🚀 go-pugleaf

In article <FARID.92Dec14141233@gradient.cis.upenn.edu>
farid@gradient.cis.upenn.edu (Hany Farid) writes:
>Excuse the ignorance but in James Gleick's book title "Chaos" on page
>50 he shows signals in a 'traditional time space' and in a 'phase
>space'.  Can anyone explain in lay terms how to translate a signal
>from time space to phase space, and perhaps what exactly phase space
>is.

     The idea of a phase space is contained in most elementary books on
differential equations.  Let's take a simple example.  Suppose you have
a pair of autonomous differential equations

	  dx
	  -- = f(x,y)
	  dt

	  dy
	  -- = g(x,y)
	  dt

Because of the existence and uniqueness theorem for systems of
autonomous differential equations, there is one and only one solution
curve (trajectory) of this system passing through every point (x0,y0) of
the xy phase plane.  There are different ways to think about this
result, but the upshot is that you can simply disregard time and think
of this system as defining trajectories in the phase plane rather than
solutions in an (x,y,t) space.  You can convince yourself that the
trajectories satisfy the equation

	  dy   g(x,y)
	  -- = ------
	  dx   f(x,y)

so that time can be formally eliminated from the equations themselves.

				Marc R. Roussel
                                mroussel@alchemy.chem.utoronto.ca


P.S., to Hany:  There is no such newsgroup as sci.chaos so I'm posting
this to sci.fractals.