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Whats going on?
- Choose Periodic maps using bars at top. (keep them near the middle
for best results)
- Choose a color by clicking on the circle
- Click somewhere on the black square to reveal an orbit.
- Repeat steps 2 and 3
- You are plotting the orbit of points under a return map of the torus
that preserves area, specifically this map is called a Generalized Standard
Map. The map is determined as follows:
- Choose two periodic functions: f,g : R/Z -->
- Given any x,y define
- y' = y + f(x)
- x' = x + g(y')
- The map is (x,y) |--> (x',y')
- These maps tend to have large regions of chaos, with "islands" of order
around fixed points. Nearby each of these islands there are smaller islands
containing periodic orbits. Each of these islands then has a family of neigboring
islands as well, and so on. This is typical of area preserving maps.
- A good reference on the subject is RS MacKay, Renormalization in
area preserving maps