November 27, 2013 10:53 WSPC/103-M3AS 1340010 …

November 27, 2013 10:53 WSPC/103-M3AS 1340010

Mathematical Models and Methods in Applied SciencesVol. 24, No. 2 (2014) 327–357c© World Scientific Publishing CompanyDOI: 10.1142/S0218202513400101

ANALYSIS OF HOPF/HOPF BIFURCATIONS IN NONLOCALHYPERBOLIC MODELS FOR SELF-ORGANISED

AGGREGATIONS

P.-L. BUONO

Faculty of Science, University of Ontario Institute of Technology,Oshawa, Ontario L1H 7K4, Canada

[email protected]

R. EFTIMIE∗

Division of Mathematics, University of Dundee,Dundee DD1 4HN, UK

[email protected]

Received 30 June 2013Revised 24 July 2013

Accepted 9 August 2013Published 20 November 2013

Communicated by N. Bellomo and F. Brezzi

The modelling and investigation of complex spatial and spatio-temporal patternsexhibited by a various self-organised biological aggregations has become one of the mostrapidly-expanding research areas. Generally, the majority of the studies in this areaeither try to reproduce numerically the observed patterns, or use existence results toprove analytically that the models can exhibit certain types of patterns. Here, we focuson a class of nonlocal hyperbolic models for self-organised movement and aggregations,and investigate the bifurcation of some spatial and spatio-temporal patterns observednumerically near a codimension-2 Hopf/Hopf bifurcation point. Using weakly nonlinearanalysis and the symmetry of the model, we identify analytically all types of solutionsthat can exist in the neighbourhood of this bifurcation point. We also discuss the sta-bility of these solutions, and the implication of these stability results on the observednumerical patterns.

Keywords: Nonlocal hyperbolic model; self-organised aggregations; bifurcation andsymmetry.

AMS Subject Classification: 35L40, 35L60, 92D40

1. Introduction

The investigation of self-organised biological aggregations represents a research areathat has undergone a great expansion in the past years. The main reason for this

∗Corresponding author

327

http://dx.doi.org/10.1142/S0218202513400101


328 P.-L. Buono & R. Eftimie

rapid expansion is represented by the complex spatial and spatio-temporal patternsexhibited by various biological aggregations (from milling or travelling schools offish,1 to zigzagging flocks of birds2 or rippling behaviour in bacterial colonies3). Themathematical models derived to investigate these behaviours try to propose plau-sible biological mechanisms regarding the interactions at the microscale level (e.g.inter-individual repulsion and attraction, individual velocities, or individual inter-actions with the environment) that could explain the observed behaviours/patternsat the macroscale level (i.e. the shape, size and dynamics of the group).4–12 To thisend, the mathematical models either: (i) focus mainly on numerical simulationswith the purpose of reproducing and comparing the simulated aggregation patternswith the available data (and thus ascertaining the correctness of the microscale-levelassumptions incorporated into the models), or (ii) focus mainly on the analyticalresults (e.g. show the existence of particular solutions exhibited by these models, ortry to connect biological interactions at the microscale and macroscale levels). Thefirst approach can be found in many individual-based models,1,10,11 which some-times are too complex to be investigated analytically. The second approach can befound in kinetic models (sometimes derived from individual-based models13), forwhich the numerical simulations can become very complicated.13,14 While there aremodels for self-organised aggregations that combine both numerical and analyticalresults,15 usually these models do not investigate the bifurcation dynamics of theresulting patterns. Such an investigation is important from both a mathematical anda biological point of view, since it can help us in understanding the factors (param-eters) that can cause the transitions between different group behaviours. It can alsohelp us in understanding what kind of patterns (behaviours) we would expect tosee bifurcating from certain other patterns (behaviours). Among the few studiesthat investigate the bifurcation of spatial and spatio-temporal patterns as variousparameters (e.g. population size or magnitude of social interactions) are changed,we remark Topaz et al.16 and Eftimie et al.17 Both studies focus on codimension-1bifurcations. In particular, Topaz et al.16 focuses on a steady-state (Turing) bifurca-tion in a nonlocal parabolic equation, and shows that the aggregations (stationarypulses) bifurcate sub-critically. The study in Eftimie et al.,17 on nonlocal hyperbolicmodels for aggregation and movement for left and right moving densities, focuseson the bifurcation dynamics of stationary pulses (via a steady-state bifurcation)and travelling trains (via a Hopf bifurcation). These two patterns bifurcate alsosub-critically.

The class of nonlocal hyperbolic models introduced in Eftimie et al.17,18 canexhibit more exotic patterns: travelling (rotating) waves, standing waves, mod-ulated waves, and even zigzagging patterns and breather patterns (see Eftimieet al.18). Here we focus on a particular sub-model introduced in Eftimie et al.18

(sub-model M4), which was shown to exhibit all these exotic patterns. The spatio-temporal regularity of these patterns indicates the presence of symmetries. In fact,all sub-models (M1–M5) introduced in Eftimie et al.,18 which have periodic bound-ary conditions, are actually O(2)-symmetric19 (where the group O(2) contains a


Nonlinear Analysis of H/H Bifurcations 329

subgroup of rotations — SO(2) — and a reflection symmetry which interchangesleft and right moving densities along with a reflection in the domain). Since manyof the patterns exhibited by these nonlocal hyperbolic models (i.e. travelling waves,standing waves and modulated waves) seem to occur near O(2) Hopf/Hopf bifur-cations, in this paper we will investigate in more detail this type of codimension-2bifurcation.

While there are many studies on codimension-2 bifurcations in various physicsand engineering models, to our knowledge there are almost no such studies in modelsfor self-organised biological aggregations. The standard methods for investigatingcodimension-2 Hopf/Hopf bifurcations are centre manifold reductions20 or weaklynonlinear analysis and multiple scales.21 In this paper, we build upon the analysisperformed in Eftimie et al.,17 and focus on weakly nonlinear analysis. However, tomake sure that we are not missing any solutions, we combine this analysis with theinvestigation of model symmetries. The unfolding analysis of the O(2) Hopf/Hopfbifurcation, in particular the classification of solutions according to their symmetrygroups, follows the classifications in Chossat et al.22 and Golubitsky et al.23

We start in Sec. 2 by describing briefly the nonlocal hyperbolic model. Wealso discuss the symmetries of this model and perform a linear analysis near ahomogeneous equilibrium using group-theoretic methods. Using the expressions forthe solutions of the linear system, we obtain contour plots for these solutions andcompare them with the results of the numerical simulations performed in the neigh-bourhood of the Hopf/Hopf point. Then, in Sec. 3, we perform a weakly nonlinearanalysis of the system and classify the steady states of the resulting amplitudeequations according to the symmetry classification. We also investigate the stabil-ity of these amplitude steady states and use the results to explain the numericalsimulations. We conclude in Sec. 4 with a general discussion and a summary of theresults.

2. Model Description

The following hyperbolic model was proposed in Eftimie et al.24,18 to describe theevolution of densities of right-moving (u+) and left-moving (u−) individuals:

∂tu+(x, t) + ∂x(γu+(x, t)) = −λ+[u+, u−]u+(x, t) + λ−[u+, u−]u−(x, t), (2.1a)

∂tu−(x, t) − ∂x(γu−(x, t)) = λ+[u+, u−]u+(x, t) − λ−[u+, u−]u−(x, t), (2.1b)

u±(x, 0) = u±0 (x). (2.1c)

Here, γ is the constant speed, while λ± are functionals that describe the turningbehaviour of individuals, in response to their neighbours:

λ±[u+, u−] = λ1 + λ2f(y±r [u+, u−] − y±a [u+, u−] + y±al[u+, u−])

= (λ1 + λ2f(0)) + λ2(f(y±r − y±a + y±al) − f(0)). (2.2)

The turning function f is positive, increasing and bounded: f(u) = 0.5 + 0.5tanh(u − 2). The terms λ1 + λ2f(0) and λ2(f(y±) − f(0)) describe the baseline



random turning rate and the bias turning rate, respectively. As f is chosen suchthat f(0) 1, we can approximate the random turning by λ1 and the directed turn-ing — in response to neighbours’ density and movement direction — by λ2f(y±).These assumptions for the turning rates are biologically realistic, since, as noted byLotka,25 “the type of motion presented by living organisms . . . can be regarded ascontaining both a systematically directed and also a random element”. Finally, theterms y±r , y±a and y±al describe the repulsive, attractive and alignment social inter-actions.18 Since individuals in an ecological aggregation can interact — via visual,sound or olfactory signals — with neighbours positioned further away,26 we assumethat these social interactions are nonlocal. Throughout this paper, we assume thata reference individual positioned at (x, t)(u±(x, t)) interacts socially only with itsneighbours moving towards it (see Fig. 1). This could be the case when individ-uals use directional signals to communicate with each other (e.g. tactile signals ordirectional sound signals18). This situation corresponds to model M4 introduced inEftimie et al.18 The diagram in Fig. 1 can be translated into mathematical equationsas follows:

y±r,a = qr,a

∫ ∞

0

Kr,a(s)(u∓(x± s, t) − u±(x∓ s, t))ds, (2.3a)

y±al = qal

∫ ∞

0

Kal(s)(u∓(x± s, t) − u±(x ∓ s, t))ds. (2.3b)

Here, qr, qa and qal are the magnitudes of the repulsive, attractive and alignmentsocial interactions. The kernels

Kj(s) =1

2πm2j

e−(s−sj)2/(2m2

j ), j = r, a, al and mj = sj/8, (2.4)

describe the spatial ranges over which the social interactions take place, with sj ,j = r, a, al being half the range of the repulsive, attractive and alignment interac-tions. Finally, parameters mj describe the width of the interaction kernels. Notethat the width of these interaction ranges and interaction kernels depends on thephysiological characteristics of each species (e.g. the interaction ranges for bacteriaare much smaller than the interaction ranges for birds, for example).

