Consider the system $(x,y)\to((\cos a) x-(\sin a) y,(\sin a) x+(\cos a) y)$. If $a=2\pi/k$ for some integer $k$, this autonomous 2D system has period-$k$ orbits.
For example, $(x,y)\to(-x,-y)$ has period-2 orbits and $(x,y)\to(-y,x)$ has period-4 orbits.
Edit: The edited version of the question seems to ask about trajectories of dynamical systems intersecting themselves. This is an entirely different problem, and the mention of "period-2 orbit" may seem unfortunate in this context. Thus, consider a dynamical system $x'(t)=f(x(t),y(t))$, $y'(t)=g(x(t),y(t))$. The configuration of the picture is impossible since the point $(x^*,y^*)$ at the intersection would be such that $(f(x^*,y^*),g(x^*,y^*))$ takes two different values at the same time. In fact, if $(x(t+s),y(t+s))=(x(t),y(t))$ for some given $t$ and $s\gt0$, one has $(x(t+s+u),y(t+s+u))=(x(t+u),y(t+u))$ for every $u\geqslant0$, that is, the trajectory is a cycle.