Universality Classes of Saturating Oscillators: Asymptotic Depletion Near Saturation Separatrices
Year: 2026
Type: preprint
Abstract: We classify the asymptotic behaviour of the time-averaged mixing scalar f(r) = ⟨M⟩_V(r), M = 1 − w²_eff, near the saturation separatrix ε = ½ − r → 0⁺ for symmetric oscillators with saturating potential V. The main result (Classification Theorem) establishes that the depletion exponent α in f(ε) ~ C·εᵅ (up to logarithmic corrections) is determined entirely by the asymptotic tail V_max − V(φ) as φ → ∞. For the exponential class (V_max − V ~ exp(−aφᵖ)): f ~ L_p·√ε·|ln ε|^((p−1)/p), verified for specific potentials (tanh² gives C = 2√2 exactly; V = ½(1−e^{−φ}) gives Q₁ ≈ 2.40 numerically). The exponent is class-universal; the prefactor L_p is potential-specific. For the algebraic class (V_max − V ~ φ^{−n}): α(n) = min(1, ½ + 1/n) with logarithmic correction at n = 2, proved rigorously via the Beta function identity I_D = φ_A·ε^{−1/2}·β_n where β_n = √π·Γ(½+1/n) / [n·Γ(1+1/n)]. The fixed point f(0) = ½ holds for all symmetric potentials with quadratic minimum. All results are stated with explicit epistemic levels (Layer A: proved; Layer B: analytical argument with error bounds). The paper includes a complete Watson's lemma error analysis for the exponential class (Appendix B), giving a relative error of O(1/|ln ε|).
Author: Drewnisz, Michał Jerzy
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7162957269
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