Hopf bifurcation with zero frequency and imperfectSO(2) symmetry

F. Marquesa,∗, A. Meseguera, J. M. Lopezb, J. R. Pachecob,c

aDepartament de Fısica Aplicada, Universitat Politecnica de Catalunya, Girona Salgados/n, Modul B4 Campus Nord, 08034 Barcelona, Spain

bSchool of Mathematical and Statistical Sciences, Arizona State University, Tempe AZ85287, USA

cEnvironmental Fluid Dynamics Laboratories, Department of Civil Engineering andGeological Sciences, University of Notre Dame, Notre Dame, Indiana 46556, USA

Abstract

Rotating waves are periodic solutions in SO(2) equivariant dynamical sys-tems. The precession frequency changes with parameters, and it may changesign, passing through zero. When this happens, the dynamical system is verysensitive to imperfections that break the SO(2) symmetry, and the waves maybecome trapped by the imperfections, resulting in steady solutions existing ina finite region in parameter space, the so called pinning phenomenon. In thisstudy we analyze the breaking of the SO(2) symmetry in a dynamical systemclose to a Hopf bifurcation whose frequency changes sign along a curve in pa-rameter space. The problem is more complex than expected, and the completeunfolding is of codimension six. A detailed analysis of different imperfections al-lows to conclude that a pinning region appear in all cases, surrounded by infiniteperiod bifurcation curves. Complex bifurcational processes, strongly dependenton the specifics of the symmetry breaking, appear very close to the intersectionof the Hopf bifurcation and the pinning region. Scaling laws of the pinnig regionwidth, and partial breaking of SO(2) to Zm, are also considered. The resultsagree with previous experimental and numerical studies.

Keywords: Bifurcations with symmetry; imperfections; pinning; homoclinicand heteroclinic dynamics.

1. Introduction

Dynamical systems theory plays an important role in many areas of math-ematics and physics because it supplies the building blocks that allow us tounderstand the changes many physical systems experience in their dynamicswhen parameters are varied. These building blocks are the generic bifurcations

∗Corresponding authorEmail address: marques@fa.upc.edu (F. Marques)

Preprint submitted to Elsevier June 24, 2010

(saddle-node, Hopf, etc.) that any arbitrary physical systems must experienceunder parameter variation, regardless of the physical mechanism underlying thedynamics. When one single parameter of the system under consideration is var-ied, codimension-one bifurcations are expected. If the system depends on moreparameters, higher codimension bifurcations are going to appear, and they actas organizing centers of the dynamics.

The presence of symmetries changes the nature and type of the bifurcationsthat a dynamical system may undergo. Symmetries play an important role inmany idealized situations, when simplifying assumptions and the considerationof simple geometries result in dynamical systems equivariant under a certainsymmetry group. Bifurcations with symmetry have been widely studied [13, 15,6, 14, 7, 9]. However, in any real system, the symmetries are only approximatelyfulfilled, and the breaking of the symmetries, due to the presence of noise,imperfections and/or other phenomena, is always present. There are numerousstudies of how imperfect symmetries lead to dynamics that are unexpected inthe symmetric problem, [e.g., 17, 5, 18, 16, 10, 20]. However, a complete theoryis currently unavailable.

One observed consequence of imperfections in systems that support propa-gating waves is that the waves may become trapped by the imperfections [e.g.,see 17, 31, 28, 29]. In these various examples, the propagation direction is typ-ically biased, however, a more recent problem has considered a case where arotating wave whose sense of precession changes sign is pinned by symmetry-breaking imperfections [1]. We are unaware of any systematic analysis of theassociated normal form dynamics for such a problem and this motivates thepresent study.

When a system is invariant to rotations around an axis (invariance under theSO(2) symmetry group), Hopf bifurcations result in rotating waves, consistingof a pattern that rotates around the symmetry axis at a given precession fre-quency without changing shape. This frequency is parameter dependent, and inmany problems, when parameters are varied, the precession frequency changessign along a curve in parameter space. What has been observed in differentsystems is that in the presence of imperfections, the curve of zero frequency be-comes a band in parameter space. Within this band, the rotating wave becomesa steady solution. This is the so-called pinning phenomenon. It can be under-stood as the attachment of the rotating pattern to some stationary imperfectionof the system, so that the pattern becomes steady, as long as its frequency issmall enough so that the imperfection is able to stop the rotation. This pin-ning phenomenon bears some resemblance to the frequency locking phenomena,although in the frequency locking case we are dealing with a system with twonon-zero frequencies and their ratio becomes constant in a parameter region (aresonance horn).

In the present paper, we analyze the breaking of the SO(2) symmetry ina dynamical system close to a Hopf bifurcation whose frequency changes signalong a curve in parameter space. The analysis shows that breaking the SO(2)symmetry is much more complex than expected, resulting in a bifurcation ofcodimension six. Although it is not possible to analyze in detail such a complex

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and high-codimension bifurcation, we present here the analysis of five differentways to break the SO(2) symmetry, by introducing into the normal form allthe possible terms, up to and including second order, that break the symmetry.In all cases we find that a band of pinning solutions appears around the zerofrequency curve from the symmetric case, and the band is limited by curvesof infinite-period bifurcations. The details of what happens when the infinite-period bifurcation curves approach the Hopf bifurcation curve are different inthe five cases, and involve complicated dynamics with several codimension-twobifurcations occurring in a small region of parameter space as well as severalglobal bifurcations.

Interest in the present analysis is twofold. First of all, although the detailsof the bifurcational process close to the zero-frequency Hopf point are verycomplicated and differ from case to case, for all cases we observe the appearanceof a pinning band limited by infinite-period bifurcations of homoclinic type.Secondly, some of the scenarios analyzed are important per se because theycorrespond to the generic analysis of a partial breaking of the SO(2) symmetry,so that after the introduction of perturbations, the system still retains a discretesymmetry (the Z2 case is analyzed in detail).

The paper is organized as follows. In section §2 the properties of a Hopf bi-furcation with SO(2) symmetry with the precession frequency crossing throughzero are summarized, and the general unfolding of the SO(2) symmetry break-ing process is discussed. The next sections explore the particulars of breakingthe symmetry at order zero (§4), one (§3) and two (§5 and §5.3). Section §3 isparticularly interesting because it corresponds to the symmetry-breaking pro-cess SO(2) → Z2 that can be realized experimentally. Some considerations ofthe SO(2) → Z3 symmetry-breaking are presented in §5.3. Section §7 extractsthe general features of the pinning problem from the particular analysis carriedout in the earlier sections. Finally, §7.1 presents comparisons with experimentsand numerical computations in a real problem in fluid dynamics, illustratingthe application of the general theory developed in the present study.

2. Hopf bifurcation with SO(2) symmetry and zero frequency

The normal form for a Hopf bifurcation is

z = z(µ+ iω − c|z|2), (1)

where ω and c are functions of parameters, but generically at the bifurcationpoint (µ = 0) both are different from zero. It is the non-zero character of ωthat allows one to eliminate the quadratic terms in z in the normal form. Thisis because the normal form z = P (z, z) satisfies

P (e−iωtz, eiωtz) = e−iωtP (z, z). (2)

If ω = 0 this equation becomes an identity and P cannot be simplified. The caseω = 0 is a complicated bifurcation and depends on details of the double-zero

3

eigenvalue linear part L. If L is a not completely degenerate,

L =(

0 10 0

)(3)

and then we have the well-studied Takens–Bogdanov bifurcation; if it is com-pletely degenerate,

L =(

0 00 0

)(4)

and it is a high-codimension bifurcation that has not been completely analyzed.If the system has SO(2) symmetry, it must also satisfy

P (eimθz, e−imθ z) = eimθP (z, z), (5)

where Zm is the discrete symmetry retained by the bifurcated solution; whenthe group Zm is generated by rotations of angle 2π/m around an axis of m-foldsymemtry, as is usually the case with SO(2), then the group is also called Cm.Equations (2) and (5) are completely equivalent, and have the same implicationsfor the normal form structure. Advancing in time is the same as rotating thesolution by a certain angle (ωt = mθ); the bifurcated solution is a rotating wave.Therefore, if ω becomes zero by varying a second parameter, we still have thesame normal form (1), due to (5), with ω replaced by a small parameter ν:

z = z(µ+ iν − c|z|2). (6)

The Hopf bifurcation with SO(2) symmetry and zero frequency is, in this sense,trivial. Introducing the modulus and phase of the complex amplitude z = reiφ,the normal form becomes

r = r(µ− ar2),

φ = ν − br2, (7)

where c = a + ib. We assume c 6= 0, and in fact, by scaling z, we will consider|c| = 1, i.e.

c = a+ ib = ie−iα0 = sinα0 + i cosα0, b+ ia = eiα0 , (8)

which helps simplify subsequent expressions.The bifurcation frequency in (7) is now the small parameter ν. The bifur-

cated solution RWm exists only for µ > 0, and has amplitude r =√µ/a and

frequency ω = ν − bµ/a. The limit cycle RWm becomes an invariant set ofsteady solutions along the straight line µ = aν/b (labeled L in figure 1) wherethe frequency of RWm goes to zero; the angle between L and the horizontal(µ = 0) axis is α0. The bifurcation diagram, and a schematic of the bifurcationsalong a one dimensional path, is shown in figure 1.

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H− H+

µ

ν

µ = aν/bL

A

C− C+

P0

AH− H+L

A

−− ++ −−P0

(a) (b)

α0

Figure 1: Hopf bifurcation with SO(2) symmetry and zero frequency. (a) Bifurcation diagram;thick lines are bifurcation curves. (b) Bifurcations along the path A shown in (a). The fixedpoint curve is labeled with the signs of its eigenvalues. C± are the limit cycles born at theHopf bifurcations H− and H+, that rotate in opposite senses. L is the line where the limitcycle becomes an invariant curve of fixed points.

