"If $\lambda$ is greater than than 1, the system explodes." Why does the system explode? David Romer in his textbook Advanced Macroeconomics (Third Edition) writes regarding the speed of convergence of the Diamond model the following:
(Pg. 83)

Equation (2.60) [$k_{t+1}={1\over{(1+n)(1+g)}}{1\over{2+\rho}}(1-\alpha)k_t^\alpha$] gives $k_{t+1}$ as a function of $k_t$. The economy is on its balanced growth path when these two are equal, that is $k^*$ is defined by
$k^*={1\over{(1+n)(1+g)}}{1\over{2+\rho}}(1-\alpha)k^{* \alpha}\space\space\space\space\space\space\space\space\space\space\space\space$ (2.61)
Solving this expression for $k^*$ yields
$k^*=\left[{(1-\alpha)\over{(1+n)(1+g)(2+\rho)}}\right]^{1 \over{1-\alpha}}\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space$ (2.62)
[...] We can also find out how quickly the economy converges to the balanced growth path. To do this, we again linearize around the balanced growth path. [...] Thus:
$k_{t+1} \simeq k^*+\left( {{dk_{t+1}}\over{dk_t}}\Big{|}_{k_t=k^*}\right) \left(k_t-k^*\right)\space\space\space\space$ (2.64)
Let $\lambda$ denote $dk_{t+1}/dk_t$ evaluated at $k_t=k^*$. With this definition, we can rewrite (2.64) [...]
$k_t-k^* \simeq \lambda^t (k_0-k^*)\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space$ (2.65)
Where $k_0$ is the initial value of $k$

My Question
(Pg 84)

[...] if $\lambda$ is greater than 1, the system explodes

What does it mean the system explodes when $\lambda$ is greater than 1? Why does it explode when $\lambda$ is greater than 1?
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

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