Question on discrete time dynamic system I have the following dynamic system in discrete time 
For p is price, d is demand and s is supply.
$$p_{t+1}-p_t= a(d_t-s_t)$$
$$s_{t+1}-s_t=bp_ts_t-ws_t$$
$$d_t= k-gp_t$$
I have to linearize this system around the nontrivial optimal equilibrium point. In fact, in order to determine asymptotically stability conditions of the phase diagram for linearized supply demand function. But I am stacked at determining asymptotically stability conditions. 
—- 
What I did attempt is  
In the equilibrium, $p_{t+1}=p_t=p^*$ and $s_{t+1}=s_t=s^*$ and $d_t=d^*$
So the first equation implies that $d_t=s_t$. Thus, $d^*=s^*=Q^*$
So, by the second equation, $p^*=w/b$ 
And the third equation implies that $s^*=d^*=Q^*=k-(gw/b)$
Now let’s linearize the second question because first and third equations are already linear. 
$$s_{t+1}=[(bp^*-w+1)s^*]+[(bp^*-q+1)(s_t-s^*)]+[bs^*(p_t-p^*)]$$
$$s_{t+1}=s_t+(bk-gw)p_t -kw+(gw^2/b)$$
When I asked this question, I learn the following below way which I should do in order to determining stability condition. But I could not calculate the transformation $$ x’(t)=F(x_t)$ for my supply and demand curve. 
“you have to turn the equation St-1=St+... into differential equation of form x’(t)=F(xt) where all equilibria are given by F(xt)=0 and they are locally stable if F’<0 and not stable if F’>0 once you have the differential equation set up you can build phase diagram from that”
Please help me to apply the following hint in my question (supply demand functions). I really cannot do it. Thanks a lot. 
 A: Putting $d_t= k-gp_t$ in the 1st equation gives the following system
$$\left\{\begin{array}{rcl}p_{t+1}-p_t&=& a(k-gp_t-s_t)\\
s_{t+1}-s_t&=&bp_ts_t-ws_t\end{array}\right.$$
Clearly the nontrivial equilibrium point is 
$$ (p^*,s^*)=(\frac{w}{b},k-\frac{gw}{b}). $$
Let $(x,y)=(p-p^*,s-s^*)$ and the system becomes
$$ \left\{\begin{array}{rcl}x_{t+1}&=&(1-ag)x_t-ay_t,\\
y_{t+1}&=&(bk-gw)x_t+y_t+bx_ty_t.
\end{array}\right.$$
The linearized system is
$$ \left\{\begin{array}{rcl}x_{t+1}&=&(1-ag)x_t-ay_t,\\
y_{t+1}&=&(bk-gw)x_t+y_t,
\end{array}\right.$$
which has the coefficient matrix
$$ A=\left[\begin{matrix}1-ag&-a\\
bk-gw&1
\end{matrix}\right]. $$
Clearly $A$ has the eigenpolynomial 
$$ P(\lambda)=\lambda^2+(ag-2)\lambda+(abk-ag-agw+1).$$
If the roots $\lambda_{1,2}$ of $P(\lambda)$ satisfies $|\lambda_{1,2}|<1$, then $(p^*,s^*)$ is asymptotically stable. But $|\lambda_{1,2}|<1$ if and only if
$$ |ag-2|<1+(abk-ag-agw+1)<2 $$
from which you can obtain the conditions for $a,g,k,w$. I omit the detail.
