Interactive particle systems with pair-wise interactions: linear (in)stability analysis and simulations
Y. Yao (TU Delft - Electrical Engineering, Mathematics and Computer Science)
D. Balagué Guardia – Mentor (TU Delft - Numerical Analysis)
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Abstract
In this project we study the stability of stationary solutions of interactive particle systems with shortrange repulsion and long-range attraction. Firstly we describe the model and discuss ring steady state solutions. We give a comprehensive and detailed proof of Theorem 2.1 done in [2]. This theorem gives the conditions for the (in)stability of stationary ring solutions. Omitting details of the computations, we present Theorem 3.1 in [2] to find the conditions for (un)stable stationary solutions in the situation when the mode is sufficiently large. Having these conditions altogether we find the region of the parameters in which the stationary solutions are stable. Finally, we use a steepest descent method for the simulations of the model. Figures of equilibrium states of the particles corresponding to various families of interaction forces will be shown.