The beauty of the Lorenz Attractor lies both in the mathematics and in the visualization of the model. Mathematically, the Lorenz Attractor is simple yet results in chaotic and emergent behavior. Visually, when the values given by the equations are plotted in two or three dimensional space the behavior is reminiscent of an orbit of an object around two central origins, when looked at from the appropriate dimension the pattern appears similar to a figure eight see below. There are three Lorenz equations that comprise the Lorenz Attractor, each of which can be though of as the x, y, or z component of a given three dimensional location in space: The Lorenz Equations.
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It is a nonlinear system of three differential equations. With the most commonly used values of three parameters, there are two unstable critical points. The solutions remain bounded, but orbit chaotically around these two points. The program "lorenzgui" provides an app for investigating the Lorenz attractor. The resulting 3-D plot looks like a butterfly. They started the field of chaos. Small changes in the initial conditions have a big effect on the solution. Lorenz is famous for talking about the butterfly effect.
A butterfly flying in Brazil can cause a tornado and Texas is a flamboyant version of a talk he gave. The equations are almost linear. The equations come out of a model of fluid flow. These are the most commonly used parameters. Y dot equals Ay. It looks linear except A depends upon y. This helps me study the differential equations in this form. This matrix form is convenient for finding the critical points.
Put a parameter eta in place of y2. Try to make the matrix singular. That happens when eta is beta times the square root of rho minus 1. And then the null vector is the critical point. If we take this vector as the starting value of the solution, then the solution stays there.
Y prime is 0. This is an unstable critical point. And values near this solution deviate the solution. And I used the Lorenz attractor as an example. Set the parameters. Set the initial value of the matrix A. Here is the critical point. Here is an initial value near the critical point. Integrate from 0 to Use ODE Give it a function called the Lorenz equation. Capture the values t and y and then plot the solution. And it continuously, every time it called, it modifies the matrix A updates it with the new values of y2.
Here is the three components of the Lorenz attractor. Time series is functions of t. I want to write a program called Lorenz GUI. Lorenz Graphic User Interface. OK, I hit the Start button. Here are the two critical points in green. We started near the critical point. We oscillate around the critical point. And here is the orbit. This is just going back and forth. It oscillates around one critical point then decides to go over and oscillate around the other for a while.
It continues around like this forever. This is not periodic. It never repeats. Now, the butterfly is associated with Lorenz in two ways. One is the butterfly effect on the weather. Also, this plot looks like a butterfly. I can grab this with my mouse and rotate it in three dimensions.
So I can get different views of the orbit. And I can look at it from different points of view to get some notion of how this is proceeding in three dimensions. It almost lives in two dimensions, but not quite. Just going like this forever. This is completely determined by the initial conditions. I can clear this out and see the orbit continue to develop.
Press Stop. Now I have a choice. This pull down menu here allows me to choose other values of rho. Let me choose rho equal to and clear and restart. Now, after an initial transient, this is now periodic. So this is not chaos. This is a periodic solution. Product Focus.
L'attracteur de Lorenz
The Butterfly: Rotate, pan or zoom! Two butterflies starting at exactly the same position will have exactly the same path. There is nothing random in the system - it is deterministic. Two butterflies that are arbitrarily close to each other but not at exactly the same position, will diverge after a number of times steps, making it impossible to predict the position of any butterfly after many time steps. Any approximation, such as approximate measurements of real life data, will give rise to unpredictable motion. This behavior can be seen if the butterflies are placed at random positions inside a very small cube, and then watch how they spread out. Press the "Small cube" button!
Programming the Lorenz Attractor
Interactive Lorenz Attractor
Attracteur de Lorenz