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Network topology plays an important role in governing the collective dynamics. Partial synchronization (PaS) on regular networks with a few non-local links is explored. Different PaS patterns out of the symmetry breaking are observed for different ways of non-local couplings. The criterion for the emergence of PaS is studied. The emergence of PaS is related to the loss of degeneration in Lyapunov exponent spectrum. Theoretical and numerical analysis indicate that non-local coupling may drastically change the dynamical feature of the network, emphasizing the important topological dependence of collective dynamics on complex networks.
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We study the relationship between the partial synchronous (PaS) state and the coupling structure in general dynamical systems. By the exact proof, we find the sufficient and necessary condition of the existence of PaS state for the coupling structure. Our result shows that the symmetry of the coupling structure is not the equivalent condition which is supposed before but only the sufficient condition. Furthermore, for the existence of the PaS state, the general structure is the equal-degree random.
By numerical simulations, we investigate the onset of synchronization of networked phase oscillators under two different weighting schemes. In scheme-I, the link weights are correlated to the product of the degrees of the connected nodes, so this kind of networks is named as the weight-degree correlated (WDC) network. In scheme-II, the link weights are randomly assigned to each link regardless of the node degrees, so this kind of networks is named as the weight-degree uncorrelated (WDU) network. Interestingly, it is found that by increasing a parameter that governs the weight distribution, the onset of synchronization in WDC network is monotonically enhanced, while in WDU network there is a reverse in the synchronization performance. We investigate this phenomenon from the viewpoint of gradient network, and explain the contrary roles of coupling gradient on network synchronization: gradient promotes synchronization in WDC network, while deteriorates synchronization in WDU network. The findings highlight the fact that, besides the link weight, the correlation between the weight and node degree is also important to the network dynamics.
Relay (or remote) synchronization between two not directly connected oscillators in a network is an important feature allowing distant coordination. In this work, we report a systematic study of this phenomenon in multiplex networks, where inter-layer synchronization occurs between distant layers mediated by a relay layer that acts as a transmitter. We show that this transmission can be extended to higher order relay configurations, provided symmetry conditions are preserved. By first order perturbative analysis, we identify the dynamical and topological dependencies of relay synchronization in a multiplex. We find that the relay synchronization threshold is considerably reduced in a multiplex configuration, and that such synchronous state is mostly supported by the lower degree nodes of the outer layers, while hubs can be de-multiplexed without affecting overall coherence. Finally, we experimentally validated the analytical and numerical findings by means of a multiplex of three layers of electronic circuits.the analytical and numerical findings by means of a multiplex of three layers of electronic circuits.
We show that the synchronized states of two systems of identical chaotic maps subject to either, a common drive that acts with a probability p in time or to the same drive acting on a fraction p of the maps, are similar. The synchronization behavior of both systems can be inferred by considering the dynamics of a single chaotic map driven with a probability p. The synchronized states for these systems are characterized on their common space of parameters. Our results show that the presence of a common external drive for all times is not essential for reaching synchronization in a system of chaotic oscillators, nor is the simultaneous sharing of the drive by all the elements in the system. Rather, a crucial condition for achieving synchronization is the sharing of some minimal, average information by the elements in the system over long times.
Synchronization on multiplex networks have attracted increasing attention in the past few years. We investigate collective behaviors of Kuramoto oscillators on single layer and duplex spacial networks with total cost restriction, which was introduced by Li et. al [Li G., Reis S. D., Moreira A. A., Havlin S., Stanley H. E. and Jr A. J., {it Phys. Rev. Lett.} 104, 018701 (2010)] and termed as the Li network afterwards. In the Li network model, with the increase of its spacial exponent, the networks structure will vary from the random type to the small-world one, and finally to the regular lattice.We first explore how the spacial exponent influences the synchronizability of Kuramoto oscillators on single layer Li networks and find that the closer the Li network is to a regular lattice, the more difficult for it to evolve into synchronization. Then we investigate synchronizability of duplex Li networks and find that the existence of inter-layer interaction can greatly enhance inter-layer and global synchronizability. When the inter-layer coupling strength is larger than a certain critical value, whatever the intra-layer coupling strength is, the inter-layer synchronization will always occur. Furthermore, on single layer Li networks, nodes with larger degrees more easily reach global synchronization, while on duplex Li networks, this phenomenon becomes much less obvious. Finally, we study the impact of inter-link density on global synchronization and obtain that sparse inter-links can lead to the emergence of global synchronization for duplex Li networks just as dense inter-links do. In a word, inter-layer interaction plays a vital role in determining synchronizability for duplex spacial networks with total cost constraint.
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