Simulation on Birds Swarms

Solving ODE with Runge-Kutta method and automatic differentiation.

Models

Assume N birds flying in swarms. Let x be positon of i-th birds, and v0 be preferred speed. Newton's equation of each bird is

$\dot v_i = \gamma (v_0^2 - v_i^2) v_i + \sum_{j \neq i } f(x_j - x_i) + \sum_{j \neq i } g(x_j - x_i, v_j - v_i)$

where

$f(x) = - \nabla V(x)$

$V(x) = A_1 \exp (-(x/R_1)^2) - A_2 \exp (-(x/R_2)^2)$

$g(x, v) = A_v \exp (-(x/ R_v)^2) v$

V is Mexican-hat potential and f expresses preferred distance between birds, whereas g aligns their velocity with that of neighborhood.

Results

Reference

Gros C.(2015). Complex and adaptive dynamical systems , Springer