- #1
Rasalhague
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I'm wondering which of the following definitions of a flow is best. Is there one primary, rigorous, general definition of which the others are informal shorthands, or are the differences no more then superficial differences in convention?
(1) Arnold, in Ordinary Differential Equations, defines a (phase) flow1 as a pair (M,{gt}) where M is a set called the phase space, and {gt} the set of all t-advance mappings gt : M --> M. And gt = 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 flow2 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 flow3 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 flow4 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 flow5 to a global real dynamical system itself.
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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 gt 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?
(1) Arnold, in Ordinary Differential Equations, defines a (phase) flow1 as a pair (M,{gt}) where M is a set called the phase space, and {gt} the set of all t-advance mappings gt : M --> M. And gt = 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 flow2 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 flow3 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 flow4 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 flow5 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 gt 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?