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Logical starting point:
A dynamical system is a manifold M called the phase (or state) space endowed with a family of smooth evolution functions \(\phi^t\) that for any element of t ∈ T, the time, map a point of the phase space back into the phase space. The notion of smoothness changes with applications and the type of manifold. There are several choices for the set T. When T is taken to be the reals, the dynamical system is called a flow; and if T is restricted to the non-negative reals, then the dynamical system is a semi-flow. When T is taken to be the integers, it is a cascade or a map; and the restriction to the non-negative integers is a semi-cascade.
I'm going to work through some of this here. Feel free anyone to add any comments that come to mind (the notes can be interleaved).