Background
I have the following system of ODEs:
$\dfrac{\mathrm{d}x}{\mathrm{d}t}=x\dfrac{q}{Q}-x\dfrac{x+y}{M}\quad$ (Eq. 1)
$\dfrac{\mathrm{d}q}{\mathrm{d}t}=y(1-\dfrac{q}{Q})(1-c)(1-v)-aq-y\dfrac{q}{Q}v(1-c)-x\dfrac{q}{Q}v\quad$ (Eq. 2)
$\dfrac{\mathrm{d}y}{\mathrm{d}t}=y(1-c)-y\dfrac{x+y}{M}\quad$ (Eq. 3),
where all variables and parameters are positive, $M\gg 0$, and $Q\gg 0$.
Setting these three equations equal to zero and solving for $(x, q, y)$ reveals that the system exhibits an equilibrium, $(x^{*},q^{*},y^{*})$, where $x^{*}>0$, $q^{*}>0$, and $y^{*} >0$. Under certain conditions, this equilibrium exhibits damped oscillations.
Goal
I am hoping to approximate the system's solution using the KBM method, which is described in this post and, most relevantly, in this paper. In the latter paper, what is noted is that, to approximate variable $y$'s dynamics, we want a second-order differential equation of the following form:
$\dfrac{\mathrm{d}^2y}{\mathrm{d}t^2}+2b\dfrac{\mathrm{d}y}{\mathrm{d}t}+r^2y=\epsilon f^{(1)}\left(y,\dfrac{\mathrm{d}y}{\mathrm{d}t}\right)+\epsilon^2 f^{(2)}\left(y,\frac{\mathrm{d}y}{\mathrm{d}t}\right)+...$ (Eq. 4),
where $\epsilon$ is a small parameter, $b \geq 0$, and $r>0$.
What I have so far
To acquire the desired second-order ODE, I first took the derivative of Eq. 3 to acquire:
$\dfrac{\mathrm{d}^2y}{\mathrm{d}t}=-\dfrac{y\frac{\mathrm{d}x}{\mathrm{d}t}+((-1+c)M+x+2y)\frac{\mathrm{d}y}{\mathrm{d}t}}{M}$ (Eq. 5).
To simplify, I then let $Q=M$ and took the limit of Eq. 5 as $M \rightarrow \infty$, which gives:
$\dfrac{\mathrm{d}^2y}{\mathrm{d}t}=-\left((-1+c)\dfrac{\mathrm{d}y}{\mathrm{d}t}\right)$ (Eq. 6).
I then rewrote Eq. 6 as:
$\dfrac{\mathrm{d}^2y}{\mathrm{d}t}+2(c-1)\dfrac{\mathrm{d}y}{\mathrm{d}t}+y(c-1)^2=(c-1)\dfrac{\mathrm{d}y}{\mathrm{d}t}+y(c-1)^2$,
where here $b=(c-1)$, $r=(c-1)$, and $\epsilon=(c-1)$.
Questions
Is my approach to acquiring the general form of Eq. 4 valid?
More broadly, how can I proceed to analytically approximate the system's oscillatory solution?
Any help would be greatly appreciated!