Variational Geometry of Limit Cycles in Planar Dynamical Systems

This work develops a variational geometric framework for planar dynamical systems based on the symmetric part of the Jacobian and its induced metric deformation. A loop-integrated variational circulation index is introduced to measure net tangential stretching along closed curves. The index provides necessary conditions and geometric sufficiency criteria for the existence of limit cycles, relates the order of vanishing to multiplicity, and yields algebraic bounds for polynomial vector fields. The approach is time-independent and formulated without return-map constructions. Examples including Hopf and Liénard systems illustrate the method.