  A Riemannian metric with a local contraction property can be used to prove
existence and uniqueness of a periodic orbit and determine a subset of its
basin of attraction. While the existence of such a contraction metric is
equivalent to the existence of an exponentially stable periodic orbit, the
explicit construction of the metric is a difficult problem.
  In this paper, the construction of such a contraction metric is achieved by
formulating it as an equivalent problem, namely a feasibility problem in
semidefinite optimization. The contraction metric, a matrix-valued function, is
constructed as a continuous piecewise affine (CPA) function, which is affine on
each simplex of a triangulation of the phase space. The contraction conditions
are formulated as conditions on the values at the vertices.
  The paper states a semidefinite optimization problem. We prove on the one
hand that a feasible solution of the optimization problem determines a CPA
contraction metric and on the other hand that the optimization problem is
always feasible if the system has an exponentially stable periodic orbit and
the triangulation is fine enough. An objective function can be used to obtain a
bound on the largest Floquet exponent of the periodic orbit.
