Chaotic atomic population oscillations between two coupled Bose-Einstein   condensates with time-dependent asymmetric trap potential
  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. 
Chaohong Lee, Lei Shi, Xiwen Zhu, Kelin Gao, Wenhua Hai, Yiwu Duan,   Wing-Ki Liu