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Transfer function derivation of my system

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s55

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For my system, I need to derive a transfer function, and I'm stuck at the derivation process. Could you please help answer to my question?

Here is the system model I have, and a(t) can be different values depending on value of x(t) as below.

y(t) = (x(t)-a(t)) + (x(t)-a(t))^3 --- (eq.1)

where
eq.jpg

--- (eq.2)

Here I need to apply a specific signal, x(t) = cos(wt), then y(t) would be
y(t) = (3*a(t)^2 + 7/4)*cos(wt) + (-(3*a(t))/2)*cos(2*wt) + cos(3*wt)/4 - a(t)^3 - (5*a(t))/2 ---(eq.3)

If there is no condition shown in (eq.2), the derivation from (eq.1) to (eq.3) is correct. No question on this.

Here is my question.
Considering the condition of (eq.2), is it still correct the derivation from (eq.1) to (eq.3)?
Basically, is (eq.3) correct, assuming x(t)=cos(wt) and (eq.2)?

If I'm wrong, could you please advise how to fix it? or the suggestion of reference materials would be very helpful for me.
 

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