Networks of delay-coupled delay oscillators

The analysis of time-delayed dynamics on networks may help to understand many systems from physics, biology, and engineering, such as coupled laser arrays, gene-regulatory networks and complex ecosystems. Beside the complexity due to the network structure, the analysis is further complicated by the presence of the delays. Delay systems are in general infinite dimensional and thus can display complex dynamics as oscillations and chaos. The mathematical difficulties related to the delays hinders the analysis of delay networks. Thus, little is known yet about basic relations between network structure and delay dynamics. It has been shown that networks without delays can be studied efficiently with the generalized modeling approach, which analyzes the stability of an assumed steady state by a direct parametrization of the Jacobian matrix. In this thesis, I demonstrate the extension of the generalized modeling approach to delay networks and analyze networks of delay-coupled delay oscillators, with delayed auto-catalytic growth on the nodes and delayed transport between nodes. For degree-homogeneous networks (DHONs), in which each node has the same number of links, the bifurcation lines that border the stable areas can be calculated analytically, where the topology of the network is described only by the eigenvalues of the adjacency matrix. For undirected networks, the stability pattern in the parameter space of growth and transport delay is governed by two periodic sets of tongues of instability, which depend on the largest positive and the smallest negative eigenvalue. The direct relation between the eigenvalue and the bifurcation lines allows us to predict stability patterns for networks with certain topological properties. Thus, bipartite networks display a characteristic periodicity of tongues. In order to analyze the stability of degree-heterogeneous networks (DHENs), I apply a numerical sampling method based on Cauchy\'s Argument Principle. The stability patterns of these networks resembles the pattern of DHONs, which is governed by the two periodic sets. For networks with sufficiently many links, one set disappears, and the stability of DHENs can be approximates by the stability of a fully-connected network with the same average degree. However, random DHENs tend to be more stable than DHONs, and DHENs with a broad degree-distribution tend to be more stable than DHENs with a narrow distribution. Thus, such networks are more likely to give rise to amplitude death, i.e. the stabilization of an unstable steady state through diffusive coupling. The stability pattern of DHENs can be qualitatively different than the pattern in DHONs. However, for small growth delays, close to the critical delay of the single node system, the bifurcation lines of all DHENs with the same average degree coincide. This, is particularly interesting, because there the stability depends on a global property of the network, which suggests a diverging interaction length. In summary, the extension of generalized modeling to time-delay networks reveals basic relations between the delay dynamics and the topology. The generality of our model should allow to apply these results to a large class of real-world systems