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C.T. van de Kamp
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On discrete and continuous state adaptive network models
With an application to self-organisation in swarming systems
Bachelor thesis
(2019)
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Carsten T. van de Kamp, Timon Idema, Johan Dubbeldam, Jos Thijssen, Tina Nane
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.
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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.