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Oscillations in three-reaction quadratic mass-action systems

Murad Banaji, Balázs Boros, Josef Hofbauer

TL;DR

This work investigates which small mass-action networks can exhibit oscillations, focusing on rank-two, three-reaction, quadratic systems with bimolecular sources but potentially higher-molecularity targets. It shows that isolated periodic orbits cannot occur in trimolecular systems with three reactions and bimolecular sources, and it classifies all networks in this class that admit a periodic orbit, revealing that nearby orbits are also periodic and identifying a new center in the Lifted LVA family that undergoes a vertical Andronov–Hopf bifurcation. The paper additionally characterizes all two-species, three-reaction, bimolecular-sourced networks admitting an Andronov–Hopf bifurcation, discovering two Hopf families that yield stable limit cycles only when a target complex has molecularity at least four, with exactly four tetramolecular networks. Overall, the results constrain the search for oscillations in small mass-action systems and connect these phenomena to higher-target molecularities and structured bifurcations, such as Hopf and Bogdanov–Takens, through trimolecular reductions.

Abstract

It is known that rank-two bimolecular mass-action systems do not admit limit cycles. With a view to understanding which small mass-action systems admit oscillation, in this paper we study rank-two networks with bimolecular source complexes but allow target complexes with higher molecularities. As our goal is to find oscillatory networks of minimal size, we focus on networks with three reactions, the minimum number that is required for oscillation. However, some of our intermediate results are valid in greater generality. One key finding is that an isolated periodic orbit cannot occur in a three-reaction, trimolecular, mass-action system with bimolecular sources. In fact, we characterise all networks in this class that admit a periodic orbit; in every case all nearby orbits are periodic too. Apart from the well-known Lotka and Ivanova reactions, we identify another network in this class that admits a center. This new network exhibits a vertical Andronov--Hopf bifurcation. Furthermore, we characterise all two-species, three-reaction, bimolecular-sourced networks that admit an Andronov--Hopf bifurcation with mass-action kinetics. These include two families of networks that admit a supercritical Andronov--Hopf bifurcation, and hence a stable limit cycle. These networks necessarily have a target complex with a molecularity of at least four, and it turns out that there are exactly four such networks that are tetramolecular.

Oscillations in three-reaction quadratic mass-action systems

TL;DR

This work investigates which small mass-action networks can exhibit oscillations, focusing on rank-two, three-reaction, quadratic systems with bimolecular sources but potentially higher-molecularity targets. It shows that isolated periodic orbits cannot occur in trimolecular systems with three reactions and bimolecular sources, and it classifies all networks in this class that admit a periodic orbit, revealing that nearby orbits are also periodic and identifying a new center in the Lifted LVA family that undergoes a vertical Andronov–Hopf bifurcation. The paper additionally characterizes all two-species, three-reaction, bimolecular-sourced networks admitting an Andronov–Hopf bifurcation, discovering two Hopf families that yield stable limit cycles only when a target complex has molecularity at least four, with exactly four tetramolecular networks. Overall, the results constrain the search for oscillations in small mass-action systems and connect these phenomena to higher-target molecularities and structured bifurcations, such as Hopf and Bogdanov–Takens, through trimolecular reductions.

Abstract

It is known that rank-two bimolecular mass-action systems do not admit limit cycles. With a view to understanding which small mass-action systems admit oscillation, in this paper we study rank-two networks with bimolecular source complexes but allow target complexes with higher molecularities. As our goal is to find oscillatory networks of minimal size, we focus on networks with three reactions, the minimum number that is required for oscillation. However, some of our intermediate results are valid in greater generality. One key finding is that an isolated periodic orbit cannot occur in a three-reaction, trimolecular, mass-action system with bimolecular sources. In fact, we characterise all networks in this class that admit a periodic orbit; in every case all nearby orbits are periodic too. Apart from the well-known Lotka and Ivanova reactions, we identify another network in this class that admits a center. This new network exhibits a vertical Andronov--Hopf bifurcation. Furthermore, we characterise all two-species, three-reaction, bimolecular-sourced networks that admit an Andronov--Hopf bifurcation with mass-action kinetics. These include two families of networks that admit a supercritical Andronov--Hopf bifurcation, and hence a stable limit cycle. These networks necessarily have a target complex with a molecularity of at least four, and it turns out that there are exactly four such networks that are tetramolecular.
Paper Structure (4 sections, 4 theorems, 16 equations, 1 figure)

This paper contains 4 sections, 4 theorems, 16 equations, 1 figure.

Table of Contents

  1. The Lifted LVA
  2. Main result
  3. Networks with the reaction
  4. Conclusions

Key Result

Theorem 10

Assume that a three-reaction, quadratic, trimolecular, mass-action system with no trivial species has a periodic orbit. Then, up to a permutation of the species, the differential equation is one of the following. Note that In particular, mass-action systems satisfying the conditions of the theorem admit no isolated periodic orbits.

Figures (1)

  • Figure 1: The list of all three-reaction, quadratic, trimolecular $(n,3,2)$ networks whose mass-action system has a periodic orbit for some rate constants. There are sixteen such networks. Four are members of the family \ref{['eq:case_7']}, eight are derived from these by adding a trivial species, and two are obtained by adding two trivial species. The latter two are the only ones with four species. The Ivanova reactions and the Lifted LVA complete the list. Notice that the only ones that are bimolecular are the Lotka reactions \ref{['eq:lotka_only']} and the Ivanova reactions \ref{['eq:ivanova_only']}.

Theorems & Definitions (8)

  • Theorem 10
  • proof
  • Lemma 11
  • proof
  • Lemma 12
  • proof
  • Lemma 13
  • proof