Dynamics in the 1:2 spatial resonance with broken ...tardis.berkeley.edu/reprints/papers/PoKn_PD201_2005.pdfDynamics in the 1:2 spatial resonance with broken reﬂection symmetry

Physica D 201 (2005) 318–344

Dynamics in the 1:2 spatial resonance with brokenreflection symmetry

J. Portera,∗, E. Knoblocha,b

a Department of Applied Mathematics, University of Leeds, Woodhouse Lane, Leeds LS2 9JT, UKb Department of Physics, University of California, Berkeley, CA 94708, USA

Received 9 March 2004; received in revised form 7 September 2004; accepted 11 January 2005

Communicated by C.K.R.T. Jones

Abstract

The dynamics near the 1:2 spatial resonance in systems with broken O(2) symmetry are described. The system is assumed toremain SO(2) invariant, and attention is focused on the behavior that replaces the structurally stable heteroclinic cycles presentin the O(2)-symmetric system. The symmetry-breaking destroys these cycles, and introduces a variety of new bifurcations intothe system, some which are associated with complex behavior. The possible scenarios with increasing symmetry-breaking aredescribed and possible applications of the results are discussed.© 2005 Elsevier B.V. All rights reserved.

PACS:05.45.−a; 47.20.Ky; 02.30.Hq

Keywords:Symmetry-breaking; Mode interaction; Global bifurcations; Heteroclinic cycles

1. Introduction

The 1:2 spatial resonance is responsible for the observed behavior in a variety of systems with O(2) symmetry.Early analyses focused on the interaction of two modes with wavenumbers in the ratio 1:2 in systems with reflectionsymmetry and Neumann boundary conditions[1,2]. These boundary conditions play an important role in thetheory since they are responsible for the existence of steady modes with a well-defined wavenumber. As shownby Armbruster and Dangelmayr[3,4], following earlier work of Fujii et al.[5], such systems can be embedded inmode-interaction problems with O(2) symmetry, the symmetry of rotations and reflections of a circle, which enjoya large number of applications in their own right[6–10].

∗ Corresponding author. Tel.: +44 113 343 5149.E-mail address:[email protected] (J. Porter).

0167-2789/$ – see front matter © 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.physd.2005.01.001

J. Porter, E. Knobloch / Physica D 201 (2005) 318–344 319

The steady-state 1:2 resonance with O(2) symmetry was first studied by Dangelmayr[11]. Subsequent work byArmbruster et al.[12] and by Jones and Proctor[13,6]showed that this analysis was incomplete. In particular, theseauthors identified the presence of attracting structurally stable heteroclinic cycles in open regions of parameterspace, connecting two states with wavenumberk = 2 and related by a 180◦ rotation. Corresponding to eachpoint in this space, there is an open set of initial conditions which circulate among the equilibria connected bythe cycle, spending ever longer time near each of them. Remarkably, this type of behavior is also observed inhigher dimensional systems, and indeed in partial differential equations where it was in fact discovered[14].In these spatially extended systems, the equilibria correspond to spatially periodic steady states, and the 180◦rotation connecting them corresponds to translation by half a wavelength. Since then, dynamics that appear to beassociated with the presence of attracting structurally stable heteroclinic cycles have been observed in experimentswith premixed flames on a circular porous plug burner[15] and in a number of partial differential equations[16–18]. One finds that in general the switching time between successive visits to a steady state saturates at afinite value, resulting either in a long-period periodic orbit or sometimes a chaotic orbit[17]. The origin of thisbehavior in the partial differential equations is not entirely clear but is believed to be due to numerical error, sincesimilar behavior is also found in the normal form for the 1:2 spatial resonance[19]. It should be emphasizedthat there is at present noproof that cycles of this type are actually present in the partial differential equationsconcerned.

These results indicate that structurally stable heteroclinic cycles are in fact quite fragile, and are easily destroyedif, for example, the numerical scheme does not respect the invariance of certain subspaces that is required in orderto prove their existence. In addition, the cycles lose stability and then disappear as one moves away from themode-interaction point. The mechanism by which this occurs has been studied recently, and is now well understood[20].

This paper is devoted to the study of the effects of small symmetry-breaking perturbations on the dynamicsnear the heteroclinic cycles. