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Bifurcation theory

For
$$
\dot r=r(\mu+\lambda r^2-r^4),\qquad \lambda>0,
$$
nonzero periodic orbits have
$$
r_\pm^2=\frac{\lambda\pm\sqrt{\lambda^2+4\mu}}2.
$$
They are born together at $\mu=-\lambda^2/4$; the inner unstable orbit then vanishes in a subcritical Hopf bifurcation at $\mu=0$, while the outer orbit is stable.


Quintic radial Hopf equation

Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 31A / b / iii / Solution

Quintic radial Hopf equation

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Quintic radial Hopf equation

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