is as follows. In this section, we briefly review both experimental and theoretical developing of BEC, and also show the purpose of the paper. The effective particle model for the oscillations of the fractional population imbalance is derived from the two-mode GPE in the next section. In the case of small time-dependent trap asymmetry and small damping, using the analytical method \[20,21\] developed by us, the chaotic dynamics of the fractional imbalance near the separatrix solution is analyzed in details in the third section. The conditions for chaotic oscillation and criteria for the onset of chaos are also obtained. And the regions of regular and chaotic oscillation are showed. In the fourth section, the population oscillations are simulated with numerical approach for arbitrary time-dependent trap asymmetry and damping. In the last section, a brief precise summary and discussion are given out.

The effective particle model for the oscillations of the fractional population imbalance
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Ignoring the damping and finite-temperature effects, the problem of coupled Bose-Einstein condensates in a double well trap can be described with the following nonlinear two-mode dynamical equations $$\begin{aligned}
i\hbar \frac{\partial \psi _{_{1}}}{\partial t} &=&[E_{1}+U_{1}\left| \psi
_{_{1}}\right| ^{2}]\psi _{_{1}}-K\psi _{_{2}},  \nonumber \\
i\hbar \frac{\partial \psi _{_{2}}}{\partial t} 