definition 9.30 Hyperbolic equilibrium

open in the book · parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1149 · p. 285

ground object -- no derivation owed

Rests on

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 definition 3.43 Map ¶
  - 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 5.5 Linear combination ¶
      - depends_on definition 5.7 Vector subspace ¶
      - proves proof ch:03-linear-algebra-representations@proof-3 ¶
    - depends_on definition 5.14 Linear independence ¶
      - depends_on definition 5.5 Linear combination ¶ ↺
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 4.8 Commutativity; abelian structure ¶
      - depends_on definition 4.32 Field ¶
      - depends_on definition 4.31 Module ¶
- 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 ¶

Supports

depends_on theorem 9.32 Hartman–Grobman; quoted ¶
- 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.

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`	→	Functions of class $C^{1}$	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1154
`depends_on`	→	Eigenvector, eigenvalue, eigenspace	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1154
`depends_on`	←	Hartman–Grobman; quoted	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1223