Braids and Knots in Driven Oscillators

Abstract

We consider the application of braid and knot theory to single-degree-of-freedom driven oscillators, giving emphasis to the braids of periodic orbits contained in horseshoes. Using such concepts as braid type, relative rotations, Nielsen equivalence, knot polynomials, the reduced Burau representation and positive, regular and ambient isotopy, we illustrate how these can be put together to gain some understanding of bifurcation structure.

Topological fluid mechanics of stirring

“…whether the flow is a two-dimensional Stokes flow or a flow at higher Reynolds number (although the requirement of a periodically repeatable sequence of flows will arise). The results come almost directly from a theorem in topology that is the cornerstone of so-called Thurston-Nielsen theory (Thurston 1988;‡ Fathi, Laudenbach & Poenaru 1979;Casson & Bleiler 1988;Handel 1985; for a review of dynamical systems applications see Boyland 1994, see also McRobie & Thompson 1993; for a general-audience write-up of the mathematical contributions of Jakob Nielsen see Lundsgaard-Hansen, 1993). Many of the mathematical concepts used in and required by Thurston-Nielsen theory appear, upon proper 'translation', to have both an intuitive immediacy and a potential utility in discussing issues of fluid advection, stirring, quality of mixing, and the like.…”

Locating oscillatory orbits of the parametrically-excited pendulum

“…The first two classes imply little about the global dynamics of the system, whilst pseudo-Anosov braids have positive topological entropy and imply a countable infinity of (unstable) periodic orbits. 'A pseudo-Anosov braid implies chaos' [19] is the 2-dimensional analogy of the famous 'Period three implies chaos' statement for 1-dimensional maps [16]. None of the period-3 solutions produce pseudo-Anosov braids; indeed we have to go to period-5 before any such solutions are located.…”

“…The parametrically-excited pendulum is an example of a simple one degree of freedom nonlinear system which can exhibit a plethora of nonlinear phenomena. The equation of motion is given by 0+c9 +(I + p cos cot) sin 9 = 0 (1) in which 6 is the angle of rotation, c is a damping constant taken as 0.1 throughout, p is the scaled parametric excitation amplitude, to is the scaled frequency of excitation and a dot represents differentiation with respect to the scaled time, t. Equilibria, oscillatory and rotating orbits exist and the latter two cases possess multiple attractors and can undergo symmetry-breaking and period-doubling bifurcations, with the possibility of chaotic motion [19]. Much of the bifurcational behaviour has been determined by Bryant and Miles [5] in an earlier rigorous study.…”

Inverted oscillations of a driven pendulum

“…Furthermore, if the solutions are displayed with time plotted modulo T , then the solutions begin to look more distinct. Indeed, if we construct braid diagrams (Birman 1974;McRobie & Thompson 1993) from the time histories, it is apparent that there is a difference in the number of crossings: the first solution has five crossings while the second has only three. This has many important consequences when it comes to determining the possible bifurcations of different pairs of orbits (McRobie 1992), and is a much stronger topological invariant than number of nods.…”