Need pointer on getting started

I'm 10 years out of engineering, and haven't really had to use the math part of my education since.

I have a new problem at work, and I could use a pointer as to where I need to start looking (linear algebra, discrete math?).

I have an equation of the form

x[t] = x[t-1]*y[t] / (y[t-1] - y[t])

For some initial inputs of x and y, this equation becomes harmonic...it repeats every handful of iterations. I would like to develop some guidelines for how to determine what types of data will cause the equation to become harmonic vs. converge to some steady-state value.

I used to be pretty good with math, just very rusty. I appreciate any advice pointing me in the right direction.

Thanks!