theorem 32.25 Hopf bifurcation, quoted

open in the book · parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:593 · p. 1087

Rests on

depends_on definition 32.22 Bifurcation ¶
- 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 32.15 Hartman–Grobman, restated from Part II ¶
  - 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 theorem 7.38 Taylor's theorem with Lagrange remainder ¶
      - depends_on proposition 7.30 Leibniz rule ¶
      - depends_on theorem 7.34 Rolle ¶
      - proves proof ch:05-real-analysis@proof-22 ¶
  - depends_on definition 6.7 Homeomorphism ¶
    - depends_on definition 3.47 Bijective map ¶
    - depends_on definition 3.51 Inverse map ¶
      - depends_on definition 3.47 Bijective map ¶ ↺
      - depends_on definition 3.49 Composition map ¶
      - depends_on definition 3.50 Identity map ¶
    - depends_on definition 6.6 Continuous map ¶
      - depends_on definition 3.53 Preimage ¶
      - depends_on definition 6.2 Open set ¶
      - depends_on definition 6.1 Topological space ¶
  - depends_on theorem 9.32 Hartman–Grobman; quoted ¶
    - depends_on definition 9.30 Hyperbolic equilibrium ¶
      - depends_on definition 7.98 Functions of class $C^{1}$ ¶
      - depends_on definition 5.69 Eigenvector, eigenvalue, eigenspace ¶
    - depends_on definition 6.7 Homeomorphism ¶ ↺
    - depends_on theorem 9.23 Solution of a constant-coefficient system ¶
      - depends_on corollary 9.9 Linear equations: existence on the whole interval ¶
      - depends_on proposition 9.22 The exponential and its derivative ¶
      - proves proof ch:07-odes-sturm-liouville@proof-12 ¶
depends_on proposition 32.17 Classification of planar fixed points ¶
- depends_on definition 32.12 Linearization ¶ ↺
- 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 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 proposition 5.49 Injective, surjective, invertible ¶
    - depends_on definition 5.39 Kernel, image, nullity, rank ¶
      - depends_on definition 5.37 Linear transformation ¶ ↺
      - depends_on definition 5.7 Vector subspace ¶
      - proves proof ch:03-linear-algebra-representations@proof-12 ¶
    - 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 ¶ ↺
      - 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 5.39 Kernel, image, nullity, rank ¶ ↺
      - depends_on lemma 5.38 Exchange and completion ¶
      - 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 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
`cites`	→	Abzweigung einer periodischen Lösung von einer stationären Lösung eines Differentialsystems	derived	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:601
`depends_on`	→	Bifurcation	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:602
`depends_on`	→	Classification of planar fixed points	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:602
`depends_on`	←	The Hopf normal form	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:615