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Roger L. Costello wrote:
>>A Dynamical System consists of some "variables" and
>>some "equations of motion" or "dynamical equations".
>>The variables are any things which can vary with time.
>>They can be multiple or single, continuous or discrete.
>>They must be chosen in such a way that complete knowledge
>>of all the variables determines uniquely the "state" of
>>the system at one time.
>
>
> Thanks Sergio. This is very useful information.
>
Notice that -
1) "Dynamic" in no way correlates with "nonlinear". There can be both
linear and nonlinear dynamic systems.
2) "Dynamic" in no way correlates with "Chaotic". There can be chaotic
and non-chaotic dynamical systems.
3) The variables _completely_ specify all possible knowledge of the
system. There are no "hidden variables" containing additional state
information.
From MathWorld (http://mathworld.wolfram.com/Chaos.html)
"A dynamical system is chaotic if it
1. Has a dense collection of points with periodic orbits,
2. Is sensitive to the initial condition of the system (so that
initially nearby points can evolve quickly into very different states), and
3. Is topologically transitive."
Note that there is nothing here about randomness. The random aspects of
a queue (which Roger suggested in a previouspost) do not qualify it as
such a system.
>
> Consider the situation of a stylesheet processing an XML document:
>
> XML (input) ---> XSLT ---> XML (output)
>
> An XML document is a "model" of a system.
Well, I would say that an xml document is a representation of a model.
The model involves some kind of "entities" and relationships between
them (and possibly their environment).
> For example, a Fitness Center
> XML document is a model of a fitness center. Thus, an input XML document
> represent the "current state" of the system, the output XML document
> represents the "next state", and the stylesheet is simply a mechanism
> for implementing the transition from state (i) to state (i+1). Thus,
> the "variables" of the system correspond to elements in an XML document.
>
> Which elements in an XML document represents the system variables? For
> example, in the following Fitness Center XML document, which elements
> correspond to the "system variables"?
>
> <?xml version="1.0"?>
> <FitnessCenter>
> <Member level="platinum">
> <Name>Jeff</Name>
> <Phone type="home">555-1234</Phone>
> <Phone type="work">555-4321</Phone>
> <FavoriteColor>lightgrey</FavoriteColor>
> </Member>
> <Member level="gold">
> <Name>Stacey</Name>
> <Phone type="home">888-1234</Phone>
> <Phone type="work">888-4321</Phone>
> <FavoriteColor>purple</FavoriteColor>
> </Member>
> </FitnessCenter>
>
> Are the <Member> elements the "system variables"? Or, are the <Name>,
> <Phone>, and <FavoriteColor> elements the "system variables"?
>
> My answer to this question is: the <Member> elements. The <Member>
> elements represent a "building block" that is presumably the product of
> evolution, i.e., over time a collection of <Name>, <Phone>, and
> <FavoriteColor> elements coalesced to create a useful building block
> (this is analogous to molecules aggregating together to form a useful
> building block - a cell). It is at the building block level that
> competition and collaboration occur (and this competition/collaboration
> activity is physically implemented by the XSLT stylesheet).
>
> Do you agree with this analysis?
Not really ... it is impossible to know what the state variables are
since we do not know what the model is yet. Maybe a Member is a state
variable and maybe it is not.
Again quoting from MathWorld, about Dynamical Systems, to complement
Sergio's quote, a dynamical system is -
"A means of describing how one state develops into another state over
the course of time."
So what is the state of the Fitness Center, and how does it develop over
time? Now that will be interesting. Notice that the system state is
something that can in principle be computed if you know the state
variables. The variables are clearly data... the computation is like a
procedure... so it would seem that the dynamical system is like an
object, with both data and a method.
Normally the XML contains only the data.
It seems that Roger is moving towards inventing an XML-Simula...
Cheers,
Tom P
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