1-D map for the double scroll family

The 1-D map π* (an approximation of the 2-D Poincare map) is used to study the periodic windows of the double scroll family. First, an algorithm based on the kneading theory is used to determine the structure and the order of appearance of periodic orbits in the 1-D map π*. This information is used to find the structure and the period of the corresponding orbits of the 3-D system. The results show that although the 1-D map π* is an approximation of the 2-D Poincare map, it gives much information about the periodic windows of the double scroll family. It is conjectured that the periodic orbits of the system are unknotted knots. Since they are equivalent to the trivial knot, it should be possible to obtain every periodic orbit from those of period-1 by making only an appropriate number of twists