Behavior of Totally Positive Differential Systems Near a Periodic Solution
Abstract
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.
Keywords
Cite
@article{arxiv.2101.06027,
title = {Behavior of Totally Positive Differential Systems Near a Periodic Solution},
author = {Chengshuai Wu and Lars Gruene and Thomas Kriecherbauer and Michael Margaliot},
journal= {arXiv preprint arXiv:2101.06027},
year = {2021}
}