[Screen shot of kicked rotor dynamics.]

Week 6 Notes: Event Models

We study ODE models and show how they can be used to detect hard disk collisions in models such as the ideal gas and Newton's cradle.  We also introduce discrete maps and Poincare sections. 

Kicked Rotor Map

Kicked Rotor Model shows the dynamics of a rotating bar (rotor) that is kicked periodically. The bar rotates uniformly between kicks and each kick changes the angular momentum of rotor p by the sine of the position angle q multiplied by a kick-strength K.

p'= p + K sin(q)

Angular position and angular momentum are recorded at each kick to produce a Poincare section.  This model is a prototype of a classical nonlinear system with regular and chaotic behavior.

At low speed, the Kicked Rotor window displays a marker trail showing the position of the end of the rod as the bar rotates between kicks. The green markers show the current motion and the smaller maroon markers show the motion before the kick. The high speed option disables the marker trail in order to compute and draw the Poincare section as quickly as possible.

Instructions:

Click within the Poincare section to set the initial conditions. Both periodic and chaotic motion can be observed with a kick strength K=1. 

References

Numerous kicked rotor computer experiments are described in the paper and book by Korsch.

• H. J. Korsch, E. M. Graefe, and H. -J. Jodl. "The kicked rotor: Computer-based studies of chaotic dynamics," Am. J. of Phys., 76 (4) p498-503, (2008).
• Chaos: A Program Collection for the PC by H. J. Korsch, H-J Jodl, and Timo Hartmann, Springer (2008).

The kicked rotor map is also known as the standard map or the Chirikov map and is discussed in Chapter 6 of:

• Chaos and Nonlinear Dynamics by Robert Hilborn,  Oxford University Press (2000)

Related Models

The following models use ODE events to detect collisions between hard objects (disks or spheres).

• Bouncing Balls
• Two Ball Bounce
• Elastic Collision in 2D
• Hard Disk Gas
• Balls In Box

The following models explore iterative maps:

• Iterative Map
• Iterative Cobweb
• Iterative Bifurcation

The following dynamical systems are investigated using Poincare maps:

• Kicked Rotor Map
• Two Ball Bounce Map
• Ball Bounce in Wedge (advanced)

Additional models may be be posted for self-study.

Credits:

The Kicked Rotor Map 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/>.