Solution

Past exam of the mathematics course of the University of Cambridge / 2008 / ii / Paper 3 / 14A / b / Solution

For $\dot x=y+ax-bx^3$, $\dot y=x^3-x$, the function $H=y^2/2+x^2/2-x^4/4$ obeys $\dot H=(x-x^3)(ax-bx^3)$. Every nonconstant <periodic orbit> lies in $|x|<1$: at a maximum above one or a minimum below minus one, $\ddot x=x^3-x$ has the wrong sign; equality would force the solution to be an <equilibrium> by uniqueness. If $a\ne0$ and $b/a<1$, then $\dot H=a x^2(1-x^2)(1-(b/a)x^2)$ has the strict sign of $a$ except at $x=0$, preventing a nonconstant <periodic orbit>.

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 Incoming links: Energy obstruction for a cubic planar oscillator