Basically what you're saying is that the force $F(t)$ is always directed radially, but its magnitude varies arbitrarily. Actually you didn't say $M > 0$, and in fact negative "mass" may be required in some cases. Then the answer is yes.
In polar coordinates $(r,\theta)$, your assumption says that $2 \dot{r} \dot{\theta} + r \ddot{\theta} = 0$ Basically this is conservation of angular momentum. If $r = f(\theta)$, the chain rule says $\dot{r} = f'(\theta) \dot{\theta}$, so $2 f'(\theta) \dot{\theta}^2 + f(\theta) \ddot{\theta} = 0$. For any given function $f$ with $f > 0$, a solution of the differential equation $2 f'(\theta) \dot{\theta}^2 + f(\theta) \ddot{\theta}$ with, say, $\theta(0) = 0$ and $\dot{\theta}(0)=1$ gives us a motion that starts at $\theta = 0$ and stays on the curve $r = f(\theta)$. Now the existence and uniqueness theorems for differential equations say that (if $f$ is smooth and $f > 0$ everywhere) a unique solution exists in some interval of time $t$. Moreover, the only way such a solution will stop existing is that $\theta$ goes off to $\infty$ in finite time. But we're assuming $f$ is periodic, and because of the conservation of angular momentum you have to come back to the starting position with the same speed as you started, so that won't happen.