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Past exam of the mathematics course of the University of Cambridge

2024

Paper 2

The only equilibrium is $(0,0)$. Its linearization is
$$
\begin{pmatrix}\dot x\\\dot y\end{pmatrix}
=
\begin{pmatrix}0&1\\-1&0\end{pmatrix}
\begin{pmatrix}x\\y\end{pmatrix},
$$
whose <eigenvalues> are $\pm i$, so the linearized system has a centre.

The exact radial identity from part (i) gives
$$
\dot r=kr^3.
$$
Thus the origin is asymptotically stable for $k<0$, neutrally stable for $k=0$, and unstable for $k>0$. Moreover,
$$
\dot\theta=\frac{x\dot y-y\dot x}{r^2}=-1,
$$
so trajectories rotate clockwise.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 2 / 7A / ii / Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 2 / 7A / ii / Solution

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Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 2 / 7A / ii / Solution

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