  We have investigated the chaotic atomic population oscillations between two
coupled Bose-Einstein condensates (BEC) with time-dependent asymmetric trap
potential. In the perturbative regime, the population oscillations can be
described by the Duffing equation, and the chaotic oscillations near the
separatrix solution are analyzed. The sufficient-necessary conditions for
stable oscillations depend on the physical parameters and initial conditions
sensitively. The first-order necessary condition indicates that the Melnikov
function is equal to zero, so the stable oscillations are Melnikov chaotic. For
the ordinary parameters and initial conditions, the chaotic dynamics is
simulated with numerical calculation. If the damping is absent, with the
increasing of the trap asymmetry, the regular oscillations become chaotic
gradually, the corresponding stroboscopic Poincare sections (SPS) vary from a
single island to more islands, and then the chaotic sea. For the completely
chaotic oscillations, the long-term localization disappears and the short-term
localization can be changed from one of the BECs to the other through the route
of Rabi oscillation. When there exists damping, the stationary chaos
disappears, the transient chaos is a common phenomenon before regular stable
frequency locked oscillations. And proper damping can keep localization
long-lived.