2.1. Unfolding the Hopf bifurcation with zero frequencyIf the SO(2) symmetry in the normal form (6) is completely broken, and

no symmetry remains, then the restrictions imposed on the normal form by (5)disappear completely and all the terms in z and z missing from (6) will reappearmultiplied by small parameters. This means that the normal form will be

z = z(µ+ iν − c|z|2) + ε1 + ε2z + ε3z2 + ε4zz + ε5z

2, (9)

where additional cubic terms have been neglected because we assume c 6= 0and that cz|z|2 will be dominant. As the εi are complex, we have a problemwith 12 parameters. Additional simplifications can be made in order to obtainthe so-called hypernormal form; this method is extensively used in Kuznetsov[19], for example. These simplifications include an infinitesimal translation of z(two parameters), a time reparametrization depending on the quadratic termsin z (three parameters), and an arbitrary shift in the phase of z (one param-eter). Using these transformations the twelve parameters can be reduced tosix. A complete analysis of a normal form depending on six parameters, i.e. acodimension-six bifurcation, is completely beyond the scope of the present pa-per. In the literature, only codimension-one bifurcations have been completelyanalyzed. Most of the codimension-two bifurcations for ODE and maps have alsobeen analyzed, except for a few bifurcations for maps that remain outstanding[19]. A few codimension-three and very few codimension-four bifurcations havealso been analyzed [8, 12]. To our knowledge, there is no systematic analysis ofbifurcations of codimension greater than two.

In the following sections, we consider the five cases, ε1 to ε5, separately. Acombination of analytical and numerical tools allows for a detailed analysis ofthese bifurcations. Then we will extract the common features of the differentcases when εi

√µ2 + ν2, which capture the relevant behavior for weakly

breaking the SO(2) symmetry. In particular, the case ε2 exhibits very interestingand rich dynamics that may be present in some practical cases, when the SO(2)

5

symmetry group is not completely broken, but a Z2 symmetry group, generatedby the half turn, remains.

Some general comments can be made here on these five cases, which are ofthe form

z = z(µ+ iν − c|z|2) + εzq zp−q, (10)

for integer values 0 ≤ q ≤ p ≤ 2, excluding the case p = q = 1 which is SO(2)equivariant. By changing the origin of the phase of z, we can modify the phaseof ε so that it becomes real and positive. Then, by re-scaling z, the time t, andthe parameters µ and ν as

(z, t, µ, ν)→ (εδz, ε−2δt, ε2δµ, ε2δν), δ =1

3− p , (11)

we obtain (10) with ε = 1, effectively leading to codimension-two bifurcations inall five cases. From now on ε = 1 will be assumed, and we can restore the explicitε dependence by reversing the transformation (11). We expect complex behaviorfor µ2 + ν2 . 1, when the three parameters are of comparable size, while theeffects of small imperfections breaking SO(2) will correspond to µ2 + ν2 1.

3. Symmetry breaking of SO(2) to Z2, with an εz term

The ε1z term in (9) corresponds to breaking SO(2) symmetry in a way thatleaves a system with Z2 symmetry, corresponding to invariance under a halfturn. The normal form to be analyzed is (10) with p = 1, q = 0 and ε = 1:

z = z(µ+ iν − c|z|2) + z. (12)

The new normal form (12) is still invariant to z → −z, or equivalently, thehalf-turn φ → φ + π. This is all that remains of the SO(2) symmetry group,which is reduced to Z2, generated by the half-turn. In fact, the Z2 symmetryimplies that P (z, z) in z = P (z, z) must be odd: P (−z,−z) = −P (z, z), whichis (5) for θ = π, the half turn. Therefore, (12) is the unfolding corresponding tothe symmetry breaking of SO(2) to Z2.

Writing the normal form (12) in terms of the modulus and phase of z = reiφ

givesr = r(µ− ar2) + r cos 2φ,

φ = ν − br2 − sin 2φ.(13)

3.1. Fixed points and their bifurcationsThe normal form (12), or (13), admits up to five fixed points. One is the

origin r = 0, the trivial solution P0. The other fixed points come in two pairsof Z2-symmetric points: one is the the pair P+ = r+eiφ+ and P ∗+ = −r+eiφ+ ,and the other pair is P− = r−eiφ− and P ∗− = −r−eiφ− . Coefficients r± and φ±are given by

r2± = aµ+ bν ±∆, φ± = (α0 ± α1)/2, (14)

∆2 = 1− (aν − bµ)2, eiα1 = aν − bµ− i∆. (15)

6

P ∗+

P−P+

P ∗−

P0+−−−

PF+ PF−A

P ∗−

P ∗+

P+

P−P0

SN

B

SN’

−−+−++

+−−−

−−+−++

+−−−

µ

ν

α0

SN’B

I III

II

IA

PF+

PF−SN

(a) (b) (c)

Figure 2: Schematic of an imperfect Hopf bifurcation: (a) regions in parameter space delimitedby the fixed points and their steady bifurcations, (b) and (c) bifurcations along the paths Aand B shown in (a), respectively.

The details of the computations are given in Appendix A. There are threedifferent regions in the (µ, ν)-parameter plane:

Region II: µ2 + ν2 < 1,Region III: µ2 + ν2 > 1, and |aν − bµ| < 1, and aµ+ bν > 0, (16)

with Region I being the remaining parameter space. These three regions areseparated by four curves along which steady bifurcations between the differentfixed points take place, as shown in figure 2.

In region I only P0 exists, in region II three fixed points P0, P+ and P ∗+exist, and in region III all five points P0, P+, P ∗+, P− and P ∗− exist. Along thesemicircle

PF+ : µ2 + ν2 = 1 and aµ+ bν < 0, (17)

the two symmetrically-related solutions P+ and P ∗+ are born in a pitchforkbifurcation of the trivial branch P0. Along the semicircle

PF− : µ2 + ν2 = 1 and aµ+ bν > 0, (18)

the two symmetrically-related solutions P− and P ∗− are born in a pitchforkbifurcation of the trivial branch P0. Along the two half-lines

SN : µ = (aν − 1)/b, SN’ : µ = (aν + 1)/b, both with aµ+ bν > 0, (19)

a saddle-node bifurcation takes place. It is a double saddle-node, due to the Z2

symmetry; we have one saddle-node involving P+ and P−, and the Z2-symmetricsaddle-node between P ∗+ and P ∗−. Bifurcation diagrams along the paths A andB in figure 2(a) are shown in parts (b) and (c) of the same figure.

We can compare with the original problem with SO(2) symmetry, corre-sponding to ε = 0. In order to do that, the ε dependence will be restored inthis paragraph. The single line L (µ = ν tanα0) where ω = 0 and nontriv-ial fixed points exist in the perfect problem, becomes a region of width 2ε inthe imperfect problem, where up to four fixed points exist, in addition to the

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SN’

TB

H+TB−

TB+

DP+

DP−SN

H0

H−

SN’L

PF−

PF+

PF+

PF−

Figure 3: Local bifurcations of fixed points in the symmetry breaking of SO(2) to Z2 case.Codimension-one bifurcation curves: Hopf H+, H0 and H−, pitchfork PF+ and PF−, saddle-node SN and SN’. Codimension-two bifurcation points: degenerate pitchfork DP+ and DP−,Takens–Bogdanov TB, TB+ and TB−. L is the zero-frequency curve in the SO(2) symmetriccase.

base state P0; they are the remnants of the circle of fixed points in the originalproblem. Solutions with ω = 0, that existed on a single line in the absence ofimperfections, now exist in a region bounded by the semicircle F+ and the half-lines aν− bµ = ±ε; this region will be termed the pinning region. It bears somerelationship with the frequency-locking regions appearing in Neimark-Sacker bi-furcations, in the sense that here we also have frequency locking, but with ω = 0.The width of the pinning region is proportional to ε, a measure of the breakingof SO(2) symmetry due to imperfections.

3.2. Hopf bifurcations of the fixed pointsIn the absence of imperfections (ε = 0) the P0 branch looses stability to a

Hopf bifurcation along the curve µ = 0. Let us analyze the stability of P0 in theimperfect problem. Using Cartesian coordinates z = x+ iy in (12) we obtain(

xy

)=(µ+ 1 −νν µ− 1

)(xy

)− (x2 + y2)

(ax− bybx+ ay

). (20)

The eigenvalues of P0 are the eigenvalues of the linear part of (20), λ± =µ ±√1− ν2. There is a Hopf bifurcation (=λ± 6= 0) when µ = 0 and |ν| > 1,i.e. on the line µ = 0 outside region II; the Hopf frequency is ω = sign(ν)

√ν2 − 1.

The sign of ω is the same as the sign of ν, from the φ equation in (13). Therefore,we have a Hopf bifurcation with positive frequency along H+ (µ = 0 and ν > 1)and a Hopf bifurcation with negative frequency along H− (µ = 0 and ν < 1).The bifurcated periodic solutions are stable limit cycles C+ and C−, respectively.

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The Hopf bifurcations of the P± and P ∗± points can be studied analogously.The eigenvalues of the Jacobian of the right-hand side of (20) at a fixed pointcharacterize the different bifurcations that the fixed point can undergo. Let Tand D be the trace and determinant of J . The eigenvalues are given by

λ2 − Tλ+D = 0 ⇒ λ =12

(T ±√Q), Q = T 2 − 4D. (21)

A Hopf bifurcation takes place for T = 0 and Q < 0. The equation T = 0 at thefour points P± results in the ellipse (see Appendix A for details) µ2 − 4abµν +4a2ν2 = 4a2. This ellipse is tangent to SN’ at the point (µ, ν) = (2b, (b2−a2)/a).The condition Q < 0 is only satisfied by P+ and P ∗+ on the elliptic arc H0 from(µ, ν) = (0,−1) to (2b, (b2 − a2)/a):

µ = 2abν + 2a√

1− a2ν2, ν ∈ [−1, (b2 − a2)/a]. (22)

The elliptic arc H0 is shown in figure 3. Along this arc a pair of unstable limitcycles C0 and C∗0 are born around the fixed points P+ and P ∗+, respectively. Wehave assumed for definitiveness that a and b are both positive, which correspondsto the fluid dynamics problem that motivated the present analysis; for othersigns of a and b, analogous conclusions can be drawn.

