Circulation Functionals and Deformation Measures in Planar Polynomial Vector Fields

This preprint studies circulation functionals derived from the symmetric part of the Jacobian of planar vector fields. These functionals quantify averaged tangential deformation along closed curves and provide scalar diagnostics related to limit-cycle behavior. Three results are established. First, an explicit relation is derived between the symmetric Jacobian and the nontrivial Floquet multiplier of a hyperbolic limit cycle. Second, for planar polynomial vector fields, the asymptotic growth of the circulation functional along large circles is determined, yielding a sufficient condition that excludes large-radius limit cycles detectable by this functional. Third, for degree-n polynomial systems and a fixed probing family, a quadratic upper bound is obtained for the number of isolated simple zeros of the circulation functional for generic coefficients. The results clarify what dynamical information circulation-based quantities encode and situate these quantities within the classical framework of planar dynamics.