Consider the two variables $x_{1},x_{2}$ which are functions of time $t$, we refer $\dot{x_{1}}$ as the time derivative of $x_{1}$.
Let $\alpha$ be a constant parameter, and consider the following system of equations:
\begin{align*} \dot{x_{1}} &= \alpha x_{1} - x_{2} - x_{1}(x_{1}^2 + x_{2}^2)\\ \dot{x_{2}} &= x_{1} + \alpha x_{2} - x_{2}(x_{1}^2 + x_{2}^2). \end{align*}
Now we transform the above system to polar $(\rho,\theta)$ coordinates. I think after the transformation the above equations lead to the following:
\begin{align*} \dot{\rho} &= \rho(\alpha - \rho^2)\\ \dot{\theta} &= 1. \end{align*}
Where $x_{1} = \rho \cos(\theta)$ and $x_{2} = \rho \sin(\theta)$. If I substitute then I am thinking about the partial derivatives involved and I think which variable I should differentiate with respect to the time? I can observe that the $x_{1}^2 +x_{2}^2 = 1$ which could simplify a lot but how should I deal with $\dot{x_{1}}, \dot{x_{2}}$ which can help me in getting the polar form of the above system?