3.3. Codimension-two pointsThe local codimension-one bifurcations of the fixed points are now com-

pletely characterized. There are two curves of saddle-node bifurcations, twocurves of pitchfork bifurcations and three Hopf bifurcation curves. These curvesmeet at five codimension-two points. The analysis of the eigenvalues at thesepoints, and of the symmetry of the bifurcating points (P0, P+ and P ∗+), charac-terizes these points as two degenerate pitchforks DP+ and DP−, two Takens–Bogdanov bifurcations with Z2 symmetry, TB+ and TB−, and a double Takens–Bogdanov bifurcation TB, as shown in figure 3.

The degenerate pitchforks, DP+ and DP−, correspond to the transition be-tween supercritical and subcritical pitchfork bifurcations. At these points, asaddle-node curve is born, and only fixed points are involved in the neighboringdynamics. The only difference between DP+ and DP− is the stability of thebase state P0, stable outside the circle µ2 + ν2 = 1 at DP+, and unstable atDP−. Schematics of the bifurcations along a one-dimensional path in parameterspace around the DP+ and DP− points are illustrated in figure 4. The maindifference, apart from the different stability properties of P0, P+ and P ∗+, is theexistence of the limit cycle C− surrounding the three fixed points in case (b),DP−.

The Takens–Bogdanov bifurcation with Z2 symmetry has two different sce-narios [8], and they differ in whether one or two Hopf curves emerge from thebifurcation point. In our problem, bifurcation point TB+ has a single Hopfcurve, H+, while the TB− point has two Hopf curves, H− and H0, emergingfrom the bifurcation point. The scenario TB+ is depicted in figure 5, showingthe bifurcation diagram and also the bifurcations along a closed one-dimensional

9

P ∗+

P ∗−

P0

P+

P−

+−−−

PF−

DP+

SN

PF− −−−−

−−

P0+−+−

PF− PF− SNAA A

(a)

P ∗+

P ∗−

P0

P+

P−

+−++

PF+

DP−

SN

PF+++++

++

P0+−+−

PF+ PF+ SN’B

B

B

(b)

C−

Figure 4: Schematics of the degenerate pitchfork bifurcations (a) DP+ and (b) DP−. Onthe left, bifurcation curves emanating from DP± in parameter space are shown, along with aclosed one-dimensional path (dashed); shown on the right are schematics of the bifurcationsalong the closed path, starting and ending at A (DP+) and B (DP−). The fixed point curvesare labeled with the signs of their eigenvalues. C− is the periodic solution born at the curveH−.

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SN TB+ TB+ Het H+SNAA

−−

+− ++ −−+−+−

P0

P+

P ∗+

P−

P ∗−

−−

−−P−

P ∗− +−

C+a bcTB+

PF−

PF+ DP+

SN

PF−

Het

A

d

a: µ = 0.05, ν = 0.9 b: µ = 0.1, ν = 1.1

c: µ = 0.025, ν = 1.2 d: µ = −0.05, ν = 1.1

Figure 5: Takens–Bogdanov bifurcation with Z2 symmetry TB+. On the top left, bifurcationcurves emanating from TB+ in parameter space, with a closed one-dimensional path; on thetop right, schematics of the bifurcations along the closed path, starting and ending at A.The fixed point curves are labeled with the signs of their eigenvalues. C+ is the periodicsolution born at the curve H+. The region inside the dashed curve on the right contains thestates locally connected with the bifurcation TB+. The bottom panels show four numericallycomputed phase portraits, at points labeled a, b, c and d, for the specified parameter values.F Change order of SN and H+ in the schematics

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AA

−−

−−+−

P0

P+

P ∗+

−− −−+−

C−

CF

P0P ∗

+

P+

H0 GluPF+ PF+H−

C0+

C∗0+

C0

++

a b c

TB−

PF+

PF+

Glu

H0

A

H−CF

a: ν = −0.68 b: ν = −0.66 c: ν = −0.65

Figure 6: Takens–Bogdanov bifurcation with Z2 symmetry TB−. Top left, bifurcation curvesemanating from TB− in parameter space, with a closed one-dimensional path; top right,schematics of the bifurcations along the closed path, starting and ending at A. The fixedpoint curves are labeled with the signs of their eigenvalues. C− is the periodic solution bornat the curve H−; C0+ and C∗

0+ are the unstable cycles born simultaneously at the Hopfbifurcation H0 around the fixed points P+ and P ∗+; and C0 is the cycle around both fixedpoints that remains after the gluing bifurcation. The bottom row shows three numericallycomputed phase portraits, at points labeled a, b and c, for µ = 0.5 and ν as indicated.F Addmore phase portraits?

path around the codimension-two point. We have also included the P+ and P ∗+solutions, that merge with the P− and P ∗− fixed points along the saddle-node bi-furcation curve SN, although they are not locally connected to the codimension-two point, in order to show all the fixed points in the phase space of (12). Acurve of global bifurcations, a heteroclinic cycle Het connecting P− and P ∗−, isborn at TB+. The heteroclinic cycle is formed when the limit cycle C+ simul-taneously collides with the saddles P− and P ∗−.

The scenario TB−, is depicted in figure 6, showing the bifurcation dia-gram and also the bifurcations along a closed one-dimensional path around thecodimension-two point. Two curves of global bifurcations are born at TB−. Onecorresponds to a gluing bifurcation Glu, when the two unstable limit cycles C0+

and C∗0+ , born at the Hopf bifurcation H0 around P+ and P ∗+, simultaneouslycollide with the saddle P0; after the collision a large cycle C0 results, surround-ing the three fixed points P0, P+ and P ∗+. The second global bifurcation curve

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AASN’ H0 Hom

C−

P0

C−

P0++

A

SN’

H0

Hom

TB

SN’

SN’

P+

P−

C0+

++−−+−

P ∗+

P ∗−

C∗0+

++

+−

−−

Figure 7: Double Takens–Bogdanov bifurcation TB. Top left, bifurcation curves emanatingfrom TB in parameter space, with a closed one-dimensional path; top right, schematics of thebifurcations along the closed path, starting and ending at A.

corresponds to a saddle-node of cycles, where C0 and C− collide and disappear.A generic Takens–Bogdanov bifurcation (without symmetry) takes place at

the TB point on the SN’ curve. At the same point in parameter space, butseparate in phase space, two Z2 symmetrically related Takens–Bogdanov bifur-cations take place, with P+ and P ∗+ being the bifurcating states. A schematicof the bifurcations along a one-dimensional path in parameter space around theTB point is shown in figure 7. Apart from the states locally connected to bothTB bifurcation, there also exist the base state P0 and the limit cycle C−.

3.4. Global bifurcationsIn the analysis of the local bifurcations of fixed points we have found three

curves of global bifurcations, a gluing curve Glu and a saddle-node of cyclesemerging from TB− (CF, cyclic-fold), a heteroclinic loop born at TB+ (Het),and a homoclinic loop emerging from TB (Hom). One wonders about the fateof these global bifurcation curves, and about possible additional global bifurca-tions. Numerical simulations of the solutions of the normal form ODE system(12), or equivalently (20), together with dynamical systems theory considera-tions have been used to answer these questions, and a schematic of all local andglobal bifurcation curves is shown in figure 8.

The gluing bifurcation Glu born at TB− and the two homoclinic loops emerg-ing from the two Takens–Bogdanov bifurcations TB (bifurcations of the sym-metric fixed points P+ and P ∗+) meet at the point PGl on the circle PF−, wherethe base state P0 undergoes a pitchfork bifurcation (see figure 9a). At thatpoint, the two homoclinic loops of the gluing bifurcation, both homoclinic atthe same point on the stable P0 branch, split when the two fixed points P−and P ∗− bifurcate from P0 (see figure 9b). The two homoclinic loops are thenattached to the bifurcated points and separate along the curve Hom. The large

13

SN’Het’

TB

H+

CF

Glu

TB−

PGL

Hom

Het0

CFH

TB+

DP+

DP−

SN

SN

HetH0

H−

Figure 8: Global bifurcations in the symmetry breaking of SO(2) to Z2 case. Codimension-one bifurcation curves: Gluing Glu, cyclic-fold CF, homoclinic collision Hom and heteroclinicloops Het, HetS and HetU . Codimension-two bifurcation points: pitchfork-gluing bifurcationPGl and cyclic-fold heteroclinic bifurcation CFH.

unstable limit cycle C0, after the pitchfork bifurcation PF−, collides simultane-ously with both P− and P ∗−, forming a heteroclinic loop along the curve Het’(see figure 9c). Both curves Hom and Het’ are born at PGl and separate, leav-ing a region in between where none of the cycles C0+ , C∗0+ and C0 exist. Theunstable periodic solution C0 merges with the stable periodic solution C− thatwas born in H− and existed in region I, resulting in a cyclic-fold bifurcation ofperiodic solutions CF. Phase portraits around PGl are shown in figure 9(d).

The curve Hom born at PGl ends at the double Takens–Bogdanov pointTB. Locally, around both Takens–Bogdanov bifurcations at TB, after crossingthe homoclinic curve the limit cycles C0+ and C∗0+ disappear, and no cyclesremain. The formation of a large cycle C0 at Het’ surrounding both fixedpoints P+ and P ∗+ is a global bifurcation involving simultaneously both P− andP ∗− unstable points. It is the re-injection induced by the presence of the Z2

symmetry that is responsible for this global phenomenon [2, 3, 27, 19]. The twoglobal bifurcation curves Het’ and CF become very close when leaving the PGlneighborhood, and merge at some point in a CFH (Cyclic-Fold–Heterocliniccollision) codimension-two global bifurcation. After CFH, the stable limit cycleC−, instead of undergoing a cyclic-fold bifurcation, directly collides with thesaddle points P− and P ∗− along the Het’ bifurcation curve (see figure 8).