(a) (b)

Fig. 1. The movement of a reference individual positioned at x, and its neighbours positionedat x + s and x− s. The continuous arrows describe the movement directions of individuals. (a) Aright-moving individual (u+); (b) A left-moving individual (u−). The reference individual at xwill change its direction of movement (with rate λ±) only after perceiving neighbours movingtowards it.



Throughout this paper, we consider a finite domain of length L0. For numericalsimulations we choose [0, L0] = [0, 10]. At the boundaries, we consider periodicboundary conditions:

u+(0, t) = u+(L0, t), u−(L0, t) = u−(0, t). (2.5)

Due to these boundary conditions, we also choose the values of mj to ensure thaton a finite domain (with wrap-around boundary conditions for the integrals in(2.3)), more than 98% of the mass of the kernels is inside the domain.24 Also, mj

are chosen such that the kernels overlap for less than 2% of their mass. For thisreason (i.e. the separation of spatial ranges for the repulsive/alignment/attractiveinteractions), we were able to assume that the effect of the social interactions wasadditive (see Eq. (2.2)). To simplify the analysis, throughout the rest of the paperwe ignore the alignment interactions (i.e. qal = 0).

Remark 2.1. Model (2.1) is the classical prototype of a 1D hyperbolic modelthat can describe the movement in opposite directions of cells27,28 or pedestrians.29

However, the majority of these models are local, incorporating the assumptionthat cells/individuals can interact only with other cells/individuals at the sameposition in space. A first nonlocal model, which was the starting point for model(2.1)–(2.3)24 investigated in this paper, was introduced by Pfistner28 to study themovement dynamics of myxobacteria. A more detailed discussion on the varioustypes of local and nonlocal hyperbolic models for collective behaviour, and of thevarious assumptions incorporated in these models, can be found in Eftimie.13

Remark 2.2. The models for collective behaviour can be classified as: (i) metric-distance models, in which individuals interact with all neighbours within a cer-tain distance, and (ii) topological-distance models, in which individuals interactonly with a fixed number of neighbours. Both types of model assumptions aresupported by empirical evidence: topological interactions in starling flocks,7 andmetric-distance interactions (given by the interaction zones) in fish shoals30 orflocking surf scoters.10 The idea of topological interactions might be connected tothe idea of communication mechanisms, by assuming that individuals can interactonly with those neighbours that they can perceive. Although experimental evidencesupporting this idea is lacking at the moment — due to the difficulty of gatheringdata regarding group-level communication — it is very plausible that individualsrespond only to those neighbours that “catch their attention”. (Partan31 showedthat at behavioural level, the mechanisms that lead to the integration of signalsfrom neighbours involve not only communication but also perception and atten-tion). Ballerini et al.7 do suggest that the number of 6–7 neighbours that wereshown to influence the movement of a reference individual in a flock of starlings,is likely the result of cortical incorporation of the visual input. Therefore, model(2.1)–(2.3) combines topological-like interactions (via communication) with metric-distance interactions (via interaction zones).



Since throughout the rest of the paper we use the symmetry theory to investigatethis hyperbolic model, let us now consider the following action on functions u =(u+, u−):

θ . u±(x, t) = u±(x+ θ, t), θ ∈ [0, L0),

κ · (u+(x, t), u−(x, t)) = (u−(L0 − x, t), u+(L0 − x, t)).(2.6)

These symmetry elements generate a group isomorphic to O(2). We have recentlyshown19 that all models introduced in Eftimie et al.18 (and hence model (2.1)) aresymmetric with respect to the action (2.6); that is, for any solution u(x, t) of (2.1),then θ . u(x, t) for all θ ∈ [0, L0) and κ . u(x, t) are also solutions of (2.1). Thus wesay that Eq. (2.1) is O(2)-symmetric or O(2)-equivariant. The next section outlinesthe symmetry methods used to study the linear system at an equilibrium solution.

2.1. Linear stability analysis

As previously shown by Eftimie et al.,17,18 the hyperbolic model (2.1) displaysat least one spatially homogeneous steady state, where individuals are evenlyspread over the domain: (u+, u−) = (u∗, u∗), with u∗ = A/2. Here, A = (1/L0)∫ L0

0(u+(x, t)+u−(x, t))dx is the total population density on a finite domain [0, L0].

In addition, system (2.1) could display also a homogeneous steady state where thereare more individuals facing one direction: (u+, u−) = (u+

∗ , u−∗ ), with u+

∗ = u−∗ (seeEftimie et al.18). For simplicity, in the following we will ignore the second homoge-neous state and focus only on (u∗, u∗). The linear stability of this steady state canbe investigated easily by considering perturbations of the form

u±(x, t) = u∗ + u±1 , with u±1 ∝ eσt+ikx, (2.7)

where σ is the growth rate and k is the wave number. Since the domain L0 is finite,the wave number is discrete: kn = 2πn/L0, n ∈ N+. (As shown previously,17 basedon the conservation of the total density, k0 = 0 is not an allowable wave number,and thus n ∈ N+.) By substituting (2.7) into Eqs. (2.1), we obtain the followinglinear operator:

L(u1) =

(∂t + γ∂x + L1 −R2K

−∗ −L1 +R2K+∗

−L1 +R2K−∗ ∂t − γ∂x + L1 −R2K

+∗

)(u+

1

u−1

)= 0, (2.8)

with u1 = (u+1 , u

−1 ) given by (2.7). The convolutions K±∗ are defined as follows:

K± ∗ u = qrK±r ∗ u− qaK

±a ∗ u and

K±r,a ∗ u =

∫ ∞

−∞Kr,a(s)u(x± s)ds. (2.9)

Finally, the constants L1 and R2 are defined as

L1 = λ1 + λ2f(0), R1 = λ2f′(0), R2 = 2u∗R1. (2.10)



Let us now introduce the matrix operators

Ld :=

(∂x 0

0 −∂x

)and L0 :=

(−L1 +R2K

− ∗ · L1 −R2K+ ∗ ·

L1 −R2K− ∗ · −L1 +R2K

+ ∗ ·

).

Then, the linear operator L is written as

L(u1) =

(∂t 0

0 ∂t

)(u+

1

u−1

)+ L

(u+

1

u−1

),

where L = −γLd + L0. The O(2)-equivariance of system (2.1) implies the O(2)-equivariance of the linear operator L. Let us write now the linearized system as anabstract differential equation

∂tu = L(u, µ), (2.11)

with µ a vector of parameters. As L0 is a bounded operator on Lp([0, L0],R2) (seeTheorem 3.4 in Hillen32), it follows that the linear operator L with domain

X := u = (u+, u−) ∈ W 1,p([0, L0],R2) |u±(0) = u±(L0),generates a strongly continuous semigroup on Lp([0, L0],R2). We use X as thephase space to (2.11). The symmetries (2.6) act on X , and they induce an isotypicdecomposition of X described in the following theorem.19

Theorem 2.1. For all n ≥ 1, let kn = 2πn/L0 and consider the subspaces

Xn = aeknxi + c.c. | a = (a+, a−) ∈ C2 ⊂ X

are isomorphic to C2, O(2)-invariant and decompose as Xn = X1n ⊕X2

n, where

X1n = (v0eknix + v0e

−knix)f1 | v0 ∈ C, f1 = (1, 1)Tand

X2n = (v1eknix + v1e

−knix)f2 | v1 ∈ C, f2 = (−1, 1)T are O(2)-irreducible representations of C dimension 1. The action (2.6) induces theaction

θ . (v0, v1) = (ekniθv0, ekniθv1),

κ . (v0, v1) = (v0,−v1),

on C2.

The O(2) symmetry of L leaves each Xn invariant, and therefore we can defineLn := L|Xn . Applying now the convolutions (2.9) to exponentials eknix leads to thefollowing Fourier transforms:

K± ∗ eknix = eknix

∫ ∞

−∞K(s)eknisds := K±(n)eknix.



In particular, we have K−(n) = K+(n). In the coordinates given by the decompo-sition Xn = X1

n ⊕X2n, we obtain

Ln(u

v

)eknix =

(0 −kniγ

R2(K+(n) − K+(n)) − kniγ −2L1 +R2(K+(n) + K+(n))

)

·(u

v

)eknix. (2.12)

The eigenvalues of Ln are

σn,± = −(L1 −R2 Re(K+(n)))

±√

(L1 −R2 Re(K+(n)))2 − (k2nγ

2 − γknR2 Im(K+(n))). (2.13)

For σn = iω, the nontrivial solution of(iω −kniγ

R2(K+(n) − K+(n)) − kniγ iω + (2L1 −R2(K+(n) + K+(n)))

)(u

v

)= 0,

leads to ωu − knγv = 0, and so the eigenspace is C(knγ, ω)T. The eigenval-ues (2.13) are roots of the dispersion relation

σ2 + σB(k) + C(k) = 0, (2.14)

with

B(k) = 2L1 −R2 Re(K+(n)), (2.15a)

C(k) = γ2k2 − γikR2 Im(K+(n)). (2.15b)

Figure 2(a) shows the dispersion relation (2.14) for (qa, qr) = (0.63, 3.6), whentwo wave numbers, k3 and k4, become unstable at the same time. Figure 2(b) showsthe neutral stability curves (σ(k) = 0) corresponding to wave numbers k3 (contin-uous line) and k4 (dashed line). We note that for small qr,a both wave numbers arestable (i.e. Re(σ(k3,4)) < 0), and for large qr,a both wave numbers are unstable (i.e.Re(σ(k3,4)) > 0).

