Mass action systems: two criteria for Hopf bifurcation without Hurwitz
arxiv(2024)
摘要
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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