The heteroclinic loop Het’ born at CFH becomes asymptotic to the SN’ curvefor µ2+ν2 1. From a practical point of view, away from the circle µ2+ν2 = 1,the SN bifurcation is very closely followed by the heteroclinic collision of the limit

14

H0

Hom

PGl

Glu CF

PF−

12

34

Het0

HomGluPGl

C−

P0

P+

P ∗+

P−

P ∗−

+− ++

−−

−−

+−+−

Het0GluPGl

C−

P0

P+

P ∗+

P−

P ∗−

+− ++

−−

−−

+−+−

(c)(a) (b)

(d) 1: (µ, ν) = (0.94,−0.35) 2: (µ, ν) = (0.987,−0.2985)

3: (µ, ν) = (0.936,−0.34) 4: (µ, ν) = (0.936,−0.352)

Figure 9: Pitchfork-gluing bifurcation PGl. Top left, bifurcation curves emanating from PGlin parameter space; top right, schematics of the bifurcations along the line Glu–Hom. Thebottom row shows four numerically computed phase portraits, at points labeled 1, 2, 3 and 4,for the specified parameter values.

15

(a) (b)

2.01 2.02 2.03 2.04 2.05µ

100

150

200

250

300

350Pe

riod

Global log fitGlobal sqrt fitSN bifurcation

2.01 2.02 2.03 2.04 2.05µ

100

150

200

250

300

350

Peri

od

Local log fitLocal sqrt fitSN bifurcation

Figure 10: Square root and logarithmic fits to the periods of C− approaching the Hom curveat ν = 0.6. (a) show fitted curves in the complete range µ ∈ [2.0135, 2.2]; (b) fits with rangesµ ∈ [2.0135, 2.027] for the log fit, and µ ∈ [2.018, 2.2] for the square root fit.

cycle C− with the saddles P− and P ∗−, and it becomes almost indistinguishablefrom a saddle-node on an invariant circle (SNIC) bifurcation. The scaling laws ofthe periods when approaching a heteroclinic or a SNIC bifurcation are different,having logarithmic or square root profiles:

THet =1λ

ln1

µ− µc +O(1), TSNIC =k√

µ− µc +O(1). (23)

where λ is the positive eigenvalue of the saddle, and k a constant. We havenumerically computed the period of C− at ν = 0.6, for decreasing µ values ap-proaching SN’ in the range µ ∈ [2.0135, 2.2]. Figure 10(a) shows both fits usingthe values of the period in the whole computed range; the log fit overestimatethe period, while the square-root fit underestimates it, and this underestimategets larger as the heteroclinic collision is approached. Figure 10(b) shows bothfits but using values close to the collision for the log fit, and for values far awayfrom the collision using the square-root fit. Both fits are now very good approx-imations of the period in their corresponding intervals, and together cover allthe values numerically computed. When the interval between the SN’ bifurca-tion and the heteroclinic collision (in figure 10, µSN = 2.01420 and µc = 2.01336respectively) is very small, and it cannot be reached experimentally (or even nu-merically, if we are dealing with an extended systems with millions of degrees offreedom, as is the case in fluid dynamics), the square-root fit looks good enough,because away from the SN’ point, the dynamical system just feels the ghost ofthe not yet formed saddle-node pair, and does not distinguish between whetherthe saddle-node appears on the limit cycle or very close to it. However, if weare able to resolve the very narrow parameter range between the saddle-nodeformation and the subsequent collision with the saddle, then the log fit matchesthe period in this narrow interval much better.

16

µ

ν

µ = aν/bLα0

u

v

µ = −bν/a

Figure 11: Coordinates (u, v) in parameter space adapted to the line L of zero frequency ((inthe unperturbed SO(2) case).

Due to the presence of two very close bifurcations (Het and SN), the scalinglaws become cross-contaminated, and from a practical point of view the onlyway to distinguish between a SNIC and a Homoclinic collision is by computingor measuring periods extremelly close to the infinit-period bifurcation point.We can also see this from the log fit equation in (23); when both bifurcationsare very close, λ, the positive eigenvalue of the saddle, goes to zero (it is exactlyzero at the saddle-node point), so the log fit becomes useless, except when theperiods are very large.

We can estimate the width of the pinning region as a function of the magni-tude of the imperfection ε, and the distance d to the bifurcation point, w(d, ε).The distance d will be measured along the line L, and the width w(d) will bethe with of the pinning region measured transversally to L, at a distance d fromthe origin. It is convenient to introduce coordinates (u, v) in parameter space,rotated an angle α0 with respect to (µ, ν), so that the parameter u along L isprecisely the distance d:(

uv

)=(a b−b a

)(µν

)=(aµ+ bνaν − bµ

),

(µν

)=(a −bb a

)(uv

), (24)

where a = sinα0, b = cosα0; in the present problem, d =√µ2 + ν2 and along

L (of equation v = 0), d = u. Figure 11 shows the relationship between thetwo coordinate systems. If the pinning region is limited by a curve of equationv = ±h(u), then w = 2h(u) = 2h(d). With an imperfection of the form εz,the case analyzed in this section, the pinning region is of constant width two.By restoring the dependence in ε, we obtain a width of value w(d, ε) = 2ε,independent of the distance to the bifurcation point; the width of the pinningregion is proportional to ε.

4. Symmetry breaking of SO(2) with an ε term

The normal form to be analyzed is (10) with p = q = 0 and ε = 1:

z = z(µ+ iν − c|z|2) + 1. (25)

17

C+

µ

ν

C−

SN0

SN

SN’

I

I

I

II

L

α0

Figure 12: Steady bifurcations of the fixed points corresponding to the normal form (25). SN,SN’ and SN0 are saddle-node bifurcation curves and C± are cusp bifurcation points. In regionI and II there exist one and three fixed points, respectively.

The normal form (25), in terms of the modulus and phase z = reiφ, is

r = r(µ− ar2) + cosφ,

φ = ν − br2 − 1r

sinφ.(26)

4.1. Fixed points and their steady bifurcationsThe fixed points of (26) are given by a cubic equation in r2, so that we

do not have convenient closed forms for the corresponding roots (the Tartagliaexplicit solution is extremely involved). The parameter space is divided intotwo regions I and II, with one and three fixed points respectively, separated bya saddle-node curve given by (see Appendix B)

(u, v) =(3 + 3s2, 2

√3s)

(2 + 6s2)2/3, s ∈ (−∞,+∞), (27)

in (u, v) coordinates (24), and shown in figure 12. The saddle-noddle curve isdivided in three different arcs SN, SN’ and SN0 by two codimension-two cuspbifurcation points, Cusp±. SN’ corresponds to values s ∈ (−∞, 1), SN0 tos ∈ (−1,+1) and SN to s ∈ (1,+∞). The cusp points Cusp± have valuess = ±1. The curves SN and SN’ are asymptotic to the line L, and region II isthe pinning region in this case. The two fixed points that merge on the saddle-node curve have phase space coordinates z2 = r2eiφ2 , and the third fixed pointis z0 = r0eiφ0 , where

r0 =( 4

1 + 3s2)1/3

, r2 =(1 + 3s2

4

)1/6

, (28)

and the phases are obtained from sinφ = r(ν − br2), cosφ = r(ar2 − µ).

18

(a) (b)

Cusp+

TB−TB+

H+H−

SN

SN’

µ

ν

ν0.43 0.511.665

1.695

µCusp−

TB−

H−

SN0

SN’

Figure 13: Bifurcations of the fixed points corresponding to the normal form (25). SN, SN’and SN0 are saddle-node bifurcation curves, H± are Hopf bifurcation curves, Cusp± arecusp bifurcation points and TB± are Takens–Bogdanov bifurcation points. (b) is a zoomof (a), showing that Cusp− and TB− are different. The parameter range in (a) is (ν, µ) ∈[−3, 6]× [−1, 6], for α = 45o.

Other bifurcations of the fixed pointsThe Hopf bifurcations of the fixed points can be obtained by imposing the

conditions T = 0 and D > 0, where T and D are the trace and determinant ofthe Jacobian of (25). These conditions result in two curves of Hopf bifurcations(see Appendix B for details):

(µ, ν) = a1/3(1− s2)1/3(

2 ,b

a+

s√1− s2

),

H− : s ∈(− 1,−

√(1− b)/2

), H+ : s ∈

(√(1 + b)/2,+1

).

(29)

For s→ ±1 both curves are asymptotic to the µ = 0 axis (ν → ±∞), the Hopfcurve for ε = 0; the stable limit cycles born at these curves are termed C− andC+ respectively. The other ends of the H± curves are on the saddle-node curvesof fixed points previously obtained, and at these points T = D = 0, so theyare Takens–Bogdanov points TB±, as shown in figure 13. The TB− and Cusp−codimension-two bifurcation points are very close, as shown in the zoomed-infigure 13(b). In fact, depending of the angle α0, the Hopf curve H− is tangentto and ends at either SN’ or SN0; for α0 > 60o, it ends at SN0, and for α0 = 60o

Cusp− and TB− coincide, and H− ends at the cusp point, a very degeneratecase.

From the Takens–Bogdanov points, two curves of homoclinic bifurcationsemerge, resulting in global bifurcations around these points. There are alsosome additional global bifurcations around these points that we have explorednumerically and using dynamical systems theory; these are summarized in fig-ure 14. There are six codimension-two points organizing the dynamics of thenormal form (25). Apart from the cusp and Takens–Bogdanov points alreadyfound, there are three new points: Bo, Hom-SNIC and SNCF. On H−, beforecrossing the SN0 point, the Hopf bifurcation becomes subcritical at the Bautin

19

H−

H+

SN’

SN’

SN0

Hom

Hom0

Hom’TB−

TB+

Bo

Cusp−

Cusp+

CF

SN0

SN

SNCF

H0

Figure 14: Schematic of the bifurcations of the normal form (25). There are four curves ofglobal bifurcations, Hom, Hom0, Hom’ (collisions of a limit cycle with a saddle) and CF (acyclic-fold), and six codimension-two points (black circles). The regions around (Cusp−,TB−)and (Cusp+,TB+) have been enhanced for clarity.

point Bo, and from this point a curve of cyclic-folds CF appears. This curveis the limit of the subcritical region where two periodic solutions exist, C− andC0, and they merge on CF; C0 is the unstable limit cycle born in the branch ofH− between Cusp− and TB−, termed from now on the Hopf curve H0. Insidethe pinning region these two periodic solutions disappear when they collide witha saddle fixed point along the curves Hom0 (C0 collision) and Hom’ (C− colli-sion). The homoclinic curve Hom0 is born at the Takens–Bogdanov point TB−,and the curve Hom’ becomes asymptotic to SN’. So, away from the origin, SN’closely followed by Hom’ become indistinguishable from a SNIC bifurcation, inthe same sense as discussed in section §3.4. The two curves Hom’ and Hom0

come closer together on approaching SN0, and on SN0 they meet together withthe CF curve at the point SNCF. At the point SNCF, the two limit cycle collidewith the saddle point born at the SN0 saddle-node bifurcation. The two limitcycles also coincide at the collision point. The point SNCF is then a SNIC bi-furcation (a saddle-node appearing on a limit cycle), but this SNIC bifurcationtakes place precisely when the limit cycle is born at a cyclic-fold bifurcation CF.

