This mapping nicely illustrates the difference between conservative and dissipative dynamics. There are many interesting cases to consider, e.g.,

and you can see from the way that the points fill in that the mapping is conservative. (Areas are conserved and there is no basin of attraction.)

Figure 6 shows the results of this mapping for 10,000 points.
We emphasize that this is a * single* orbit. A case showing
dissipation is given by

and, as the iterations proceed, the orbit spirals inward, collecting around the eight oval-shaped objects surrounding the center. Figure 7

shows the mapping of this chaotic orbit for 10,000 points. Check
sensitivity to initial conditions. Also, consider the orbits resulting from
other combinations of **a** and **b**.

Figure 6 Conservative mapping [6] for Eqs. (18) and (19) (, ). A fortran program was used to perform 10,000 iterations, and the results were plotted with xmgr.

Figure 6: Conservative mapping for above equations.