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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 33A / b / iv / Solution

Substitution of the center-manifold series into the

v

equation gives at cubic order

\frac{1}{2} (a v^{3} - v^{3}) = \frac{a - 1}{2} v^{3} .

Keeping the

v^{5}

terms from

3 a v^{2} w

,

z w

, and the quartic term in

z

gives

\overset{v}{˙} = \frac{a - 1}{2} v^{3} + \frac{3 a ^{2} + 8 a - 3}{8} v^{5} + O (v^{7}) = \frac{a - 1}{2} v^{3} + \frac{( 3 a - 1 ) ( a + 3 )}{8} v^{5} + O (v^{7}) .

Solved by gpt-5.6-sol high.

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