proposition 32.23 Saddle-node bifurcation

open in the book · parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:529 · 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 theorem 32.13 Linear stability ¶
- depends_on definition 32.12 Linearization ¶ ↺
- depends_on theorem 5.77 Criterion for diagonalizability ¶
  - depends_on definition 5.76 Diagonalizable operator ¶
    - depends_on definition 5.56 Endomorphism ¶
      - depends_on definition 5.37 Linear transformation ¶
    - depends_on equation 5.74 eq:lin-matrix-rep ¶
  - depends_on definition 5.69 Eigenvector, eigenvalue, eigenspace ¶ ↺
  - depends_on proposition 5.75 Eigenvectors for distinct eigenvalues are independent ¶
    - depends_on definition 5.69 Eigenvector, eigenvalue, eigenspace ¶ ↺
    - depends_on definition 5.14 Linear independence ¶
      - depends_on definition 5.5 Linear combination ¶
    - proves proof ch:03-linear-algebra-representations@proof-29 ¶
  - proves proof ch:03-linear-algebra-representations@proof-30 ¶
- proves proof ch:15-nonlinear-dynamics-chaos@proof-4 ¶
proves proof ch:15-nonlinear-dynamics-chaos@proof-7 ¶

Supports

depends_on proposition 32.95 Intermittency: the scaling of the laminar phase ¶

Neighborhood

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Edges

type	direction	node	provenance	where
`depends_on`	→	Bifurcation	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:537
`depends_on`	→	Linear stability	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:537
`depends_on`	←	Intermittency: the scaling of the laminar phase	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:2935
`proves`	←	ch:15-nonlinear-dynamics-chaos@proof-7	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:540