larger.

Numerical simulation
====================

In general, the atomic population oscillation is far away from the separatrix solution, and then the oscillation dynamics can not be obtained from the previous analytical method. In this section, using the fourth Runge-Kutta method with variable step-width, the chaotic population oscillations are simulated by straightforward numerical integration of the motion equations of the dimensionless model of system (4) with constant parameter $K$, and the trap asymmetry is in form of $\Delta E=\Delta
E_{0}+\Delta E_{1}\sin \omega t$, and the damping of the population oscillation is $-\eta z(t)$. In the time -independent symmetric trap $%
(\Delta E=0)$, because of the damping, both Rabi oscillation and MQST reach an equilibrium state with zero population imbalance, see Fig. 4 (B) and (F), for the time-independent asymmetric case, the equilibrium state departure from the zero population imbalance, see (D) and (H) of Fig.4. Ignoring the damping effects, the oscillations are regular, they contain two different kinds, Rabi oscillation and MQST,see (A), (E), (C) and (G) of Fig.4.

For the time-dependent asymmetric trap potential, the chaotic oscillation emerges out. Sampling a single trajectory every period of the varying of the trap asymmetry, then we can obtain the stroboscopic Poincare section (SPS). 