2.2. Linear modes at bifurcation

Consider the linear system

∂tu(x, t) = Lnu(x, t), (2.16)

where u(x, t) = (u+(x, t), u−(x, t)) with solutions of the form

u(x, t) = Re

((v0

v1

)eknixeσt

)= Re(v0eknixeσt)f1 + Re(v1eknixeσt)f2.



σ

3

σIm( )

Re( )

kk 4

σ( )

Re

kσ( )<0ReRe kσ( )>0

Re

σ( )<0

k

k3,4σ( )

k

k

H/H

σ( )>0

Re >0

4

Re <03,4

4

3

3

(a) (b)

Fig. 2. (a) The real (continuous curve) and imaginary (dotted curve) parts of the dispersionrelation σ(k). (b) Neutral stability curves σ(k3) = 0 (continuous line) and σ(k4) = 0 (dashed line),as functions of two parameters, qa and qr. At (qa, qr) = (0.637, 3.607) the two stability curvesintersect and a Hopf/Hopf bifurcation occurs. The homogeneous equilibrium is asymptoticallystable in the lower-left region delimited by the bifurcation curves. The rest of parameter valuesare: qal = 0, sr = 0.25, sa = 1, γ = 0.1, L0 = 10.0.

For σ = ±iω, the eigenspaces are C(knγ, ω)T and C(knγ,−ω)T, respec-tively. Define Φn1 = knγf1 and Φ2 = ωf2, where f1 and f2 are described in Theorem2.1. Note that

Φn1 + Φ2 =

(knγ − ω

knγ + ω

)and Φn1 − Φ2 =

(knγ + ω

knγ − ω

). (2.17)

Then, the solutions of the linear system (2.16) can be written as

u(x, t) = Re(w1eknixeiωt)(Φn1 + Φ2) + Re(w2e

knixe−iωt)(Φn1 − Φ2).

The O(2) group acts on Φn1 and Φ2 as follows:

κ . (Φn1 + Φ2) = Φn1 − Φ2

and θ acts trivially on Φn1 and Φ2.The general real solution at the 3 : 4 H/H bifurcation point is

u(x, t) = Re(w1ek3ixeiω3t)(Φ3

1 + Φ2) + Re(w2ek3ixe−iω3t)(Φ3

1 − Φ2)

+ Re(w3ek4ixeiω4t)(Φ4

1 + Φ2) + Re(w4ek4ixe−iω4t)(Φ4

1 − Φ2). (2.18)

We rearrange the terms as follows:

u(x, t) = w1eiω3t(ek3ix(Φ3

1 + Φ2)) + w2eiω3t(e−k3ix(Φ3

1 − Φ2))

+w3eiω4t(ek4ix(Φ4

1 + Φ2)) + w4eiω4t(e−k4ix(Φ4

1 − Φ2)) + c.c.,

where “c.c.” stands for “complex conjugate”. Let

hn1 = eknix(Φn1 + Φ2) and hn2 = e−knix(Φn1 − Φ2).



Thus, u(x, t) is written in terms of the basis h31, h

32, h

41, h

42. The action of O(2) on

the elements of the basis is

κ . hn1 = hn2 , θ . hn1 = ekniθhn1 and θ . hn2 = e−kniθhn2 .

This basis is the standard one for the classification of isotropy subgroups in theO(2) Hopf/Hopf bifurcation.23 The superposition of linear modes at the Hopf/Hopfbifurcation is now given by (in the new coordinates zi)

u(x, t) = (z1h31 + z2h

32)e

iω3t + (z3h41 + z4h

42)e

iω4t + c.c. (2.19)

The action on (z1, z2, z3, z4) is

θ . (z1, z2, z3, z4) = (ek3iθz1, e−k3iθz2, ek4iθz3, e−k4iθz4),

κ . (z1, z2, z3, z4) = (z2, z1, z4, z3).(2.20)

For the parameter values chosen in this paper (see caption of Fig. 2), the fre-quencies at the Hopf/Hopf bifurcation point are estimated at ω3 ≈ 0.2461539166and ω4 ≈ 0.3454406196. Hence, it is safe to assume that they are not rationallydependent. Therefore a T2 = S1 × S1 action generated (after a convenient rescal-ing) by e(2πi/ω3L0)ψ1ω3 ⊕ e(2πi/ω4L0)ψ2ω4 , with ψ1, ψ2 ∈ [0, L0), acts on the criticaleigenspace C2 ⊕ C2 as follows:

(ψ1, ψ2) . (z1, z2, z3, z4)

= (e2πiψ1/L0z1, e2πiψ1/L0z2, e

2πiψ2/L0z3, e2πiψ2/L0z4). (2.21)

Together, (2.20) and (2.21) generate the action of O(2) × T2 on the criticaleigenspace, and thus on the linear patterns given by u(x, t). The classification ofsolutions with respect to their symmetry groups is achieved by computing “isotropysubgroups”, and their general form is obtained by computing “fixed point sub-spaces”. If we consider the action of a group Γ on a vector space V , then theisotropy subgroup of the point v ∈ V is

Γv := ρ ∈ Γ | ρ . v = v.For an isotropy subgroup Σ ⊂ Γ, the fixed point subspace of Σ is

Fix(Σ) = v ∈ V |σ . v = v, for all σ ∈ Σ.The importance of the isotropy subgroups and the fixed-point subspaces is in thefollowing flow-invariance property23: If f is a Γ-equivariant (possibly nonlinear)mapping and Σ is an isotropy subgroup of Γ, then f : Fix (Σ) → Fix (Σ). We willreturn to this property in Sec. 3.0.3, when we will discuss the equilibrium solutionsfor the amplitude equations obtained via the weakly nonlinear analysis.

Linear patterns at isotropy subgroups. We now consider the isotropy subgroupclassification of possible solutions near the O(2) Hopf/Hopf bifurcation, and foreach solution we describe the linear mode u(x, t). First, consider the subgroup



S(k, l,m) = (kψ, ψ,mψ) |ψ ∈ S1 of O(2)×T2. The action (2.20) and (2.21) hasthe largest isotropy subgroups for n = 3, 4,

SO(2) = S(0, 0, 1)× S(1, n, 0) and H × Z2(κ) × Z(L0/2n, 0, L0/2),

with H = S(0, 0, 1) and H = S(0, 1, 0) (each with two-dimensional fixed pointsubspaces). The bifurcating solutions in these fixed point subspaces are rotatingwaves and standing waves, respectively.23 Figure 3 shows linear patterns for thesesolutions near the bifurcation point.

Next, we focus on possible tori solutions, and consider the following subgroupsof O(2) × T2 (see also Chossat et al.22,23):

Z(θ, ψ1, ψ2) = 〈(θ, ψ1, ψ2)〉 ⊂ SO(2) × T2,

Zκ(θ, ψ1, ψ2) = 〈(κθ, ψ1, ψ2)〉.These subgroups are found in the four-dimensional fixed point subspaces of theO(2) × T2 action. Table 1 shows the isotropy subgroups and the fixed point sub-spaces defining conditions for these subgroups.23 Here we follow standard notationfor these subgroups,23 but we keep in mind that = 3 and m = 4.

(a) (b) (c)

Fig. 3. Rotating wave with (a) = 3, (b) m = 4 and (c) standing waves.

Table 1. Table of isotropy subgroups with four-dimensional fixed point subspaces of theO(2) × T

2 action at Hopf/Hopf bifurcation. Recall that = 3 and m = 4.

Isotropy subgroup Conditions u(x, t)

S(0, 0, 1) × Z(L0/2, L0/2, 0) z3 = z4 = 0 (z1h31 + z2h3

2)eiω3t + c.c.

S(0, 1, 0) × Z(L0/2m, 0, L0/2) z1 = z2 = 0 (z3h41 + z4h4

2)eiω4t + c.c.

S(1, , m) z1 = z3 = 0 z2eiω3th32 + z4eiω4th4

2 + c.c.

S(1, ,−m) z1 = z4 = 0 z2eiω3th32 + z3eiω4th4

1 + c.c.

Zκ2 × Z(L0/2, L0/2, mL0/2) z1 = z2, z3 = z4 z1Ω3 + z3Ω4 + c.c.

Ωi := eiωit(hi1 + hi

2),

i = 3, 4

Zk(0, 0, L0/2) × Z(L0/2, L0/2, mL0/2) z1 = z2, z3 = −z4 z1Ω3 + z3eiΩ4t(h41 − h4

2) + c.c.



Table 2. Table of isotropy subgroups with four-dimensional fixed point subspaces of the

O(2) × T2 action, near the Hopf/Hopf bifurcation point. Recall that = 3 and m = 4.

Isotropy subgroup u(x, t)

S(0, 0, 1) × Z(L0/2, L0/2, 0) 2(k3γ)eiω3t(z1eik3x + z2e−k3ix) + c.c.

S(0, 1, 0) × Z(L0/2m, 0, L0/2) 2(k4γ)eiω4t(z3ek4ix + z4e−k4ix) + c.c.

S(1, , m) 2γ[k3z2e−k3ixeiω3t + k4z4e−ik4xeiω4t] + c.c.

S(1, ,−m) 2γ[k3z2e−k3ixeiω3t + k4z3eik4xeiω4t] + c.c.

