H-1)/\Lambda ^{2}}\sec h\xi 
\begin{array}{lll}
& , & 
\end{array}
\xi =C_{0}+t\sqrt{\Lambda H-1}.$$ Where, constant $C_{0}$ is determined by the initial conditions. Considering the physical constrain, the amplitude of the fractional atomic population imbalance oscillations must be little than one, i.e.,$\left| z\right| _{\max
}\leqslant 1$ , so the abstract value $z$ of at points with the lowest potential energy must be little than one too, this requires the parameters satisfy $\sqrt{2(\Lambda H-1)/\Lambda ^{2}}\leqslant 1$, the atoms completely localize on one of the two condensates when $\sqrt{2(\Lambda
H-1)/\Lambda ^{2}}=1$. And if the amplitude of the separatrix solution is lager than one, i.e., $2\sqrt{(\Lambda H-1)/\Lambda ^{2}}>1$, then there only exists MQST.

Based upon the above analysis, we know that the pitchfork bifurcation occurs at the point with $\Lambda H=1$ for the time-independent symmetric trapping potential, that is to say the equilibrium point at the origin changes stability type and two new additional equilibrium points are created. For the asymmetric and time-dependent trapping potential, the bifurcation becomes more complex, the regular oscillations become chaotic through the route of period doubling.

Chaotic oscillations near the separatrix solution with small trap asymmetry
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It is very interesting to investigate the dynamics of the fractional population imbalance near 