\[18\]. With a slowly varying trap, the nonlinear resonances and chaos exist in the oscillations of the fractional imbalance. The conditions for chaotic macroscopic quantum tunneling phenomena were obtained with the use of the Melnikov function approach, and the chaotic oscillations depend on the frequency and modulation amplitude sensitively. For the rapidly varying case, the averaged system was given out by using the multiscale expansion method. They also considered the macroscopic quantum tunneling and resonances in coupled BECs with oscillating atomic scattering length \[19\]. The chaotic oscillations in the relative atomic population due to the overlaps between nonlinear resonances were showed. And the possibility of stabilization of the unstable -mode regime was derived from the analyzing of the oscillations in the rapidly varying case.

We know the laser barrier position and the laser beam intensity of the laser beam in the trap can be modified in experiments, so the trap asymmetry and the amplitude of the tunneling between the coupled BECs can be time-dependent. In our present paper, we will analyze the chaotic oscillation of the fractional population imbalance between two Josephson coupled BECs with time-dependent asymmetric trap potential, using both analytical and numerical approach. The structure of this paper 