The DM3 Operator - Explicit Toy Model and Global Dynamic Analysis

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.