How the cosmological scale factor $a(t)$ will have maximum value followed by a contracting phase of the universe? The scale factor satisfies the following equation - 
$$\frac{\dot a^{2}}{a^{2}} + \frac{k}{a^{2}} = \frac{8\pi G}{3} \rho(t)$$ where $k$ is constant proportional to total energy of the dynamical system. Now  numerical value of $k$ can be absorbed in the definition of $a(t)$ by rescaling it hence $k$ can be treated having one of the three values $(0,\pm1)$. From this equation how can we show that cosmological scale factor $a(t)$ will have maximum value followed by a contracting phase to the universe if $k=1$ and $\Omega >1$, where $\Omega = \frac{\rho}{\rho_{c}}$.
I tried following - 
if $k=1$ and $\Omega >1$, we get -
$$\frac{\dot a^{2}}{a^{2}} + \frac{1}{a^{2}} = \frac{8\pi G}{3} \rho_{c}\cdot(\Omega>1) >\frac{8\pi G}{3} \rho_{c}\cdot\Omega$$ 
$\implies \dot a^{2} + 1 > \frac{8\pi G}{3} \rho_{c}\cdot\Omega \cdot a^{2} \implies a < \bigg[ \frac{3(\dot a^{2} + 1)}{8\pi G\rho_{c}\cdot\Omega}\bigg]^\frac{1}{2} \implies a_{max} = \bigg[ \frac{3(\dot a^{2} + 1)}{8\pi G\rho_{c}\cdot\Omega}\bigg]^\frac{1}{2}$
But, from here how will I prove that this maximum value will be followed by a contracting phase to the universe?
To find the form of $a(t)$ we need to know $\Omega(t)$, Could somebody please tell what kind of models are available to determine $\Omega(t)$?
Ref: Theoretical Astrophysics, T.Padmanabhan, Vol 3, pg - 4
 A: From:
$$
\frac{\dot a^{2}}{a^{2}} + \frac{k}{a^{2}} = \frac{8\pi G}{3} \rho(t)
$$
If you set $k=1$ and $\Omega = \frac{\rho}{\rho_{c}}$:
$$
\dot a^{2} + 1 = \frac{8\pi G}{3} \rho_{c}\cdot\Omega \cdot a^{2}.
$$
At $a= a_{max}$, the expansion rate is zero $\dot{a}=0$, therefore
$$
1 = \frac{8\pi G}{3} \rho_{c}\cdot\Omega \cdot a^{2}_{max}.
$$
Now the solution is dependent on how  $\Omega$ scales with $a$. Let's for example examine a matter only universe, i.e. $\Omega = 1/a^3$, therefore:
$$
1 = \frac{8\pi G}{3} \rho_{c}\cdot \frac{1}{a^3_{max}} \cdot a^{2}_{max}.
$$
Et voila:
$$
a_{max}= \frac{8\pi G}{3} \rho_{c}.
$$

how will I prove that this maximum value will be followed by a contracting phase to the universe?

Well if you plug any value of $a > a_{max}$ into the Friedmann equation, you will get an negative value of $\dot{a}^2$. Hence we know that $a$ can not exceed $a_{max}$. But you might wonder whether $a$ can stay at $a_{max}$ without contracting? Hint: for the above example you can prove (as a home work for you)
$$
\ddot{a}|_{a = a_{max}} < 0.
$$
A: You don't know $\Omega(t)$. How $\Omega$ varies with time depends on the specific nature of $\Omega$, e.g. $\Omega_r \propto a^{-4}$, but $\Omega_m \propto a^{-3}$. A universe with different amounts of radiation and matter will have different behaviors for $\Omega(t)$.
However you do know that after this maximum value there will be a contracting phase. That comes from the definition of a maximum point - all nearby points have lower values than the maximum. So both approaching the maximum and going away from it, $a(t)$ must decrease. If it starts increasing again, there must be a minimum, which does not exist (from the differential equation you solved). Hence it keeps decreasing, until the assumptions that went into the calculation break down.
