What question did this study set out to answer?

The aim is to develop a geometric framework for dissipative dynamical systems with unique characteristics, emphasizing limit cycles and stability.

May 25, 2026Open Access

Generative Contact Mechanics: A Geometric Framework for Dissipative Systems with Structured Limit Cycles

Key Points

The aim is to develop a geometric framework for dissipative dynamical systems with unique characteristics, emphasizing limit cycles and stability.
Introduced the dm3 system as a smooth Riemannian manifold with a hyperbolic limit cycle and a Lyapunov function.
Constructed a complete operator algebra to describe system properties like expansion and unification.
Formalized the framework based on eight axioms governing the system's behavior.
The dm3 system demonstrates a unique exponentially stable hyperbolic limit cycle with stochastic stability below a threshold τ.
The category of dm3 systems is closed under a unification operator, indicating that unification reduces noise tolerance.
The operator algebra exhibits C¹-structural stability with a calculated stability radius ε₀.

Abstract

Title: Generative Contact Mechanics: A Geometric Framework for Dissipative Systems with Structured Limit Cycles Authors: Pablo Nogueira Grossi Description (Abstract): We introduce a geometric framework for dissipative dynamical systems whose limit cycles carry structured phase, multi-scale coherence, and stochastic stability. The central object is the dm3 system — a smooth Riemannian manifold equipped with a hyperbolic limit cycle, a Lyapunov function, and a stochastic extension — formalized through eight axioms. On this foundation we construct a complete operator algebra (the g-, L-, R-, and U-operators) governing expansion, coherence, resonance, and unification, and we embed the entire structure in contact geometry. Four main results are established. Theorem A: any dm3 system admits a unique exponentially stable hyperbolic limit cycle with a topological invariant (the winding integral) and stochastic stability below an embodiment threshold τ. Theorem B: the category dm3 is closed under a unification operator, with the quantitative prediction τ₁₂ ≤ min (τ₁, τ₂): unification weakens noise tolerance. Theorem C: every dm3 system near its limit cycle is locally contact-diffeomorphic to a universal three-parameter normal form (μₘax, ω, β). Theorem D: the full operator algebra is C¹-structurally stable with explicit stability radius ε₀ = |μₘax| / 2 (1 + sup‖Hess V‖). Complete verification on an explicit toy model appears in the companion paper. Keywords: contact geometry, limit cycles, dissipative dynamical systems, operator algebra, structural stability, resonance, stochastic stability, nonequilibrium dynamics, Lyapunov functions, generative contact mechanics, embodiment threshold, dm3 system MSC codes: 37C10, 37C27, 37D30, 53D10, 60H10, 18A30 License: Creative Commons Attribution Non Commercial No Derivatives 4. 0 International (CC BY-NC-ND 4. 0) Publication date: 2026-03-17 Journal title: Journal of Geometric Mechanics Status: Submitted Notes: Preprint. Submitted to Journal of Geometric Mechanics. Companion paper: The dm3 Operator: Explicit Toy Model and Global Dynamical Analysis. Part of the Principia Orthogona / GCM research program. G6LLC, Newark NJ, 2026.

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