definition 32.19 Lyapunov function

open in the book · parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:445 · p. 1086

ground object -- no derivation owed

Rests on

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 3.43 Map ¶
  - 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 4.33 Vector space ¶
    - depends_on definition 5.7 Vector subspace ¶
      - depends_on definition 5.5 Linear combination ¶ ↺
    - 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 13.71 Differentiable curve ¶
      - depends_on definition 13.45 Differentiable map on a topological space ¶
      - depends_on definition 13.48 Differentiable manifold ¶
  - depends_on definition 9.1 Ordinary differential equation ¶

Supports

depends_on theorem 32.20 Lyapunov's direct method ¶
- depends_on example 32.21 The damped pendulum ¶

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`	→	Partial derivative; gradient	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:454
`depends_on`	→	Fixed point	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:454
`depends_on`	←	Lyapunov's direct method	declared	parts/03-classical-mechanics/15-nonlinear-dynamics-chaos.tex:460