Asymptotic expansion on 3 nonlinear ordinary differential equations

Question

The 3 nonlinear differential equations are as follows \begin{equation} \epsilon \frac{dc}{dt}=\alpha I + \ c (-K_F - K_D-K_N s-K_P(1-q)), \nonumber \end{equation}
\begin{equation} \frac{ds}{dt}= \lambda_b P_C \ \epsilon \ c (1-s)- \lambda_r (1-q) \ s, \nonumber \end{equation} \begin{equation} \frac{dq}{dt}= K_P (1-q) \frac{P_C}{P_Q} \ \ c - \gamma \ q, \nonumber \end{equation} I want to use asymptotic expansion on $c, s$ and $q$. And values of parameters are:

$K_F = 6.7 \times 10^{-2},$

$K_N = 6.03 \times 10^{-1}$

$K_P = 2.92 \times 10^{-2}$,

$K_D = 4.94 \times 10^{-2}$,

$\lambda_b= 0.0087$,

$I=1200$

$P_C = 3 \times 10^{11}$

$P_Q = 2.304 \times 10^{9}$

$\gamma=2.74 $

$\lambda_{b}=0.0087 $

$\lambda_{r}= 835$

$\alpha=1.14437 \times 10^{-3}$

For initial conditions:

\begin{equation} c_0(0)= c(0) = 0.25 \nonumber \end{equation} \begin{equation} s_0(0)= cs(0) = 0.02 \nonumber \nonumber \end{equation} \begin{equation} q_0(0)=q(0) = 0.98 \nonumber \nonumber \end{equation} and \begin{equation} c_i(0)= 0, \ i>0\nonumber \end{equation} \begin{equation} s_i(0)= 0, \ i>0 \nonumber \nonumber \end{equation} \begin{equation} q_i(0)=0, i>0. \nonumber \nonumber \end{equation}

=> i started with the expansions : \begin{equation} c= c_0+ \epsilon c_1 + \epsilon^2 c_2+......... \nonumber \end{equation} \begin{equation} s= s_0+ \epsilon s_1 + \epsilon^2 s_2+......... \nonumber \end{equation} \begin{equation} q= q_0+ \epsilon q_1 + \epsilon^2 q_2+......... \nonumber \end{equation} we are only interseted in up to fisrt power of $\epsilon$. so, we should get total 6 approximate differential equations to get answer for $\dfrac{dc_0}{dt}, \dfrac{ds_0}{dt}, \dfrac{dq_0}{dt}, \dfrac{dc_1}{dt}, \dfrac{ds_1}{dt}$ and $\dfrac{dq_1}{dt}$

but i think $\dfrac{dc_1}{dt}$ will disappear while expanding and equating the up to first power of $\epsilon$, do i need to go further up to $\epsilon{^2}$ because $\dfrac{dc_1}{dt}$ is very important to find and we need 6 approximate differetial equations in total. what can i do? please some one help me.

Answer 1

This is an example of a singular perturbation problem and so the type of perturbative expansion you are attempting is not likely to give you anything useful. I would recommend reading about multi-scale analysis, which can be found in most graduate books on nonlinear differential equations (Glenddings book has an introduction).
For a detailed exposition i would refer you to Kervokian and Coles book.

The system you have presented has $3$ time scales; fast time $\epsilon^{-1}t$, 'slow' time $t$, and 'really slow' time $\epsilon t$. On the fast time scale, $c$ will rapidly approach the slow manifold, ie the fixed point at $$\dot{c} =0 \implies c(q,s) = \frac{I\alpha}{(K_F+K_D+K_Ns+K_P(1−q))}.$$

You can then plug this into the equations for $\dot{s}$ and $\dot{q}$ to produce a two dimensional nonlinear equation for the motion on this slow manifold. Note that in this reduced system, there is still $\epsilon$ floating around. Since it is not a coefficient of a derivative term, the perturbative expansion attempted in the original post will work.