On Periodic Orbits of Vector Fields in Arbitrary Dimension Via Autonomous and Nonautonomous Inverse Jacobi Multipliers

We investigate periodic orbits of C¹ C 1 autonomous vector fields in Rⁿ R n using inverse Jacobi multipliers that may depend explicitly on time. We establish a localization principle for T -periodic orbits in arbitrary dimension, extending known planar results and deriving nonexistence conditions through the relation between the time-slices V(0, · ) V ( 0 , · ) and V(T, · ) V ( T , · ) . We further characterize hyperbolicity and orbital stability, including a decomposition of characteristic multipliers along invariant surfaces associated with autonomous inverse Jacobi multipliers. A test for the algebraicity of periodic orbits in 3-dimensional vector fields is given based on non-autonomous inverse Jacobi multipliers. The interplay between normalizers, inverse Jacobi multipliers and invariants is analyzed, with applications to the Lorenz and Rössler systems.