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?