It's not clear when Ghys made the slides to which you have linked. The only date I could find in those was 1963 (referring to the Lorenz equations), which would make the bound on "recent" rather generous. Here's a quick summary of relatively recent concrete activity in this area that might be interesting to you.
In 1983, various reasonable conjectures were made by Birman and Williams (see BW1 and BW2) after considerable experimentation with the Lorenz equation at various parameter values. The basic line of investigation aimed to find answers to the following question:
What types of knots and links can occur as periodic orbits of stable flows on $S^3$?
For instance, it was conjectured in BW1 that no flow supported all knots as periodic orbits. The conjecture stood uncontested for just over a decade, until Ghrist constructed a flow supporting all links as periodic orbits! His paper G is merely six pages long, picture-filled, and eminently readable.
The entire story is summarized in Bob Williams' article W which contains a wealth of other information including references.
This is really not my area of expertise, but here are two potential research problems. I'm not sure how tractable or elementary this would be, but that's the nature of the beast...
Given a triangulated knot $K$, construct small triangulations $T$ of $S^3$ and piecewise-linear flows $\phi:T \to T$ which contain $K$ as a periodic orbit.
And somewhat harder-seeming,
Given a family $F$ of knots, construct a flow on $S^3$ which supports all knots as periodic orbits except the ones in $F$.
BW1: Birman and Williams, Knotted periodic orbits in dynamical systems-I: Lorenz's equations, Topology 22 , 1 (1983), 47 - 82.
BW2: Birman and Williams, Knotted periodic orbits II: Knot holders for fibered knots, Low Dimensional Topology, Contemporary Mathematics 20, A.M.S. (1983), 1-60.
G: Ghrist, Flows on $S^3$ supporting all links as orbits, Electronic Research Announcements of the AMS, 1(2), 91-97 (1995).
W: Williams, The universal templates of Ghrist, BAMS 35, No. 2, 145-156 (1998).