 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2021

Paper 2

Eliminating $y$ gives
$$
\ddot x+x-F=\mu(1-x^2)\dot x.
$$
For $\mu\ll1$, take $x=F+A\cos(t+\delta)$ over one fast period. The oscillator energy satisfies
$$
\dot E=\mu(1-x^2)\dot x^2,
$$
and averaging gives
$$
\langle\dot E\rangle
=\frac{\mu A^2}{2}\left(1-F^2-\frac{A^2}{4}\right).
$$
Thus, if $F^2<1$, amplitudes approach
$$
A=2\sqrt{1-F^2},
$$
producing a stable weakly nonlinear limit cycle. If $F^2>1$, every nonzero small amplitude loses energy and trajectories approach the stable equilibrium.

Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 33A / b / Solution

Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 33A / b / Solution

 Ancestors (11)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 33A / b / Solution

 Ancestors (11)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)