Symmetry in dynamical systems

I am trying to find out ,if possible, the origin of symmetry in physical laws. I am using a naive approach, letting the computer generate random formulas for two dimensional discrete dynamical systems of the type:

   x = f(x,y)
   y = g(x,y)

Then I will look in detail to the symmetric ones that I get, trying to understand where the symmetry comes from. In the formula I allow only the four arithmetic operations and to get something in a finite time, I have put a limit of 50 on the operators that may be present in each formula. I know that the easiest way to get symmetry is by using complex numbers or by using transcendental functions like sin or cos (this is I presume because the complex product is a rotation, and the sin,cos functions produce rotations), but I am not interested in this way of getting symmetry. I would like instead to consider completely general formulas and try to understand, if there is the possibility that like the deterministic chaos, the symmetry arises from some universal property of algorithms. It is very difficult to find with this brute force method symmetric chaotic limit sets a la Golubitsky. I managed to find only a few with a simple bilateral symmetry, the simplest being(465):

    x = -y
    y = x^3-x-y

Something interesting is the following: the fact that I choose a square (so a highly symmetric) window for the escape time plot has effect on the symmetry of this image. A very common case is that a asymmetric chaotic attractor has a symmetric basin of attraction. Click here to have a more detailed analysis on the same problem.

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