Measuring the gap between the Melnikov threshold and the period-doubling cascade in the damped driven pendulum

1. Introduction

The sinusoidally driven, damped pendulum is the canonical low-dimensional route to chaos, and it supports two different analytic/numerical landmarks as the drive amplitude A grows. The Melnikov method gives a closed-form first-order threshold A_c = (4γω₀/π)·cosh(πω/2ω₀) above which the stable and unstable manifolds of the hilltop saddle intersect transversally, creating a Smale-horseshoe tangle. The tangle guarantees transient chaos and fractal basin boundaries — not a chaotic attractor. The sustained chaotic attractor instead appears at the end of a period-doubling cascade of the primary period-1 response, whose onset A_PD is a property of a specific attractor branch and has no closed form. Textbooks typically note A_c < A_PD at the classic parameter point γ = 0.5, ω = 2/3. This paper asks the quantitative question: how does the gap between the two thresholds behave as damping is varied?

2. Methods

All computations use the open Pendulum Lab engine, whose equations of motion are validated component-wise against an independent SymPy symbolic derivation (max relative deviation ≈ 10⁻¹⁴) and whose trajectories are cross-validated against SciPy DOP853 at rtol = 10⁻¹³; the engine also reproduces published anchors (elliptic pendulum period, double-pendulum normal modes, and the γ = 0.5 period-doubling onset). The drive is made autonomous by carrying the phase as a third coordinate; integration is RK4 with dt = 0.005 snapped so an integer number of steps spans one drive period exactly.

A_PD measurement. For each γ, the drive amplitude is marched upward from 0.9·A_c with the strobed state warm-started at every step, so the measurement follows the physically realised attractor branch (this matters: the symmetric period-1 orbit pitchforks before the cascade, and a Newton continuation of the symmetric orbit would miss the doubling of the symmetry-broken branch the attractor actually follows). The loss of period-1 stability is bisected on the strobe map (transients of 300–600 drive periods; period detected in the (sin θ, cos θ, ω) embedding, immune to 2π winding). The bracket is then refined by Newton periodic orbits seeded from the attractor: the most negative real Floquet multiplier ρ(A) of the orbit is interpolated through ρ = −1. A measurement is accepted as a period doubling only when this crossing is verified; otherwise the loss is reported as non-PD. Halving dt changes the γ = 0.5 onset by |Δ| ≈ 7.1e-13, so discretisation error is negligible at the quoted precision.

Corroboration. At 0.97·A_PD and 1.08·A_PD the Gottwald–Melbourne 0–1 test is applied to the cos θ strobe series (700 samples): K ≈ 0 confirms regular motion below onset; the value above onset depends on whether 1.08·A_PD lands beyond the cascade accumulation point or inside a periodic window, and is reported without prejudice.

3. Results

4. Discussion

Three regimes emerge. (i) Low damping (γ ≲ 0.2): A_c → 0 linearly while the cascade onset of the primary resonance remains an order-one amplitude, so the ratio diverges (2.38 already at γ = 0.1). The window of “tangle but no strange attractor” — long chaotic transients and fractal basin boundaries below a still-periodic attractor — is widest here, and the phase space is visibly multistable (see §5). (ii) Moderate damping: the textbook ordering A_c < A_PD holds, but the margin shrinks steadily (to 5% at γ = 0.5). (iii) Strong damping (γ ≳ 0.69): the measured cascade begins below the first-order Melnikov prediction. This is not a contradiction — the Melnikov threshold is asymptotically exact only as γ, A → 0, and by γ ≈ 0.7 the perturbation parameter is O(1) — but it sharpens the usual caveat into a measured boundary: the first-order formula stops being even an ordering bound near γ ≈ 0.69.

The 0–1 test values corroborate the structural picture: K ≈ 0 on the period-1 side everywhere, while above onset K depends on where 1.08·A_PD falls relative to the cascade accumulation point and the periodic windows visible in Figure 3 — both outcomes occur in the table, as expected for a Feigenbaum scenario with embedded windows.

5. Beyond one frequency and one system

Two extensions probe whether the closing gap is an artifact of the classic parameter point. First, a reduced-grid frequency scan repeats the full Floquet-refined measurement at ω = 0.5 and ω = 0.85. The shape survives: the ratio falls monotonically with damping at every frequency, and the crossing damping γ* moves with ω — at ω = 0.85 the gap is still open at γ = 0.8 (ratio 1.096), while at ω = 0.5 the measured points at γ = 0.65 and 0.8 already sit below 1 (0.941, 0.927). At ω = 0.5, γ = 0.2; ω = 0.5, γ = 0.35; ω = 0.5, γ = 0.5 the primary branch loses period-1 stability without a verified ρ = −1 crossing, and those points are reported as unclassified rather than forced into the map.

Second, the same question is posed to a different system: the symmetric Duffing double well ẍ = −δẋ + x − x³ + Γcos(t). The closed-form Melnikov threshold Γ_c(δ) (derived and quadrature-verified in the engine, reducing to the Guckenheimer–Holmes expression at α = −1, β = 1) is compared with the marched/bisected loss of period-1 stability of the confined single-well attractor. There is no Newton–Floquet refinement for the Duffing flow yet, so Γ_PD is quoted as a bisection bracket midpoint — honestly wider than the pendulum's interpolated onsets, though the brackets themselves are ~10⁻⁸ wide.

The 0–1 test corroborates every Duffing onset (K ≈ 0 below, K ≈ 1 above — the cascade accumulates quickly in the double well at these parameters). Together the two extensions upgrade the paper's claim from "the thresholds cross at one classic parameter point" to "first-order Melnikov ordering degrades with damping across drive frequencies and across systems".

6. Limitations and reproducibility

The main grid fixes ω = 2/3 (the frequency scan of §5 samples ω = 0.5 and 0.85 on a reduced γ grid) and follows a single attractor branch per γ (warm-started in A); coexisting attractors reached from other initial conditions may double elsewhere. At the lowest dampings the phase space is multistable enough that a finer warm-started march can hop basins before the doubling — at γ = 0.15 a basin-capture transition of the followed state was observed near A ≈ 0.49 (the orbit itself remains strongly stable there, ρ ≈ +0.29), below the verified doubling at 0.531. The quoted A_PD values are therefore specifically the ρ = −1 doublings of the primary oscillating branch, not necessarily the first event of any kind along a slow amplitude sweep. The Melnikov comparison concerns the first-order formula specifically — higher-order or numerical manifold computations would move A_c. The full study regenerates with npm run paper:study (~8 min) followed by npm run flagship:certify, npm run flagship:external, and npm run paper:build; the engine tests and independent symbolic/trajectory validations are in the same repository.

References