theorem 32.30 Poincaré–Bendixson, restated from Part II

open in the book · parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:687 · p. 1088

Rests on

depends_on corollary 32.5 Trajectories do not cross ¶
- depends_on proposition 32.4 The flow is a one-parameter group ¶
  - 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 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:15-nonlinear-dynamics-chaos@proof-1 ¶
- depends_on theorem 9.8 Picard–Lindelöf ¶ ↺
- proves proof ch:15-nonlinear-dynamics-chaos@proof-2 ¶
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 9.34 Poincaré–Bendixson; quoted ¶
- depends_on definition 9.1 Ordinary differential equation ¶ ↺
- depends_on definition 6.9 Compact set ¶ ↺
- depends_on theorem 9.8 Picard–Lindelöf ¶ ↺

Supports

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/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:693
`cites`	→	Mémoire sur les courbes définies par une équation différentielle (seconde partie)	derived	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:693
`depends_on`	→	Trajectories do not cross	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:696
`depends_on`	→	Compact set	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:696
`depends_on`	→	Poincaré–Bendixson; quoted	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:696
`depends_on`	←	No chaos in a planar flow	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:724