Optimal and Poor Synchronizations of Directionally Coupled Phase-Coherent Chaotic Oscillators

We study directionally coupled phase-coherent chaotic oscillators in complex networks. We introduce an adjusted Lyapunov function that incorporates the frequencies of the oscillators and the interaction structure. Using the well-known Rössler system as an example, we address two optimization problems: frequency allocation and network design. Through numerical experiments, we demonstrate that the systematic synchrony can be effectively enhanced or inhibited by minimizing or maximizing the objective function, respectively. We then delve into the relationship between the structural and dynamical properties that lead to optimal synchronization. Interestingly, we observe a positive correlation between nodal in-degrees and frequency magnitudes, indicating that nodes with higher in-degrees tend to exhibit larger frequency magnitudes. On the other hand, we also find a negative correlation between nodal frequency and adjacent in-frequencies, suggesting that nodes with higher frequencies tend to be surrounded by nodes with lower frequency values. Finally, we explore the connections between degree correlations and optimal synchronization. We find that when minimizing the objective function, the presence of degree correlations always inhibits the systematic synchrony for frequency allocation, while the act of network design causes the correlations to become negative.