G. X. Qi

Fast synchronization in neuronal networks

We study the fast synchronization in complex networks of coupled Hindmarsh-Rose neurons. The relation between the maximal Lyapunov exponent corresponding to the least stable transverse mode and the speed of synchronization is given, based on which we can obtain an optimal value of global coupling strength, with which the network synchronizes with the minimal synchronization time. The speed limits of several kinds of complex networks (small-world, scale-free, and modular) with different eigenratios are studied. Finally, we extend to the modified Hodgkin-Huxley neuron case.

Dynamical symmetry and synchronization in modular networks

The effects of dynamical symmetry on the chaotic pattern synchronization in modular networks have been studied. It is found that the topological and the coupling symmetries between modules (subnetworks) can both enhance and speed up the chaotic pattern synchronization between modules. The calculation of Lyapunov exponent shows that this dynamical symmetry is a necessary condition for complete chaotic pattern synchronization in both modular networks composed by identical oscillators and heterogeneous modular networks if the states of nodes are much different from one another.

Searching good indicators for predicting the synchronizability of heterogeneous networks and beyond

In searching the indicators of synchronizability of complex networks, the maximal betweenness centrality is usually proposed as a good indicator. However, we find that a better indicator for synchronizability in heterogeneous networks is the maximal degree from both the average results and the individual realization of a network, which usually makes more sense in practice. Both the largest eigenvalue and the eigenratio are found, in a wide range of heterogeneity, to hold linear relations with the maximal degree. Our results may provide some clues to mathematically solve the relation between the synchronizability and the network degree, also to the optimal strategies to enhance synchronization.

Time-delay–induced synchronization in complex networks: Exploring the dynamical mechanism

We show that delayed coupling could induce or enhance stable chaotic synchronization in complex networks, where no or weak synchrony would exist for the usual instantaneous coupling. The mechanism behind this phenomenon reveals that the phase structure of the coupled chaotic oscillator plays the main role. Numerical results for Rossler and Lorenz oscillators as network nodes confirm the generality of this phenomenon. Together with our previous findings, we highlight the importance of taking the dynamical structure into account when studying or designing large-scale networks for stable synchronization.

Manipulating synchronous states by dynamical flow in complex networks

We show that if the dynamical flow, i.e., the non-vanishing coupling term, exists between nodes in synchronized networks, a wide variety of stable synchronous states of complex networks may occur, which may differ substantially from the dynamics of an individual isolated node. Stability analysis of the dynamics of Hindmarsh-Rose and foodweb networks shows that controlling this dynamical flow can greatly enhance the synchronization and generate both chaotic and regular synchronous states for whatever state of an isolated node. Our results provide a possibility for the control of synchronization in complex networks by the manipulation of the dynamical flow.

SIZE INSTABILITIES IN THE RING AND LINEAR ARRAYS OF CHAOTIC SYSTEMS

We investigate the dynamical stabilities of ring and linear arrays of chaotic oscillators with asymmetric nearest-neighbor and long-range couplings. It is shown that the instabilities of complete chaotic synchronization occur as the numbers of oscillators are increased beyond critical values which depend on the coupling schemes and coupling parameters of the systems. Based on the master stability function and eigenvalue analysis methods, we give the semi-analytical relations between the critical values and the coupling parameters. Results are demonstrated with numerical simulations in a set of coupled Lorenz oscillators.