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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?

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The standard approach is to come up with a function $V(x,y)$ which does not increase in time. – user53153 Jan 1 '13 at 22:23

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.

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