Given $\varphi$ a periodic orbit of $X=(X_1,X_2)$, $C^1$ in $\Delta\subset \mathbb{R^2}$. If $X_\theta=\left[ \begin{array}{cc} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{array} \right] \left[ \begin{array}{cc} X_1 \\ X_2 \end{array} \right]$, prove that:
$a)$ Prove that exists $\delta>0$, $|\theta|<\delta$, that $X_\theta$ has a periodic orbit $\varphi_\theta$, which $\varphi_\theta \to \varphi$ when $\theta \to 0$.
$b)$ Prove that $\varphi_\theta$ are all disjoint and $$\bigcup_{|\theta|\leq \delta}\varphi_\theta$$ is an annular region.
So, $a)$ is ok because if $\theta\in (0,1/n)$ and, if $n\to \infty$ then $\theta\to 0$. $X_\theta$ is linear and cos, sin is continuous, then $X_\theta \to X_0$, hence $\varphi_\theta \to \varphi_0$. For the second, i tried considered poincaré maps and see their graphics (everything makes me concluded that is an annular region, geometrically), how can i write this argument?
edit: I don't know how write the annular region yet... someone can help me?