For $\alpha, \beta>0$ the differential equation, I am trying to solve, is given by
$$\begin{pmatrix}\dot x_1\\\dot x_2\end{pmatrix}=\alpha\sin(x_1^2+x_2^2)\begin{pmatrix}x_2\\-x_1\end{pmatrix}+\beta\sin(2x_1^2+2x_2^2)\begin{pmatrix}x_1\\x_2\end{pmatrix},$$ where $x_1,x_2:\mathbb{R}\rightarrow \mathbb{R}$ are functions depending on the time $t\in\mathbb{R}$. I tried to tranform this equation by using
$$x_1=r\cos\theta\\x_2=r\sin\theta$$ with $\theta\in\mathbb{R}$ and $r>0$.
First I computed $$\dot x_1=\frac{d(r\cos\theta)}{dt}=\dot r\cos\theta-r\dot\theta\sin\theta\\ \dot x_2=\frac{d(r\sin\theta)}{dt}=\dot r\sin\theta+r\dot\theta\cos\theta.$$ Putting this in the differential equation leads to $$\dot r =2\beta r \sin(r^2)\cos(r^2)r=\beta r \sin(2r^2)\\ \dot \theta =-\alpha \sin(r^2)\\ \frac{dr}{d\theta}=\frac{dr}{dt}\Big(\frac{d\theta}{dt}\Big)^{-1}=\frac{-\beta r \sin(2r^2)}{\alpha \sin(r^2)}=-\frac{2\beta}{\alpha}r\cos(r^2).$$
The second equation leads to $$\theta(t)=\theta(t_0)-\alpha\int\limits_{t_0}^t \sin(r(t)^2)dr(t)=\theta(t_0)-\alpha\int\limits_{t_0}^t \sin(r(t)^2)\dot r (t) dt \\=\theta(t_0)-\alpha\beta\int\limits_{t_0}^t \sin(r(t)^2)\sin(2r(t)^2)r(t) dt.$$
Now I need to find all constant and time-periodic solutions and also say something about the other solutions (not constant or time-periodic).
Constant solutions: By using $\dot r=0,\dot \theta=0,r(t)\equiv r$ and $\theta(t)\equiv \theta$ I get $$\beta r \sin(2r^2)=0\Leftrightarrow r=\pm \Big( \frac{\pi k}{2}\Big)^\frac{1}{2}\\-\alpha \sin(r^2)=0\Leftrightarrow r=\pm (\pi k)^\frac{1}{2}$$ for $k\in\mathbb{Z}$. Since both equations need to hold, if $\theta$ and $r$ are constant, and $r\in\mathbb{R}_{>0}$, $r$ needs to be $(\pi k)^\frac{1}{2}$ for $k\in\mathbb{N}$. But what can I say about $\theta$ (is there any restriction)? And how can i understand time-periodic solutions?
The trouble is that there is no standard closed forms for the integrals involved. So, the solution cannot be expressed on explicite form, but only with integrals in the implicit equations.