Presentation on theme: "Pendulum without friction"— Presentation transcript:
1Pendulum without friction Limit cycle in phase space: no sensitivity to initial conditions
2Pendulum with friction Fixed point attractor in phase space: no sensitivity to initial conditions
3Pendulum 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.
4Pendulum with friction Adding a third dimension of potential energy: the basin of attraction as a gravitational well.
5Inverted Pendulum: ball on flexible rod flops to one side or the other Basin of attraction in phase space: two fixed points.
6Inverted Pendulum: ball on flexible rod Potential energy plot shows the two fixed points as the “landscape” of the basin of attraction.
7Driven Pendulum with friction Horizontal version:Chaotic behavior in time
8Driven Pendulum with friction Horizontal version:Chaotic attractor in phase space
9Double PendulumVery simple device, but its motion can be very complex (here an LED is attached in a time exposure photo)Simulation at
10Logistic 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: xtPopulation next year: xt+1Rate of population increase: RPositive Feedback Loop: xt+1= R*xtNegative Feedback Loop: 1-xt (if x gets big, 1-x gets small)
11Logistic Equation: a period-doubling route to chaos Positive Feedback Loop: xt+1= R*xt
12Logistic MapStarting 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.
13Logistic MapStarting at xt = 0.2 and R= 3.1: limit cycle of “period two” (because it oscillates between two values).
14Logistic 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 pointAttractor.Animation:
15Logistic MapStarting at xt = 0.2 and R = we double the period (“bifurcation”): a limit cycle of four values.
16Logistic MapIncreasing R continues to double the period. Starting at xt = 0.2 and R = 4 we see a chaotic attractor. The values will never repeat.
17Bifurcation MapWhere does x “settle to” for increasing R values?
18Bifurcation MapThe 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)
19Water drop modelPlotting 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+1TnTwo frequency dripOne-frequency dripThe period-doubling route to chaos: eventually the dripping faucet produces a strange attractor: