Dynamics and Synchronization of Complex-Valued Ring Networks

Networks of coupled oscillators have been used to model various real-world self-organizing systems. However, the dynamics, especially chaos and bifurcation, of complex-valued networks are rarely investigated. In this paper, a ring network of interacting complex-valued van der Pol oscillators is studied to model the formation of ring dynamics. Although there are only stable limit cycles in a complex-valued van der Pol oscillator, chaos, hyperchaos, and coexisting chaotic attractors are observed from the ring network, which are analyzed by using the Lyapunov exponent spectrum, bifurcation diagram and 0-1 test. In addition, complexity analysis on nonlinear coefficients and coupling strengths illustrates that the range of parameters within the chaotic (hyperchaotic) region has positive correlation with the number of oscillators. It is shown that the chaotic bifurcation path is highly robust against the size variation of the ring network, which always evolves to chaos directly from period-1 and quasi-periodic states, respectively. Moreover, it is demonstrated that complete synchronization and phase synchronization of oscillations are stable in a large-scale ring network, while chaotic phase synchronization is unstable in a small-scale network.