1.   It transpires that the most important bifurcations occurring in the equations are associated with the possibility that there may be homoclinic orbits to the origin for some parameter values.

2.   A homoclinic orbit is a trajectory which tends, in both forwards and backwards time, towards an unstable stationary point.

3.   They produce a picture showing some of the r and b parameter values for which homoclinic orbits to the origin can be seen.

4.   There are, in fact, infinitely many other families of homoclinic orbits winding any number of times around the z-axis.

5.   Of course, the various families of homoclinic orbits to the origin cannot cross and the dotted lines representing homoclinic orbits to the points also cannot cross.

6.   In other words, homoclinic orbits are possible.

7.   The analysis still only concerns a small region contained within thin tubes around the homoclinic orbit.

8.   Finally, and importantly, the study of homoclinic orbits can be undertaken relatively easily and cheaply on a computer.