Let the system
\begin{align*}
\dot{x} &= y^3 -4x\\
\dot{y} &= y^3-y-3x
\end{align*}
(a) Prove that the line $y=x$ is an invariant set.
(b) Prove that $|x(t) – y(t)| \to 0$ when $t \to \infty$, for all other trajectories.
I have some questions of this exercise. For (a) I proposed the function $V(x,y)=x-y$, then I calculated $\dot{V}(x,y)$ and evaluate the function in the points $y=x$. The result I got was that $$\dot{V}(x,y)|_{x=y}=0$$
but then I do not know how to conclude. Is this right? For (b) I do not know well how to proceed.
Any hint?
Best Answer
First question. We subtract the two equations and we get:
$ x'(t)-y'(t)=-(x(t)-y(t))$. We define $z(t)=x(t)-y(t)$ then we get :
$z'(t)=-z(t)$ which gives $z'(t)+z(t)=0$ multiply by $e^{t}$ and get:
$(e^{t}z(t))'=0$ which gives $e^{t}z(t)=c$. If for some $t_{0}$ we have $x(t_{0})=y(t_{0})$ then $z(t_{0})=0$ and
$c=0$ and hence $z(t)=0$ for all $t$, i.e.
$x(t)=y(t)$ for all $t$. So $x=y$ is invariant.
Second question. By the same equation for $c\neq\,0$ we obtain $z(t)=ce^{-t}$ and clearly
$|z(t)|=|x(t)-y(t)|\to 0$ when $t\to +\infty$.