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**Pendulum without friction**

Limit cycle in phase space: no sensitivity to initial conditions

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**Pendulum with friction**

Fixed point attractor in phase space: no sensitivity to initial conditions

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**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.

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**Pendulum with friction**

Adding a third dimension of potential energy: the basin of attraction as a gravitational well.

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**Inverted Pendulum: ball on flexible rod flops to one side or the other**

Basin of attraction in phase space: two fixed points.

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**Inverted Pendulum: ball on flexible rod**

Potential energy plot shows the two fixed points as the “landscape” of the basin of attraction.

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**Driven Pendulum with friction**

Horizontal version: Chaotic behavior in time

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**Driven Pendulum with friction**

Horizontal version: Chaotic attractor in phase space

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Double Pendulum Very simple device, but its motion can be very complex (here an LED is attached in a time exposure photo) Simulation at

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**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)

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**Logistic Equation: a period-doubling route to chaos**

Positive Feedback Loop: xt+1= R*xt

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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.

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

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**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:

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Logistic Map Starting at xt = 0.2 and R = we double the period (“bifurcation”): a limit cycle of four values.

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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.

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Bifurcation Map Where does x “settle to” for increasing R values?

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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)

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

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