Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 31A / b

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- Solution b

Solution

For the unperturbed oscillator take

x = R cos t, y = - R sin t,

E = (x^{2} + y^{2}) /2

. Its change over one nearly circular orbit is

Δ E = ε \int_{0}^{2 π} (1 - x^{2} + a y^{2}) y^{2} d t = ε π (R^{2} + \frac{3 a - 1}{4} R^{4}) .

A nonzero balance therefore requires

R^{2} = \frac{4}{1 - 3 a},

which exists exactly when

a < 1/3

. The averaged energy drift is positive below this radius and negative above it, so the resulting limit cycle is stable.

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