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A great way to approach it is to think of two variable linear dynamical systems, like

x_n+1 = A x_n + B y_n y_n+1 = C x_n + D y_n

where A, B, C, D are constants (real if you like). This kind of model is very useful, such as with two species predator-prey.

The solution of the system is just the powers of the matrix [ A B ; C D ]. That is easily expressed in terms of the eigenvalues.

When the eigenvalues are complex, the resulting formulas for x_n and y_n involve trig functions. They will be:

x_n = u cos(n theta) + v sin(n theta)

and similar for y_n. u and v are certain constants that are a combination of the initial values x_0, y_0, A, B, C, D and the real and imaginary parts of the eigenvalues.

This is well explained in Fred Marotto's book

http://www.amazon.com/Introduction-Mathematical-Modeling-Discrete-Dynamical/dp/0495014176