Short description:

The Rössler map is a diffeomorphism of the plane given by

R(x,y)=(R1(x,y),R2(x,y)), R1(x,y)=3.8x(1-x)-0.1y, R2(x,y)=0.2(y-1.2)(1-1.9x).

Numerical simulation of the discrete dynamical system induced by the Rössler map shows the existence of an attracting set with very complicated and chaotic dynamics. It is easy to prove the following

Lemma. Define W=[0.01,0.99]´[-0.33,0.27]. The set R(W) is a subset of the set W.

Since W is a compact set the previous Lemma implies the existence of a compact, connected, invariant set - an attracting set.

Theorem. The six iteration of the Rössler map is semicojugated to the full shift on two symbols. Moreover, the preimage of an arbitrary periodic sequence of symbols contains a periodic point with the same basic period.

Applet settings:

clear button - clears the Graphics Frame,
default values button - sets the drawing region on [0,1]´[-0.5,0.5] and clears the Graphics Frame.
Wimage button - shows the iterations of the boundary of W, i.e. Rn(W).
show trajectory button - shows the trajectory of a point. You can choose a start point from the set W and the number of iterations.
These operations are usually long-lasting, so the title of the Graphics Frame is changed on wait.

Zoom in: In the Graphics Frame you can select some subset of the drawing region. Then click clear and show trajectory buttons.