the initial conditions and the physical parameters, and the first necessary condition indicates that the Melnikov function of system is equal to zero, so the stable oscillations are Melnikov chaotic. The stable curves are presented out for different initial conditions, the sensitive dependence exhibits implicitly in the figures. Varying the damping strength, the regions of chaotic oscillations and regular oscillations can be changed into the other.

However, the general oscillations are not close to the separatrix solution, and the usual parameters are not in perturbative regime, in such case, the numerical method is very useful. Using the fourth Runge-Kutta method with variable step-width, the chaotic population oscillations are simulated by straightforward numerical integration of the dimensionless motion equations. When the damping disappears, with the increasing of $\Delta E_{1}$, the regular oscillations gradually become chaotic, and in the completely chaotic regime, the long-lived localization or MQST disappears. In the SPS, the single regular island is separated into many little islands, and then all islands are submerged into the chaotic sea. In the completely chaotic oscillations, the long-term localization or MQST disappears and the short-term localization or MQST can be changed from one of the BECs to the other through the tunnel 