Pattern prediction in networks of diffusively coupled nonlinear systems

Journal article (2018)

Authors

K. Rogov Eindhoven University of Technology
A. Pogromsky Eindhoven University of Technology
E. Steur Team Bart De Schutter - Mechanical, Maritime and Materials Engineering
Wim Michiels Katholieke Universiteit Leuven
H. Nijmeijer Eindhoven University of Technology
Research Group
Team Bart De Schutter (Mechanical, Maritime and Materials Engineering) (TU Delft)
Theory Applications of Complex Dynamical Networks Bifurcations in Chaotic or Complex Systems Limit Cycles in Networks of Oscillators
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Published Date

2018

Language

English

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Faculty

Mechanical, Maritime and Materials Engineering

Department

Delft Center for Systems and Control

Research Group

Team Bart De Schutter

Abstract

In this paper, we present a method aiming at pattern prediction in networks of diffusively coupled nonlinear systems. Interconnecting several globally asymptotical stable systems into a network via diffusion can result in diffusion-driven instability phenomena, which may lead to pattern formation in coupled systems. Some of the patterns may co-exist which implies the multi-stability of the network. Multi-stability makes the application of common analysis methods, such as the direct Lyapunov method, highly involved. We develop a numerically efficient method in order to analyze the oscillatory behavior occurring in such networks. We show that the oscillations appear via a Hopf bifurcation and therefore display sinusoidal-like behavior in the neighborhood of the bifurcation point. This allows to use the describing function method in order to replace a nonlinearity by its linear approximation and then to analyze the system of linear equations by means of the multivariable harmonic balance method. The method cannot be directly applied to a network consisting of systems of any structure and here we present the multivariable harmonic balance method for networks with a general system's structure and dynamics.