Difference equation with non-linear term

Busi

Hi all--

I can't figure out how to approach the following difference equation:

ax_{t}+f(x_{t-1})+bx_{t-2}=e_{t}

where a, b are constants, e_t is a known function and f(x_t-1) is a convex, u-shaped function that goes through the origin.

(Sorry Tex would not want to work)

To begin with, I considered f linear and solved the equation. Exactly one of the roots of the corresponding homogenous equation lies within the unit circle, so I set the free coefficients in the general solution to zero to obtain a bounded solution and derive the particular solution.

Does anyone know a way to treat a nonlinear function f?