  Using the Calogero-Moser model and the Nakamura equations for a multi-partite
quantum system, we prove an inequality between the mean bi-partite entanglement
rate of change under the variation of a critical parameter and the
level-curvature. This provides an upper bound for the rate of production or
distraction of entanglement induced dynamically. We then investigate the
dependence of the upper bound on the degree of chaos of the system, which in
turn, through the inequality, gives a measure of the stability of the entangled
state. Our analytical results are supported with extensive numerical
calculations.
