🚀 go-pugleaf

In article <1991Jun20.194552.15875@cunews.carleton.ca> rdb@scs.carleton.ca
(Robert D. Black) writes:
>I recently read that the chaotic logistic equation
>    u(t+1) = 4u(t)(1-u(t))      u(0) in 0..1,
>                                t = 0,1,2,...
>has an ANALYTIC SOLUTION: 
>    u(t) = sin**2 (2**(t-1) arccos(1-2u(0)))
>
>    Reference "Differential Equations" by Walter G. Kelly and
>    Alan C. Peterson, Academic Press 1991, p184.
>
>This is CONFUSING!  Wasn't it the case that solvable systems 
>are by definition predictable and hence not chaotic?  Here you
>can find the value of the system at any time t without computing
>intermediate values.

     The problem with the term "chaos" is that it has a substantially
different technical meaning from its common meaning.  Chaotic systems
are defined as systems whose long-term properties are unpredictable
given some initial condition, except perhaps in a statistical sense.  If
you stuck some initial condition u(0) into your analytic solution and
found (say) u(10000) in single precision and then in double precision,
I'll wager that the two answers would be substantially different.
Analogously (and perhaps more importantly), if you took some u(0) and
another initial condition u(0)+du, where du is very small, you would
more than likely find significant differences in u(10000).  Look at the
way t enters into the solution.  Small differences in u(0) (or
differences in the precision of the arithmetic used) will be
greatly amplified by the 2**(t-1) term.  The unpredictability in chaotic
dynamical systems is quite apparent.  Given an initial condition
specified to finite accuracy, you can't say where exactly you'll wind up
after a very long time.

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