Hello, I have a problem in a non-linear equation analysis.

I have a non-linear equation:
x'' + kx(1+ax^2)+bx' = A*sin(wt)

that I already transformed into the following system
x' = y
y' = -kx(1+ax^2) - by + A*sin(wt)

Now I have to obtain the transfer function, so I suppose the only way
to do it is to first linearize (approximating) the second equation. Do
I have to use Taylor series to linearize around the zero (which I
calculated to be (0,0))? If so, how?

My attempt was to posticipate the "linearizing": I let a particular
solution to be H*sin(w1t) (where w1 <> w),
then I substitute it in the main equation (the first that I wrote), but
I have a term
ka*H^3*(sin(w1t))^3
that blocks me from calculating H as a function of the other terms. I
guess I have to linearize that term,
but now what is the sense in linearizing around the zero??

I think you are stuck with the case 'a' << 1. That might give results
that are good for short times.

I think the above equation might better
be solved by other methods. Given that it's 2D there may be closed-form
solutions for it. It looks similar to a Van der Pol type, except
driven. You need to start searching the literature and, probably, the
web (Google Scholar, for example).

I think you are stuck with the case 'a' << 1. That might give results
that are good for short times.

Yes, sorry, I forgot to say that I can assume a small. But what does it
imply?
May I remove the whole term kaH^3(sin(wt))^3 ? I guess not..

I think the above equation might better
be solved by other methods. Given that it's 2D there may be closed-form
solutions for it. It looks similar to a Van der Pol type, except
driven. You need to start searching the literature and, probably, the
web (Google Scholar, for example).

I don't know what you mean by closed-form solutions,

but the system is
actually 3d since time has its influence in the dynamic, due to the
presence of the drive force. Indeed the trajectory in the phase space
can overlap to itself.