Zκ2 × Z(L0/2, L0/2, mL0/2) z1eiω3t4 cos(k3x) + z3eiω4t4 cos(k4x) + c.c.

Zk(0, 0, L0/2) × Z(L0/2, L0/2, mL0/2) z1eiω3t4 cos(k3x) + z3eiω4t4i sin(k4x) + c.c.

(a) (b)

(c) (d)

Fig. 4. Total density u(x, t) for submaximal isotropy subgroup solutions: (a) S(0, 0, 1),(b) S(1, , m), (c) Z

κ2 × Z(L0/2, L0/2, mL0/2) and (d) Zk(0, 0, L0/2) × Z(L0/2, L0/2, mL0/2).

Table 2 shows u(x, t) = u+(x, t) + u−(x, t) for each isotropy subgroup (notethat there is no contribution to u(x, t) from the Φ2 terms). Finally, Fig. 4 showsthe contourplots of u(x, t) as given by the eigenfunctions, for the isotropy sub-groups S(0, 0, 1), S(1, ,m), Zκ2 × Z(L0/2, L0/2,mL0/2) and Zk(0, 0, L0/2) ×Z(L0/2, L0/2,mL0/2).



0 2 4 6 8 10 2200

2220

2240

2260

2280

2300

2.005

2.01

2.015

2.02

2.025

2.03

2.035

tota

l den

sity

(u)

space

time

0 2 4 6 8 10 2200

2220

2240

2260

2280

2300

0.995

1

1.005

1.01

1.015

1.02

1.025

space

time

righ

t−m

ovin

g de

nsity

(u

)+

0 2 4 6 8 10 2200

2220

2240

2260

2280

2300

1 1.002 1.004 1.006 1.008 1.01 1.012 1.014 1.016 1.018 1.02

space

time

left

−m

ovin

g de

nsity

(u

)

(a) (a′) (a′′)

0 2 4 6 8 10 1600

1620

1640

1660

1680

1700

2.0199

2.01995

2.02

2.02005

2.0201

2.02015

2.0202

2.02025

tota

l den

sity

(u)

time

space0 2 4 6 8 10

1600

1620

1640

1660

1680

1700

1.00992 1.00994 1.00996 1.00998 1.01 1.01002 1.01004 1.01006 1.01008 1.0101 1.01012 1.01014

space

time

righ

t−m

ovin

g de

nsity

(u

)+

0 2 4 6 8 10 1600

1620

1640

1660

1680

1700

1.0099

1.00995

1.01

1.01005

1.0101

1.01015

space

time

left

−m

ovin

g de

nsity

(u

)

(b) (b′) (b′′)

0 2 4 6 8 10 2990

2995

3000

3005

3010

3015

3020

1 1.2 1.4 1.6 1.82 2.2 2.4 2.6

time

tota

l den

sity

(u)

space0 2 4 6 8 10

2990

2995

3000

3005

3010

3015

3020

0 0.2 0.4 0.6 0.81 1.2 1.4 1.6 1.82 2.2

time

space

righ

t−m

ovin

g de

nsity

(u

)+

0 2 4 6 8 10 2990

2995

3000

3005

3010

3015

3020

0.3

0.4

0.5

0.6

0.7

0.8

0.9

space

time

left

−m

ovin

g de

nsity

(u

)(c) (c′) (c′′)

0 2 4 6 8 10 1070

1080

1090

1100

1110

1120

1130

0.5

1

1.5

2

2.5

3

3.5

4

time

space

tota

l den

sity

(u)

0 2 4 6 8 10 1070

1080

1090

1100

1110

1120

1130

0.3 0.4 0.5 0.6 0.7 0.8 0.91 1.1 1.2 1.3

space

time

left

−m

ovin

g de

nsity

(u

)

0 2 4 6 8 10 2010 2020 2030 2040 2050 2060 2070 2080 2090 2100

1.9952

2.005 2.01 2.015 2.02 2.025 2.03 2.035 2.04 2.045

time

space

tota

l den

sity

(u)

(d) (d′) (d′′)

Fig. 5. Spatio-temporal patterns observed near the 3:4 H/H bifurcation point. Panels (a)–(d)show the total density u = u+ +u−. Panels (a′)–(c′) show the density of right-moving individuals

u+. Panels (a′′)–(c′′), (d′) show the density of left-moving individuals. (a)–(a′′) Standing waves(SW), corresponding to k4, for qa = 0.72, qr = 3.55; (b)–(b′′) modulated standing waves (MSW),for qa = 0.66, qr = 3.56; (c)–(c′′) rotating waves (RW), for qa = 0.72, qr = 3.55; (d), (d′)modulated rotating waves (MRW), for qa = 0.76, qr = 3.56; (e) spatially homogeneous solutions(SH), for qr,a in the parameter region where Re(σ(k3,4)) < 0: qa = 0.51, qr = 3.53. The rest ofthe parameters are: qal = 0, λ1 = 0.2, λ2 = 0.9, γ = 0.1, L0 = 10.



2.3. Numerical simulations

Figure 5 shows the patterns (and the absence of spatio-temporal patterns) observednear the H/H bifurcation point identified in Fig. 2. For numerical simulations, weused a simple, first-order up-wind scheme that incorporates periodic boundary con-ditions. The simulations were also checked with a second-order McCormack scheme.The integrals were discretised using the Simpson’s rule. The initial conditions forthe simulations are small random perturbations of spatially homogeneous steadystates (u+, u−) = (u∗, u∗), with u∗ = 1.0.

The numerical results in Fig. 5 show four types of spatio-temporal patternsthat can emerge around the H/H point: standing waves corresponding to one wavenumber, here k4 (panels (a), (a′), (a′′)), modulated standing waves with isotropy sub-groups Z2(κ)×Z(L0/2, L0/2,mL0/2) or Zk(0, 0, L0/2)×Z(L0/2, L0/2,mL0/2),with = 3, m = 4 (panels (b), (b′), (b′′); similar to Fig. 4(c), (d)), rotating waves(panels (c), (c′), (c′′)) and modulated rotating waves (panels (d), (d′)). The smallinitial perturbations can also decrease in amplitude, and in this case the dynamicsapproaches a spatially homogeneous steady state (panel (e)). Note that for rotat-ing waves and modulated rotating waves all populations move in the same direction(i.e. the u− populations, which are small here, move in the direction of the largerpopulation, the u+ population here). Also, these two waves seem to occur furtheraway from the H/H point (i.e. rotating waves for qa ≥ 0.72 and modulated rotatingwaves for qa ≥ 0.75).

Multiple simulations with various parameter values show that near the H/Hpoint, the spatio-temporal patterns are usually unstable, with the dynamicsapproaching the spatially homogeneous states (e) (either the initial (u∗, u∗) state orthe state (u+

∗ , u−∗ ) mentioned in the previous section). These results raise the ques-

tion of whether it is possible to have stable spatio-temporal patterns (at least insome restricted parameter space near the H/H point). In the next section, we startthe analytical investigation of these patterns. We identify the amplitude curves forall bifurcating solutions that could possibly emerge near this 3:4 H/H point, anddiscuss the stability of these curves.

3. Weakly Nonlinear Analysis

In the following, we investigate the types of patterns that can arise around the3:4 Hopf/Hopf bifurcation point identified previously, as we vary two parame-ters, qa and qr, around their critical values q0a = 0.637 and q0r = 3.607. Atthis bifurcation point, the imaginary eigenvalues of the dispersion relation areσ(k3; q0a, q

0r) = ±iω(k3) and σ(k4; q0a, q

0r) = ±iω(k4). As shown in Sec. 2.2, at the

linear level, the solution can be expressed as

u±(x, t) ∝ eiω(k3)t+ik3x + eiω(k3)t−ik3x + eiω(k4)t+ik4x + eiω(k4)t−ik4x + c.c. (3.1)

In the neighbourhood of the critical point (q0a, q0r), we have

qr = q0r + νrε2, qa = q0a + νaε

2, (3.2)



where νr,a = ±1 and 0 < ε 1. The dispersion relation can now be written as apower series about the critical point, as follows:

σ(kn; qa, qr) = σ(kn; q0a, q0r) +

∂σ

∂qaε2νa +

∂σ

∂qrε2νr. (3.3)

Thus, we can re-write the exponential terms that appear in the solution (3.1) as

eσ(kn;qa,qr)t±iknx ≈ α(ε2t; νa, νr)eσ(kn;q0a,q0r)t±iknx. (3.4)

Therefore, the solution evolves on two different time scales: a fast time scalet∗ = t (in the exponential term) and a slow time scale T = ε2t (in the amplitudeterm). In the limit ε → 0, we treat these two scales as being independent.33 Forsimplicity, we can drop the asterisk from the fast time scale. If we expand thesolution u±(x, t, ε, T ) of (2.1) in powers of ε

u±(x, t, ε, T ) = u∗ + εu±1 + ε2u±2 + ε3u±3 +O(ε4) (3.5)

and then substitute it back into the nonlinear system (2.1). This system can thenbe written as

0 = N

∑j≥1

εjuj

≈

∑j≥1

Nj(uj) =∑j≥1

L(uj) −Nj − Ej . (3.6)

Here uj = (u+j , u

−j ), L is the linear operator defined by (2.8), Nj is the nonlinear

operator (at O(εj)) which contains the terms u±j−1, u±j−2, etc. and Ej is the nonlinear

operator (at O(εj)) which contains the slow-time derivatives (∂u±j /∂T ) and theterms multiplied by νa, νr.

