It has been suggested by some that these equations correspond to the rotational velocity of the waterwheel, the difference in the volume of water between the right half and the left half, and the difference in volume between the top half and bottom half. We must now consider what conditions have to be met in order for the functions for "total water in the right half" and "total water in the left half" to be correct. The three-parameter Lorentzian function indicated is not, in general, a probability density function, since it does not integrate to 1, except in the special case where =. This sensitivity is "Chaos." Changing the leak rate was possible, but tedious, and working with only two parameters at a time simplifies things.We ran twenty tests and kept all of the data with the exception of the tests that exhibited rolling motion. The design was or a vertical wheel with swiveling buckets like the seats on a ferris wheel. You can see the trajectory being pulled toward a center even if the center isn�t very clear. If the parameters of the wheel are set correctly, the wheel will exhibit chaotic motion: rather than spinning in one direction at a constant speed, the wheel will speed up, slow down, stop, change directions, and oscillate back and forth between combinations of behaviors in an unpredictable manner.The design for our waterwheel went through many changes. Lorenz was running a computer program that modeled weather patterns and created forecasts when he noticed, by chance, that the system he had programmed was severely sensitive to initial conditions. The network gathers more than 350 organisations in Europe and world-wide. Because we created functions like this:We can create inflow versus time in the following manner:This means we have a specific inflow versus time function for each position versus time function. Therefore, when the wheel�s acceleration is zero, the affects of the water must be zero. The process is exactly as described above:� Use velocity versus time to create position versus time.� Use position versus time to create inflow versus time.� Use inflow versus time along with some leakage rate to create left, right, top and bottom functions.� Combine left, right, top, and bottom functions as necessary.Considering that our initial conditions were off, the results were quite good. Unpredictability, (Strange) Atttactors, Lorentzian Waterwheel, Butterfly Effect, Self Similarity, Multi-stability, Logistic Map, Bifurcation Diagram, On the Edge of Chaos, Description of a Complex System. Lorenzian waterwheel Experience the practice of chaos theory and try to predict the motions of the waterwheel. There is an obvious relationship between the frequency of the oscillations in the functions we created and those of the acceleration function.

We achieved chaotic motion at every level of tilt. We noticed the importance of the system being driven during EVERY second. If we screw one of these up, our new equations will make no sense. This behavior of this system is analogous to that of a Lorenz attractor. With 25 years of designing and manufacturing efficient, off grid, solar powered water pumps and water applications, LORENTZ are the global leaders in the solar water pumping market.
Cumulative distribution function. With this in mind, our design evolved into a tilted wheel model much like one constructed by Stogatz (seen to the right).With such a design, we could easily change the tilt of the wheel (thus changing how much the weight in the buckets contributes to the spin of the wheel). � Explore the parameter space of the wheel. Water pours into the top bucket and leaks out of each bucket at a fixed rate. After trying several different tilt angles without noticing any different type of behavior, we decided to use the larger wheel and as many cups along the perimeter as possible. The logic behind it is this: If we have data for velocity versus time, we can obtain position versus time by integrating. From our experiment and the rough sketch we have of the parameter space, we can notice that rolling motion is associated with very large water flow rates and very small water flow rates. Matlab solved our simplified version of the acceleration equation and returned sinusoidal functions. Interact on desktop, mobile and cloud with the free.In Lorenz's water wheel, equally spaced buckets hang in a circular array.

We put only 8 cups around the perimeter and ran a few preliminary tests.

W'=(-vW+pi*g*rad*A)/ u The states are not placed precisely and the graph is not to scale. A phase space plot of the three Lorenz equations produces an image known as a "Strange Attractor." � Using Matlab, solve the system and examine plots. He accidentally changed an initial condition by what could be compared to the effect of the flap of a butterfly�s wings on the global weather pattern. The conditions are as follows. � Build a Lorenzian Waterwheel.

By numerically solving this equation, we would be able to easily prove its chaotic nature.Sadly, our attempt at numerically solving our 2nd order equation as a system of two first order equations did not work out. We had access to a rapid-prototyping machine thanks to the college of Mechanical Engineering and we planned to use the machine to manufacture the buckets for our wheel. We ended up having to make quite a few simplifying assumptions that changed our system too drastically for the result to be accurate. We decided against this design because of our inability to change as many parameters as we thought would be necessary. We decided to attempt to use numerical methods to solve a 2nd order ode that we created for the acceleration of the wheel in terms of position and time. Das klassische Modell des Lorentz-Oszillators (nach Hendrik Antoon Lorentz) beschreibt ein an den Atomrumpf gebundenes Elektron, welches durch ein elektrisches Feld zu harmonischen Oszillationen angeregt wird. The simplified version of our m file along with some of Matlab�s solutions to it is included at the end of this report.Obtain Missing Equations From Experimental Data.Because we obtained experimental data of velocity versus time graphs, we essentially have sample solutions to one of the three equations that describe our wheel.