- #1

Rasalhague

- 1,387

- 2

(1) Arnold, in

*Ordinary Differential Equations*, defines a (phase)

**flow**

_{1}as a pair (M,{g

^{t}}) where M is a set called the phase space, and {g

^{t}} the set of all t-advance mappings g

^{t}: M --> M. And g

^{t}= g(t,_), where g is the evolution function of the dynamical system, as defined by Wikipedia: Dynamical system.

(2) The aftoresaid article, in the section "General definition", gives the name

**flow**

_{2}to functions of the form g(_,x), which it calls "the flow through x".

(3) The same article, in the section "Geometric cases" tells us: "A real dynamical system, real-time dynamical system or

**flow**

_{3}is a tuple (T, M, Φ) with T an open interval in the real numbers R, M a manifold locally diffeomorphic to a Banach space, and Φ a continuous function."

(4) Wikipedia: Flow, defines a

**flow**

_{4}as the evolution function of a global real dynamical system, that is, one for which T =

**R**.

(5) And Scholarpedia: Dynamical systems gives the name

**flow**

_{5}to a global real dynamical system itself.

*

Are the definitions as a tuple, such as Arnold's, rather superfluous? By this I mean, isn't the existence of M and its relationship to the functions g or g

^{t}already part of the definition of those functions? Is that why it's okay to define a flow, or indeed a dynamical system in general, as its own evolution function? Or are the defininitions which take this approach shorthand definitions which miss out information necessary for a completely general and rigorous definition of a dynamical system?