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

Some Characteristics Of Chaos Bifurcations

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

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

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

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

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

.4mm Nozzle opened 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.

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

.4mm Nozzle opened 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.

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

All Data with.4mm Nozzle

All Data with.5mm Nozzle

All Data with.75mm Nozzle

All Data with.8mm Nozzle

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

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.

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.

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