  We consider an autonomous differential system in $\mathbb{R}^n$ with a
periodic orbit and we give a new method for computing the characteristic
multipliers associated to it. Our method works when the periodic orbit is given
by the transversal intersection of $n-1$ codimension one hypersurfaces and is
an alternative to the use of the first order variational equations. We apply it
to study the stability of the periodic orbits in several examples, including a
periodic solution found by Steklov studying the rigid body dynamics.

Dynamical Systems