32.12 Linearization ¶
- 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 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 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 7.26 Derivative of a function at a point ¶
      - depends_on proposition 7.6 Algebra of limits ¶
      - 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 ¶
      - 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 theorem 7.7 Bolzano–Weierstrass ¶
      - proves proof ch:05-real-analysis@proof-9 ¶
    - proves proof ch:05-real-analysis@proof-18 ¶
  - proves proof ch:05-real-analysis@proof-22 ¶
depends_on theorem 5.71 The eigenvalues are the roots of the characteristic polynomial ¶
- depends_on definition 5.70 Characteristic polynomial ¶
  - depends_on definition 5.56 Endomorphism ¶
    - depends_on definition 5.37 Linear transformation ¶
      - depends_on definition 4.33 Vector space ¶
  - depends_on equation 5.19 eq:lin-leibniz-det ¶
  - depends_on equation 5.74 eq:lin-matrix-rep ¶
  - proves proof ch:03-linear-algebra-representations@proof-25 ¶
- depends_on definition 5.69 Eigenvector, eigenvalue, eigenspace ¶
  - depends_on definition 5.56 Endomorphism ¶ ↺
  - depends_on definition 5.39 Kernel, image, nullity, rank ¶
    - depends_on definition 5.37 Linear transformation ¶ ↺
    - depends_on definition 5.7 Vector subspace ¶
      - depends_on definition 5.5 Linear combination ¶
    - proves proof ch:03-linear-algebra-representations@proof-12 ¶
- depends_on proposition 5.49 Injective, surjective, invertible ¶
  - depends_on definition 5.39 Kernel, image, nullity, rank ¶ ↺
  - depends_on proposition 5.48 Existence, uniqueness, and linearity of the inverse ¶
    - depends_on definition 5.47 Inverse of a linear transformation ¶
      - depends_on definition 5.37 Linear transformation ¶ ↺
    - depends_on definition 5.37 Linear transformation ¶ ↺
    - proves proof ch:03-linear-algebra-representations@proof-17 ¶
  - depends_on theorem 5.40 Rank–nullity ¶
    - depends_on definition 5.15 Basis ¶
      - depends_on definition 5.12 Subspace generated by a set of vectors ¶
      - depends_on definition 5.14 Linear independence ¶
    - depends_on definition 5.39 Kernel, image, nullity, rank ¶ ↺
    - depends_on lemma 5.38 Exchange and completion ¶
      - depends_on definition 5.15 Basis ¶ ↺
      - depends_on definition 5.12 Subspace generated by a set of vectors ¶ ↺
      - depends_on definition 5.14 Linear independence ¶ ↺
      - proves proof ch:03-linear-algebra-representations@proof-11 ¶
    - proves proof ch:03-linear-algebra-representations@proof-13 ¶
  - proves proof ch:03-linear-algebra-representations@proof-18 ¶
- proves proof ch:03-linear-algebra-representations@proof-26 ¶
proves proof ch:15-nonlinear-dynamics-chaos@proof-5 ¶

Supports

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 ¶

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`	→	Linearization	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:378
`depends_on`	→	The eigenvalues are the roots of the characteristic polynomial	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:378
`depends_on`	←	A linear centre that is really a stable focus	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:437
`depends_on`	←	Hopf bifurcation, quoted	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:602
`proves`	←	ch:15-nonlinear-dynamics-chaos@proof-5	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:381