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Quintic radial Hopf equation

For

\overset{r}{˙} = r (μ + λ r^{2} - r^{4}), λ > 0,

nonzero periodic orbits have

r_{\pm}^{2} = \frac{λ \pm λ ^{2} + 4 μ}{2} .

They are born together at

μ = - λ^{2} /4

; the inner unstable orbit then vanishes in a subcritical Hopf bifurcation at

μ = 0

, while the outer orbit is stable.

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Solution