E=\frac{E_{1}-E_{2}}{2K}+\frac{U_{1}-U_{2}}{4K}N_{T}.$$ $$\Lambda =UN_{T}/2K 
\begin{array}{lll}
& , & 
\end{array}
U=(U_{1}+U_{2})/2.$$ Because of the overlapping condensate, there exists different kind of damping for different type of overlapping. For examples, if we take into account a noncoherent dissipative current of normal-state atoms, the differential equation which describes the oscillations of the atomic population including the damping term $-\eta d\phi /dt$; and the damping has the form $-\eta z(t)$ for the two interacting condensates with different hyperfine levels in a single harmonic trap. In the case of time-independent parameter $K$, we can rescale $(\frac{2K}{\hbar })t$ to a dimensionless time $t$. The motion of the fractional population imbalance and relative phase is very similar to a nonrigid pendulum. The Hamiltonian of the unperturbed $%
(\Delta E=cons\tan t,\eta =0)$ dimensionless system is as follows $$H=\frac{\Lambda z^{2}}{2}+\Delta Ez-\sqrt{1-z^{2}}\cos \phi .$$ The corresponding canonical equations of the motion are equivalent to the equations of the oscillations of the atomic population imbalance and relative phase, their forms are $$\frac{d\phi }{dt}=\frac{\partial H}{\partial z} 
\begin{array}{lll}
& , & 
\end{array}
\frac{dz}{dt}=-\frac{\partial H}{\partial \phi }.$$ For the time-independent trapping potential, the energy of the above system is conservative.

In order to see the oscillations of the fractional population imbalance more transparently, we introduce 