In different value of free parameter in simple dynamic (for example Lorenz eq)
,Is the genus number invariant?
Is the linking number of template invariant?
(in close orbit) The self-linking number is not invariant!
The limit cycle in tori is equal to knots in real space.
If the genus number doesn't change in different value of free parameter
we can say:
if the dynamics are on surface of n-torus
with changing free parameter we can change the angle of vector that are in surface.
and This angle cast the dynamics is closed or not!
see:
Tsankov, Tsvetelin D and Gilmore, Robert. Strange attractors are classified by bounding tori.
Physical review letters, 91(13):134104, 2003.
In the work of Tsankov and Gilmore, the genus bounding the torus depend on the flow structure. Considering the Lorenz system, for the classical parameters values the solution is an attractor bounded by a genus 3 torus. It exists parameters values (increase the value of R) where the solution of the Lorenz system is two co-existing attractors. These 2 attractors are bounded by a genus 1 torus. Thus the genus number is not invariant.
If understand well your final question, you are trying to use an angle (obtained from the bounding torus) in order to decide if two distinct dynamics are "closed or not". I suggest you to have a closer look to the references below. I hope these works will convince you that return maps are a good path to compare dynamics because they contains all dynamics properties. Maybe you will find some links between the return maps and your angle?
Article Qualitative analysis of the Rössler equations: Bifurcations ...
Article Dressed return maps distinguish chaotic mechanisms
Article Templates and subtemplates of Rössler attractors from a bifu...
Article Architecture of chaotic attractors for flows in the absence ...
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