TITLE: Show that every orbit is forward bounded
QUESTION [5 upvotes]: This was a question in an exam on ODEs:
For the planar system
$$
\begin{align}
x' &= 3y\\ 
y' &=-x^3+x-4y
\end{align}
$$
show that every orbit is forward bounded.
I have no idea how to show this. What can I do?

REPLY [3 votes]: You can show this using a Lyapunov function. Define 
$$V(x,y) = -2 x^2 + x^4 + 6 y^2$$
which is a function with a single minimum at $x=-1$, $y=0$.
During the evolution, the function changes as 
$$V' = (\partial_x V)  x' + (\partial_y V) y'
= 4 x (x^2-1) 3y + 12 y(-x^3 + x - 4 y) = -48 y^2 \leq 0 $$
that is it decreases or stays at most constant.
So the motion will be forever confined in the region with $V(x,y) \leq V^*$ where $V^*$ is the Lyapunov function evaluated at the initial condition.