The other Takens–Bogdanov point TB+ do not present any additional com-plications. The homoclinic curve Hom emerging from it becomes asymptotic tothe SN curve, exactly as in the εz case.

The width of the pinning w(d, ε) region away from the origin (large s) is easy

20

to compute from (27):

d = u ∼ (3s2/4)1/3, w = 2v ∼ (16/√

3s)1/3 ⇒ w = 2/√d. (30)

Restoring the ε dependence, we obtain w(d, ε) = 2ε/√d. The pinning region

becomes narrower away from the bifurcation point, and the width is proportionalto ε, the size of the imperfection.

5. Symmetry breaking of SO(2) with quadratics terms

5.1. The εzz caseThe normal form to be analyzed in this case is

z = z(µ+ iν − c|z|2) + zz, (31)

or in terms of the modulus and phase z = reiφ,

r = r(µ− ar2) + r2 cosφ,

φ = ν − br2 − r sinφ.(32)

There are three fixed points: the origin P0 (r = 0) and the two solutions P± ofthe biquadratic equation r4 − 2(aµ+ bν + 1/2)r2 + µ2 + ν2 = 0, given by

r2± = aµ+ bν +12±√aµ+ bν +

14− (aν − bµ)2. (33)

These solutions are born at the parabola aµ+ bν + 1/4 = (aν − bµ)2, and existonly in its interior, which is the pinning region III in figure 15. The parabolais a curve of saddle-node bifurcations. It can be seen (details in Appendix C)that P+ is stable while P− is a saddle in the whole region III, so there are noadditional bifurcations of fixed points in the zz case, in contrast to the z2 case.

As the perturbation is of second order, the Jacobian at P0 is the same asin the unperturbed case, and the Hopf bifurcations of P0 take place along thehorizontal axis µ = 0. As in the unperturbed case, the Hopf frequency isnegative for ν < 0 (H−), it is zero at the origin, and becomes positive for ν > 0(H+). The Hopf curves H− and H+ extend in this case up to the origin, incontrast with the zero and first order cases examined previously, where the Hopfcurves ended in Takens–Bogdanov bifurcations without reaching the origin.

The stable limit cycles C− and C+, born at the Hopf bifurcations H− andH+, on entering region III, collide with the saddle P+ and disappear alongtwo homoclinic bifurcation curves Hom and Hom’ for small values of µ. Forlarger values of µ, the curves Hom and Hom’ collide with the parabola at thecodimension-two bifurcation points SnHom±, and for larger values of µ, thesaddle-node bifurcations take place on the parabola, and the limit cycles C−and C+ undergo SNIC bifurcations. This is different to what happened in the εand εz cases, where the homoclinic curves become asymptotic to the saddle-nodecurves. In the present case εzz (and as we shall see, also in the other quadraticcases) the curves Hom and Hom’ collide with the parabola at a finite distancefrom the origin. Figure 15 summarizes all the local and global bifurcation curvesin the zz case.

21

(a)

µ

ν

L

H− H+

SNIC

Hom’

SN

Hom

SNIC’III

I+

I

I−

SnHom’ SnHom

(b): (µ, ν) = (1.167,−0.2) (c): (µ, ν) = (1.155,−0.2)

Figure 15: Bifurcation curves corresponding to the normal form with quadratic terms in theεzz case. (b) and (c): crossing the SNIC’ curve.F Add phase portraits crossing Hom’

22

(a) (b)

µ

νH− H+

SNIC

Hom’

SN

Hom

SNIC’

IIII−

SnHom’ SnHom

TB+

TB−H0

I

I+

Figure 16: (a) Bifurcation curves corresponding to the normal form with quadratic terms inthe εz2 case. (b) Phase portrait on the curve H0 showing the degenerate Hopf bifurcation andthe associated homoclinic loop.

5.2. The εz2 caseThe normal form to be analyzed in this case is (10) with p = q = 2 and

ε = 1:z = z(µ+ iν − c|z|2) + z2. (34)

The normal form (34), in terms of the modulus and phase z = reiφ, reads

r = r(µ− ar2) + r2 cosφ,

φ = ν − br2 + r sinφ.(35)

There are three fixed points: the origin P0 (r = 0) and the two solutions P± ofthe biquadratic equation r4 − 2(aµ+ bν + 1/2)r2 + µ2 + ν2 = 0, which are thesame as in the εzz case (§5.1). In fact, the fixed points in these two cases havethe same modulus r and their phases have opposite sign (changing φ → −φin (32) results in (35)). Therefore, the bifurcation curves of the fixed points(excluded Hopf bifurcations of P±) in this case are also given by figure 15.

It can be seen (details in Appendix C) that P+ is a saddle in the wholeregion III, while P− is stable for µ > 0, unstable for µ < 0, and undergoes aHopf bifurcation H0 along the segment of µ = 0 delimited by the parabola ofsaddle-node bifurcations. The points TB± where H0 meets the parabola areTakens–Bogdanov codimension-two bifurcations. Figure 16 summarize all thelocal and global bifurcation curves just described.

In the present case, the two Takens–Bogdanov bifurcations and the Hopfbifurcations along H0 are degenerate, as shown in Appendix C. Detailed anal-ysis and numerical simulations show that the Hopf and homoclinic bifurcationcurves emerging from the Takens–Bogdanov point are both coincident with theH0 curve mentioned before. Moreover, the interior of the homoclinic loop is

23

(a) (b)µ

ν

SnHet’

SN

H− H+

HetHet’SnHet

SNIC’

SNICIII

µ

ν

SN’III

SN

H− H+

Het

Het’H0Hom0

TB+

TB−

SnHet

SnHet’

SNIC’

SNIC

Figure 17: Bifurcation curves corresponding to the normal forms with quadratic terms (36)in the εz2 case. (a) α0 > 30, (b) α0 < 30. H0 is tangent to the parabola at the Takensbogdanov points TB±, and H0, Hom0 almost coincide with SN’; in the figure the distanceshave been exagerated for clarity.

filled with periodic orbits, and no limit cycle exists on either side of H0. Thissituation is illustrated in the phase portrait in figure xx.

Finally, and exactly in the same way as in the xz examined in the previoussubsection, the stable limit cycles C− and C+ born respectively at the Hopfbifurcations H− and H+ and existing in regions I− and I+ (figure 16a), on en-tering region III, collide with the saddle P+ and disappear along two homoclinicbifurcation curves Hom and Hom’ for small values of µ. These curves collidewith the parabola at the codimension-two bifurcation points SnHom±, and forlarger values of µ the limit cycles C− and C+ undergo SNIC bifurcations onthe parabola. The two curves Hom and Hom’ emerge from the origin, as in theprevious case.

5.3. The εz2 caseThe normal form to be analyzed in this case is

z = z(µ+ iν − c|z|2) + z2, (36)

or in terms of the modulus and phase z = reiφ,

r = r(µ− ar2) + r2 cos 3φ,

φ = ν − br2 − r sin 3φ.(37)

The fixed points are the origin P0 (r = 0) and the solutions of the same bi-quadratic equation as in the two previous cases. However, there is an importantdifference: due to the factor 3 inside the trigonometric functions in (37), theP i± points come in triplets (i = 1, 2, 3), each triplet has the same radius r buttheir phases differ by 120. This is a consequence of the invariance of the gov-erning equation (36) to the Z3 symmetry group generated by rotations of 120

24

around the origin. This invariance was not present in the two previous cases.The bifurcation curves of the fixed points (excluding the Hopf bifurcations ofP i±) are still given by figure 16(a), but now a triplet of symmetric saddle-nodebifurcations take place simultaneously on the SN and SN’ curves.

It can be seen (details in Appendix C) that P i− are saddles in the wholeregion III, while P i+ are stable, except for small angles α0 < π/6, in a narrowregion close to SN’, where they are unstable. For α0 > π/6, the bifurcation dia-gram is exactly the same as in the zz case (figure 15), except that the homocliniccurves are now heteroclinic cycles between the triplets of saddles P i−. This caseis illustrated in figure 17(a). For α0 < π/6 there appear two Takens–Bogdanovbifurcation points, at the tangency points between the SN’ curve and the arcH0 of the ellipse (bµ− 2aν)2 + (aµ− 1)2 = 1, as shown in figure 17(b). The arcH0 is a Hopf bifurcation curve of P i+: three unstable limit cycles Ci0 are born,while the triplet P i+ becomes stable. These unstable limit cycles disappear whencolliding with the saddles P i−, on a curve of homoclinic collisions, Hom0, thatends at the two Takens–Bogdanov points TB+ and TB−. This situation is verysimilar of what happens in the εz analyzed in section §3, where a Hopf curve H0

appeared close to SN’ joining two Takens–Bogdanov points. In both cases theSO(2) symmetry is not completely broken, but a Zm symmetry remains. Forα0 = π/6 the ellipse H0 becomes tangent to SN’, and the two Takens–Bogdanovpoint coalesce, disappearing for α0 > π/6.

Finally, the stable limit cycles C− and C+, born at the Hopf bifurcations H−and H+, on entering region III, collide simultaneously with the saddles P i+, i=1,2and 3, and disappear along two heteroclinic bifurcation curves Het and Het’ forsmall values of µ. These curves collide with the parabola at the codimension-two bifurcation points SnHet±, and for larger values of µ the limit cycles C−and C+ undergo SNIC bifurcations on the parabola. The two curves Hom andHom’ emerge from the origin as in the previous cases.

