theorem 32.15 Hartman–Grobman, restated from Part II

open in the book · parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:334 · p. 1084

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 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 3.31 Empty set ¶
      - depends_on definition 3.35 Union, intersection, difference ¶
      - depends_on definition 3.30 Subset ¶
      - depends_on equation 3.51 eq:set-indexed ¶
  - 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 7.26 Derivative of a function at a point ¶ ↺
      - depends_on definition 5.15 Basis ¶
  - depends_on definition 5.69 Eigenvector, eigenvalue, eigenspace ¶
    - depends_on definition 5.56 Endomorphism ¶
      - depends_on definition 5.37 Linear transformation ¶
    - 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 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 definition 9.1 Ordinary differential equation ¶ ↺
    - depends_on remark 9.3 Normal form and first-order systems ¶
    - depends_on theorem 9.8 Picard–Lindelöf ¶
      - depends_on definition 9.4 Lipschitz condition ¶
      - depends_on equation 9.3 eq:slt-ode-system ¶
      - depends_on lemma 9.6 Weierstrass $M$-test; uniform limits are continuous ¶
      - proves proof ch:07-odes-sturm-liouville@proof-2 ¶
    - 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 definition 5.19 Norm ¶
      - depends_on proposition 5.125 prop:lin-matrix-algebra ¶
    - depends_on lemma 9.6 Weierstrass $M$-test; uniform limits are continuous ¶ ↺
    - depends_on theorem 7.51 Termwise differentiation ¶
      - depends_on proposition 7.47 Comparison; absolute convergence ¶
      - depends_on proposition 7.27 Differentiable implies continuous ¶ ↺
      - depends_on proposition 7.48 Ratio test ¶
      - depends_on proposition 7.6 Algebra of limits ¶ ↺
      - depends_on theorem 7.50 Power series; radius of convergence ¶
      - proves proof ch:05-real-analysis@proof-32 ¶
    - proves proof ch:07-odes-sturm-liouville@proof-11 ¶
  - proves proof ch:07-odes-sturm-liouville@proof-12 ¶

Supports

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 proposition 32.26 The Hopf normal form ¶
depends_on example 32.18 A linear centre that is really a stable focus ¶

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`	→	A lemma in the theory of structural stability of differential equations	derived	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:342
`depends_on`	→	Linearization	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:346
`depends_on`	→	Homeomorphism	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:346
`depends_on`	→	Hartman–Grobman; quoted	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:346
`depends_on`	←	Bifurcation	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:521
`depends_on`	←	A linear centre that is really a stable focus	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:437