3.0.1. Function spaces

As we will see shortly, at O(ε1), equation L(u1) = 0 has a nontrivial solution.Therefore, at each O(εj), j ≥ 2, Eq. (3.6) will have solutions if and only if Nj +Ejsatisfy the Fredholm alternative.34 To investigate this aspect, we need to ensurethat the linear operator L is compact. We consider the following Hilbert space:

Y =

u(x, τ) | (x, t) ∈ [0, L0] × [0,∞), s.t. lim

T→∞1T

∫ T

0

∫ L0

0

|u|dxdτ <∞, (3.7)

with the O(2)-invariant inner product

〈u,w〉 = limT→∞

1T

∫ T

0

∫ L0=2nπkn

0

(u+w+ + u−w−dx)dτ, (3.8)

with u = (u+, u−), w = (w+, w−). One can easily check that for all θ ∈ SO(2),

〈θ .u, θ .w〉 = 〈u,w〉 and 〈κ .u, κ .w〉 = 〈u,w〉.



Consider now the space

Vbc = u = (u+, u−) ∈ Y | (∂xu+, ∂xu−) ∈ Y, and u± satisfy (2.5), (3.9)

endowed with the norm

‖u‖2V = ‖(u+, u−)‖2

Y + ‖(∂xu+, ∂xu−)‖2

Y . (3.10)

Here, ‖ · ‖Y is the norm associated with the inner product (3.8). Note thatsince u± are bounded on L∞([0, L0]× [0, T ]),17 and L1([0, L0]× [0, T ]),35 and ∂xu±

are bounded on L1([0, L0] × [0, T ]),35 it can be shown that ∂tu± are bounded onL1([0, L0]× [0, T ]). Therefore, using the relation between the L1 and L2 norms, weobtain that the limits limT→∞ 1

T ‖(u+, u−)‖2L2([0,L0]×[0,T ]) and limT→∞ 1

T ‖(∂tu+ +γ∂xu

+, ∂tu−− γ∂xu

−)‖L2([0,L0]×[0,T ]) are finite. Thus the linear operator L : Vbc →Y given by (2.8) is bounded. Following the approach in Mallet-Paret36 or Kmit andRecke37 it can be shown that the operator L : Vbc → Y is a Fredholm operator.

3.0.2. O(εj) solutions

We start the weakly nonlinear analysis of the steady state (u∗, u∗) near the 3:4codimension-2 bifurcation point (q0a, q

0r) by identifying the coefficients of the solution

at each O(εj). At O(ε1) we write(u+

1

u−1

)= α1(T )

(v11

v12

)eiω(k3)t+ik3x + α2(T )

(v21

v22

)eiω(k3)t−ik3x

+ β1(T )

(w11

w12

)eiω(k4)t+ik4x + β2(T )

(w21

w22

)eiω(k4)t−ik4t + c.c. (3.11)

Since the nonlinear operators are N1 = 0, E1 = 0, we calculate the coeffi-cient vectors (v11, v12), (v21, v22), (w11, w12) and (w21, w22) that appear inEq. (3.11) by solving

L(u1) = 0, (3.12)

with u1 = (u+1 , u

−1 ). Note that applying the linear operator L (2.8) to solutions

u±j ∝ eiω(kn)±iknx, leads to

Lωn,±kn =

(iωn ± γikn + L1 −R2K

∓(kn) −L1 +R2K±(kn)

−L1 +R2K∓(kn) iωc ∓ γikn + L1 −R2K

±(kn)

).

(3.13)

To simplify the notation, we defined here ωn = ω(kn). We use this linear oper-ator to calculate the previous coefficient vectors (shown in Appendix A), whichcorrespond to the expressions obtained in (2.17).



At O(ε2), N2 = 0 and E2 = 0. The nonlinear equation

L

(u+

2

u−2

)= N2 (3.14)

has a nontrivial solution iff N2 + E2 satisfies the Fredholm alternative. ThereforeN2 +E2 has to be orthogonal to the bounded solution of the adjoint homogeneousproblem L(u) = 0. The adjoint operator is defined by

Lωn,±kn

=

(−iωn ∓ γikn + L1 − R2K

±(kn) −L1 +R2K±(kn)

−L1 +R2K∓(kn) −iωn ± γikn + L1 −R2K

∓(kn)

).

(3.15)

The solution of the adjoint homogeneous problem is

u = αa1(T )v∗1eiω(k3)t+ik3x + αa2(T )v∗

2eiω(k3)t−ik3x

+ βa1 (T )w∗1eiω(k4)t+ik4x + βa2 (T )w∗

2eiω(k4)t−ik4t + c.c.

The coefficients v∗1,2 and w∗

1,2 are given in Appendix A. Simple calculations showthat N2 + E2 satisfies the Fredholm alternative (that is, 〈u, N2 + E2〉 = 0).

By substituting the expressions for u±1 into the nonlinear terms of N2 +E2, wecan calculate the solution at O(ε2) (see also Appendix A):(

u+2

u−2

)= α3(T )v3e

iω3+ik3x + α4(T )v4eiω3t−ik3x + β3(T )w3e

iω4t+ik4x

+ β4(T )w4eiω4t−ik4x +G1|α1|2 +G2|α2|2 +G3|β1|2 +G4|β2|2

+G5α1α2e2iω3t +G6β1β2e

2iω4t +G7α1α2e2ik3x +G8β1β2e

2ik4x

+G9(α1)2e2iω3t+2ik3x +G10(α2)2e2iω3t−2ik3x +G11(β1)2e2iω4t+2ik4x

+G12(β2)2e2iω4t−2ik4x +G13α1β1ei(ω3+ω4)t+i(k3+k4)x

+G14α1β2ei(ω3+ω4)t+i(k3−k4)x +G15α1β1e

i(ω3−ω4)t+i(k3−k4)x

+G16α1β2ei(ω3−ω4)t+i(k3+k4 )x +G17α2β1e

i(ω3+ω4)t−i(k3−k4)x

+G18α2β2ei(ω3+ω4)t−i(k3+k4)x +G19α2β1e

i(ω3−ω4)t−i(k3−k4)x

+G20α2β2ei(ω3−ω4)t−i(k3−k4)x + c.c. (3.16)

The coefficient vectors Gj , j = 1, . . . , 20, which are shown in Appendix A, appearfrom the nonlinear operator N2.

Finally, at O(ε3), the nonlinear operators are nonzero (N3, E3 = 0; seeAppendix A). For the Fredholm alternative, we are interested only in the termse±iωjt±ikjx that appear in N3 + E3 (since all other terms are orthogonal to thesolution u ∝ e±iωnt±iknx of the adjoint homogeneous problem). Imposing the ortho-gonality condition 〈u, N3 + E3〉 = 0 leads to the following four complex amplitude



equations:

dα1(T )dT

= −X1α1 +X2α1|α1|2 +X3α1|α2|2 +X4α1|β1|2 +X5α1|β2|2, (3.17a)

dα2(T )dT

= −Y1α2 + Y2α2|α1|2 + Y3α2|α2|2 + Y4α2|β1|2 + Y5α2|β2|2, (3.17b)

dβ1(T )dT

= −Z1β1 + Z2β1|α1|2 + Z3β1|α2|2 + Z4β1|β1|2 + Z5β1|β2|2, (3.17c)

dβ2(T )dT

= −Ψ1β2 + Ψ2β2|α1|2 + Ψ3β2|α2|2 + Ψ4β2|β1|2 + Ψ5β2|β2|2. (3.17d)

The coefficients Xm, Ym, Zm and Ψm, m = 1, 2, 3, 4, 5, are given in Appendix A.Note that, for the parameter values used in this paper (see the caption of Fig. 5),the calculations show the following important equalities:

X1 = Y1, Z1 = Ψ1, (3.18a)

X2 +X3 = Y2 + Y3 = 2.024, Z4 + Z5 = Ψ4 + Ψ5 = 3.781, (3.18b)

X4 +X5 = Y4 + Y5 ≈ 4.1, Z2 + Z3 = Ψ2 + Ψ3 ≈ 2.2. (3.18c)

We will return to these equalities shortly, when we will discuss the types of bifur-cating solutions.

By rewriting

α1(T ) = Λ1(T )eiΘ1(T ), α2(T ) = Λ2(T )eiΘ2(T ),

β1(T ) = Λ3(T )eiΘ3(T ), β2(T ) = Λ4(T )eiΘ4(T ),

we obtain four coupled equations for real amplitudes (Λj , j = 1, 2, 3, 4) and fourcoupled phase equations (Θj, j = 1, 2, 3, 4). Here, we ignore the phase equationsand focus only on the real amplitudes.

dΛ1(T )dT

= Λ1(T )(−Re(X1) + Λ1(T )2 Re(X2) + Λ2(T )2 Re(X3)

+ Λ3(T )2 Re(X4) + Λ4(T )2 Re(X5)), (3.19a)

dΛ2(T )dT

= Λ2(T )(−Re(Y1) + Λ1(T )2 Re(Y2) + Λ2(T )2 Re(Y3)

+ Λ3(T )2 Re(Y4) + Λ4(T )2 Re(Y5)), (3.19b)

dΛ3(T )dT

= Λ3(T )(−Re(Z1) + Λ1(T )2 Re(Z2) + Λ2(T )2 Re(Z3)

+ Λ3(T )2 Re(Z4) + Λ4(T )2 Re(Z5)), (3.19c)

dΛ4(T )dT

= Λ4(T )(−Re(Ψ1) + Λ1(T )2 Re(Ψ2) + Λ2(T )2 Re(Ψ3)

+ Λ3(T )2 Re(Ψ4) + Λ4(T )2 Re(Ψ5)). (3.19d)

