I have my vector field $X=(\frac{-y}{x^2+y^2},\frac{x}{x^2+y^2})$, which leaves me with a circular vector field with center at origin(as can be seen in the picture below)
I was wondering if it is possible to modify said vector field for it to still maintain the circular direction but for it to repeat at each integer along the x axis?
I have tried by modifying my $x$ value to a $cos(x)$ for it to repeat along the x-axis. I have tried:
$X=(\frac{-y}{cos(x*pi)^2+y^2},\frac{cos(x*pi)}{cos(x*pi)^2+y^2})$
and
$X=(\frac{-y}{cos(x)^2+y^2},\frac{cos(x)}{cos(x)^2+y^2})$
Both have not given me the results I am looking for (they did repeat the circular patterns I wanted but they did not repeat at each integer as I wanted). What I'm I missing? How should I modify my vector field? Is there an additional parametrization I have to do?