Preserving Bifurcations through Moment Closures

Christian Kuehn, Jan Mölter
Nonlinear Sciences, Adaptation and Self-Organizing Systems, Adaptation and Self-Organizing Systems (nlin.AO), Dynamical Systems (math.DS)
2023-07-06 16:00:00
Moment systems arise in a wide range of contexts and applications, e.g. in network modelling of complex systems. Since moment systems consist of a high or even infinite number of coupled equations, an indispensable step to obtain a low-dimensional representation that is amenable to further analysis is in many cases to select a moment closure. A moment closure consists of a set of approximations that express certain higher-order moments in terms of lower-order ones, so that applying those leads to an closed system of equations for only the lower-order moments. Closures are frequently found drawing on intuition and heuristics in trying to come up with quantitatively good approximations. In contrast to that, we propose an alternative approach where we instead focus on closures giving rise to certain qualitative features such as bifurcations. Importantly, this fundamental change of perspective provides one with the possibility to classify moment closures rigorously in regards to these features. This makes the design and selection of closures more algorithmic, precise and reliable. In this work, we carefully study the moment systems that arise in the mean-field descriptions of two widely known network dynamical systems, the SIS epidemic and the adaptive voter model. We derive conditions that any moment closure has to satisfy so that the corresponding closed systems exhibit the transcritical bifurcation that one expects in these systems coming from the stochastic particle model.
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