Tetration Forum

Full Version: THE problem with dynamics
You're currently viewing a stripped down version of our content. View the full version with proper formatting.
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
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