  In present work, we show how the threshold expansion formula of $N$ identical
bosons in finite volume may be derived by iterations of Faddeev-type coupled
dynamical equations. The energy shift of $N$-boson system near threshold is
dominated by zero momenta mode of $N$-body amplitudes with all particles nearly
static. The dominant zero momenta mode and sub-leading non-zero momenta mode
contributions are connected through finite volume Faddeev-type coupled
dynamical equations. Eliminating non-zero momenta modes by iterations
ultimately yields an analytic expression that can be solved by threshold
expansion.
