"If $\lambda$ is greater than than 1, the system explodes." Why does the system explode?
Eq. (2.64) can be written as (at first order)

$$
k_{t+1} - k^* = \lambda (k_t - k^*) \tag{1a}
$$ 

Define the quantity $\kappa_t$ as 

$$
\kappa_t \stackrel{\rm def}{=} k_t - k^* \tag{2}
$$

So that

$$
\kappa_{t+1} = \lambda \kappa_t \tag{1b}
$$

This can be solved very easy by realizing that

\begin{eqnarray}
\kappa_1 &amp;=&amp;\lambda \kappa_0 \\
\kappa_2 &amp;=&amp;\lambda \kappa_1 = \lambda(\lambda\kappa_0) = \lambda^2\kappa_0 \\
&amp;\vdots&amp; \\
\kappa_t &amp;=&amp; \lambda^t \kappa_0 \tag{3}
\end{eqnarray}

So the dynamics is $\kappa$ is driven by the value $\lambda^t$. You can see that if $\lambda = 1$ then $\lambda^t=\lambda$ for each $t$ and $\kappa$ remains constant. However, if $\lambda &gt; 1$ the values $k^t$ will become increasingly large (explodes) as $t$ increases, that is 

$$
\lim_{t\to \infty} \kappa_t \stackrel{\lambda &gt; 1}{=} \infty
$$

Similarly

$$
\lim_{t\to \infty} \kappa_t \stackrel{0&lt; \lambda &lt; 1}{=} 0
$$