Dynamical Systems on Three Manifolds Part I: Knots, Links and Chaos

Details Dynamical Systems on Three Manifolds Part I: Knots, Links and Chaos Dynamical Systems on Three Manifolds Part I: Knots, Links and Chaos

2007-06-16

arxiv.org/abs/0706.2416v1

Nonlinear Sciences

Chaotic Dynamics

19 pages with 15 figures, latex, to be published in International Journal of Bifurcation and Chaos

Scientific paper

In this paper, we give an explicit construction of dynamical systems (defined within a solid torus) containing any knot (or link) and arbitrarily knotted chaos. The first is achieved by expressing the knots in terms of braids, defining a system containing the braids and extending periodically to obtain a system naturally defined on a torus and which contains the given knotted trajectories. To get explicit differential equations for dynamical systems containing the braids, we will use a certain function to define a tube neigbourhood of the braid. The second one, generating chaotic systems, is realized by modeling the Smale horseshoe.

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