I'm working with a frequency-response curve of a nonlinear oscillator and came across the following equation (Kovacic & Brennan 2011, p. 179):
$$ A^2 = \frac{f^2}{4 \xi^2 \omega^2 + (\omega^2 - 1 - \frac{3}{4}A^2)^2 }. $$
Question: what is the strategy for plotting of $A$ vs. $\omega$?
I am only interested in real roots with $A > 0$. The parameters $\xi, f$ and $\omega$ are fixed and known. Furthermore $0 <\xi \ll 1$, and $0 < f, \omega$.
Set $$f_\omega(A) = A^2 - \frac{f^2}{4 \xi^2 \omega^2 + (\omega^2 - 1 - \frac{3}{4}A^2)^2 }.$$
Now look for zeros of this function. One way to do this is to use Newton's method (wiki). If there are multiple zeros for fixed $\omega$ you would have to choose different starting points $A_0$ for every $\omega$, e.g. $A_0=0,0.1,0.2,\ldots$ and repeat Newton's method for all of those. But note that then the thing you plot is not a function.