Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 33A (Dynamical Systems)

On each side of the stated parallelogram, substitution into the outward normal component ofshows that the leading $L^3$ terms point inward; the remaining terms are $O(L)$, so the region is positively invariant for sufficiently large $L$.

There is one equilibrium,Its Jacobian has determinant one and trace $\mu (1-F^2)$. If $F^2<1$ the equilibrium is unstable. A trajectory starting nearby remains in the compact trapping region, and its omega-limit set contains no stable equilibrium. The Poincare-Bendixson theorem therefore supplies a periodic orbit.

Eliminating $y$ givesFor $\mu \ll 1$, take $x=F+A\cos (t+\delta )$ over one fast period. The oscillator energy satisfiesand averaging givesThus, if $F^2<1$, amplitudes approachproducing a stable weakly nonlinear limit cycle. If $F^2>1$, every nonzero small amplitude loses energy and trajectories approach the stable equilibrium.

For $\mu \gg 1$, trajectories rapidly approach the cubic nullclineand then move slowly along its attracting outer branches $|x|>1$. For $0<F<1$, the equilibrium lies on the repelling middle branch. Slow motion to a fold, a rapid jump to the opposite outer branch, and repetition produce a relaxation oscillation. For $F>1$, the equilibrium lies on the attracting right branch; trajectories undergo any necessary fast jump and then drift to that equilibrium, so no relaxation cycle remains.