Periodic orbits of a third-order ODE are topological knots. Using results from the theory of branched 2-manifolds, we prove the existence of simple ODEs whose periodic orbit set contains every type of knot and collection of knots (link). The construction depends on the dynamics near certain configurations of Shil'nikov connections, and can be applied to, among other things, a model of a nonlinear electric circuit.
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Ghrist et al. (1996) studied this question.