Phase synchronization between nonlinearly coupled Rossler systems

Phase synchronization between nonlinearly coupled systems with 1:1 and 1:2 resonances is investigated. By introducing a concept of phase for a chaotic motion, it is demonstrated that for different internal resonances, with relatively small parameter epsilon, the difference between the mean frequencies of the two sub-oscillators approaches zero. This implies that phase synchronization can be achieved for weak interaction between the two oscillators. With the increase in coupling strength, fluctuations of the frequency difference can be observed, and for the primary resonance, the amplitudes of the fluctuations of the difference seem much smaller compared to the case with frequency ratio 1:2, even with the weak coupling strength. Unlike the enhanced effect on synchronization for linear coupling, the increase in nonlinear coupling strength results in the transition from phase synchronization to a non-synchronized state. Further investigation reveals that the states from phase synchronization to non-synchronization are related to the critical changes of the Lyapunov exponents, which can also be explained with the diffuse clouds.