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?

 

[jvicens]

Hello,

Thank you for bringing this interesting problem to our attention. Nonlinear difference equations can be challenging to solve, but there are some general approaches that can be used.

One approach is to use numerical methods, such as iteration or simulation, to approximate the solution. This can be done by breaking the equation into smaller steps and using known values to calculate the next step. This can give a good approximation of the solution, but it may not be exact.

Another approach is to use analytical methods, such as the method of successive approximations or the method of undetermined coefficients. These methods can be used to find an approximate solution or to prove the existence of a solution, but they may not be able to find an exact solution in all cases.

In addition, there are some specific techniques that can be used for certain types of nonlinear difference equations. For example, if the equation is a quadratic difference equation, it can be solved using the quadratic formula.

I hope this helps. Good luck with solving your equation!