On discrete and continuous state adaptive network models

with an application to self-organisation in swarming systems

Bachelor thesis (2019)

Authors

C.T. van de Kamp Applied Sciences

Contributors

T. Idema BN/Timon Idema Lab - AS (supervisor 1)
J.L.A. Dubbeldam Mathematical Physics - EEMCS (supervisor 1)
J.M. Thijssen QN/Thijssen Group - AS (supervisor 2)
G.F. Nane Applied Probability - EEMCS (supervisor 2)
Faculty
Electrical Engineering, Mathematics and Computer Science
Copyright: © 2019 Carsten T. van de Kamp
Modelling Differential Equations Partial Differential Equations Adaptive Networks Bifurcation Analysis Complex Networks Complex Adaptive Systems Swarming Self-organisation Collective motion
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Published Date

14-07-2019

Language

English

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Faculty

Electrical Engineering, Mathematics and Computer Science

Abstract

We consider adaptive network models with discrete and continuous state sets obeying dynamical rules that enable application to swarming systems. The 2-state adaptive network contains a supercritical pitchfork bifurcation in the transition between ordered and disordered stationary solutions. We derive an adaptive network model that works on a continuous state set and apply it to swarming motion in both a mean field and a moment closure approximation. In numerical solutions of the mean field approximation the relation between the variance of the ordered stationary distributions and the system parameters is given by a square root function. Cauchy distributions form a good fit to these steady state distributions, although they are not the analytic stationary solutions. We show that in numerical solutions of the moment closure approximation a bistable region is formed, in which the initial condition determines if the system ends up in an ordered or a disordered state. Further research could focus on finding the exact details of the corresponding subcritical pitchfork and saddle-node bifurcations and comparing the derived models to real-life swarming systems.