What is the order of the following system $\frac{dx}{dt} = \sqrt{1-x^2}$

Question

$\frac{dx}{dt} = \sqrt{1-x^2}$

I’m referring “Non linear dynamics” by Steven Strogatz. Does the above mentioned system belongs to first order? One particular solution of the system is $x(t) = \sin(t)$. Then there are no fixed point in the system, rather the system exhibits limit cycle. Still I’m able to represent the system in first order format. As per Strogatz, oscillations are prohibited in first order system. What am I missing here?

Answer 1

Yes, $\frac{dx}{dt} = \sqrt{1-x^2}$ is a first order autonomous differential equation, but no, it does not have oscillatory solutions. Notice that $\frac{d}{dt}\sin(t)=\cos(t)$, but $\sqrt{1-\sin^2(t)}=|\cos(t)|$, so $\sin(t)$ is a solution to the ODE only for $t\in(-\pi/2,\pi/2)$. At $t= \pi/2$, the motion stops; $x(t)$ cannot increase beyond $x=1$, for it would make $\sqrt{1-x^2}$ imaginary, nor cannot decrease, since $\frac{dx}{dt}\geq 0$. Therefore, $x=1$ is a stable fixed point.