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Numerical Methods for Pattern and Bifurcation Analysis in Complex Networks

Boek - Dissertatie

The thesis focuses on analyzing the phenomenon of pattern formation in (delay) coupled nonlinear systems. The starting points are works of Turing on morphogenesis and pattern formation, and Smale on determining conditions under which initially globally asymptotically stable systems being interconnected, exhibit emergent oscillatory behavior, possibly in a structured way (e.g. synchronous and partial synchronous oscillatory behavior). One of the major open questions concerns the prediction or analysis of oscillatory patterns in the neighbourhood of the bifurcation point of complex networked dynamical systems. In such complex networks, co-existence of several patterns is not rare and this makes common methods of analysis cumbersome to use. The thesis focuses on a numerical way of analysis, starting from the multivariable harmonic balance (MHB) method developed by Iwasaki. Iwasaki's work shows that the MHB method is a powerful tool for determining oscillatory profiles. However, it does not exploit network structures, which is important with respect to scalability to large networks, neither does it provide any information about the system's behaviour if the system is not in the neighbourhood of the bifurcation point.In order to address the challenge described above, numerical approaches are developed based on the MHB method for networks of Lur'e systems with a more general structure and dynamics, which allow us to predict the oscillatory patterns appearing in the networks, even if some of them co-exist. The main contributions are:1. Equal amplitude oscillations. The thesis introduces a semi-analytical - semi-numerical approach to predict oscillations with equal amplitudes in complex networks described by regular graphs. The approach determines a harmonic profile (frequency, amplitude, phases) that is encoded in the largest eigenvalue and its corresponding eigenvector of the coupling matrix, and it successfully decouples the individual dynamics of the nodes and the topology of the network. The MHB method is efficient and provides very good approximations of oscillation patterns obtained by numerical simulation in the neighborhood of bifurcation points.2. General amplitude case. A modification and generalization of the MHB method is proposed. The new approach allows to determine oscillatory profiles without restriction on amplitudes. The problem of determining an oscillatory profile is converted into an optimization problem. The interior point method is used to find optimal solutions. The optimization problem is, typically, a non-convex problem and multiple minima may exist, corresponding to the co-existence of oscillatory patterns. In order to solve these problems, we combine the optimization procedure with a multistart approach, which is feasible because of the small number of unknowns in the MHB setting. This approach keeps the accuracy of equal amplitude case approach and deals with co-existing modes.3. Pattern analysis in networks with time delays. An MHB-method is developed for networks with time-delays. It is shown that the approach can be used as the initial data for a numerical bifurcation analysis. An oscillatory profile now consists of frequency, offset, amplitudes and phases. The optimization procedure is modified, and the convergence time is significantly decreased. In addition, it is demonstrated that the MHB approach can be used for basic bifurcation analysis and for an accurate prediction of phases and frequency of oscillations even out of the neighbourhood of the bifurcation point.4. Delay induced full synchronization. In certain networks, full synchronization can be observed only in presence of time-delay in the coupling. In this thesis, this delay induced full synchronization phenomenon is studied in ring networks. Numerical analysis of this phenomenon is performed by means of the MHB approach and the bifurcation/continuation software package DDE-BIFTOOL. The analysis provides understanding of the phenomenon and pointed out the importance of usage the MHB and bifurcation analysis together in a complementary way for analysis of the behavior of complex networks.
Jaar van publicatie:2021
Toegankelijkheid:Open