Dynamics of the function

[Sarif Hassan]

I am trying to understand the dynamics of the difference equation, f(z,w)=a/z+b/w; and the dynamical system is defined as Z_{n+1}=f(Z_{n},f{n-1}). 

Anyone can enlighten towards the study of the dynamics?

[Ben]

Sarif,
Could you please be more specific as to what you mean by the
dynamics? I can think of more than a few 'definitions' of the dynamics
of a function.

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[Sarif Hassan]

I am mainly interested to find fixed points and their stability. Also I am interested about periodicity and chaotic properties.

[Ben]

Well we don't have any functionality in Sage implemented for efficiently
dealing with non-morphisms. However, you can define such a map in Sage:

a=3
b=1
A.<z,w>=AffineSpace(QQ,2)
H=End(A)
f=H([a/z+b/w,z])
f

or more generically

R.<a,b>=PolynomialRing(QQ)
A.<z,w>=AffineSpace(QQ,2)
H=End(A)
f=H([a/z+b/w,z])


However, unless I'm mistaken, the fixed points of this map are quite
simple as the two values satisfying

a+b = z^2
z=w

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