Consider the vector field $X=(\alpha, \beta)$ in $\mathbb{T}^2$(a torus). Show that the flux given by the following differential equation $\dot{x}=X(x)$ is the translation flux and that:

a) if $\frac{\beta}{\alpha}\in\mathbb{Q}$ then all the orbits are periodic, determnie the period.

b) if $\frac{\beta}{\alpha}\in\mathbb{Q}$ then all the orbits are dense in $\mathbb{T}^2$

Consider $x\in\mathbb{R}^2$, if $\varphi:\mathbb{R}\times\mathbb{T}^2$ is the flux given by the differential equation solution, then $\varphi_t(x)=x+\alpha t(1,\frac{\beta}{\alpha})$. If I think on the identification map from a square, then $\frac{\beta}{\alpha}$ would be the slope of the trajectory from $x$ to $\varphi_t(x)$ for $t\in\mathbb{R}$ however I cannot compute the period, and I do not see the reason why if we had a irrational slope, why would the orbits be dense(I think we could not compute the period).


How should I solve the problem?

Thanks in advance!


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