Poincaré section analysis to measure synchronization in coupled Hamiltonian systems

As a novel synchronization phenomenon, measure synchronization (MS) in Hamiltonian systems is characterized by a commonly shared phase space that has invariant measure. Although there has been some discussions of the underlying mechanism, a general explanation of MS is still lacking. In this paper, we reveal the essence of MS by using Poincaré section analysis. A two coupled Bosonic Josephson junction model is employed to perform the analysis as an illustration. It is shown that coupled Hamiltonian systems exhibit separatrix crossing behavior at quasiperiodic MS transition. Moreover, using this analysis method we find that chaotic MS is the result of separatrix chaos. It is concluded that the Poincaré section technique is a general method to explain MS, and the direct correspondence between separatrix behavior and the MS transition is a universal property in coupled Hamiltonian systems.