limit cycle with a non-zero time-averaged value of the fractional population imbalance $z$, so the atoms are localized on one of the condensates. Amazedly, for large $\Delta E_{1}$ (1.700), due to the damping effects, the final $\frac{1}{6}$ FL oscillation possesses of a non-zero time-averaged value of $z$, comparing with the non-damping regime (Fig. 6), one can obtain that the proper damping can keep the MQST long-lived.

Summary and discussion
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Using both analytical and numerical methods, we analyzed the chaotic oscillations between two coupled Bose-Einstein condensates with time-dependent asymmetric trap potential. The trap asymmetry has been chosen as $\Delta E=\Delta E_{0}+\Delta E_{1}\sin \omega t$, this can be realized by varying the laser barrier position of the laser beam which possesses of fixed value of intensity. The damping of the oscillations of the fractional population imbalance is in form of $-\eta z(t)$, it commonly exists in the two interacting condensates with different hyperfine levels in a single harmonic trap.

In the perturbative regime, the population oscillations have been depicted with Duffing equation, the chaotic oscillations near the separatrix solution are detailed. The form of the general solution and the sufficient-necessary conditions for stable oscillations are obtained. These conditions sensitively depend on 