Synchronization and phase balancing in networked chaotic systems with ring lattice and small world networks

Strategies for achieving fully synchronized, phase-synchronized, and phase-balanced oscillations in networked chaotic systems with a strong 2D oscillatory component on ring and ring lattice networks are introduced. These strategies are derived from a generalized Kuramoto model of coupled phase oscillators along with a novel reshaping strategy manifested by higher harmonics of the phase differences between nearest neighbors in order to globally stabilize the desired state by destabilizing undesired states. To show that the desired state is obtained in the network of chaotic systems, two steps are employed. First, both local and Lyapunov-based stability analysis are employed on the corresponding generalized Kuramoto model along with the reshaping strategy. Second, for synchronization problems the Master Stability Function is then employed to determine the range of the coupling parameter(s) necessary to achieve the desired synchronized oscillations for the networked chaotic systems. Two strategies are suggested for achieving phase-balanced oscillations using desired relative angles and a time-inverted form in the generalized Kuramoto oscillator. Illustrative examples of networked Rössler oscillators coupled in ring and ring lattice topologies illustrate the efficacy of these methods. Finally, chaos synchronization is demonstrated for small world networks of Rössler oscillators obtained by rewiring various ring lattice topologies.