In the three quadratic cases, the pinning region is delimited by the sameparabola u = v2 − 1/4. The width of the pinning region is easy to compute,and is given by w = 2v = 2

√u+ 1/4 ∼ 2

√d. By using (11) the dependence

on ε is restored, resulting in w(d, ε) = 2ε√d. The with of the pinning region

increases with the distance to the bifurcation point, and it is proportional tothe amplitude of the imperfection ε.

6. Symmetry breaking SO(2)→ Zm, m ≥ 4

For completeness, and also for their intrinsic interest, we will explore thebreaking of the SO(2) symmetry to Zm, so that the imperfections added to thenormal form (6) preserve the Zm subgrup of SO(2) generated by rotations of2π/m (also called Cm). The lowest order monomial in (z, z) not of the formz|z|2p and equivariant under Zm, is zm−1, resulting in the normal form

z = z(µ+ iν − c|z|2) + εzm−1. (38)

25

(a) m = 4 (b) m > 4

µ

ν

α0

SNIC’

I−

III

I

SNIC

I+

H− H+

L

µ

ν

α0

SNIC’

I−

III

I

SNIC

I+

H− H+

L21

45

3

Figure 18: Bifurcation curves corresponding to the normal forms retaining a Zm symmetry(39); (a) corresponds to the m = 4 case and (b) corresponds to the m > 4 cases.

In terms of the modulus and phase of the complex amplitude z = reiφ, thenormal form becomes

r = r(µ− ar2) + εrm−1 cosmφ,

φ = ν − br2 − εrm−2 sinmφ.(39)

The cases m = 2 and m = 3 have already been examined in §3 and §5.3respectively. When m ≥ 4 the term εzm−1 is smaller than the remaining termsin (38), so the effect of the symmetry breaking is going to be small, comparedwith the previous cases analyzed in this paper. The fixed point solutions of (39),apart from the trivial solution P0 (r = 0), are very close the the zero-frequencyline L in the perfect system. Using the coordinates (u, v) along and orthogonalto L (24), the nontrivial fixed points of (39) satisfy

(r2 − u)2 = ε2r2m−4 − v2. (40)

On L, r2 = u; close to L, the fixed points P± are given by r2 ∼ u ±√ε2um−2 − v2. The pinning region, at dominant order in ε, is v = ±εu(m−2)/2.

This gives a wedge-shaped region around L for m = 4 and a horn for m > 4,as illustrated in figure 18. On the boundaries of the pinning region, the fixedpoints merge in saddle-node bifurcations that take place on the limit cycles C±.These are curves of SNIC bifurcations, exactly the same phenomena that isobserved in Neimark-Sacker bifurcations [4]. Due to the symmetry Zm, from(39) we see that at the boundaries of the wedge, m simultaneous saddle-nodebifurcations take place. By contructing a Poincare section of the limit cyclesC± it is easy to see that these saddle-node bifurcations take place precisely onthe limit cycles, exactly as in the Neimark-Sacker bifurcation. Figure 19 showshow the fixed points appear and disappear in saddle-node bifurcations on thelimit cycle C± when crossing the horn for the m = 5 case, at the five pointsin parameter space illustrated in figure 18(b). The nontrivial fixed points, from

26

(a)

(b)

1 2 3 4 5

Figure 19: Crossing the horn for m = 5. (a) Fixed point solutions of (39) at the five pointsin figure 18(b); grey points in 2 and 4 are the saddle-node points, that split in a stable point(black) and saddle (white). (b) Phase portraits corresponding to the five cases in (a).

(39) and r2 ∼ µ/a ∼ ν/b, satisfy

r2 = µ/a+ ε(µ/a)(m−1)/2 cosmφ,

r2 = ν/b− ε(ν/b)(m−1)/2 sinmφ,(41)

and the intersection of these circles modulated by the sinmφ and cosmφ termsis illustrated in figure 19(a). Phase portraits corresponding to the five points inparameter space are also schematically shown in figure 19(b).

7. Summary and Conclusions

Here we summarize the features that are common to the different perturba-tions analyzed in the previous sections. The most important feature is that thecurve of zero frequency splits into two curves with a region of zero-frequencysolutions appearing in between (the so-called pinning region). Of the infinitenumber of steady solutions that exist along the zero-frequency curve in the per-fect system with SO(2) symmetry, only a small finite number remain. Thesesteady solutions correspond to the pinned solutions observed in experimentsand in numerical simulations, like the ones to be described in §7.1. The numberof remaining steady solutions depends on the details of the symmetry-breakingimperfections, but when SO(2) is completely broken and no discrete symmetriesremain, there are three steady solutions in the pinning region III (see figure 20).One corresponds to the base state, now unstable with eigenvalues (+,+). Theother two are born on the saddle-node curves SN and SN’ delimiting region IIIaway from the origin. Of these two solutions, one is stable (the only observ-able state in region III) and the other is a saddle. There are also the two Hopfbifurcation curves H− and H+. The regions where the Hopf bifurcations meetthe infinite-period bifurcations cannot be described in general, and as has been

27

Hom’ Hom SN

H+

A A’

SN’

H−

I

I− IIII+

IB

B’

II

SN’

I− I+ I

H− HomHom’ SN H+

−−+−

−−

−−

−−

A A’I III

IIIIIIB B’

++

−−+−++

Figure 20: Imperfect Hopf under general perturbations: (a) regions in parameter space; (b)and (c), bifurcation diagrams along the two one-dimensional paths, (1) and (2), respectively.The signs (++, −−. . . ) indicate the sign of the real part of the two eigenvalues of the solutionbranch considered.

28

shown in the examples in the previous sections, will depend on the specifics ofhow the SO(2) symmetry is broken, i.e. on the specifics of the imperfectionspresent in the problem considered. These regions contain complex bifurcationalprocesses. The stable limit cycle existing outside III, in regions I±, collides withthe saddle point and disappears upon entering region III. These are saddle-loophomoclinic collisions Hom and Hom’ that occur very close to the saddle-nodebifurcations SN and SN’. These homoclinic collisions behave like a SNIC bifurca-tions, except in a very narrow region in parameter space around the saddle-nodecurves, as has been discussed in §3.4.

The width of pinnig region scales in all cases considered linearly with thestrenght of the symmetry breaking ε. In all cases we have found w(d, ε) = f(d)ε,where the function f depends on the details of the symmetry breaking terms.For lower order terms the width decreases (ε case) or remains constant (εz case)when increasing the distance to the bifurcation point. For quadratic and higherorder terms, the width increases with the distance.

7.1. Comparison with a pinning case from fluid dynamicsExperiments in small aspect-ratio Taylor–Couette flows have reported the

presence of a band in parameter space where rotating waves become steadynon-axisymmetric solutions (a pinning effect) via infinite-period bifurcations[25]. Previous numerical simulations, assuming SO(2) symemtry of the appa-ratus, were unable to reproduce these observations [21, 1]. Recent additionalexperiments suggest that the pinning effect is not intrinsic to the dynamics ofthe problem, but rather is an extrinsic response induced by the presence ofimperfections that break the SO(2) symmetry of the ideal problem. The imper-fection introduced in the experiments was a tilt of one of the endwalls [1]. Ina very recent paper, Pacheco et al. [22] conducted direct numerical simulationsof the Navier–Stokes equations including the tilt of one endwall by a very smallangle. Those simulations agree very well with the experiments and the normalform theory developed in this paper. A brief summary of those results follows.

Taylor–Couette flow consists of a fluid confined in an annular region withinner radius ri and outer radius ro, capped by endwalls a distance h apart.The endwalls and the outer cylinder are stationary, and the flow is driven bythe rotation of the inner cylinder at constant angular speed Ω. The system isgoverned by three parameters: the Reynolds number Re = Ωrid/ν, the aspectratio Γ = h/d, and the radius ratio η = ri/ro where ν is the kinematic viscosityof the fluid. Both in the experiments and the numerical simulations, the radiusratio was kept fixed at η = 0.5. Re and Γ were varied, and these correspond tothe parameters µ and ν in the normal forms studied.

In the parameter region (Re, Γ ) ∈ (300, 860)×(0.7, 1.6), there exists a steadyaxisymmetric one-vortex state that has a jet of angular momentum emergingfrom the inner cylinder boundary layer near one of the endwalls. This state, onincreasing Re, suffers a Hopf bifurcation that breaks the SO(2) axisymmetryand a rotating wave state emerges with azimuthal wave number m = 2. Forslight variations in aspect ratio, the rotating wave may precess either progradeor retrograde with the inner cylinder. Various different experiments in this

29

(a) (b)

1.00 1.05 1.10 1.15 1.20 1.25Γ

850

900

950

1000

1050

1100

Re

1.00 1.05 1.10 1.15 1.20 1.25Γ

850

900

950

1000

1050

1100

Re

Figure 21: Bifurcation diagrams for the one-cell state from (a) the experimental results ofAbshagen et al. [1] with the natural imperfections of their system, and (b) the numericalresults of Pacheco et al. [22] with a tilt of 0.1 on the upper lid. The dotted curve in both isthe numerically determined Hopf curve with zero tilt.

regime have been conducted in the nominally perfect system, i.e. with the SO(2)symmetry to within the tolerances in building the apparatus, as well as with asmall tilt of an endwall [24, 25, 23, 1]. Figure 21(a) shows a bifurcation diagramfrom the laboratory experiments of Abshagen et al. [1]; these experiments showthat without an imposed tilt, the natural imperfections of the system producea remarkable pinning region, and that the additional tilting of one endwallincreases the pinning region. Tilts of the order of 0.1 are necessary in orderfor the tilt to dominate over the natural imperfections. Figure 21(b) shows abifurcation diagram from the numerical results of Pacheco et al. [22] in the sameproblem, showing a remarkable agreement with the experimental results. Theeffects of imperfections are seen to be only important in the parameter rangewhere the Hopf frequency is close to zero. In this case, a pinning region appears,bounded by infinite period bifurcations of limit cycles. The correspondence ofthese results with the normal form theory described in the present study isexcellent, strongly suggesting that the general remarks on pinning extractedfrom the study of the five particular cases analyzed are indeed realized bothexperimentally and numerically. These to studies are the only cases we knowwhere quantitative data about the pinning region are available. And even in thiscase, the dynamics close to the intersection of the Hopf curve with the pinningregion, that acording to our analysis should include complicated bifurcationalprocesses, has not been explored, neither numerically nor experimentally. Thisis a very interesting problem that deserves further exploration.

