 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/ii/paper-4/32a/v/solution

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

The exact second-iterate equation factors as

F (F (x)) - x = x (λ x^{2} - λ - 1) (λ x^{2} - λ + 1) (λ^{2} x^{4} - λ^{2} x^{2} + 1) .

The positive nonfixed period-two points come from the final factor. If

y = x^{2}

, their squared values are

y_{\pm} = \frac{1 \pm 1 - 4/ λ ^{2}}{2}, y_{+} + y_{-} = 1, y_{+} y_{-} = \frac{1}{λ ^{2}} .

They are real and distinct for

λ > 2

F^{'} (x_{+}) F^{'} (x_{-}) = λ^{2} (1 - 3 y_{+}) (1 - 3 y_{-}) = λ^{2} (1 - 3 (y_{+} + y_{-}) + 9 y_{+} y_{-}) = 9 - 2 λ^{2} .

For

λ = 2 + μ

with

0 < μ ≪ 1

9 - 2 λ^{2} = 1 - 8 μ + O (μ^{2}),

whose modulus is less than one. The new two-cycle is therefore locally asymptotically stable.

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