I consider the curve $c(t)=(x(t),y(t))$ in $\mathbb{R}^2$ such that


$\frac{d^2x(t)}{dt^2}=-(a\sin t+b)\frac{dy(t)}{dt}$


$\frac{d^2y(t)}{dt^2}=(a\sin t+b)\frac{dx(t)}{dt}$


$a,b\in\mathbb{R}$


Is the orbit curve of solution of above equations known?



Is this curve well known?
It is not clear what you mean by "known" but this system can be solved explicitly, in quadratures of elementary functions. Set $x'=u,\; y'=v,\; g(t)=a\sin t+b$. Then your system becomes
$$u'=-gv,\quad v'=gu.$$
Multiplying the first equation on $u$ and second on $v$ and adding, we obtain
$u'u+v'v=0,$ therefore $u^2+v^2=c$. Then the first equation becomes
$(u')^2=g^2(c-u^2)$, and this is a separable equation, with an explicit integral.


