KNOT-TYPES AND BIFURCATION SEQUENCES OF HOMOCLINIC AND TRANSIENT ORBITS OF A SINGLE-DEGREE-OF-FREEDOM DRIVEN OSCILLATOR

We study the homoclinic and transient orbits of a 3-striped Smale hors eshoe using Birkhoff signatures, knot-types, relative rotations and ma nifold intervention rules to infer information about bifurcation seque nces. The motivation is the conjectured existence of such a 3-shoe, of 'spiral' configuration on the usual (periodically-sampled) Poincare' plane of a driven oscillator, in a parameter regime near to the primar y resonance. This is further work towards a contention that the analys is of the ways of creating and destroying such a 3-shoe provides some insight into the complex bifurcational behaviour of the oscillator.