Nonlinear Dynamics

Items for this website e.g. publications,
links, comments and opinions can be
e-mailed to Roumen Anguelov at
roumen.anguelov@up.ac.za

Project:  Stationary states of diffusion aggregation models

1. One dimensional simulation showing the evolution in time of the solution to the aggregation diffusion equation where the attractive kernel has compact support. The initial condition is given by a random distribution. (Comp1D)

2. Two dimensional simulations showing the side and top views of the evolution in time of the solution to the aggregation diffusion equation with a compactly supported attractive kernel. The initial condition is given by a random distribution. (Patt2Dtop and Patt2Dside)

3. Two-dimensional simulation showing the top view of the evolution in time of the solution to the aggregation diffusion equation, given a time dependent attractive kernel with infinite support. An exogenous force, given by a cluster of four Gaussian functions situated at the bottom left corner of the domain, is incorporated into the model. The final state of the simulation is a single swarm. This is a result of the attractive kernel having infinite support, preventing the formation of a stationary state with multiple swarms. (ChangingW2DInf)

4. Two-dimensional simulation showing the top view of the evolution in time of the solution to the aggregation diffusion equation, given a time dependent attractive kernel with compact support. A randomly distributed exogenous force is incorporated into the model.  The social attractive force is initiated at the start of the simulation, in absence of the exogenous force.  At time 0:37 in the video, the social attractive force is suspended and the exogenous force is initiated. At time 1:03, the attractive force is initiated again, while the exogenous force is suspended. This process allows for swarms to break away into a random distribution and then aggregate again over time, forming different patterns. The initial condition is given by a random distribution. (ChangingW2DComp)