Behavior of totally positive differential systems near a periodic solution

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Wu, Chengshuai ; Grüne, Lars ; Kriecherbauer, Thomas ; Margaliot, Michael:
Behavior of totally positive differential systems near a periodic solution.
In: Proceedings of the 2021 IEEE Conference on Decision and Control (CDC). - Austin, Texas, USA , 2021 . - pp. 3160-3165
DOI: https://doi.org/10.1109/CDC45484.2021.9683061

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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.