Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 33A / c

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- Solution c

Solution

At a fixed point with

x, y > 0

, the two bracketed factors must vanish:

μ - y^{2} - 2 x^{2} = 0, 1 - x^{2} - y^{2} = 0.

Subtracting gives

x^{2} = μ - 1, y^{2} = 2 - μ .

Hence the interior branch is

(x, y) = (μ - 1, 2 - μ), 1 < μ < 2.

Near

μ = 1

, if

ε = μ - 1

, then

x = ε, y - 1 = 1 - ε - 1 = - \frac{ε}{2} + O (ε^{2}) = - \frac{x ^{2}}{2} + O (x^{4}),

which agrees with both the centre manifold and the positive branch of the pitchfork normal form.

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