  This paper presents an algorithm for computing inner estimates of the regions
of attraction of limit cycles of a nonlinear hybrid system. The basic procedure
is: (1) compute the dynamics of the system transverse to the limit cycle; (2)
from the linearization of the transverse dynamics construct a quadratic
candidate Lyapunov function; (3) search for a new Lyapunov function verifying
maximal regions of orbital stability via iterated of sum-of-squares programs.
The construction of the transverse dynamics is novel, and valid for a broad
class of nonlinear hybrid systems. The problem of stabilization of unstable
limit cycles will also be addressed, and a solution given based on
stabilization of the transverse linearization.
