THE SMALLEST BIMOLECULAR MASS-ACTION SYSTEM WITH A VERTICAL ANDRONOVHOPF BIFURCATION MURAD BANAJI BAL AZS BOROS AND JOSEF HOFBAUER

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THE SMALLEST BIMOLECULAR MASS-ACTION SYSTEM

WITH A VERTICAL ANDRONOV–HOPF BIFURCATION

MURAD BANAJI, BAL ´

AZS BOROS, AND JOSEF HOFBAUER

Abstract. We present a three-dimensional diﬀerential equation, which ro-

bustly displays a degenerate Andronov–Hopf bifurcation of inﬁnite codimen-

sion, leading to a center, i.e., an invariant two-dimensional surface that is

ﬁlled with periodic orbits surrounding an equilibrium. The system arises from

a three-species bimolecular chemical reaction network consisting of four reac-

tions. In fact, it is the only such mass-action system that admits a center via

an Andronov–Hopf bifurcation.

1. Summary of the main results

In order to admit an Andronov–Hopf bifurcation, the underlying chemical reac-

tion network of a bimolecular mass-action system must have at least three species

and at least four reactions. It has recently been shown that there are exactly 138

nonisomorphic three-species four-reaction bimolecular reaction networks, whose as-

sociated mass-action systems admit Andronov–Hopf bifurcation [1]. These net-

works fall into 87 dynamically nonequivalent classes. Of these classes, 86 admit

nondegenerate Andronov–Hopf bifurcation for almost all parameter values on the

bifurcation set, leading to isolated limit cycles. In the remaining class, however,

the Andronov–Hopf bifurcation can only be degenerate. A representative of this

exceptional class is

Z+X2X

X+Y2Y

Y+Z02Z

κ1

κ2

κ3κ4

(1)

giving rise to the mass-action diﬀerential equation

˙x=x(κ1z−κ2y),

˙y=y(κ2x−κ3z),

˙z=z(−κ3y−κ1x)+2κ4

(2)

with state space R3

≥0, where κ1,κ2,κ3,κ4are positive parameters, called the reac-

tion rate constants. (The other member of the exceptional class is obtained from (1)

by replacing the reaction 0→2Zby 0→Z.) The question left open in [1] concerns

the behaviour of system (2). In Section 3, we prove that whenever the Jacobian

2020 Mathematics Subject Classiﬁcation. Primary 34A05, 34C25, 34C45.

BB’s work was supported by the Austrian Science Fund (FWF), project P32532.

Email: m.banaji@mdx.ac.uk, balazs.boros@univie.ac.at, josef.hofbauer@univie.ac.at.

arXiv:2210.06119v1 [math.DS] 12 Oct 2022

2 M. BANAJI, B. BOROS, AND J. HOFBAUER

matrix at the unique positive equilibrium has a pair of purely imaginary eigenval-

ues, the equilibrium is a center, i.e., there is a one parameter family of periodic

orbits that ﬁll the two-dimensional center manifold. In particular, Andronov–Hopf

bifurcations in system (2) are always vertical, i.e., all the periodic orbits occur si-

multaneously at the critical value of the bifurcation parameter. Additionally, we

prove that every positive solution converges either to one of these periodic orbits or

to the unique positive equilibrium. Further, we show that the global center man-

ifold is analytic and discuss how its closure intersects the boundary of the state

space R3

≥0.

2. Vertical Andronov–Hopf bifurcations in mass–action systems

There are two well-known small reaction networks that exhibit oscillations. The

Lotka reactions [9] (left) and the Ivanova reactions [12, page 630] (right) along with

their associated mass-action diﬀerential equations are

X2X

X+Y2Y

κ1

κ2

κ3

˙x=x(κ1−κ2y)

˙y=y(κ2x−κ3)

Z+X2X

X+Y2Y

Y+Z2Z

κ1

κ2

κ3

˙x=x(κ1z−κ2y)

˙y=y(κ2x−κ3z)

˙z=z(κ3y−κ1x)

Both the Lotka and the Ivanova networks are bimolecular (i.e., the molecularity of

every reactant and product is at most two) and have rank two (i.e., the span of

the vectors of the net changes of the species is two-dimensional). For the Lotka,

the unique positive equilibrium is surrounded by periodic orbits, the level sets of

xκ3yκ1e−κ2(x+y). For the Ivanova, the triangle ∆c={(x, y, z)∈R3

+:x+y+z=c}

is invariant for any c > 0, and the unique positive equilibrium in ∆cis surrounded

by periodic orbits, the level sets of xκ3yκ1zκ2. For both the Lotka and the Ivanova

systems, the described behaviour holds for all κ1, κ2, κ3>0, and hence, these

systems admit no bifurcation.

By [2, Theorem 4.1], the Lotka and the Ivanova systems are the only rank-two

bimolecular mass-action systems with periodic orbits. Thus, for an Andronov–Hopf

bifurcation to occur in a bimolecular mass-action system, its rank must be at least

three, and hence, it must have at least three species. Moreover, by [1, Lemma 2.3],

it must have at least four reactions.

We turn to the question of when mass-action systems admit vertical Andronov–

Hopf bifurcations. If we do not require bimolecularity then these can occur in rank-

two networks. For example, by adding the reactions Xκ5

←− 2Xκ4

−→ 3Xto the Lotka

network above, the resulting mass-action system exhibits a vertical Andronov–Hopf

bifurcation: for κ4slightly smaller than κ5the positive equilibrium is asymptotically

stable, for κ4slightly larger than κ5it is repelling, while for κ4=κ5it is a center.

Focussing on bimolecular networks, we can construct rank-three networks with

vertical Andronov–Hopf bifurcation. For instance, by inserting some intermediate

steps into the Ivanova reactions and choosing the rate constants appropriately, we

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THESMALLESTBIMOLECULARMASS-ACTIONSYSTEMWITHAVERTICALANDRONOV{HOPFBIFURCATIONMURADBANAJI,BALAZSBOROS,ANDJOSEFHOFBAUERAbstract.Wepresentathree-dimensionaldierentialequation,whichro-bustlydisplaysadegenerateAndronov{Hopfbifurcationofinnitecodimen-sion,leadingtoacenter,i.e.,aninvarianttwo-dimensional...

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THE SMALLEST BIMOLECULAR MASS-ACTION SYSTEM WITH A VERTICAL ANDRONOVHOPF BIFURCATION MURAD BANAJI BAL AZS BOROS AND JOSEF HOFBAUER

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