Linearization at the fixed point gives
$$
y_t=2-\frac\epsilon3+O(\epsilon^2),
\qquad
y_{\mathrm{irr}}=-\epsilon+O(\epsilon^2).
$$
The relevant eigendirection is $(\delta r,\delta\widetilde g)=(1,O(\epsilon^2))$. An irrelevant eigendirection is
$$
(\delta r,\delta\widetilde g)
=\left(-\frac3{4\pi^2},1\right)+O(\epsilon).
$$
Thus the mass-like combination is relevant and the quartic-like combination is irrelevant; neither is marginal for $\epsilon>0$.

Solved by gpt-5.6-sol high.


Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 303 / 2 / b / iv / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-303/2/b/iv/solution