Basin of Attraction of the Golden Angle in Douady-Couder Phyllotaxis

The Douady-Couder model established that sequential optimal packing on a cylindrical surface converges to the golden angle (≈137.5°) within a specific range of the growth parameter Γ. However, the fine structure of the attractor basin of attraction in Γ-space has not been sufficiently characterised.

This paper presents three results.

First, we prove analytically that concentric spherical shells with harmonic spacing (inter-shell gaps proportional to r²) produce a strictly constant Γ under spherical inversion r → R²/r (Theorem 1).

Second, a systematic 150-point numerical sweep over Γ ∈ [0.01, 1.0] reveals that the golden-angle basin of attraction consists of an unstable lower region (Γ ≈ 0.22–0.28, containing internal distichous interruptions) and a stable upper band (Γ ≈ 0.50–0.84, with near-zero variance), and that sharp, first-order phase-transition-like transitions occur at both ends of the upper band.

Third, we show that under a specific non-constant Γ sequence producing a strongly inner-dense/outer-sparse distribution, convergence to the golden angle is achieved despite a basin residence fraction of only 31%, and propose a history-dependent mechanism whereby pattern establishment in the initial stages persists thereafter.

Finally, we pose the analytic determination of the basin boundaries as an explicit open problem and discuss connections to Arnold tongues and KAM theory.