Collective dynamics in a non-dissipative two-coupled pendulum system

We study the collective dynamics of a non-dissipative two-coupled pendulum system, including phase synchronization (PS) and measure synchronization (MS). We find that as the coupling intensity between the two pendulums increases, the PS happens prior to the MS. We also present a three-dimensional phase space representation of MS, from which a more detailed information about evolution can be obtained. Furthermore, the order parameters are introduced to describe the phase transition between PS and MS. Finally, through the analysis of the Poincare sections, we show that the system exhibits separatrix crossing behavior right at the MS transition point, and as the total initial energy increases, the Hamiltonian chaos will arise with separatrix chaos at the chaotic MS transition point.