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

Chaotic behavior in time

Horizontal version: http://www.youtube.com/watch?v=0LSPxDB8OPM

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/watch?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 diagramStarting at xt = 0.2 and R= 3.1: limit cycle of “period two” (because it oscillates between two values).

Graphing xt on horizontal; xt+1 on vertical

In each iteration there are two steps. The first gives you a seed value on the x axis. The second step gives you xt+1 on the y axis, where it hits the parabola. In the next iteration, xt = is “reset” to xt+1, so it lands on the diagonal. Now you repeat the process.

It will always move between the diagonal and parabola.

xt

xt+1

Logistic Map: cobweb diagramStarting at xt = 0.2 and R= 3.1: limit cycle of “period two” (because it oscillates between two values).

Note that eventually it settles into a limit cycle. As we increase R, the shape of the parabola gets higher, and we see other behaviors.

xt

xt+1

Animation: http://lagrange.physics.drexel.edu/flash/logistic

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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

One-frequency drip Two frequency drip

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

The period-doubling route to chaos: eventually the dripping faucet produces a strange attractor: