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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2024/ii/paper-3/31a/a/solution

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Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 31A / a / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-06

X (t)

gives

\dot{Φ} (t) = D f (X (t)) Φ (t), Φ (0) = I .

(1)

One Floquet multiplier of

Φ (T)

1

, corresponding to a phase shift along the orbit. Liouville's formula gives

det Φ (T) = exp (\int_{0}^{T} \nabla \cdot f (X (t)) d t) .

(2)

In the plane this determinant is the other, transverse multiplier. The periodic orbit is transversely asymptotically stable when the displayed quantity is less than one, equivalently when the divergence integral is negative, and unstable when it is greater than one.

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