theorem 32.20 Lyapunov's direct method

open in the book · parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:457 · p. 1086

Rests on

depends_on definition 32.19 Lyapunov function ¶
- 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 7.2 Absolute value ¶
      - depends_on definition 7.9 Real function ¶
    - 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 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 definition 32.11 Lyapunov stability ¶
- depends_on definition 32.10 Fixed point ¶ ↺
proves proof ch:15-nonlinear-dynamics-chaos@proof-6 ¶

Every logical edge within two steps of this node.

type	direction	node	provenance	where
`depends_on`	→	Lyapunov function	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:460
`depends_on`	→	Lyapunov stability	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:460
`depends_on`	←	The damped pendulum	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:509
`proves`	←	ch:15-nonlinear-dynamics-chaos@proof-6	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:463