THE problem with dynamics

[tommy1729]

THE problem with dynamics is multivariable dynamics.

So far we only considered real or complex single variable dynamics.

Example of 2 variable dynamics problem :

For x >= 0 ,
Find Pairs of analytic functions f,g such that

f(x+1) = 2 f(x)^2 + 3 g(x)^2 + 4 f(x) + 5 g(x) + 6
g(x+1) = f(x)^2 + 7 g(x)^2 + 8 f(x) + 9 g(x) + 10


I have considered this idea with mick, but without results.

Regards

Tommy1729

[tommy1729]

Well after Some thinking it appears that all multivariable problems reduce to analogues of single variable dynamics , univariate diff equations , delay differential equations and PDE ( partial ( = multivariable ) differential equations.

For instance there are analogue fractals where the half-iteration is not defined.
And analogue Koenig functions.

( if you compute the half-iterate of a random polynomial of degree 2 by using koenigs , you have a problem within the Julia set ( the fractal ) of that polynomial almost surely )

I thank mick and Sheldon for realizing it " completely " now.
Although they did not actively help their past ideas did.
Im not going to define completely here.


There is no reason to assume a multivariable difference equation can be expressed by a univariate difference equation easier or more often than a PDE can be expressed in univar diff equations and vice versa.

Im unaware of a Satisfying formal statement and formal proof of that though.

Regards

Tommy1729