Mass action systems: two criteria for Hopf bifurcation without Hurwitz

TL;DR

This work presents two sufficient criteria for Hopf bifurcation in mass-action networks that circumvent the computational burden of Hurwitz tests. By expressing the Jacobian as $Jac = A D$ in Clarke's convex coordinates and leveraging $D$-stability and $P^-$ matrix theory (including Fisher–Fuller results), the authors prove that a stable $B(\bar{\mathbf{v}})$ that is not $P^-_0$ or an unstable $P^-_{FF}$ guarantee the existence of parameter choices yielding periodic oscillations, under a global Hopf framework. The paper provides interpretable mechanisms in terms of positive and negative feedbacks, and offers concrete, large-network examples to illustrate each criterion. It also establishes an independent result for fully-open networks, linking Hopf capacity to the presence of a complex-conjugate eigenvalue pair with positive real part. Collectively, the results advance practical, structurally-informed criteria for oscillations in mass-action systems beyond Hurwitz-based methods, with implications for analysis of large biochemical networks and design of oscillatory motifs.

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.

Figures (2)

• Figure 1: Numerical simulations for system \ref{['eq:ex1']}. The steady-state flux vector has been chosen as $\bar{\mathbf{v}}=(1,1,1,1,1)$. The values of the (unique) unstable steady-state are $\bar{x}=(1,1,1,10,10)$, which imply a chice of rates $(k_1,k_2,k_3,k_4,k_5)=(1,1,1,0.1,0.1)$. Initial conditions have been chosen $x(0)=(1.1,0.9,1,10,10)$. The plot shows convergence to a stable limit cycle.
• Figure 2: Numerical simulations for system \ref{['eq:ex2']},with $n=10$. The steady-state flux vector has been chosen as $\bar{\mathbf{v}}=(3,1,1,...,1,1)$. The values of the (unique) unstable steady-states are $\bar{x}=(1,1,...,1,0.5)$, which imply a choice of rates $(F,k_1,...,k_{12})=(3,1,1,...,1,2)$. We have opted to choose $\bar{x}_B\neq\bar{x}_C$ because the two trajectories would fully overlap otherwise. Initial conditions have been chosen $x(0)=(1.1, 1,1,...,1,0.5)$. The plot shows convergence to a stable limit cycle. For graphical simplicity, we only include trajectories for $A_1,B,C$.

Theorems & Definitions (17)

• Definition 4.1: Inertia of a matrix
• Definition 4.2: (in)stability, $D$-(in)stability
• Lemma 4.3
• Definition 4.4: $P^-$ and $P^-_0$ matrices
• Proposition 4.5
• Definition 4.6: Fisher and Fuller $P^-_{FF}$ matrices
• Theorem 4.7: Theorem 1' in Fisher72simple
• Corollary 4.8: of Proposition \ref{['prop:Pf']}
• Corollary 4.9: of Theorem \ref{['thm:FF']}
• Theorem 5.1: (4.7) from Fiedler85PhD