Pendulum without friction
Time series: angle and velocity are 180o out of phase
Pendulum without friction
http://webphysics.davidson.edu/applets/anim
ator4/demo_pendulum.html
Phase space: plot angle vs velocity. This shows a “limit
cycle” in phase space.
Pendulum without friction
Larger initial angles put you in a larger limit cycle.
Pendulum with friction: fixed point
http://www.mcasco.com/Order/oap2dpsd.html
“Fixed point” attractor in phase space: gradually
spirals in to one point.
Pendulum with friction: basin of
attraction
Different starting positions end up in the same fixed
point. Its like rolling a marble into a basin. No matter
where you start from, it ends up in the drain.
Pendulum with friction
Adding a third dimension of potential energy: the
basin of attraction as a gravitational well.
Inverted Pendulum: ball on flexible rod
flops to one side or the other
Basin of attraction in phase space: two fixed points.
Inverted Pendulum: ball on flexible rod
Potential energy plot shows the two fixed points as
the “landscape” of the basin of attraction.
Bi-polar emotions as phase space
If someone flips between “manic” and “depressive”
moods, we can model as two basins of attraction.
Fixed point attractors would eventually settle.
Driven Inverted Pendulum
Move the inverted pendulum back and forth:
At first period 2, then period 4, then… chaos!
Chaotic attractor impossible to predict in the long run
because of sensitivity to initial conditions
Emotional phase space
Imagine a wolf captured for a zoo: it might cycle
between fear and aggression unpredictably.
Behavioral “modes” for fish schools
A. Swarm: low P, low M
B. Torus: low P, high M
C. Parallel: high P, low M
P = Group polarization -- how much of group is aligned
M = Group angular momentum -- how much of group rotates about center
Fish school behavior: basins of attraction
Swarm: low P, low M
Torus: low P, high M
Parallel: high P, low M
P = Group polarization -- how much of group is aligned
M = Group angular momentum -- how much of group rotates about center
Human school behavior: basins of attraction
Phase space of cliques (roles, stereotypes) as basins of attraction for
high school
Network behavior and basins of attraction
Stuart Kaufman modeled
genetic networks as
Boolean nodes (1 or 0). N
nodes with K (max)
inputs and one output.
At every time-step, each
node changes its state
(“lights up”) depending
on its internal rules.
Since there are 2n
possible states, we might
assume it just cycles
through some huge
number before it repeats
Network behavior and basins of attraction
But in fact it quickly
settles into just a few
basins of attraction
http://www-users.cs.york.ac.uk/susan/cyc/n/nk.htm.
Network behavior and basins of attraction
Kauffman speculates that
each basin of attraction
could roughly correspond
to a different cell type.
His model predicts a
power law for how
number of cell types rises
with number of genes.
The actual rise in number
of cell types with number
of genes is not far off
(although this is
controversial).
Social networks and basins of attraction
Hubs are one way to think about basins of attraction in social networks:
highly connected nodes are “attractive” (recall the theory of
“preferential attachment” to explain scale-free degree distribution).
But web links are relatively slow to change -- a more dynamic model
would ask about basins of information flow (think of photo memes)