theorem 9.32 Hartman–Grobman; quoted

open in the book · parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1211 · p. 286

Rests on

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 7.16 Limit ¶
      - depends_on definition 7.2 Absolute value ¶
      - depends_on definition 7.9 Real function ¶
    - depends_on equation 7.7 eq:ana-limit-left ¶
    - depends_on equation 7.5 eq:ana-limit-right ¶
  - depends_on definition 7.97 Partial derivative; gradient ¶
    - depends_on definition 7.26 Derivative of a function at a point ¶
      - depends_on definition 7.16 Limit ¶ ↺
      - depends_on definition 7.9 Real function ¶ ↺
    - 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 5.69 Eigenvector, eigenvalue, eigenspace ¶
  - depends_on definition 5.56 Endomorphism ¶
    - depends_on definition 5.37 Linear transformation ¶
      - depends_on definition 4.33 Vector space ¶
  - 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 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 9.23 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 ¶
      - depends_on definition 7.20 Continuity at a point ¶ ↺
      - depends_on proposition 7.47 Comparison; absolute convergence ¶
      - proves proof ch:07-odes-sturm-liouville@proof-1 ¶
    - 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 definition 4.33 Vector space ¶ ↺
    - depends_on proposition 5.125 prop:lin-matrix-algebra ¶
      - depends_on definition 5.121 Algebra ¶
      - depends_on definition 5.37 Linear transformation ¶ ↺
      - depends_on equation 5.150 eq:lin-ctesesc2 ¶
      - depends_on equation 5.74 eq:lin-matrix-rep ¶
      - depends_on proposition 5.45 prop:lin-matrix-unique ¶
      - proves proof ch:03-linear-algebra-representations@proof-55 ¶
  - 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 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 7.48 Ratio test ¶
      - depends_on proposition 7.47 Comparison; absolute convergence ¶ ↺
      - depends_on proposition 7.46 Geometric series ¶
      - depends_on theorem 7.8 Cauchy criterion ¶
      - proves proof ch:05-real-analysis@proof-29 ¶
    - depends_on proposition 7.6 Algebra of limits ¶ ↺
    - depends_on theorem 7.50 Power series; radius of convergence ¶
      - depends_on axiom 7.1 Completeness of $\R$ ¶
      - depends_on proposition 7.47 Comparison; absolute convergence ¶ ↺
      - depends_on proposition 7.46 Geometric series ¶ ↺
      - proves proof ch:05-real-analysis@proof-31 ¶
    - 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 theorem 32.15 Hartman–Grobman, restated from Part II ¶
- 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.

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partly declared
a check failed
not graded
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inferred from structure

Edges

type	direction	node	provenance	where
`cites`	→	A lemma in the theory of structural stability of differential equations	derived	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1219
`depends_on`	→	Hyperbolic equilibrium	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1223
`depends_on`	→	Homeomorphism	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1223
`depends_on`	→	Solution of a constant-coefficient system	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1223
`depends_on`	←	Hartman–Grobman, restated from Part II	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:346