Emergence of Synchronization in Coupled Dynamical Systems:Influence of Network Topology and Coupling Delay
Final Report Abstract
We investigated the influence of network topology and time delay upon the formation of synchronized patterns on coupled cell networks. Network topology gives rise to robust synchrony patterns that are independent of specific forms of vector fields, but determined by the network coupling structure alone. These synchrony patterns are key to understanding organic behavior of networks that arise in different application context but share similar network dynamics. This resembles symmetry and symmetry patterns in equivariant dynamical systems, where many of the dynamics from symmetrically coupled systems are governed by symmetry alone. As part of the project, we tried to adopt ideas from equivariant analysis, especially from equivariant degree theory, to better understand the role of network topology in network dynamics. This resulted in the construction of two types of equivariant degrees: the lattice equivariant degree and the interior equivariant degree; and their application in network bifurcations caused by symmetric aspect of network topology: quotient symmetry and interior symmetry. Classifications of bifurcating branches by their symmetry and synchrony patterns were obtained using this method. They can be useful in many areas of network analysis such as neuroscience where individual neurons whose membrane potentials can be modeled by nonlinear dynamical systems, are inter-connected with each other forming webs of neural networks in nervous systems. Synchrony patterns arise from such networks may describe learning procedure and help understanding memory formations in human brain, since the huge amount of information processed every second by a human brain can hardly be stored in specific neural units, but rather by structured coordinations among them. We investigated the influence of time delay upon pattern formation in networks using a bi-directional ring where cells are connected to equal number of cells on the left as on the right. We gave a detailed bifurcation analysis of such models in relation to its network topology, coupling strength and time delay. This includes a bifurcation diagram and a list of all bifurcating branches (for stable and unstable) together with their symmetry and synchrony forms. This result can be useful, for example, in understanding chimera states in coupled oscillators, where many of such patterned dynamics have been observed and of practical interest.
Publications
- Symmetry analysis of coupled scalar systems under time delay, Nonlinearity 28 (2015) 795-824
F. Atay and H. Ruan
(See online at https://doi.org/10.1088/0951-7715/28/3/795) - An equivariant degree theory for networked dynamical systems, Habilitation Thesis, Fachbereich Mathematik, Universität Hamburg, 2016. XIII, 124 S.
H. Ruan
- Synchrony and elementary operations on coupled cell networks, SIAM J. Appl. Dyn. Syst., 15(1) (2016) 322-337
M. Aguiar, A. Dias and H. Ruan
(See online at https://doi.org/10.1137/140980119)
