New methods to study the phase of chaos with multiple rotating center

In studying the phase synchronization of chaotic oscillators, the most fundamental is to define a proper phase variable for the chaotic oscillation. The definitions of the phase of a general chaotic trajectory are investigated in this paper. Considering a great deal of chaos whose trajectory projected on a plan is not a proper rotation, we present two simple procedures to get its instantaneous phase. One procedure called the direct - decomposition method is based on the idea of decomposition. The other called the tangency method is implemented by calculating angles in the tangent space. By the comparison of the numerical results for some examples, the advantages of these methods are discussed. In conclusion, the tangency method is proper for any chaotic oscillation and therefore may be widely used in the phase synchronization of chaos.