The Rotating Formation in a Selfpropelled Particle Model

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The Rotating Formation in a Self-propelled
Particle Model

• Ryan Lukeman
• Supervised by
• Dr. L Keshet, Dr. Y.X. Li

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Schooling Dynamics

• Individuals in a school tend to maintain roughly
  the same heading as other school members
  (polarized)
• Individuals do not want to collide with others,
  nor do they want to stray too far from neighbours
  (repulsion/attraction)
• Model this behaviour through interaction force,
  which acts between nearest neighbours

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A minimal modeling approach single nearest
neighbours
Attach Newtons equations of motion to each
particle
y
x
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Types of Stable Solutions
Varying autonomous self- propulsion leads to a
number of distinct solution types
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Another Stable Solution Emerges

For certain sets of initial conditions, and
certain interaction functions(), get stable
rotating solutions
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Is similar behaviour seen in nature?
"Schools change formation from one moment to the
next. A straight line becomes a parabola, which
becomes a cartwheel. Off Cape Cod one summer a
school of about 200 bluefin was milling. Then a
fish broke off the edge, followed by another, and
an echelon formed, a great diagonal line pulling
away from the mill like thread coming off a
spool. D. Whynott, Giant Bluefin
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More Basking Sharks
Offshore Nova Scotia!
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Army Ants
Separated from main columnar raid from heavy
rain, marched for 24 hours until death
(hand-drawn from observations).
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Analysis
1. How do we characterize the solution at
equilibrium?
-angular velocity
-group radius
2. What are these values? Can obtain in a
number of ways, such as a force balance in
centripetal and tangential directions.
Regardless, we get
Recall from our model equations
-drag term
- particles
-interaction function
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Stability

• Investigate the linear stability
• transform into new coordinates to gain structure
  in stability matrix
• perturb position and velocity of all particles
• solve for eigenvalues bounding eigenvalues lt 0
  gives conditions on
• g(d) for stability

n particles, transform, perturb
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Gah.
n complex quartics to solve Straightforward
calculation of eigenvalues not feasible -
numerically solve for a range of g(d) values
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Numerical solutions of eigenvalues
real part of eigenvalues vs. g(d)
n 4 case
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Simulations
Intersections give equilibrium distance b/t
particles
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g(d) outside stable region
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Moving groups
Cross a threshold where autonomous propulsion
dominates social forces, rotating group breaks
into linear group
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Questions?
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Questions?
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Questions?
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Questions?
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Questions?
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Questions?
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Questions?