Consider now the group Z42 generated by elements of the form (δ1, δ2, δ3, δ4),

where δi = ±1 for i = 1, 2, 3, 4, and Z42 acts on R4 = (Λ1,Λ2,Λ3,Λ4)



componentwise. Then, the system of equations (3.19) is Z42-equivariant. Moreover,

let us restrict ourselves to the fixed-point subspaces

Fix (Z2(δ3) × Z2(δ4)) = (Λ1,Λ2, 0, 0) and

Fix (Z2(δ1) × Z2(δ2)) = (0, 0,Λ3,Λ4) (3.20)

and consider the operation κ . (Λ1,Λ2,Λ3,Λ4) = (Λ2,Λ1,Λ4,Λ3) acting on thesesubspaces. Note that κ is not a symmetry of (3.19). Then, due to the equali-ties in (3.18), Fix(κ)∩Fix(Z2(δ3)×Z2(δ4)) and Fix(κ)∩Fix(Z2(δ1)×Z2(δ2)) areone-dimensional flow-invariant subspaces of (3.19). Because of equality (3.18c),Fix(κ) is also a flow-invariant subspace. Indeed, setting Λ1 = Λ2 and Λ3 = Λ4, (3.19)becomes

dΛ1(T )dT

= Λ1(T )(−Re(X1) + Λ1(T )2(Re(X2 +X3)) + Λ3(T )2(Re(X4 +X5))),

dΛ1(T )dT

= Λ1(T )(−Re(X1) + Λ1(T )2(Re(Y2 + Y3)) + Λ3(T )2(Re(Y4 + Y5))),

dΛ3(T )dT

= Λ3(T )(−Re(Z1) + Λ1(T )2(Re(Z2 + Z3)) + Λ3(T )2(Re(Z4 + Z5))),

dΛ3(T )dT

= Λ3(T )(−Re(Z1) + Λ1(T )2(Re(Ψ2 + Ψ3)) + Λ3(T )2(Re(Ψ4 + Ψ5))).

Thus, any solution with initial condition in Fix(κ) stays in Fix(κ).

3.0.3. Equilibrium solutions for the amplitude equations

In the following, we use a symmetry-based classification of the equilibrium solutionsof (3.19), to obtain equations for the amplitudes of the bifurcating solutions. Tothis end, we will link these equilibrium solutions to solutions (3.11) with isotropysubgroups of the O(2) × T2 action (2.20).

We begin with the symmetry-based classification of equilibrium solutionsof (3.19). The isotropy subgroups of the Z4

2-action are the various combinationsof δi = ±1, i = 1, 2, 3, 4. Thus, the fixed-point subspaces of isotropy subgroups inR4 are determined by which of the coordinates Λ1, . . . ,Λ4 are equal to zero (seefor instance (3.20)). Recall that fixed-point subspaces of the Z4

2 action are flow-invariant for (3.19).

Table 3. Labelling of isotropy subgroups of the O(2) × T2 action.

Σ1 = S(0, 0, 1) × S(1, 3, 0) Σ2 = S(0, 1, 0) × S(1, 0, 4)

Σ3 = S(0, 0, 1) × Z2(κ) × Z

“L06

, L02

, 0”

Σ4 = S(0, 1, 0) × Z2(κ) × Z

“L08

, 0, L02

”Σ5 = S(0, 0, 1) × Z

“L06

, L02

, 0”

Σ6 = S(0, 1, 0) × Z“

L08

, 0, L02

”Σ7 = S(1, 3, 4) Σ8 = S(1, 3,−4)

Σ9 = Z2(κ) × Z“

L02

, 3L02

, 4L02

”Σ10 = Zκ

“0, 0, L0

2

”× Z

“L02

, 3L02

, 4L02

”


346 P.-L. Buono & R. EftimieTable

4.

Solu

tions

u± 1

obta

ined

from

the

equilib

rium

solu

tions

ofth

eam

plitu

de

equations.

The

isotr

opy

subgro

ups

are

list

edin

Table

3.SH

:sp

ati

ally

hom

ogen

eous,

RW

:ro

tati

ng

wav

e,SW

:st

andin

gw

ave,

MSW

:m

odula

ted

standin

gw

ave,

MRW

:m

odula

ted

rota

ting

wav

e.N

ote

that

for

9and

10,th

e+

sign

isfo

rΣ

9and

the

min

us

for

Σ10.

#Solu

tion

Isot.

sub.

Defi

nin

gco

ndit

ions

Wea

kly

-nonlinea

rso

luti

on

u1(x

,t)

0SH

O(2

)×

T2

Λi=

0,i=

1,2

,3,4

.0

1RW

Σ1

Λ2 1

=X

1X

2,Λ

2=

Λ3

=Λ

4=

0.

α1(T

)v1ei

ω1t+

k3ix

2RW

Σ2

Λ2 4

=Ψ

1Ψ

5,Λ

1=

Λ2

=Λ

3=

0.

β2(T

)w2ei

ω2t+

k4ix

3SW

Σ3

Λ2 1

=Λ

2 2=

Re(X

1)

Re(X

2)+

Re(X

3),Λ

3=

Λ4

=0.

α1(T

)(v

1ei

ω1t+

k3ix

+v

2ei

ω2t−

k3ix

)

4SW

Σ4

Λ2 3

=Λ

2 4=

Re(Z

1)

Re(Z

4)+

Re(Z

5),Λ

1=

Λ2

=0.

β1(T

)(w

1ei

ω2t+

k4ix

+w

2ei

ω2t−

k4ix

)

5M

SW

Σ5

Λ3

=Λ

4=

0,Λ

2 1=

Λ2 2,

α1(T

)v1ei

ω1t+

k3ix

+α

2(T

)v2ei

ω1t−

k3ix

Re(

X2)

Re(

X3)

Re(

Y2)

Re(

Y3)

! Λ

2 1

Λ2 2

! =

Re(

X1)

Re(

X1)!

6M

SW

Σ6

Λ1

=Λ

2=

0,Λ

2 3=

Λ2 4,

β1(T

)w1ei

ω2t+

k4ix

+β2(T

)w2ei

ω2t−

k4ix

Re(

Z4)

Re(

Z5)

Re(

Ψ4)

Re(

Ψ5)!

Λ2 3

Λ2 4

! =

Re(

Z1)

Re(

Z1)!

7M

RW

Σ7

Λ1

=Λ

3=

0,

α2(T

)v2ei

ω1t−

k3ix

+β2(T

)w2ei

ω2t−

k4ix

Re(

Y3)

Re(

Y5)

Re(

Ψ3)

Re(

Ψ5)!

Λ2 2

Λ2 4

! =

Re(

X1)

Re(

Z1)!

8M

RW

Σ8

Λ1

=Λ

4=

0,

α2(T

)v2ei

ω1t−

k3ix

+β1(T

)w1ei

ω2t+

k4ix

Re(

Y3)

Re(

Y4)

Re(

Ψ3)

Re(

Ψ4)!

Λ2 2

Λ2 3

! =

Re(

X1)

Re(

Z1)!

9,10

MRW

Σ9,1

0Λ

1=

Λ2,Λ

3=

±Λ4

α1(T

)(v

1ei

ω1t+

k3ix

+v

2ei

ω2t−

k3ix

) R

e(X

2+

X3)

Re(

X4

+X

5)

Re(

Z2

+Z

3)

Re(

Z4

+Z

5)

! Λ

2 1

Λ2 3

! =

Re(

X1)

Re(

Z1)!

+β1(T

)(w

1ei

ω2t+

k4ix±w

2ei

ω2t−

k4ix

)



In Table 4, we list the (conjugacy classes of) isotropy subgroups of the O(2)×T2

action (third column; see also Table 3), the defining conditions for the equilibriaof (3.19) (fourth column) and the form of the solutions (3.11) (fifth column). Recallthat two subgroups H1 and H2 of a group G are conjugate in G if there exists anelement g ∈ G such that g−1H1g = H2. As a consequence, if x1 ∈ Fix(H1), theng−1x1 ∈ Fix(H2). Therefore, the existence of a solution for an isotropy subgroupH1 automatically implies a solution with a conjugate isotropy subgroup H2. Forthis reason, we do not list in Table 4 all possible solutions (RW, SW, etc.) of thefull system (2.1).

The case with only one Λi = 0, say Λ1 = 0, yields Λ21 = Re(X1)/Re(X2) (see

solution #1 in Table 4). The other cases with only one nonzero Λi are similar (seesolution #2 in Table 4). For exactly two nonzero amplitudes Λi,j = 0 (e.g. Λ1 =Λ2 = 0 as for solution #3 in Table 4), we distinguish the cases of (3.20) (i.e. solutions#3 and #4) from the other cases because of the proper flow-invariant subspace ofdimension 1 fixed by κ inside Fix(Z2(δ3)×Z2(δ4)) and inside Fix(Z2(δ1)×Z2(δ2)).For solutions #5–#10, our calculations (with the parameter values investigatedin this paper) show that the matrices are non-singular. Note that the right-handsides of the matrix equations for cases 5–10 are nonzero, and so these solutionsdo not bifurcate directly from the Hopf/Hopf point, which is the generic case.The modulated standing waves (MSW) and the modulated rotating waves (MRW)emerge as secondary bifurcations from standing waves (SW) and rotating waves(RW) (see Chossat et al.22 or Golubitsky et al.23 for details).

