Hi
> I don't know about these equations,
I show explanation by examples. This is discrete dynamical system based
on polynomials.
f(z,c):=z*z+c;
F(n, z, c) :=
if n=1 then f(z,c)
else f(F(n-1, z, c),c);
m(n):=diff(F(n,z,c),z,1);
-------
period 1:
e1:z=F(1,z,c);
e2:m(1)=%e^(%i*t);
to_poly_solve([e1, e2], [z, c]);
[[z=%e^(%i*t)/2,c=-(%e^(2*%i*t)-2*%e^(%i*t))/4]]
First curve:
c1:-(%e^(2*%i*t)-2*%e^(%i*t))/4]
--------------------
period 2 :
e1:z=F(2,z,c);
e2:m(2)=%e^(%i*t);
to_poly_solve([e1, e2], [z, c]);
[[z=%e^((%i*t)/2)/2,c=-(%e^(%i*t)-2*%e^((%i*t)/2))/4],[z=-%e^((%i*t)/2)/2,c=-(%e^(%i*t)+2*%e^((%i*t)/2))/4],
[z=(sqrt(1-%e^(%i*t))-1)/2,c=(%e^(%i*t)-4)/4],[z=-(sqrt(1-%e^(%i*t))+1)/2,c=(%e^(%i*t)-4)/4]]
second curve:
c2:(%e^(%i*t)-4)/4;
Now we can draw :
plot2d ([[parametric, realpart(c1), imagpart(c1),
[t,0,2*%pi],[nticks,80]],[parametric,
realpart(c2), imagpart(c2), [t,0,2*%pi],[nticks,80]]])$
(If there is a better method let me know. It would be nice to draw like
....parametric,c1,c2 ... )
--------------------
period 3:
e1:z=F(3,z,c);
e2:m(3)=%e^(%i*t);
load(topoly_solver);
C:/PROGRA~1/MAXIMA~1.1/share/maxima/5.16.1/share/contrib/topoly_solver.mac
(%i7) to_poly_solve([e1, e2], [z, c]);
`algsys' cannot solve - system too complicated.
#0: to_poly_solve(e=[z = ((z^2+c)^2+c)^2+c,8*z*(z^2+c)*((z^2+c)^2+c) =
%e^(%i*t)],vars=[z,c])
-- an error. To debug this try debugmode(true);
(:-(
Method fails although period tends to infinity.
-------------------------------------------
Adam