🚀 go-pugleaf

In article <00962123.79FB9060@128.104.39.173> sprott@128.104.39.173 writes:
>3) Is there a mapping or system of ODE's involving only polynomials whose
>solution attracts to:
>     a) a 2-torus

It's easy in 4 dimensions, e.g. the ODE system
     x' = y + x (1 - x^2 - y^2)
     y' = -x + y (1 - x^2 - y^2)
     z' = a w + z (1 - z^2 - w^2)
     w' = - a z + w (1 - z^2 - w^2)
where a is a positive irrational constant.  The attractor is the surface
x^2 + y^2 = 1, w^2 + z^2 = 1.

A bit more complicated to make a 3-dimensional example.  Try the
following ODE system:

   x' = x (- c z + p - p^3 - p z^2) + a y
   y' = y (- c z + p - p^3 - p z^2) - a x
   z' = 2 (p + b) (c p + z - p^2 z - z^3)

where p = x^2 + y^2 - b, a, b, c are constants, and b > 1.  Except for
values of the constants that create periodic orbits, the attractor is
the 2-torus  p^2 + z^2 = 1.  

>     b) a 3-torus

Easy in 6 dimensions, analogous to my first example.
-- 
Robert Israel                            israel@math.ubc.ca
Department of Mathematics             or israel@unixg.ubc.ca
University of British Columbia
Vancouver, BC, Canada V6T 1Y4