In this dynamical system, a point hops around inside a rectangle. We call the rectangle the phase space. Our rule of motion is given by the so called Standard Map Function. If we were studying billiards, there would also be a phase space and a point would hop around inside it. But there, the rule of motion in the phase space is given by the billiard motion rather than by the Standard map.
Produce another orbit by choosing a different starting point. Click with the mouse anywhere in the rectangle. This point will be the new starting point. Now press repeatedly on the "1 more" button to see the orbit.
Choose several different initial points and generate their orbits. The picture inside the rectangle is pretty complicated!
If you would like to draw the orbit of an initial point more quickly, you can enter a number of iterations, let's say 100, in the textfield by clicking in the textfield, typing in the number, and then pressing return. Now when you click in the rectangle, the 100 iterations will be drawn one after another very quickly. Depending on your computer, these points may be drawn so fast that you can not see the individual points being drawn. Now you can click on the "100 more" or "1 more" button to continue iterations of the same inital point. Click on other starting points. Choose points spread out over the rectangle. This way, you can generate a complete picture of the phase space motion.
You can change the rule of motion by varying the value of the parameter k. (See STANDARD MAP for an explanation).
Try k=0.0 What do you notice about the motion? ANSWER
Here are some other interesting k values:
Try k=0.3 How does the motion compare to the k=0 case? ANSWER
Try k=0.8 What does the phase space picture look like now? ANSWER
Try k=5.0 Is the motion chaotic or regular? ANSWER
Try k=6.0 Is there any regular motion for this system? ANSWER
The number k that appears in these formulas is a parameter that you can change. For example, the program is initially set to k=0. For this k value, 7 the formulas become
When a point moves off the right side of the rectangle, it reappears on the left side. And when it goes off the top of the rectangle, it comes back on the bottom. Mathematically, this means we are taking the numbers modulo 1: if a number becomes bigger than 1, we throw away its integer part and take only its fractional part:
Many mathematicians have studied this Standard Map system. It provides an excellent example of the transition from a regular system (k=0) to a mixture of regular and chaotic motion (k=5) to a completely chaotic system (k =6).
Many important and difficult questions remain to be answered about this system. For example, the computer picture shows that for k=6, the motion is chaotic: a point will move all over the rectangle. Mathematicians have not been able to give a mathematical proof that this happens.
Related Materials: Professor Meiss at the University of Colorado in Boulder has written a fancy version of the phase space game. You can get a copy of his program (for free!) from the web site ?????