Global Stability of a Nonlinear Self-Adjoint Operator Flow via a Tracial Lyapunov Functional

This self-contained note explores a nonlinear flow on self-adjoint operators within a Hilbert space, built as an anticommutator-symmetrized version of a planar rotator with isotropic cubic damping. By applying a faithful tracial state, we build a strict Lyapunov functional whose time derivative is strictly non-positive and only vanishes at the origin. This proves the Lyapunov stability of the zero equilibrium, as well as global asymptotic stability in finite dimensions. Additionally, the commutative limit is solved exactly, demonstrating that the amplitude decays following a power-law rate rather than exponentially. Numerical tests validate these results for both commutative and non-commuting cases, proving that non-commutativity does not speed up decay into exponential behavior. The note clearly outlines its precise, modest scope and explicitly separates itself from older, unverified drafts.