Figures 6 and 7 show the (qa, qr) parameter regions where the real solutions#0–#10 exist and are stable/unstable. (For the stability, we used Maple to calcu-late the eigenvalues ej , j = 1, 2, 3, 4 of (3.19) corresponding to the steady states#0–#10.) For a direct comparison of the parameter regions where solutions exist

1,2,3,4Λ =0 2

(a) Solution #0 (a′) Solution #0

Fig. 6. (a)–(e) Parameter region where the nonzero amplitude states #0–#4 can exist (i.e. where(Λj)2 ≥ 0, j = 1, . . . , 4). The states exist in the coloured regions, and do not exist in the whiteregions. (a′)–(e′) Parameter region where the states #0–#4 are stable (white regions: eigenvaluesej < 0, for all j = 1, 2, 3, 4) or unstable (coloured regions: eigenvalues ej > 0 for some j). Forinstability, the colour is given by the largest eigenvalue. The parameter values are similar to thoseshown in the caption of Fig. 5.



Λ >0

Λ <0

2

2

1

1

(b) Solution #1 (b′) Solution #1

4Λ <0

4Λ >0

2

2

(c) Solution #2 (c′) Solution #2

Λ <0

Λ >01,2

2

2

1,2

(d) Solution #3 (d′) Solution #3

3,4Λ <0

Λ >03,4

2

2

(e) Solution #4 (e′) Solution #4

Fig. 6. (Continued )



(a) Solution #5 (a′) Solution #5 (b) Solution #6 (b′) Solution #6

(c) Solution #7 (c′) Solution #7 (d) Solution #8 (d′) Solution #8

(e) Solution #9 (e′) Solution #9 (f) Solution #10 (f′) Solution #10

Fig. 7. (a)–(f) Parameter region where the nonzero amplitude states #5–#10 can exist (i.e.where (Λj)

2 ≥ 0, j = 1, . . . , 10). The states exist in the overlapped coloured regions, and donot exist in the white regions, or the non-overlapped regions. (a′)–(f′) Parameter region wherethe states #5–#10 are stable (white regions: eigenvalues ej < 0 for all j = 1, 2, 3, 4) or unstable(coloured regions: eigenvalues ej > 0 for some j). For instability, the colour is given by the largesteigenvalue. The parameter values are similar to those shown in the caption of Fig. 5.

and the parameter regions where solutions are stable/unstable, we show solutions#0–#4 in Fig. 6 and solutions #5–#10 in Fig. 7.

As expected, the zero amplitude solution (#0) has its stability region determinedby the neutral stability curves Re(σ(k3)) = 0 and Re(σ(k4)) = 0 (compare Fig. 6(a′)with Fig. 2(b)). All other solutions are unstable in the parameter region around theH/H point where they exist, i.e. where Λ2

j ≥ 0, j = 1, . . . , 4 (see Figs. 6(b)–(e′)and Figs. 7(a)–(f ′)). In particular, this shows that the rotating waves (RW) andthe standing waves (SW) bifurcating from the Hopf bifurcation curve boundingthe stability region do so subcritically. This result is consistent with the numericalsimulations, which show unstable spatio-temporal patterns around the 3:4 H/Hpoint (see Fig. 5).



Regarding the RW #1 and #2 (Figs. 6(b) and 6(c)), we note that they cannotco-exist in the parameter region where the zero-amplitude solution is unstable (i.e.for large qa and small qr, or for small qa and large qr). Neither the SW #3 and #4(Figs. 6(c) and 6(d)) can co-exist in the parameter region where this zero-amplitudesolution is unstable. The MSW #5 can exist for large qa and small qr (see Fig. 7(a),where the two coloured regions overlap). Thus for large qa and small qr, the MSW#5 bifurcates from the SW #3 (which exists in this region). Similarly, for small qaand large qr, the MSW #6 bifurcates from the SW #4. The MRW #7 and #8 canexist only for small qa and small qr (see Figs. 7(c) and 7(d)). However, the numericalpatterns shown in Figs. 5(d) and 5(d′) were obtained for large qa (i.e. qa ≥ 5).These analytical and numerical results are not in contradiction, since the numericalpatterns likely bifurcate secondary or tertiary from some rotating waves, and theweakly nonlinear analysis (which gives the cubic amplitude equations (3.19)) canpredict only the primary bifurcations occurring near the bifurcation point.

Finally, solutions #9 and #10 also seem to exist only in a very restricted param-eter space (where the coloured surfaces in Figs. 7(e) and 7(f) overlap). The fact thatwe obtained them numerically also in other regions (see Fig. 5) could mean thatthose numerical patterns are part of a solution branch that bifurcates in this region.

4. Summary and Discussion

In this paper we focused on a nonlocal hyperbolic model for biological aggregations,and investigated in detail the types of patterns that can arise near a codimension-2Hopf/Hopf bifurcation point. Both the weakly nonlinear analysis and the numericalresults showed that near the 3:4 Hopf/Hopf point, all patterns are unstable (i.e.bifurcate sub-critically). However, the weakly nonlinear analysis is valid only nearthe bifurcation point. Therefore, the results of the numerical simulations showingmodulated rotating waves further away from the bifurcation point (i.e. for qa ≥ 0.78,qr = 3.56), do not contradict the analytical results.

By investigating the parameter spaces where the various steady states of theamplitude equations can exist, we showed that the standing waves #3 exist in thesame parameter space as the rotating waves #1, and the standing waves #4 exist inthe same parameter space as the rotating waves #2. The modulated standing waves#5 and #6 can coexist in a small parameter space where qa ≥ 0.63 and qr ≤ 3.6,and where the state #0 is stable. Also, the modulated rotating waves #7–#10 cancoexist in some narrow parameter regions, where they are unstable but the state#0 is stable (and the dynamics thus approaches a spatially homogeneous steadystate). Thus, our analysis of model (2.1) identified the parameter spaces where mul-tiple patterns/behaviours can exist simultaneously. Biologically, this means that forexactly the same parameter values (i.e. same speed, turning rates, magnitudes ofsocial interactions), the groups can display different behaviours. Moreover, the tran-sitions between different behaviours (e.g. from standing waves to rotating waves)do not involve changes in the parameter values that characterise the individual



movement or the magnitude of inter-individual interactions. Previous studies1 haveshown that transitions between different aggregation patterns can be obtained bychanging various parameters (e.g. the size of interaction zones, or group polariza-tion1). Our study supports the idea that in some cases, such transitions can also beintrinsic to the aggregations (and do not require changes in the parameter values).

As so many studies focus only on the bifurcation of various spatial and spatio-temporal patterns in biological systems, we emphasise here the importance of con-sidering the model symmetries when deriving the amplitude equations. First, notethat if we would have ignored the reflectional symmetries (i.e. α2 = β2 = 0), atO(ε3) we would have obtained a system of two coupled equations for the evolu-tion of the amplitudes α1(T ) and β1(T ) on the slow time scale. This system wouldhave had only four steady states corresponding to homogeneous solutions (withzero amplitudes), two rotating wave solutions and standing wave solutions. Thus,we would have missed the modulated standing wave solutions and the modulatedrotating wave (tori) solutions. Second, if we would have ignored the symmetries ofthe model when investigating the equilibria of the amplitude equations (3.19), wewould have obtained a very large number of steady states. However, the symmetriesof the model restricted the number of possible equilibria.

To conclude, in this study we investigated the symmetries and the bifurcationof spatio-temporal patterns displayed by a macroscopic model for self-organisedbiological aggregations. However, this model has been previously derived via a cor-related random-walk approach from a microscopic model for the probabilities ofleft-moving and right-moving individuals.13 An individual-based model correspond-ing to this correlated random walk approach has been derived by Gerda de Vries(personal communication). A possible extension of the analysis performed in thispaper could see an investigation of the symmetries of the individual-based modeland a comparison with the symmetries of the macroscopic model.

Appendix A. Details of the Calculations

The coefficients that appear in the solution u±1 (Eq. (3.11)) are:

v11 = γk3 − ω3, v12 = γk3 + ω3, v21 = ω3 + γk3, v22 = γk3 − ω3,

w11 = γk4 − ω4, w12 = γk4 + ω4, w21 = ω4 + γk4, w22 = γk4 − ω4.

The coefficients of the solution u of the adjoint linear system are:

v∗11 = −iω3 + γik3 +R2K+(k3) −R2K

−(k3),

v∗12 = iω3 + γik3 +R2K+(k3) −R2K

−(k3),

v∗21 = −iω3 − γik3 +R2K−(k3) −R2K

+(k3),

v∗22 = iω3 − γik3 +R2K−(k3) −R2K

+(k3),

w∗11 = −iω4 + γik4 +R2K

+(k4) −R2K−(k4),



w∗12 = iω4 + γik4 +R2K

+(k4) −R2K−(k4),

w∗21 = −iω4 − γik4 +R2K

−(k4) −R2K+(k4),

w∗22 = iω4 − γik4 +R2K

−(k4) −R2K+(k4).

At O(ε2), the nonlinear operator is

N2 =

(R1(K−u+

1 −K+u−1 )(u+1 + u−1 )

−R1(K−u+1 −K+u−1 )(u+

1 + u−1 )

).

