Geometric Completeness of Variational Circulation for Hyperbolic Limit Cycles

This work studies the geometric completeness of variational invariants in planar dynamical systems. Using the symmetric part of the Jacobian and its induced metric deformation, a variational circulation index is defined to measure averaged tangential stretching along closed curves. It is shown that every hyperbolic limit cycle necessarily appears as a transverse zero of this index along a suitable transverse foliation. Consequently, hyperbolic limit cycles cannot be geometrically invisible to this invariant. The framework unifies existence, uniqueness, stability, and exact counting of hyperbolic limit cycles within a single geometric perspective. Applications to polynomial vector fields yield an explanation for finiteness and multiplicity phenomena, and limitations beyond the hyperbolic and planar settings are discussed.