Celebratio Mathematica — Williams

An Autobiography

Works connected to Philip John Holmes

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P. Holmes and R. F. Williams: “Knotted periodic orbits in suspensions of Smale’s horseshoe: Torus knots and bifurcation sequences,” Arch. Rational Mech. Anal. 90 : 2 (1985), pp. 115–194. MR 798342 Zbl 0593.58027 article

We construct a suspension of Smale’s horseshoe diffeomorphism of the two-dimensional disc as a flow in an orientable three manifold. Such a suspension is natural in the sense that it occurs frequently in periodically forced nonlinear oscillators such as the Duffing equation. From this suspension we construct a knot-holder or template — a branched two-manifold with a semiflow — in such a way that the periodic orbits are isotopic to those in the full three-dimensional flow. We discuss some of the families of knotted periodic orbits carried by this template. In particular we obtain theorems of existence, uniqueness and non-existence for families of torus knots. We relate these families to resonant Hamiltonian bifurcations which occur as horseshoes are created in a one-parameter family of area preserving maps, and we also relate them to bifurcations of families of one-dimensional ‘quadratic like’ maps which can be studied by kneading theory. Thus, using knot theory, kneading theory and Hamiltonian bifurcation theory, we are able to connect a countable subsequence of “one-dimensional” bifurcations with a subsequence of “area-preserving” bifurcations in a two parameter family of suspensions in which horseshoes are created as the parameters vary. One implication is that infinitely many bifurcation sequences are reversed as one passes from the one dimensional to the area-preserving family: there are no universal routes to chaos