Stability Analysis of Flock and Mill rings for 2nd Order Models in Swarming

Stability Analysis of Flock and Mill rings for 2nd Order Models in Swarming.

read more: http://epubs.siam.org/doi/abs/10.1137/13091779X

We study the stability of flock ring and mill ring solutions for the following class of IBMs models, $$\left\{\begin{array}{l}\dot{x}_i=v_i\\\dot{v}_i=\left(\alpha – \beta|v_i|^2\right)v_i+\frac{1}{N}\sum_{j\neq i}\nabla W(x_j-x_i)\end{array}\right.\qquad i=1,\ldots, N; $$

where

$$W(x-y)=k(r)= \frac{r^a}{a}- \frac{r^b}{b},\qquad  \textrm{with }\quad r=|x-y|.$$

We are able to characterize stable region of parameters for flock ring and mill ring solutions.

Mill Ring Flock Ring
Mill Ring solution Flock Ring solution

Out of these stable regions, we recover different pattern and stable solutions. The aim of the next sections is to give a qualitative description of the emerging patterns.

Fattening Flock

Let consider the case of flock ring solutions, we fix $a=5$ and we take different values of $b=0.5, 0.8, 0.9$, as consequence flock rings are not stable for this set of parameters and   fat flock patterns emerge.

$N=1000$ $b=0.5$ $b=0.8$ $b=0.9$
$a=5$


Mill Ring  solution tuning $b$ and $|u_0|=\sqrt{\alpha/\beta}$.

 

Tuning $b$ $|u_0|$
N=1000

Fat Flock and  Fat Mill Emerging patterns

We take a random initial data in space and velocity, and we take $a=4$ and strong repulsion, i.e. $b$ small. After a similar dynamical evolution  what we observe is the emergence of different stable solutions a fat mill and a fat flock solution.

$N=100$ $b=0.005$ $b=0.001$
$a=4$