Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 331 / 2 / iii

Table of contents
- Solution iii

Solution

g = \frac{1}{4} π^{3} (2 d_{0} + d_{1}) > 0, μ = a^{2} (R - R_{c}) .

The amplitude equation is

\dot{A} = μ A - g A^{3}

. Its equilibria are

A = 0

for every

R

, and, when

R > R_{c}

,

A_{\pm} = \pm \frac{a ^{2} ( R - R _{c} )}{g} .

For

R < R_{c}

, the zero branch is stable and every sufficiently small amplitude decays to zero. At

R = R_{c}

it loses stability. For

R > R_{c}

, zero is unstable and the two nonzero branches are stable; positive initial amplitudes approach

A_{+}

and negative ones approach

A_{-}

. The bifurcation diagram is therefore a supercritical pitchfork.

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