Can you write a function in two variables that exhibits chaotic behaviour?

Question by Mechoman: Can you write a function in two variables that exhibits chaotic behaviour?

I recall analysis of the Newton-Raphson method for iteratively solving polynomials – depending on the initial value chosen, the equation would ‘home in’ on a function.

At certain values, f(a), the equation would head off to inifity, and for infinitesimally different values to this, f(a ± ?a), a solution would be found.

Can you list any others functions that exhibit chaos in two variables?

Best answer:

Answer by Scythian1950
It depends on what you’re looking for. An easy way to put together a function that has a random output is through combining a rational with an irrational number, such as Sin(x) + Sin(?x). However, this is still differentiable everywhere, so it wouldn’t be a fractal. It’s difficult to devise a function in terms of elementary functions that’s a fractal, because it’s the same thing as attempting to write a function in terms of elementary functions that one CANNOT differentiate. Good luck with that. But if one is allowed to use infinite sums or products or limiting case functions in constructing such functions, yes, it’s possible. A variation of Dirichlet’s function (see wolfram) could get you such a function that is not only non-differentiable everywhere, but is chaotic as well. Another good example would be a variation of the Weierstrass function, see wiki.

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