Ober­sem­in­ar "Nu­mer­ics for PDEs": Sören von der Gracht (Uni­versität Pader­born) "Mod­el in­de­pend­ent fea­tures and di­men­sion re­duc­tion of net­work dy­nam­ic­al sys­tems"

29 January 2026, Tuesday, 14:15, in seminar room J2.138

Title:
Model independent features and dimension reduction of network dynamical systems

Abstract:
Many dynamical systems in fields such as neuroscience (the workings of the brain), systems biology (metabolic networks), and robotics (robot swarms) exhibit the structure of a network: they consist of nodes (neurons, proteins, robots) interconnected by edges. The specific architecture of these interactions can give rise to remarkably complex dynamics that go far beyond the behavior of individual nodes. Prominent examples include synchronization and intricate branching patterns in bifurcations—phenomena that do not arise in general dynamical systems lacking a network structure.

Modern dynamical systems theory seeks to characterize model-independent phenomena—geometrical or topological features of trajectories that are robust to the specific form of the equations of motion. Unfortunately, network dynamical systems remain poorly understood in this framework, as most classical tools fail to distinguish networks from arbitrary dynamical systems. Recently, new mathematical approaches tailored to network problems have emerged—and strikingly, they all share a common foundation: the algebraic nature of network structure.

In this talk, I will explain how structural features of a network give rise to (non-classical) symmetries in the governing equations. In the second part, I will demonstrate how these symmetries can be systematically exploited to achieve dimension reduction in bifurcation problems. This reduction enables a tractable, model-independent analysis of the generic bifurcation behavior associated with a given interaction structure—offering a powerful pathway to understanding complex network dynamics.