From Morse-Smale to all Knots and Links

We analyse the topological (knot-theoretic) features of a certain codimension-one bifurcation of a partially hyperbolic xed point in a ow onR 3 originally described by Shil'nikov. By modifying how the invariant manifolds wrap around themselves, or \pleat," we may apply the theory of templates, or branched two-manifolds, to capture the topology of the ow. This analysis yields a class of ows which bifurcate from a Morse-Smale ow to a Smale ow containing periodic orbits of all knot and link types. The goal of this paper is to show that ows in R 3 with the most complex knot structure possible are accessible from the set of Morse-Smale ows via a comdimension one bifurcation. Main Theorem: There exists a codimension-one bifurcation of vector elds X having the property that for < 0, X is a Morse-Smale vector eld, and for > 0, X is a Smale vector eld whose closed orbits span every possible knot and link type. The starting point for this bifurcation is a partially hyperbolic saddle-node bifurcation, originally described by Shil'nikov [27]. In this bifurcation, a bouquet of homoclinic curves to a saddle-node xed point is fattened up to a nontrivial hyperbolic invariant set. This bifurcation was signi cant in that it showed that a multi-dimensional system might change from simple to complex behavior via a codimension one bifurcation and it has been of interest in applications because it is believed to play a role in intemittency [24]. An explicit system of ODE's exhibiting the Shil'nikov bifurcation was produced in [8]. Recall that a Morse-Smale vector eld is a vector eld whose chain-recurrent set consists solely of a nite collection of hyperbolic xed points and closed orbits; hence, in such a eld, the link of closed orbits is particularly simple. On the other hand, a Smale vector eld is one whose chain-recurrent set breaks into a nite collection of xed points, closed orbits, and nontrivial saddle sets, all of which are hyperbolic. In a Smale ow which is not Morse-Smale, there are an in nite number of distinct closed orbits in each saddle set, and the associated link is topologically complex [10]. The question of existence of Smale ows having all knots and links as closed orbits was originally raised by M. Hirsch [30] and later popularised in a conjecture of Birman and Williams [3]. The issue was resolved in [12], where several examples were produced using the theory of templates. In [13], a 1-parameter family of ODEs leading to all knots and links was described; however, the bifurcation was by no means instantaneous, but rather, the knot types built up gradually. The results of this paper show that such a continuous gradation is unnecessary. Several authors have considered the knot-theoretic data associated to Morse-Smale ows [21, 29, 7] and bifurcations thereof [6]. For de nitions of dynamical terms, see any of the excellent modern texts, including [15, 25]. Techniques in global bifurcation theory used in x2 can be found in [1, 23]. Comprehensive introductions to knot theory and template theory are, respectively, [26] and [14]. Following a seminar on the Shil'nikov bifurcation given by T.Y. at Georgia Tech in 1992, K. Mischiakow proposed that this bifurcation might be analyzed for the knot structure of its periodic orbits.