30

7.2. Codimension-two bifurcations of limit cyclesThe bifurcations that a limit cycle can undergo have been an active sub-

ject of research since the dynamical systems theory was born. Even in thecase of isolated codimension-one bifurcations, a complete classification was notcomplete until fifteen years ago [30], when the blue-sky catastrophe was found.The seven possible bifurcations are: the Hopf bifurcation, where a limit cy-cle shrinks to a fixed point, and the length of the limit cycle reduces to zero.Three bifurcations where both the length and period of the limit cycle remainfinite: the saddle node of cycles (or cyclic fold), the period doubling and theNeimark-Sacker bifurcations. Two bifurcations where the length remains finitybut the period goes to infinity: the collision of the limit cycle with an externalsaddle forming a homoclinic loop, and the appearance of a saddle node of fixedpoints on the limit cycle (the SNIC bifurcation). And finally the blue sky bi-furcation, where both the length and period go to infinity, corresponding to theappearance of a saddle node of limit cycles transversally to the given limit cycle.The seven bifurcations are described in many books on dynamical systems, e.g.Shil’nikov et al. [26], Kuznetsov [19]; they are also described in the web pagehttp://www.scholarpedia.org/article/Blue-sky catastrophe maintainedby A. Shil’nikov and D. Turaev.

Of the seven bifurcations, only four (Hopf, cyclic fold, homoclinic collisionand SNIC) are possible in planar systems, as is the case in the present study,and we have foound the four of them in the different scenarios explored. Wehave also found a number of codimension two bifurcations of limit cycles. Forthese bifurcations a complete classification is still lacking, and it is interestingto list them because some of the bifurcations obtained are not very common.The codimension-two bifurcations of limit cycles associated to codimension-twobifurcations of fixed points can be found in many dynamical systems texbooks,and include Takens-Bogdanov bifurcations (present in almost all cases consid-ered) and the Bautin bifurcation (in the ε case). Codimension-two bifurcationsof limit cycles associated only to global bifurcations are not so common. Wehave obtained four of them, that we briefly summarize here.

CFH A cyclic fold and a homoclinic (or heteroclinic) collision happening atthe same time. See §3.

PGL A gluing bifurcation with the saddle point undergoing a pitchfork bifur-cation; it may happens in systems with Z2 symmetry; see §3.

SNCF A cyclic fold and a saddle node happening at the same time, with thesaddle node appearing on the limit cycle when it is born at the cyclic foldbifurcation. See §4.

SnHom A SNIC bifurcation and a homoclinic collision happening at the sametime. See §5.

The last case is particularly important in our problem because separate thetwo possible scenarios when entering the pinning region: the stable limit cycleoutside may disappear in a homoclinic collision or a SNIC bifurcation.

31

AcknowledgmentsThis work was supported by the National Science Foundation grants DMS-

05052705, CBET-0608850, and the Spanish Government grant FIS2009-08821.

Appendix A. Symmetry breaking of SO(2) to Z2: computations

Fixed pointsThe fixed points of the normal form (12) are given by r = φ = 0. One

solution is r = 0 (named P0). The other fixed points are the solutions of

cos 2φ = ar2 − µ,sin 2φ = ν − br2,

⇒ (µ− ar2)2 + (ν − br2)2 = 1, (A.1)

resulting in the bi-quadratic equation r4−2(aµ+bν)r2 +µ2 +ν2−1 = 0, whosesolutions are

r2± = aµ+ bν ±∆, e2iφ± = (aν − bµ∓ i∆)eiα0 (A.2)

∆2 = (aµ+ bν)2 + 1− µ2 − ν2 = 1− (aν − bµ)2, (A.3)

and every φ± admits two solutions, differing by π (they are related by thesymmetry Z2, z → −z discussed above). Introducing a new phase α1,

aν − bµ− i∆ = eiα1 , (A.4)

where α1 ∈ [−π, 0] because ∆ > 0, we immediately obtain:

e2iφ± = ei(α0±α1) ⇒ φ± = (α0 ± α1)/2, (A.5)

with the other solution being (α0 ± α1)/2 + π; α1 is a function of (µ, ν) whileα0 is a fixed constant. We have obtained two pairs of Z2 symmetric points,P+ = r+eiφ+ and P ∗+ = −r+eiφ+ , and P− = r−eiφ− and P ∗− = −r−eiφ− .

Hopf bifurcations of fixed pointsThe Jacobian of the right-hand side of (20) is given by

J =(µ+ 1− 3ax2 − ay2 + 2bxy −ν + bx2 + 3by2 − 2axyν − 3bx2 − by2 − 2axy µ− 1− ax2 − 3ay2 − 2bxy

). (A.6)

The invariants of the Jacobian are the trace T , the determinant D and thediscriminant Q = T 2 − 4D. They are given by

T = 2(µ− 2ar2), (A.7)

D = µ2 + ν2 − 1− 4(aµ+ bν)r2 + 3r4 + 2(a(x2 − y2)− 2bxy

), (A.8)

Q = 4(

1− ν2 + 4bνr2 + (1− 4b2)r4 − 2(a(x2 − y2)− 2bxy

)). (A.9)

32

The eigenvalues of the Jacobian matrix (A.6) in terms of the invariants areλ± = 1

2 (T ± √Q). For example, a Hopf bifurcation takes place iff T = 0 andQ < 0. For the fixed points Ps and P ∗s , where s = ±, we obtain

T (Ps) = T (P ∗s ) = 2((b2 − a2)µ− 2abν − 2as∆

), (A.10)

D(Ps) = D(P ∗s ) = 4s∆r2s , (A.11)

Q(Ps) = Q(P ∗s ) = 4((µ− 2ar2s)

2 − 4s∆r2s). (A.12)

As a result, Q(P−) = Q(P ∗−) > 0 and P− and P ∗− never experience a Hopfbifurcation. After some computations, T (P+) = 0 results in the ellipse

µ2 − 4abµν + 4a2ν2 = 4a2, (A.13)

centered at the origin, contained between the straight lines SN and SN’ andpassing through the points (µ, ν) = (0,±1), the ends of the horizontal diameterof the circle µ2 + ν2 = 1. This ellipse is tangent to SN and SN’ at the points(µ, ν) = ±(2b, (b2− a2)/a). The condition Q < 0 is only satisfied on the ellipticarc from (µ, ν) = (0,−1) to (2b, (b2 − a2)/a) with µ > 0; along this arc P+

and P ∗+ undergo a Hopf bifurcation. We have assumed that a and b are bothpositive. The properties of the ellipse are:

major semiaxis (1− `−)µ = 2abν, length 2a/√`−, (A.14)

minor semiaxis (1− `−)ν = −2abµ, length 2a/√`+, (A.15)

where 2`± = 1 + 4a2 ±√1 + 8a2. The eccentricity e is given by

2e2

= 1 +1 + 4a2

√1 + 8a2

. (A.16)

Codimension-two bifurcations of fixed pointsThe Jacobian evaluated at the three points TB+, TB− and TB is:

J(TB+) =(

1 −11 −1

), (A.17)

J(TB−) =(

1 1−1 −1

), (A.18)

J(TB) =(

1 (1 + b)/a(b− 1)/a −1

). (A.19)

The three matrices have double-zero eigenvalues and are of rank one, so the threeof them correspond to Takens–Bogdanov bifurcations. The state that bifurcatesat the TB point is P+, without any symmetry, so that is an ordinary Takens–Bogdanov bifurcation, although the Z2 symmetric state P ∗+ also bifurcates atthe same point in parameter space (but removed in phase space) at anotherordinary Takens–Bogdanov bifurcation. The state that bifurcates at the TB±

33

points is P0. This state is Z2 symmetric, therefore we are dealing with Takens–Bogdanov bifurcations with Z2 symmetry.

The Jacobian evaluated at the two points DP+ and DP− is

J(DP+) =(

1− b −aa −1− b

), J(DP−) =

(b+ 1 a−a b− 1

). (A.20)

The corresponding eigenvalues are λ+ = −2b and λ− = 0 for DP+, and λ+ = 2band λ− = 0 for DP−. Both points are pitchfork bifurcations, and in orderto determine if they are degenerate, their normal form needs to be computedin order to verify that the cubic term is zero. However, since in a degeneratepitchfork bifurcation a curve of saddle-node bifurcations emerges that is tangentto the pitchfork bifurcation curve, from figure 3 it is immediate apparent thatboth DP+ and DP− are degenerate pitchfork bifurcations.

Appendix B. Symmetry breaking of SO(2) with an ε term: compu-tations

Fixed pointsThe fixed points of the normal form (25) are given by r = φ = 0, i.e.

cosφ = r(ar2 − µ),sinφ = r(ν − br2),

⇒ r2[(µ− ar2)2 + (ν − br2)2] = 1, (B.1)

resulting in the cubic equation f(ρ) = ρ3 − 2(aµ+ bν)ρ2 + (µ2 + ν2)ρ− 1 = 0,where ρ = r2. This equation has always a real solution with ρ > 0, and in someregions in parameter space may have three solutions. The curve separatingboth behaviors will be a curve of saddle-node bifurcations, where a couple ofadditional fixed points are born. This saddle-node curve is given by f(ρ) =f ′(ρ) = 0; from these equations we can obtain (µ, ν) as a function of ρ. In orderto describe the curve is better to use a rotated reference frame (u, v) where theu axis coincides with the line L:

u = aµ+ bνv = −bµ+ aν

or

(uv

)=(

sinα0 cosα0

− cosα0 sinα0

)(µν

). (B.2)

The saddle-node curve is given by

(u, v) =1

2ρ22

(1 + 2ρ3

2,±√

4ρ32 − 1

), ρ2 ∈ (2−2/3,+∞), (B.3)

where ρ2 is the double root of the cubic equation f(ρ) = 0. The third rootρ0 is given by ρ0ρ

22 = 1. If there is any point where the three roots coincide

(i.e. a cusp bifurcation point, where two saddle-node curves meet), it mustsatisfy f(ρ) = f ′(ρ) = f ′′(ρ) = 0. There are two such points Cusp±, givenby ρ2 = 1 and (u, v)Cusp± = (3/2,±√3/2), dividing the saddle-node curve intothree branches: SN0, joining Cusp+ and Cusp−, and to unbounded branches SN

34

and SN’, starting at Cusp+ and Cusp− respectively, and becoming asymptoticto the line L. Along SN0, ρ2 < 1 < ρ0, while along SN and SN’, ρ0 < 1 < ρ2.At the cusp points, the three roots coincide and their common value is +1.

