The dm³ Operator: Explicit Toy Model and Global Dynamical Analysis

Title: The dm3 Operator: Explicit Toy Model and Global Dynamical Analysis

Authors: Pablo Nogueira Grossi 

Description (Abstract): We construct and analyze a complete explicit instantiation of the Generative Contact Mechanics framework on the two-dimensional system: ṙ = r(1−r²) + 2(r−1)e^{−z}, θ̇ = 1, ż = r² − 2(r−1)²e^{−z}, on the contact manifold M = R²₍>0₎ × R. Every definition, operator, and boundary from the framework is instantiated explicitly and verified by direct computation. Four main results are established. Theorem A: the global attractor of the full system is the resonant orbit Γ₁₂. Theorem B: the invariant torus conjecture holds for the 1:2 resonant case, with a normally hyperbolic invariant circle and transverse Lyapunov exponent −3. Theorem C: the system undergoes four bifurcations — contact Hopf, saddle-node of limit cycles, Neimark–Sacker, and slow-fast crossover — as parameters vary. Theorem D: the stationary SDE measure concentrates on Γ for noise amplitude below the embodiment threshold τ = 2 and spreads for amplitude above τ. The canonical invariant triple is (T*, μ_max, τ) = (2π, −2, 2) and the stability radius is ε₀ = 1/3.

Keywords: contact geometry, limit cycles, bifurcation theory, invariant measures, structural stability, resonance, normal form, stochastic stability, global attractor, dm3 operator, toy model, embodiment threshold

MSC codes: 37C10, 37C27, 37G15, 37H10, 53D10, 60H10

License: Creative Commons Attribution Non Commercial No Derivatives 4.0 International (CC BY-NC-ND 4.0)

Publication date: 2026-03-17

Journal title: SIAM Journal on Applied Dynamical Systems Status: Submitted

Notes: Preprint. Submitted to SIAM Journal on Applied Dynamical Systems.

Companion paper: Generative Contact Mechanics (GCM). Part of the Principia Orthogona / GCM research program. G6LLC, Newark NJ, 2026.

Abstract. We construct and analyze a complete explicit instantiation of the generative contact mechanics framework of [1] on the two-dimensional system r˙ = r(1 − r 2 ) + 2(r − 1)e −z , ˙θ = 1, z˙ = r 2 − 2(r − 1)2 e −z , on the contact manifold M = R2 >0 ×R. Every definition, operator, and boundary from [1] is instantiated explicitly and verified by direct computation. Four main results are established: Theorem A: the global attractor of the full system is the resonant orbit Γ12; Theorem B: the invariant torus conjecture of [1] holds for the 1:2 resonant case, with a normally hyperbolic invariant circle and transverse Lyapunov exponent −3; Theorem C: the system undergoes four bifurcations (contact Hopf, saddle-node of limit cycles, Neimark–Sacker, and slow-fast crossover) as parameters vary; Theorem D: the stationary SDE measure concentrates on Γ for noise amplitude below the embodiment threshold τ and spreads for amplitude above τ.