  We study the spread complexity in two-mode Bose-Einstein condensations and
unveil that the long-time average of the spread complexity $\overline{C}_{K}$
can probe the dynamical transition from self-trapping to Josephson oscillation.
When the parameter $\omega$ increases over a critical value $\omega_c$, we
reveal that the spread complexity exhibits a sharp transition from lower to
higher value, with the corresponding phase space trajectory changing from
self-trapping to Josephson oscillation. Moreover, we scrutinize the
eigen-spectrum and uncover the relation between the dynamical transition and
the excited state quantum phase transition, which is characterized by the
emergence of singularity in the density of states at critical energy $E_{c}$.
In the thermodynamical limit, the cross point of $E_{c}(\omega)$ and the
initial energy $E_{0}(\omega)$ determines the dynamical transition point
$\omega_c$. Finally, we show that the different dynamical behavior for the
initial state at a fixed point can be distinguished by the long-time average of
the spread complexity, when the fixed point changes from unstable to stable.