A better parametrization of the saddle-node curve is obtained by introducings = ±

√(4ρ3

2 − 1)/3, so that now s ∈ (−∞,+∞), Cusp± corresponds to s = ±1and

(u, v) =(3(1 + s2), 2

√3s)

[2(1 + 3s2)]2/3, ρ0 =

( 41 + 3s2

)2/3

, ρ2 =(1 + 3s2

4

)1/3

. (B.4)

Hopf bifurcations of the fixed pointsUsing Cartesian coordinates z = x+ iy in (25) we obtain(

xy

)=(

10

)+(µ −νν µ

)(xy

)− (x2 + y2)

(ax− bybx+ ay

), (B.5)

where we have set ε = 1. The Jacobian of the right-hand side of (B.5) is givenby

J =(µ− 3ax2 − ay2 + 2bxy −ν + bx2 + 3by2 − 2axyν − 3bx2 − by2 − 2axy µ− ax2 − 3ay2 − 2bxy

). (B.6)

The invariants of the Jacobian are given by

T = 2(µ− 2ar2), D = µ2 + ν2 − 4(aµ+ bν)r2 + 3r4. (B.7)

A Hopf bifurcation takes place iff T = 0 and D > 0. When T = 0, 4a2D =4(aν − bµ)2 − µ2. The fixed points satisfying T = 0 are given by f(ρ) = 0 andµ = 2aρ, resulting in the curve T in parameter space

4aµν(aν − bµ) = 8a3 − µ3, (B.8)

that can be parametrized as

(µ, ν) = a1/3(1− s2)1/3(

2 ,b

a+

s√1− s2

), s ∈ (−1,+1). (B.9)

For s→ ±1, µ = 0 and ν → ±∞, and the curve is asymptotic to the µ = 0 axis,the Hopf curve for ε = 0. Along the T curve, the determinant D is given by

D = a2/3(1− s2)−1/3(

2s2 − 1− 2b

as√

1− s2), (B.10)

resulting in two Hopf bifurcation curves (when D > 0):

H− : s ∈(− 1,−

√(1− b)/2

), H+ : s ∈

(√(1 + b)/2,+1

). (B.11)

The end points of these curves have T = D = 0, and are Takens–Bogdanovbifurcation points TB±. They are precisely on the saddle-node curve (B.4),

35

where both curves are tangent. The coordinates of the four points Cusp± andTB± are:

(µ, ν)Cusp+=

32

(a− b√

3, b+

a√3

), (µ, ν)Cusp− =

32

(a+

b√3, b− a√

3

),

(B.12)

(µ, ν)TB+ =(2a, 2b+ 1)(2(1 + b)

)1/3 , (µ, ν)TB− =(2a, 2b− 1)(2(1− b))1/3 . (B.13)

As TB± are on the saddle-node curve, its s± parameter, according to (B.4), canbe computed. The result is sδ = δ

√(1− δb)/3(1 + δb), with δ = ±′. Therefore

TB+ ∈SN0, closer to Cusp+ than to Cusp−; TB− ∈SN’ if α0 < 60o, TB− ∈SN0

when α0 > 60o and TB− = Cusp− for α0 = 60o.

Appendix C. Symmetry breaking of SO(2) with quadratic terms:computations

We can deal with the tree cases (32), (35) and (37) by considering the normalform

r = r(µ− ar2) + r2 cosmφ,

φ = ν − br2 + r sinmφ,(C.1)

where m = 1 for the εz2 case (§5.2), m = −1 for the εzz case (§5.1) and m = −3for the εz2 case (§5.3). The fixed points are given in the three cases, apart fromthe trivial solution P0 (r = 0), by the biquadratic equation r4 − 2(aµ + bν +1/2)r2 + µ2 + ν2 = 0, with solutions P±

r2± = aµ+ bν + 1/2± (aµ+ bν + 1/4− (aν − bµ)2)1/2

. (C.2)

The phases φ of the P± fixed points can be recovered from

cosmφ = ar − µ/r, sinmφ = br − ν/r. (C.3)

For m = ±1 the solution is unique; for m = 3 the solutions come in triples,differing by 2π/m. It is convenient to use phase space coordinates adapted tothe tilt of the line L (zero frequency curve in the unperturbed SO(2) symmetriccase), by rotating the (µ, ν):

u = aµ+ bνv = aν − bµ

, and the inverse tranform

µ = au− bvν = bu+ av

. (C.4)

In terms of these coordinates, r2± = u + 1/2 ±√u+ 1/4− v2, and the fixedpoints P± exist only in the interior of the parabola u = v2 − 1/4, whose axis isthe line L. On the parabola, these points are born in saddle-node bifurcations.In order to explore additional bifurcations of these points, we compute theJacobian matrix of the normal form (C.1),

J =(µ− 3ar2 + 2r cosmφ −mr2 sinmφ−2br + sinmφ mr cosmφ

), (C.5)

36

whose trace and determinant, for the P± points, are easily computed:

T (P±) = (m− 1)ar2± − (m+ 1)µ,

D(P±) = −2m√u+ 1/4− v2

(√(u+ 1/2)2 − u2 − v2 ± (u+ 1/2)

).

(C.6)

Therefore signD(P±) = ∓ signm, and for m > 0 (m < 0) only P+ (P−) mayundergo a Hopf bifurcation.

The εzz case (§5.1). Here m = −1 and T (P±) = −2ar2 < 0 so there are noHopf bifurcations. Moreover, D(P+) > 0 so it is always stable, and D(P−) < 0so it is a saddle. The only exception is when r = 0, and this only happens atµ = ν = 0, the degenerate high-codimension point at the origin.

The εz2 case (§5.2). Here m = 1, and T = −2µ is zero on the line µ = 0inside the parabola. On this line H0, P+ undergoes a Hopf bifurcation, andthe points of contact with the parabola have D = T = 0 so they are Takens–Bogdanov bifurcations (see figure 16b). The Hopf and Takens–Bogdanov bifur-cations are degenerate, as will be discussed in §Appendix C.1.

The εz2 case (§5.3). Here m = −3 and T = 2µ− 4ar2. P− is a saddle, butP+ undergoes a Hopf bifurcation when T = 0. The condition T = 0 for P+

gives√aµ+ bν + 1/4− (aν − bµ)2 = − 1

2a

((a2 − b2)µ+ 2abµ+ a

)> 0. (C.7)

By squaring and simplifying, we obtain the ellipse (bµ− 2aν)2 + (aµ− 1)2 = 1which is tangent to the line µ = 0 at the origin, with its center at (µ, ν) =(2a, b)/(2a2), and whose elements are:

major semiaxis parallel to (1− `−)µ = 2abν, length 1/√`−, (C.8)

minor semiaxis parallel to (1− `−)ν = −2abµ, length 1/√`+, (C.9)

where 2`± = 1 + 4a2 ± √1 + 8a2. This ellipse has much in common with theone found in the εz case, and the eccentricity e is given by the same expression(A.16). For α0 > π/6, the ellipse is located in the interior of the parabola ofsaddle-nodes, for α0 = π/6 it becomes tangent to the parabola at a single point,and for α0 < π/6 it becomes tangent at the two points

µ =1a

(1− 2a2 − sb

√1− 4a2

),

ν =2a2

√1− 4a2

(b√

1− 4a2 − s(1− 2a2)), s = ±1.

(C.10)

These are the points TBs in figure 17(b). Only the points on the elliptic arc H0

joining these two points satisfy (C.7), and along this arc P+ undergoes a Hopfbifurcation.

Appendix C.1. A degenerate Takens-Bogdanov bifurcationIn the εz2 case, numerical simulations of the normal form (34) show that

the Hopf bifurcation H0 and the Takens–Bogdanov points TB± are degenerate.

37

This can also be found by direct computation. Let us work out the details forthe TB− point.

The coordinates of TB− in parameter space are (µ, ν) = (0,−0.5/(1 + b)),where P± are born in a saddle-node bifurcation, and the fixed points are givenby

r2± =1

2(1 + b), z± =

c+ i2(1 + b)

. (C.11)

In order to obtain the normal form corresponding to the Takens–Bogdanovpoint, a translation of the origin plus a convenient rescaling of all variables ismade:

t = 4τ cos2(α/2), ζ = 2(z− z±) cos(α/2), µ+ iν = 4(µ+ iν) cos2(α/2). (C.12)

and substituting in (34) results in

ζ = −(µ+ iν)ζ + ie−2iαζ + eiα/2ζ2 + 2e−3iα/2|ζ|2 − ie−iαζ|ζ|2. (C.13)

In order to obtain the normal form up to and including quadratic terms, we cantake the critical values (µ, ν) = (0,−1). Introducing the real variables

ζ = (y + 2ix)e−iα, (C.14)

the linear part of the ODE is transformed into Jordan form, and we obtain(xy

)=(y0

)+ cos(α/2)

(2xy

4x2 + 3y2

)+ sin(α/2)

(−2x2 − 3y2/24xy

). (C.15)

Now we can reduce the quadratic terms to normal form by near-identity quadratictransformation. Closely following [32] (exemple 2.2.2 in pages 217-219), allquadratic terms can be eliminated with the exception of the 4x2 term:

x = yy = 4x2 cos(α/2) +O(3)

, (C.16)

and the xy term in the normal form of the Takens–Bogdanov bifurcation ismissing, resulting in a degenerate case. The unfolding of this degenerate casehas been analyzed in detail in Dumortier et al. [11], and is of codimension three.

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