Gaussian Collatz Dynamics

Computational investigation of Collatz-like dynamical systems on the Gaussian integers Z[i], focusing on two contrasting variants:

Variant	Character	Key result
C	Balanced / globally stable	Average Lyapunov exponent λ ≈ −0.66; stable for k > √2
E	Chaotic / fractal	Period-40 cycle (proved stable); basin boundary D ≈ 1.61

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Publication

This code accompanies the research paper:

Garg, A. (2026). Collatz-Type Dynamics on Gaussian Integers: Fractal Stability Islands, Long Cycles, and Dyadic Rational Structure (Version 1.0). Zenodo. https://doi.org/10.5281/zenodo.19078383

Abstract: We introduce and study a family of Collatz-type dynamical systems defined on the Gaussian integers. Our computational investigation reveals three principal findings: (i) a periodic orbit of length 40, proved locally asymptotically stable with contraction factor 2^-45, (ii) orbits live in the ring of dyadic Gaussian rationals Z[1/2][i], and (iii) the stability boundary is fractal with dimension D ≈ 1.61.

Paper: View on Zenodo

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Repository layout

gaussian-collatz-dynamics/
├── paper/
│   ├── collatz_arxiv.tex       LaTeX source
│   └── figures/                All figures (PNG)
├── code/
│   ├── variant_c.py            Variant C step function + orbit iterator
│   ├── variant_e.py            Variant E step function + orbit iterator
│   ├── exact_cycle.py          Exact rational arithmetic; proves 40-cycle stability
│   ├── fractal_dimension.py    Box-counting dimension for Variant E boundary
│   ├── lyapunov.py             Lyapunov exponent distribution for Variant C
│   ├── initial_survey.py       Evaluation of stability over lattice points
│   └── visualizations.py       Driver — regenerates all paper figures
├── data/
│   ├── 40cycle_exact.txt       Exact rational coordinates of the 40-cycle
│   ├── stability_islands.csv   Converged/diverged grid for Variant E
│   └── lyapunov_results.csv    Per-orbit Lyapunov exponents for Variant C
├── requirements.txt
└── LICENSE

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Quick start

pip install -r requirements.txt
cd code
python visualizations.py        # regenerates all figures and data files

Individual modules can also be run standalone:

python exact_cycle.py           # verify 40-cycle, write data/40cycle_exact.txt
python lyapunov.py              # compute Lyapunov distribution
python fractal_dimension.py     # compute box-counting dimension

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Map definitions

Both variants operate on z = a + bi ∈ Z[i] and branch on the parities of a and b.

Variant C (balanced)

Condition	Rule
a even	z / k (default k = 2)
a odd, b even	(1+i)z + 1
a odd, b odd	(1+i)z + i

Contraction probability 1/2, expansion factor √2 per step. Theoretical stability threshold: k > √2 ≈ 1.414.

Variant E (chaotic)

Condition	Rule
a even, b even	z / 4
a odd, b even	(1+i)z + 1
a even, b odd	(1−i)z + i
a odd, b odd	(1+i)z + (1+i)

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Key results

Variant C — global convergence

• Empirical convergence rate ≈ 100 % on Z[i] ∩ [−100, 100]².
• Mean Lyapunov exponent λ ≈ −0.66 (all 1 000 sampled orbits negative).
• Critical threshold k★ = √2 confirmed analytically and numerically.

Variant E — period-40 cycle

• Cycle starts at z₀ = −120 + 66i.
• Proved locally asymptotically stable with contraction factor 2^−45.
• All 40 elements lie in the dyadic Gaussian rationals Z[i, 1/2]: denominators are powers of 2, peaking beyond 2^1000 mid-cycle.
• Verified with zero floating-point error via fractions.Fraction.

Variant E — fractal basin boundary

• Box-counting dimension D ≈ 1.61 (R² ≈ 0.966).
• Boundary computed on a 10,000 × 10,000 grid over [−50, 50]².
• Uses continuous orbit iteration in ℂ.

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Figures

File	Description
variant_e_julia.png	Stability islands of Variant E, coloured by cycle ID
fractal_fit.png	Log-log regression for box-counting dimension
denom_growth.png	Denominator growth along random Variant E orbits
lyapunov_dist.png	Lyapunov exponent histogram for Variant C

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License

MIT — see LICENSE.