 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ii/paper-2/33a/b/iv/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

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}) .

 New to topics? Read the docs here!