# Conditions to the existence of periodic orbits of non vanishing vector fields on $\mathbb{T}^2$

I'm doing a research about Filippov systems on $\mathbb{S}^3$ with discontinuities on $\displaystyle\frac{1}{\sqrt{2}}\cdot\mathbb{T^2} =\left\{\displaystyle\frac{1}{\sqrt{2}}x ; \ x \in \mathbb{S}^1\times\mathbb{S}^1\subset\mathbb{R}^4\right\}$.

Briefly, we define the function $h:\mathbb{S}^3 \to \mathbb{R}$, $h(x_1,x_2,x_3,x_4) = x_1^2 + x_2^2 -1/2,$ and given two smooth fields $X,Y$ on $\mathbb{T}^2$, we try to or understand the behavior of solutions of the differential inclusion

$$\dot{x} \in F(x)= \left\{\begin{array}{ll} X(x)& \mbox{if } h(x) >0, \\ Y(x)& \mbox{if } h(x)<0,\\ \text{Conv}[X(x),Y(x)] & \mbox{if } h(x)=0, \end{array} \right.$$

where $$\text{Conv}[X(p),Y(p)] = \left\{\frac{1+\lambda}{2} X(p) + \frac{1-\lambda}{2} Y(p); \ \lambda \in [-1,1] \right\}.$$

I.e; understand the behavior of the functions $\varphi: I\to \mathbb{T}^2;$ such that $\varphi$ is absoute continous and $\varphi'(t) \in F(\varphi(t))$ for almost every $t$ $\in$ $I$.

At one point, I faced the following question that I've been having problems to solve /finding papers or books that have this type of result

Question: Let $Z$ be a smooth non-vanishing vector field on $\mathbb{T}^2,$ there are any conditions/equivalences regarding the vectors field $Z$, to $Z$ admits (or not) at least one periodic orbit?

The best that I could find was: if $Z$ is a smooth non-vanishing vector field on $\mathbb{T}^2$ and has no periodic orbit then $Z$ is equivalent to the irrational flow on $\mathbb{T}^2$ ("Introduction to the Modern Theory of Dynamical Systems - Katok" on page 458). But once the equivalence is just a continuous function I don't know how $Z$ "looks like".

• There is a complete description of the direction fields on $\mathbb{T}^2$ with and without periodic orbits in Chapter IV of the book Dynamical Systems on Surfaces by Claude Godbillon. I am in the process of restating the main result for the format of MathOverflow currently. The idea is to reduce a direction field on $\mathbb{T}^2$ with a periodic orbit to an orientable direction field on an annulus. – Michael Lee Jun 29 '18 at 9:40