The Lorenz attractor with the embedded AABB unstable periodic orbit.
The Lorenz attractor with the embedded AABB unstable periodic orbit.
The butterfly-like Lorenz attractor is one of the best known images of chaos. The attractor is derived by solving the following three coupled differential equations.
Because Edward Lorenz derived these equations as a simplified model of two-dimensional convective fluid flow, the a and b parameters are referred to as the Prandtl number the Rayleigh number, respectively. The c parameter is geometric factor. The xyz-coordinates are the convective intensity, temperature difference, and temperature profile. The solution to these equations and its geometric structure can, however, be appreciated without this physics interpretation. We follow Lorenz and use a=10, b=28, and c=8/3 as default conditions.
Embedded in the Lorenz attractor are unstable periodic orbits described by Viswanath and this model computes a number of these orbits. Each periodic orbit is classified by the number of times the trajectory orbits the A and B fixed points before it repeats. Note that because the attractor is chaotic and because of numerical errors and the finite precision of the initial conditions, errors accumulate and the trajectory leaves the vicinity of a periodic orbit after a half dozen cycles.
A hallmark of chaos theory is sensitivity to initial conditions. Select an unstable periodic orbit, change the initial value of x by one part in 103, and estimate the time it takes for the changed orbit to diverge from the periodic orbit. Repeat with with changes of one part in 104, 105 and 106. Does increasing the accuracy of the initial condition by a factor of 10 increase the divergence time by the same factor? Does your conclusion hold if you vary the accuracy of y or z?
The Lorenz equations are not always chaotic. Run the model with b=99.96 and observe the orbit. Shift-drag within the 3D view to move the camera for a better view. Note that the trajectory only shows the last 2000 points so that initial transients will disappear after the the trajectory reaches its steady state.
The Lorenz Attractor Orbits Model was created by Wolfgang Christian using the Easy Java Simulations (EJS) version 4.1 authoring and modeling tool.
You can examine and modify a compiled EJS model if you run the model (double click on the model's jar file), right-click within a plot, and select "Open EJS Model" from the pop-up menu. You must, of course, have EJS installed on your computer. Information about EJS is available at: <http://www.um.es/fem/Ejs/> and in the OSP comPADRE collection <http://www.compadre.org/OSP/>.