definition 32.22 Bifurcation

open in the book · parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:514 · p. 1086

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 ¶
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 definition 7.12 Product of functions ¶
      - depends_on equation 7.14 eq:ana-derivh ¶
      - depends_on proposition 7.27 Differentiable implies continuous ¶
      - 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 theorem 7.24 Extreme value theorem ¶
      - proves proof ch:05-real-analysis@proof-18 ¶
    - 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 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 7.20 Continuity at a point ¶
      - depends_on definition 7.97 Partial derivative; gradient ¶
    - 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 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 definition 9.2 Linear equation; homogeneity ¶
      - depends_on remark 9.3 Normal form and first-order systems ¶
      - depends_on theorem 9.8 Picard–Lindelöf ¶
      - proves proof ch:07-odes-sturm-liouville@proof-3 ¶
    - depends_on proposition 9.22 The exponential and its derivative ¶
      - depends_on definition 9.21 Matrix exponential ¶
      - depends_on lemma 9.6 Weierstrass $M$-test; uniform limits are continuous ¶
      - depends_on theorem 7.51 Termwise differentiation ¶
      - proves proof ch:07-odes-sturm-liouville@proof-11 ¶
    - proves proof ch:07-odes-sturm-liouville@proof-12 ¶

Supports

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 ¶

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:521
`depends_on`	→	Hartman–Grobman, restated from Part II	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:521
`depends_on`	←	Saddle-node bifurcation	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:537
`depends_on`	←	Transcritical and pitchfork bifurcations	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:564
`depends_on`	←	Hopf bifurcation, quoted	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:602