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Chaos In A Dripping Faucet

Chaos In A Dripping Faucet. What Is Chaos?. Chaos is the behavior of a dynamic system which exhibits extreme sensitivity to initial conditions. Mathematically, arbitrarily small variations in initial conditions produce differences which vary exponentially over time. Chaos is not randomness.

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Chaos In A Dripping Faucet

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  1. Chaos In A Dripping Faucet

  2. What Is Chaos? • Chaos is the behavior of a dynamic system which exhibits extreme sensitivity to initial conditions. • Mathematically, arbitrarily small variations in initial conditions produce differences which vary exponentially over time. • Chaos is not randomness

  3. Some Characteristics Of Chaos Bifurcations

  4. Some Characteristics Of Chaos Strange Attractors Discrete (Poincare Plot) Continuous (Lorentz Attractor)‏

  5. How A Dripping Faucet Can Lead To Chaos • After each drop, the water at the tip oscillates • These oscillations affect the initial conditions of the next drop • As flow rate increases, these variations in initial conditions become significant and lead to chaos

  6. Supply Tank Valve-Regulated Tank Dropper Laser Photosensor Computer Experimental Set-Up

  7. We experimented with nozzles of four different diameters: • 0.4mm • 0.5mm • 0.75mm • 0.8mm

  8. 0.4mm Nozzlebeginning at a slow drip rate • Period-1 attractor: 0.393s • Point of attraction increases over time: probably due to decreased pressure

  9. .4mm Nozzleopened a little more • Appears to have two periods. • However, considering our device measured 700 times per second, or every ~.0014s, there is probably still only a single period.

  10. .4mm Nozzleopened a little more • Has undergone a bifurcation. • The difference in density is due to the different drop sizes in the cycle.

  11. .4mm Nozzleopened a little more • The two periods now have about the same density of observations. • The wide range of times clustered around each period may indicate further bifurcations have occurred.

  12. .4mm Nozzleas open as possible without producing a stream • Appears to have descended into chaos.

  13. All Data with .4mm Nozzle

  14. All Data with .5mm Nozzle

  15. All Data with .75mm Nozzle

  16. All Data with .8mm Nozzle

  17. Time-Delay Graphs .75mm, 2.3 drops/s .40mm, 8.5 drops/s .80mm, 14 drops/s .40mm, 22.9 drops/s

  18. Conclusions • The time between drops begins as a period-1 attractor at low flow rate. • As the flow rate increases, it becomes a period-2 or period-3 attractor. • Each period bifurcates further, resulting in two branches. • Eventually the system approaches chaos, as evident in the time-delay graphs.

  19. Improving the Experiment • Being able to accurately measure the setting on the valve would let us quantitatively compare the behavior of different nozzles. • A better processor that can handle a high sample rate would allow for more accurate observation of fine-level bifurcations. • Dying the water might reduce the number of unobserved and accidental counts.

  20. Acknowledgments • K.Dreyer and F.R. Hickley Chaos In A Dripping Faucet. 1990. • S.N. Rasband. Chaotic Dynamics of Nonlinear Systems. 1990. • J.R. Taylor Classical Mechanics. 2005.

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