  We theoretically study binary Bose-Einstein condensates trapped in a
single-well harmonic potential to probe the dynamics of collective atomic
motion. The idea is to choose tunable scattering lengths through Feshbach
resonances such that the ground-state wave function for two types of the
condensates are spatially immiscible where one of the condensates, located at
the center of the potential trap, can be effectively treated as a potential
barrier between bilateral condensates of the second type of atoms. In the case
of small wave function overlap between bilateral condensates, one can
parametrize their spatial part of the wave functions in the two-mode
approximation together with the time-dependent population imbalance $z$ and the
phase difference $\phi$ between two wave functions. The condensate in the
middle can be approximated by a Gaussian wave function with the displacement of
the condensate center $\xi$. As driven by the time-dependent displacement of
the central condensate, we find the Josephson oscillations of the collective
atomic motion between bilateral condensates as well as their anharmonic
generalization of macroscopic self-trapping effects. In addition, with the
increase in the wave function overlap of bilateral condensates by properly
choosing tunable atomic scattering lengths, the chaotic oscillations are found
if the system departs from the state of a fixed point. The Melnikov approach
with a homoclinic solution of the derived $z,\,\phi$, and $\xi$ equations can
successfully justify the existence of chaos. All results are consistent with
the numerical solutions of the full time-dependent Gross-Pitaevskii equations.

Quantum Gases