NeuroGenesysAn expository note. The results below are correct and, in this exact form, appear to be unpublished — but they are a worked example of the weak-noise Lyapunov-exponent expansion (Arnold–Papanicolaou–Wihstutz 1986 and the weak-disorder localization literature), not a new theorem or a new phenomenon. It is offered as a tidy, fully explicit special case with a pleasant golden-ratio corollary, and honestly labelled as such.
1. Setup
Take a second-order linear recurrence and perturb its two coefficients in opposite directions by the same amount, so their sum is conserved ("balanced" transfer), where the perturbation is a memory-filtered noise:
with A, B > 0, 0 ≤ a < 1, and q_n a bounded, zero-mean, stationary process with spectral
measure σ_q. The memory is a one-pole (AR(1)) filter: d_n is q_n colored to the Lorentzian
spectrum M_a(ω) = 1/(1 + a² − 2a cos ω).
We study the Lyapunov exponent Λ(ε) = lim_{n} (1/n) log|x_n|.
2. The result
Let λ = (A + √(A²+4B))/2 be the dominant root of z² = Az + B, and β = B/λ². Then
with the kernel factoring as memory filter × recurrence geometry:
Because a occurs only in M_a, memory decides how strongly each frequency is felt; the
recurrence geometry P decides whether it helps or hurts. The sign of the response flips at
Golden-ratio corollary (A = B = 1)
Then λ = φ, β = 1/φ², and everything collapses to clean constants:
where a_c is the memory strength at which the flat-forcing response ∫_{-π}^{π} W_a(ω)\,dω
changes sign: below a_c broadband forcing suppresses growth, above it enhances growth.
It generalizes to every order (same shape)
For a p-th order balanced recurrence x_{n+1} = Σ_k (c_k + d_n δ_k) x_{n-k} with
Σ_k δ_k = 0, the kernel keeps the same skeleton:
where the product runs over the subdominant roots μ_i of the characteristic polynomial and
N_δ is a low-degree polynomial fixed by the perturbation direction δ. The poles are pinned
at the eigenvalue ratios — nothing about them is free. When a subdominant pair is complex the
kernel develops a broad peak near their oscillation frequency (ordinary near-mode resonance; it
only sharpens as |μ|/λ → 1).
3. Where this sits
The general statement — the Lyapunov exponent of a weakly, randomly perturbed linear recurrence
has the form Λ₀ + ε²·(kernel integrated against the forcing spectrum) + O(ε³), with the linear
term vanishing — is classical:
- Arnold, Papanicolaou & Wihstutz, Asymptotic analysis of the Lyapunov exponent and rotation number of the random oscillator, SIAM J. Appl. Math. 46 (1986) 427–450, gives the ε-expansion for all 2×2 systems; the "close to the identity" regime is our small-ε regime (arXiv:1208.6430).
- The weak-disorder localization literature expresses the second-order Lyapunov correction as
an integral against the disorder's power spectrum / pair-correlation function (Thouless formula);
colored disorder — our AR(1) memory
M_a— is a standard instance. - The object itself (random Fibonacci / random recurrences) is studied by Viswanath (1999) and Embree–Trefethen.
What this note adds is only the explicit closed form for the balanced companion family: the
factorization into M_a·P, the sign frequency ω_c, and the golden constants ω_c,a_c. These
follow mechanically from the classical linear-response formula (the poles are eigenvalue ratios,
the numerator is set by δ); the general-p case is a corollary in the same key.
The A+B=1 "invariant" (where Λ ≡ 0 for any forcing) is elementary: then
(A+d)+(B-d)=1, so the constant sequence is a fixed point.
4. Numerical verification
Both claims are checked by direct simulation (renormalized transfer-matrix / state-vector product) against the closed form, run once with pre-registered parameters.
verify_theorem.py (the p=2 theorem):
| Test | Claim | Result |
|---|---|---|
| Λ(0) = log φ | baseline | ✓ 0.481212 |
| pure ε² term | Λ′(0)=0 | ✓ coefficient flat across ε (2.7%) |
| C_num(ω) = k·W_a(ω) | full kernel + factorization | ✓ k=1.0000, corr=1.0000, resid=0.000 |
| ω_c = 1.763 rad, a-independent | sign frequency | ✓ 1.7635 / 1.7631 |
| a_c = 1/(3φ) | flat-forcing threshold | ✓ 0.20601 (formula & simulation) |
probe_nth_order.py (the general-p claim):
| Test | Claim | Result |
|---|---|---|
| memory factors out (p=3) | C_num/M_a is a-independent | ✓ 0.0002 |
| spectrum-pinned rational kernel (p=3, real roots) | poles = eigenvalue ratios | ✓ resid 1e-4 |
| survives complex subdominant roots | closed form + broad peak | ✓ resid 4e-4 |
5. Honest bottom line
| Status | |
|---|---|
| Is the math correct? | Yes — verified to 1e-4, all tests. |
| Is the general result new? | No — weak-noise Lyapunov expansion, 1986. |
| Is colored-noise / memory shaping new? | No — standard correlated-disorder result. |
| Is the explicit balanced kernel + golden constants published? | Not found — a new worked example. |
| Is there a new mechanism at higher order? | No — poles pinned by spectrum; resonance is ordinary. |
A referee in dynamical systems would classify this as an application/example of a known framework. That is exactly what it is — a clean one, worth writing down as exposition, not as a discovery.
References
- Arnold, Papanicolaou, Wihstutz, SIAM J. Appl. Math. 46 (1986) 427–450 — https://epubs.siam.org/doi/10.1137/0146030
- "The Lyapunov exponent of products of random 2×2 matrices close to the identity" — https://arxiv.org/pdf/1208.6430
- "Weak disorder expansion of Liapunov exponents in a degenerate case", J. Stat. Phys. — https://link.springer.com/article/10.1007/BF01026489
- Embree & Trefethen, "Growth and Decay of Random Fibonacci Sequences" — https://people.maths.ox.ac.uk/trefethen/publication/PDF/1999_86.pdf
- Viswanath, "Random Fibonacci Sequences and the Number 1.13198824…" (1999)
