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The inheritance of local bifurcations in mass action networks

Murad Banaji, Balázs Boros, Josef Hofbauer

TL;DR

This work addresses whether local bifurcations of equilibria in mass action CRNs persist when the network is enlarged. It develops an inheritance framework based on a codimension-$c$ bifurcation $B$ unfolded transversely by rate constants and proves a main theorem: any enlargement built from six elementary operations $E_1$–$E_6$ preserves $B$ in an appropriately extended positive stoichiometric class. The proof relies on regular and singular perturbation theory, employing Fenichel slow manifolds to handle added species and reactions, and is complemented by a suite of worked examples including fold, Hopf, Bautin, and Bogdanov–Takens scenarios. The results enable inference of complex dynamical behaviours in large biological networks from simpler subnetworks, with potential extensions to other kinetics and to bifurcations of periodic orbits.”

Abstract

We consider local bifurcations of equilibria in dynamical systems arising from chemical reaction networks with mass action kinetics. In particular, given any mass action network admitting a local bifurcation of equilibria, assuming only a general transversality condition, we list some enlargements of the network which preserve its capacity for the bifurcation. These results allow us to identify bifurcations in reaction networks from examination of their subnetworks, extending and complementing previous results on the inheritance of nontrivial dynamical behaviours amongst mass action networks. A number of examples are presented to illustrate applicability of the results.

The inheritance of local bifurcations in mass action networks

TL;DR

This work addresses whether local bifurcations of equilibria in mass action CRNs persist when the network is enlarged. It develops an inheritance framework based on a codimension- bifurcation unfolded transversely by rate constants and proves a main theorem: any enlargement built from six elementary operations – preserves in an appropriately extended positive stoichiometric class. The proof relies on regular and singular perturbation theory, employing Fenichel slow manifolds to handle added species and reactions, and is complemented by a suite of worked examples including fold, Hopf, Bautin, and Bogdanov–Takens scenarios. The results enable inference of complex dynamical behaviours in large biological networks from simpler subnetworks, with potential extensions to other kinetics and to bifurcations of periodic orbits.”

Abstract

We consider local bifurcations of equilibria in dynamical systems arising from chemical reaction networks with mass action kinetics. In particular, given any mass action network admitting a local bifurcation of equilibria, assuming only a general transversality condition, we list some enlargements of the network which preserve its capacity for the bifurcation. These results allow us to identify bifurcations in reaction networks from examination of their subnetworks, extending and complementing previous results on the inheritance of nontrivial dynamical behaviours amongst mass action networks. A number of examples are presented to illustrate applicability of the results.
Paper Structure (12 sections, 4 theorems, 69 equations)

This paper contains 12 sections, 4 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Local bifurcations of equilibria
  3. Persistence of bifurcations in perturbed systems
  4. The inheritance of bifurcations in CRNs
  5. Preliminaries on reaction networks and mass action systems
  6. The main theorem
  7. A worked example illustrating all six enlargements
  8. Some examples
  9. A supercritical Bautin bifurcation
  10. The homogenised Brusselator: a multitude of bifurcations
  11. Lifting a degenerate Andronov--Hopf bifurcation
  12. Discussion and conclusions

Key Result

Lemma 2.5

Let $I\subseteq \mathbb{R}$ be an open interval with $0 \in I$, and consider the ODE where $(x,\kappa,\varepsilon) \in V_x \times V_\kappa \times I$, and $\hat{f}$ is $C^{k+s}$ ($s \geq 1$). We assume that (eqregpert) is a regular perturbation of (eqgeneral), namely, $\hat{f}(x,\kappa,0) = f(x,\kappa)$. Then, there exists $\varepsilon_0>0$ such that $(-\varepsilon_0, \varepsilon_0)

Theorems & Definitions (7)

  • Lemma 2.5: Persistence of bifurcations under regular perturbations
  • proof
  • Lemma 2.6: Persistence of bifurcations under singular perturbations
  • proof
  • Theorem 3.2
  • Corollary 3.5: Fully open CRNs and the induced subnetwork partial ordering
  • proof