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 |

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.

$N=100$ | $b=0.005$ | $b=0.001$ |

$a=4$ |