DFG-RSF: Complex dynamical networks: effects of heterogeneity, adaptivity, and delays

点击翻译按钮获取中文摘要

Two types of network models have been mostly considered so far in the literature: static and dynamic networks. In static networks, only the structure of connections is of importance. In contrast, in dynamical networks the state of each unit depends on time, and its dynamics is influenced by the interaction with other units. In spite of the large number of interesting and significant results, the general theory of dynamical networks is only at the stage of early development, which is due to the extreme complexity and broadness of the subject. This project addresses a fundamental problem of the theory of complex nonlinear dynamical networks. More specifically, it considers the following aspects of complexity: inhomogeneous topological structure, adaptive connections, and the presence of coupling delays. The types of networks to be studied include but are not limited to neural and neuromorphic networks, wireless sensor networks, and networks of electronic oscillators. The main objectives include: I. Development of the definition of the term ''complexity'' for dynamical networks, as well as the qualitative criteria and quantitative measures for their complexity. This direction is very important since the term ''complex network'' has not been strictly defined so far, although its wide use in nonlinear dynamics has long been calling for elaboration of its definition.II. Development of neuromorphic network models with complex modular structure and analytic methods of study of such models. We will construct models reproducing structural properties of real neuronal networks, primarily their modularity, and study the link between the connectivity characteristics and their function. III. Study of the influence of coupling delays and adaptivity of connections on the dynamics of pulse-coupled networks of phase oscillators. The main objective is to study the suppression of the global synchronization and emergence of phase patterns and other complex dynamical regimes. We aim to understand how delayed coupling and plasticity affect the bifurcation mechanisms for the desynchronization in pulse-coupled systems. IV. Development and study of electronic prototypes of complex networks with coupling delays demonstrating nontrivial dynamical regimes. Particularly, we plan to implement a network of oscillators interacting via long chains of excitable units and demonstrating novel types of chimera-like behavior.