Simple proof of invariant sets

Question

Let

How to prove the unit circle is an invariant set?

My way is that:

At $t_1$, $x_1(t_1)^2 + x_2(t_2)^2 = 1$, so the system of equations becomes:

Since both $x_1$ and $x_2$ are functions of $t$, so solve it and obtain:

Then how to verify it?

Thanks!!

Answer 1

Consider $s(t)=x_1^2(t)+x_2^2(t)$. You know that $s(\bar{t})=1$. Now find the derivative: $$ \dot s=2(x_1\dot x_1+x_2\dot x_2)\\ =2(x_1x_2+x_1^2(1-s)-x_2x_1 +x_2^2(1-s))\\ 2s(1-s). $$ Since, one more time, $$ \dot s=s(1-s), $$ then $s=1$ is an equilibrium, hence $s=1$ is a solution, therefore $$ s(t)\equiv 1 $$ for any $t$.

Answer 2

Consider the system $\dot{r} = r(1 - r^2)$, $\dot\phi = \sin^2 \phi + a$ around $a = -1$. Note that $\dot{r}$ can only vanish for $r = 0, 1$, implying that the origin is an equilibrium and that the unit circle is an invariant set. Q.E.D.