Dear Igor, the reason of oscillation of your map for y < -0.25 is the following. Suppose to have convergence of Yn ; then for sufficiently large values of n we have Yn+1 very close to Yn, so that (defined u = Yn) from map definition it must be

u2 + u - y = 0

real solution u to this equation exist if discriminant = 1 + 4 y >= 0, therefore y >= -1/4 = -0.25.  Gianluca

Thank you, Gianluca, but that was the starting point. That's where the effective resistance becomes real, that is unphysical. 

Igor, what do you intend for "effective resistence" and "unphysical"? Thanks.  Gianluca

That is the simple well known problem: a network consists of reactive elements (capacitors and inductances). The impedance (resistance) of the finite network is naturally imaginary, i.e. reactive. The infinite network solved by effective resistance method has an active component, i.e. a real part. It is unphysical and funny. 

But the problem persists in     the effective medium method of the composite media and that is not funny any more. 

First 1000 iterations for y = -0.26

The iteration scheme

x_{n+1} = 1/2(-1/x_n + x_n)

which is the solution of the equation x^2 + 1 = 0

converges for all x_0 except real.

I find that curious.