Pendulum without friction
Limit cycle in phase
space: no sensitivity
to initial conditions
Pendulum with friction
Fixed point attractor in phase space: no sensitivity to
initial conditions
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.
Driven Pendulum with friction
Horizontal version:
http://www.youtube.com/watch?v=0LSPxDB8OPM
Chaotic behavior in time
Driven Pendulum with friction
Horizontal version:
http://www.youtube.com/watch?v=0LSPxDB8OPM
Chaotic attractor in
phase space
Double Pendulum
Very simple device, but its motion
can be very complex (here an LED is
attached in a time exposure photo)
Simulation at
http://www.youtube.com/wat
ch?v=QXf95_EKS6E
Logistic Equation: a period-doubling route
to chaos
0<x<1 (think of x as percentage of total
population, say 1 million rabbits)
Population this year: xt
Population next year: xt+1
Rate of population increase: R
Positive Feedback Loop: xt+1= R*xt
Negative Feedback Loop: 1-xt (if x gets big, 1-x gets
small)
Logistic Equation: a period-doubling route
to chaos
Positive Feedback Loop: xt+1= R*xt
Logistic Map
Starting at xt = 0.2 and R= 2: “fixed point” or
“point attractor.” All starting values are in this
“basin of attraction” so they eventually end there.
Logistic Map
Starting at xt = 0.2 and R= 3.1: limit cycle of
“period two” (because it oscillates between two
values).
Logistic Map: cobweb diagram
Starting at xt = 0.2 and R= 3.1: limit cycle of “period two” (because
it oscillates between two values).
In each iteration there are two steps. The first gives the parabola,
. The second step we “reset” xt to xt+1 which is the
straight line. We see a “fixed point
Attractor.
Animation:
http://lagrange.physics.drexel.edu/flash/logistic/
Logistic Map
Starting at xt = 0.2 and R = 3.49 we double the
period (“bifurcation”): a limit cycle of four values.
Logistic Map
Increasing R continues to double the period. Starting
at xt = 0.2 and R = 4 we see a chaotic attractor. The
values will never repeat.
Bifurcation Map
Where does x “settle to” for increasing R values?
Bifurcation Map
The logistic map is a fractal: similar structure at
different scales. Thus bifurcations happen with
increasing frequency: the rate of increase is the
Feigenbaum constant (4.7)
Water drop model
Plotting the time interval between one drip and the next: The amount of water in a drip
depends on the drip that came before it—this feedback can create complex dynamics.
Tn+1
Tn
One-frequency drip
The period-doubling
route to chaos:
eventually the dripping
faucet produces a
strange attractor:
Two frequency drip