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arXiv · 2402.18188

Mass action systems: two criteria for Hopf bifurcation without Hurwitz

Nicola Vassena
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Abstract

We state two sufficient criteria for periodic oscillations in mass action systems. Neither criterion requires a computation of the Hurwitz determinants. Instead, both criteria exploit the linear algebra concepts of $D$-stability and $P$-matrices. The criteria are complementary: the first is based on a stable matrix that is not a $P^-$ matrix, while the second is based on a $P^-$ matrix that is not stable. In analogy, a qualitatively different interpretation follows: the first criterion relates to positive feedback in the network, while the second concerns negative feedback. We present examples that showcase the applicability of both criteria. As a final independent remark, we prove that for the special case of fully-open networks, the capacity for Hopf bifurcation is just equivalent to the capacity for a steady-state with a complex pair of eigenvalues with positive-real part.

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Nicola Vassena. 2024-02-28. Mass action systems: two criteria for Hopf bifurcation without Hurwitz. https://arxiv.org/abs/2402.18188

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