Differential Nonlinear Robustness of Critical States in Fibonacci and Tribonacci Substitution Chains

A Numerical Study of Discrete Nonlinear Schrödinger Dynamics Version 4 — finite-size scaling, long-time evolution, and self-trapping threshold gap

Pablo Nogueira Grossi · G6 LLC, Newark NJ · ORCID: 0009-0000-6496-2186 Zenodo V4: 10.5281/zenodo.20075822 Concept DOI (auto-resolves to latest): 10.5281/zenodo.20026942

What this paper does

We study the discrete nonlinear Schrödinger (DNLS) equation on two quasiperiodic tight-binding chains — the Fibonacci chain (n=2) and the Rauzy–tribonacci chain (n=3) — generated by their respective substitution rules. Starting from mid-gap eigenstates of the linear Hamiltonian, we integrate the DNLS time evolution over nonlinearity strengths λ ∈ [0, 10] and times T ∈ [50, 10⁶], tracking the inverse participation ratio (IPR) as a measure of localization.

Core finding (all versions): The tribonacci chain exhibits differential nonlinear robustness relative to the Fibonacci chain. At the canonical parameters (T=50, N=500, λ=1.5), the tribonacci mid-gap state retains >95% of its linear IPR while the Fibonacci state loses ~57%, a ratio of ~8.6×. This is termed differential nonlinear robustness; its mechanism is the stronger multifractal spatial hierarchy of the Rauzy–tribonacci eigenstate (IPR ratio ~3.9× in the linear limit).

The characteristic decay constant η ≈ 1.839287 is the Perron–Frobenius eigenvalue of the tribonacci companion matrix. Its existence as the unique real root of x³ − x² − x − 1 = 0 in [1,2], the property η > 1, and the strict antitonicity of the geometric weight sequence {η⁻ᵏ} are formally verified in Lean 4 / Mathlib4 without sorry in TribonacciMeasure.lean of the AXLE repository (github.com/TOTOGT/AXLE).

To our knowledge this is the first numerical study of DNLS dynamics on a tribonacci substitution chain.

Version history

V1 (May 2026) — baseline

• T=50 baseline at N=500 across λ ∈ [0,10]
• Established differential nonlinear robustness: Fibonacci IPR drops ~57% at λ=1.5; tribonacci drops <5%
• Tribonacci linear-limit IPR ~3.91× larger than Fibonacci (IPR = 0.0820 vs 0.0210)
• Named the phenomenon; proposed multifractal hierarchy as mechanism
• Lean 4 verification of η > 1 and strict antitonicity of {η⁻ᵏ}
• Companion code: dnls_nbonacci.py; companion Lean file: TribonacciDNLS.lean

V2 — figures and LaTeX

• Added publication-quality figures and full LaTeX source

V3 — (intermediate)

• Additional figures and analysis

V4 (current) — finite-size scaling, long-time evolution, self-trapping threshold gap

New in V4:

1. Finite-size scaling at T=10⁴ (Section 4.4) Across N ∈ {500, 1000, 2000} at seven values of λ, the differential ratio IPRtrib/IPRfib generally grows with N at fixed λ, supporting persistence of the differential in the thermodynamic limit. The λ=1.5 non-monotone N-dependence (peak at N=1000) is confirmed transient by the T=10⁵ verification run.

2. Long-time evolution to T=10⁶ (Section 4.5) At N=1000, λ=1.5: the ratio peaks at 4.18× at T=10⁴, collapses to 1.20× at T=10⁵, and from T~3×10⁵ onward both chains saturate to chain-limited IPR ≈ 0.002, with the ratio settling into an oscillatory band around 1.04 ± 0.04. The T=10⁴ peak is a transient finite-size feature. Long-time runs use the DOP853 integrator (rtol=10⁻⁹, atol=10⁻¹¹); norm conservation verified to <10⁻⁵ throughout.

3. Pre-saturation spreading rates (Section 4.6) Fitting IPR(t) ~ t⁻ᵅ in the window t > 10⁴ of T=10⁵ runs at N=2000: αtrib > αfib uniformly across λ ∈ [0.5, 2.0]. Tribonacci spreads faster toward saturation but starts from a higher initial IPR and remains more retained throughout. (Note: α here is a pre-saturation rate toward finite-size saturation, not an asymptotic power-law exponent.)

4. Self-trapping threshold gap (Section 4.7) Full sweep λ ∈ {1, 2, 4, 8, 10} at N=500, T=10⁵ reveals a qualitative distinction: the Fibonacci chain crosses into the self-trapped regime (α ≈ 0) in λ ∈ [4, 8], while the tribonacci chain has not self-trapped by λ=10 (αtrib ≥ 0.09 throughout). This is a difference in regime, not just magnitude.

5. Multifractal dimension extraction (Section 4.3) At natural Fibonacci lengths Fₙ ∈ {233, 377, 610, 987, 1597, 2584}: clean power law, D²fib = 0.65 ± 0.11 (R² = 0.90). Tribonacci at natural Rauzy lengths T⁽³⁾ₙ ∈ {274, 504, 927, 1705, 3136}: plateau across n=9,10,11 (IPR ≈ 0.082 constant), then discontinuous drop at n=12 (IPR = 0.041). This is a state-isolation artifact in the closest-to-E=0 selector; extraction of D²trib is deferred pending state-tracking diagnostics (new open question, Section 7).

6. Connection to companion criticality paper The spectral-gap-controls-threshold mechanism identified here (tribonacci self-trapping threshold higher than Fibonacci because η > φ) is the nonlinear-dynamics counterpart of the criticality threshold result in the companion paper (10.5281/zenodo.20077205), where λc(n) ≈ 0.958 Δₙ + 0.107 across n=2,...,5 with r=0.989.

Open questions (current status)

Files in this deposit

All long-time simulation code, raw output CSVs, analysis notebooks, and figure generators are openly available at github.com/grossi-ops/Atratores.

Lean 4 formal verification

Key analytic lemmas supporting the amplitude envelope ansatz are proved without sorry in TribonacciDNLS.lean (this deposit) and TribonacciMeasure.lean (AXLE repository):

• η_gt_one: 1 < η ✓
• η_characteristic: η³ = η² + η + 1 ✓
• w_pos: ∀ k, 0 < η⁻ᵏ ✓
• w_strictAnti: StrictAnti (k ↦ η⁻ᵏ) ✓
• w_tendsto_zero: η⁻ᵏ → 0 as k → ∞ ✓

Open Lean proof obligations (tracked in AXLE sorry roadmap): IPRtrib(0) > IPRfib(0); Lean 4 statement of the differential nonlinear robustness threshold.

Keywords

tribonacci chain · Fibonacci quasicrystal · discrete nonlinear Schrödinger equation · multifractality · inverse participation ratio · finite-size scaling · spreading exponent · self-trapping · Rauzy substitution · Lean 4 formal verification · quasiperiodic tight-binding · differential nonlinear robustness