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Would someone care to explain the basic theory of dynamical systems? i.e. an explanation of the following definition of a dynamical system:

A dynamical system is a tuple $(T,M,\Phi)$ where $T$ is a monoid, written additively, $M$ is a set, and $\Phi$ is a function. $$ \Phi : U \subset T \times M \to M$$ with

$ I(x) = \{t \in T : (t,x) \in U \}$
$ \Phi(0,x) = x $
$ \Phi(t_2, \Phi(t_1(x)) = \Phi(t_1 + t_2, x)$ for $t_1, t_2,t_1+t_2 \in I(x)$

Could someone elaborate on this? I'm not quite following the definition.

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    $\begingroup$ Wow. In my first lecture on dynamical systems this semester, I defined a (discrete) dynamical system to be a set $S$ with a function $f:S\to S$. If I had defined it the way you've presented it, I would have emptied out the classroom. $\endgroup$ – Gerry Myerson Aug 25 '13 at 1:16
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    $\begingroup$ Echoing @GerryMyerson's comment, the definition presented above doesn't even begin to speak to the geometries one might encounter, for which (at least for one sort of discrete dynamical systems) a good starting point is Clint Sprott's sprott.physics.wisc.edu/sa.htm $\endgroup$ – graveolensa Aug 25 '13 at 1:46
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    $\begingroup$ It's clearly the most general definition of a dynamical system I have ever seen! $\endgroup$ – Robert Lewis Aug 25 '13 at 2:02
  • $\begingroup$ @AnthonyPeter, could you refer a text where this definition was presented? I got curious about what can be done in this framework. $\endgroup$ – Evgeny Aug 29 '13 at 14:22
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    $\begingroup$ @Evgeny Simply the wikipedia entry for Dynamical Systems $\endgroup$ – Anthony Peter Sep 3 '13 at 5:35
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As I noted in the comments, I'm somewhat blown away by this definition, but I'll take a stab at it. You've got a set, and the points in it are moving around with the passage of time, and $\Phi$ is keeping track of the movement. Think of $T$ as giving the time. $\Phi(0,x)=x$ says that at time zero the points are at their starting places. The third condition says that if you look at where the points are after $t_1$ seconds (or days, or centuries, whatever), and then you look at where they are $t_2$ seconds later, you find out where they are after $t_1+t_2$ seconds.

Only everything's being done in greater generality. I've been thinking of $T$ as time, which is naturally an interval in the continuous case, or the integers or naturals in the discrete case, but whoever wrote that definition wants to allow $T$ to be an arbitrary monoid. And I've been thinking of $M$ as a set of points on the line, or in the plane, or 3-space, but the author intends for it to be any abstract set. So I haven't so much explained the definition, as given you one very specialized (but, I think, very important) setting it which you can interpret it.

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