theorem 9.34 Poincaré–Bendixson; quoted

open in the book · parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1259 · p. 286

Rests on

depends_on definition 9.1 Ordinary differential equation ¶
depends_on definition 6.9 Compact set ¶
- depends_on definition 6.5 Open cover ¶
  - 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 equation 3.51 eq:set-indexed ¶ ↺
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 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 proposition 7.47 Comparison; absolute convergence ¶
    - depends_on corollary A.46 Monotone convergence ¶
      - depends_on theorem A.40 Least-upper-bound property ¶
      - proves proof app:A-long-proofs@proof-33 ¶
    - depends_on definition 7.45 Series ¶
      - depends_on definition 7.2 Absolute value ¶ ↺
      - depends_on definition 7.4 Convergence ¶
    - depends_on theorem 7.8 Cauchy criterion ¶
      - depends_on corollary A.47 Cauchy completeness ¶
      - depends_on definition 7.4 Convergence ¶ ↺
      - proves proof ch:05-real-analysis@proof-5 ¶
    - proves proof ch:05-real-analysis@proof-28 ¶
  - proves proof ch:07-odes-sturm-liouville@proof-1 ¶
- proves proof ch:07-odes-sturm-liouville@proof-2 ¶

Supports

depends_on theorem 32.30 Poincaré–Bendixson, restated from Part II ¶
- depends_on corollary 32.32 No chaos in a planar flow ¶

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`	→	Sur les courbes définies par des équations différentielles	derived	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1265
`cites`	→	Mémoire sur les courbes définies par une équation différentielle (seconde partie)	derived	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1265
`depends_on`	→	Ordinary differential equation	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1268
`depends_on`	→	Compact set	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1268
`depends_on`	→	Picard–Lindelöf	declared	parts/02-mathematical-methods/07-odes-sturm-liouville.tex:1268
`depends_on`	←	Poincaré–Bendixson, restated from Part II	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:696