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THE problem with dynamics - tommy1729 - 04/02/2017
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 RE: THE problem with dynamics - tommy1729 - 04/04/2017
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 |