Consider the following dynamic model for sustained oscillations in yeast glycolysis:

$\frac{dG}{dt}=V_{in}-k_{1}GA=f_1(G,A)$

$\frac{dA}{dt}=2k_{1}GA-k_p\frac{A}{K_m+A}=f_2(G,A)$

where $G$ and A are the intracellular concentrations of glucose and ATP, respectively, $V_{in}=0\mathrm{.}36$ is the constant flux of glucose into the yeast cell, $k_1=0\mathrm{.}02$ is an enzyme activity, and the parameters $k_p=6\mathrm{.}0$ and $K_m=13\mathrm{.}0$ determine the kinetics of ATP degradation.

$($a$)$ Show that the given parameter values produce a single steady$-$state solution $\frac{?}{\text{b}\text{a}\text{r}}(G)$ and $\frac{?}{\text{b}\text{a}\text{r}}(A).$

$($b$)$ Find the linearized model at this steady$-$state point $\frac{dx}{dt}=Ax$ where $x=[G^{\text{'}}{A}^{\text{'}}{]}^T.$

$($c$)$ Determine the stability of the steady state by computing the eigenvalues of the A matrix.