  A time-varying nonlinear dynamical system is called a totally positive
differential system (TPDS) if its Jacobian admits a special sign pattern: it is
tri-diagonal with positive entries on the super- and sub-diagonals. If the
vector field of a TPDS is T-periodic then every bounded trajectory converges to
a T-periodic solution. In particular, when the vector field is time-invariant
every bounded trajectory of a TPDS converges to an equlbrium. Here, we use the
spectral theory of oscillatory matrices to analyze the behavior near a periodic
solution of a TPDS. This yields information on the perturbation directions that
lead to the fastest and slowest convergence to or divergence from the periodic
solution. We demonstrate the theoretical results using a model from systems
biology called the ribosome flow model.
