# 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".

• I think characterization from Katok's book is the optimal you can give in such generality. If you provide more details about the nature of vector field, then something more useful can be said I guess – Aknazar Kazhymurat Jun 25 '18 at 23:30
• 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
• @MichaelLee Thank you for your reference, this book has awesome theorems! You helped me a lot. – Matheus Manzatto Jun 29 '18 at 21:47