Substituting the expressions for u±1 into N2 = (N12 , N

22 )T , we obtain

N12 = Q1|α1|2 +Q2|α2|2 +Q3|β1|2 +Q4|β2|2 +Q5α1α2e

2iω3t

+Q6β1β2e2iω4t +Q7α1α2e

2ik3x +Q8β1β2e2ik4x +Q9(α1)2e2iω3t+2ik3x

+Q10(α2)2e2iω3t−2ik3x +Q11(β1)2e2iω4t+2ik4x +Q12(β2)2e2iω4t−2ik4x

+Q13α1β1ei(ω3+ω4)t+i(k3+k4)x +Q14α1β2e

i(ω3+ω4)t+i(k3−k4)x

+Q15α1β1ei(ω3−ω4)t+i(k3−k4)x +Q16α1β2e

i(ω3−ω4)t+i(k3+k4 )x

+Q17α2β1ei(ω3+ω4)t−i(k3−k4)x +Q18α2β2e

i(ω3+ω4)t−(k3+k4)x

+Q19α2β1ei(ω3−ω4)t−i(k3−k4)x +Q20α2β2e

i(ω3−ω4)t−i(k3−k4)x,

and N22 = −N1

2 . Then, the coefficients Gj , j = 1, . . . , 20 of u±2 (Eq. (3.16)) are cal-

culated by solving the following equations, with the operator Lωc,kc given by (3.13):

L0,0Gn = Qn, n = 1, 2, 3, 4,

L2ω3,0Gn = Q5, L2ω4,0G

6 = Q6, L0,2k3G7 = Q7, L0,2k4G

8 = Q8,

L2ω3,2k3G9 = Q9, L2ω3,−2k3G

10 = Q10, L2ω4,2k4G11 = Q11,

L2ω4,−2k4G12 = Q12, Lω3+ω4,k3+k4G

13 = Q13, Lω3+ω4,k3−k4G14 = Q14,

Lω3−ω4,k3−k4G15 = Q15, Lω3−ω4,k3−k4G

16 = Q16,

Lω3+ω4,−(k3−k4)G17 = Q18, Lω3+ω4,−(k3+k4)G

18 = Q18,

Lω3−ω4,−(k3−k4)G19 = Q19, Lω3−ω4,−(k3−k4)G

20 = Q20.

At O(ε3), the nonlinear operators are

N3 =

R1(K−u+1 −K+u−1 )(u+

2 + u−2 ) +R1(K−u+2 −K+u−2 )(u+

1 + u−1 )

+S1(K+u−1 −K−u+1 )2(u−1 − u+

1 )

+ 2T2(K+u−1 )(K−u+1 )(K+u−1 −K−u+

1 )

−R1(K−u+1 −K+u−1 )(u+

2 + u−2 ) −R1(K−u+2 −K+u−2 )(u+

1 + u−1 )

−S1(K+u−1 −K−u+1 )2(u−1 − u+

1 )

− 2T2(K+u−1 )(K−u+1 )(K+u−1 −K−u+

1 )



and

E3 =

−∂u

+1

∂T−R2(K+

ν u−1 ) +R2(K−

ν u+1 )

−∂u−1

∂T+R2(K+

ν u−1 ) −R2(K−

ν u+1 )

.

Constants R1 and R2 are defined by (2.15). The new terms are: S1 = λ2f′′(0)/2,

T1 = λ2f′′′(0)/6, T2 = 3T1u

∗. Finally, kernels K±ν = νrK

±r − νaK

±a .

At O(ε3), the eiωj±ikjx terms that appear in the nonlinear operator N3 +E3 =(N1

3 + E13 , N

23 + E2

3) are:

N13 + E1

3 = eiω3t+ik3x

(−dα1

dTF 1

1 − α1F21 + α1|α1|2F 3

1

+α1|α2|2F 41 + α1|β1|2F 5

1 + α1|β2|2F 61

)

+ eiω3t−ik3x(−dα2

dTH1

1 − α2H21 + α2|α1|2H3

1

+α2|α2|2H41 + α2|β1|2H5

1 + α2|β2|2H61

)

+ eiω4t+ik4x

(−dβ1

dTM1

1 − β1M21 + β1|α1|2M3

1

+ β1|α2|2M41 + β1|β1|2M5

1 + β1|β2|2M61

)

+ eiω4t−ik4x(−dβ2

dTN1

1 − β2N21 + β2|α1|2N3

1

+ β2|α2|2N41 + β2|β1|2N5

1 + β2|β2|2N61

)+ c.c.,

N23 + E2

3 = eiω3t+ik3x

(−dα1

dTF 1

2 − α1F22 + α1|α1|2F 3

2

+α1|α2|2F 42 + α1|β1|2F 5

2 + α1|β2|2F 62

)

+ eiω3t−ik3x(−dα2

dTH1

2 − α2H22 + α2|α1|2H3

2

+α2|α2|2H42 + α2|β1|2H5

2 + α2|β2|2H62

)

+ eiω4t+ik4x

(−dβ1

dTM1

2 − β1M22 + β1|α1|2M3

2

+ β1|α2|2M42 + β1|β1|2M5

2 + β1|β2|2M62

)



+ eiω4t−ik4x(−dβ2

dTN1

2 − β2N22 + β2|α1|2N3

2

+ β2|α2|2N42 + β2|β1|2N5

2 + β2|β2|2N62

)+ c.c.,

with F l2 = −F l1, H l2 = −H l

1, Ml2 = −M l

1, Nl2 = −N l

1, l = 2, 3, 4, 5, 6, and F 11 = v11,

F 12 = v12, H1

1 = v21, H12 = v22, M1

1 = w11, M12 = w12, N1

1 = w21 and N12 = w22.

Finally, the coefficients of the amplitude equations (3.17) are given by

X1 =v∗11F

21 + v∗12F

22

v∗11F 11 + v∗12F 1

2

, X2 =v∗11F

31 + v∗12F

32

v∗11F 11 + v∗12F 1

2

, X3 =v∗11F

41 + v∗12F

42

v∗11F 11 + v∗12F 1

2

,

X4 =v∗11F

51 + v∗12F

52

v∗11F11 + v∗12F

12

, X5 =v∗11F

61 + v∗12F

62

v∗11F11 + v∗12F

12

,

Y1 =v∗21H

21 + v∗22H

22

v∗21H11 + v22H1

2

, Y2 =v∗21H

31 + v∗22H

32

v∗21H11 + v22H1

2

, Y3 =v∗21H

41 + v∗22H

42

v∗21H11 + v22H1

2

,

Y4 =v∗21H5

1 + v∗22H52

v∗21H11 + v22H1

2

, Y5 =v∗21H6

1 + v∗22H62

v∗21H11 + v22H1

2

,

Z1 =w∗

11M21 + w∗

12M22

w∗11M

11 + w∗

12M12

, Z2 =w∗

11M31 + w∗

12M32

w∗11M

11 + w∗

12M12

, Z3 =w∗

11M41 + w∗

12M42

w∗11M

11 + w∗

12M12

,

Z4 =w∗

11M51 + w∗

12M52

w∗11M

11 + w∗

12M12

, Z5 =w∗

11M61 + w∗

12M62

w∗11M

11 + w∗

12M12

,

Ψ1 =w∗

21N21 + w∗

22N22

w∗21N

11 + w22N1

2

, Ψ2 =w∗

21N31 + w∗

22N32

w∗21N

11 + w22N1

2

, Ψ3 =w∗

21N41 + w∗

22N42

w∗21N

11 + w22N1

2

,

Ψ4 =w∗

21N51 + w∗

22N52

w∗21N

11 + w22N1

2

, Ψ5 =w∗

21N61 + w∗

22N62

w∗21N

11 + w22N1

2

.

Appendix B. Explicit Solutions for Cases #5–#10

• Case #5: Λ3 = Λ4 = 0 and

(Λ2)2 =−Re(Y1) + Re(Y2)

(Re(X1)Re(X2)

)−Re(Y3) + Re(Y2)

(Re(X3)Re(X2)

) , (Λ1)2 =Re(X1)Re(X2)

− (Λ2)2Re(X3)Re(X2)

.

• Case #6: Λ1 = Λ2 = 0 and

(Λ4)2 =−Re(Ψ1) + Re(Ψ4)

(Re(Z1)Re(Z4)

)−Re(Ψ5) + Re(Ψ4)

(Re(Z5)Re(Z4)

) , (Λ3)2 =Re(Z1)Re(Z4)

− Λ24

Re(Z5)Re(Z4)

.



• Case #7: Λ1 = Λ3 = 0 and

(Λ4)2 =−Re(Ψ1) + Re(Ψ3)

(Re(Y1)Re(Y3)

)−Re(Ψ5) + Re(Ψ3)

(Re(Y5)Re(Y3)

) , (Λ2)2 =Re(Y1)Re(Y3)

− (Λ4)2Re(Y5)Re(Y3)

.

• Case #8: Λ1 = Λ4 = 0 and

(Λ3)2 =−Re(Z1) + Re(Z3)

(Re(Y1)Re(Y3)

)−Re(Z4) + Re(Z3)

(Re(Y4)Re(Y3)

) , (Λ2)2 =Re(Y1)Re(Y3)

− (Λ3)2Re(Y4)Re(Y3)

.

• Cases #9, #10: Λ1 = Λ2, Λ3 = ±Λ4, with

(Λ3)2 =Re(Z1) − Re(X1)

Re(Z2)+Re(Z3)Re(X2)+Re(X3)

Re(Z4) + Re(Z5) − (Re(Z2)+Re(Z3))(Re(X4)+Re(X5))(Re(X2)+Re(X3))

,

(Λ1)2 =Re(X1)

Re(X2) + Re(X3)− (Λ3)2

Re(X4) + Re(X5)Re(X2) + Re(X3)

.

Acknowledgments

P.-L.B. would like to thank M. Kovacic and L. van Veen for useful discussions. P.-L.B. acknowledges the funding support from NSERC (Canada) in the form of a Dis-covery Grant. R.E. acknowledges fruitful discussions on the H/H bifurcations withM.A. Lewis and G. de Vries (during R.E.’s Ph.D. years at University of Alberta).

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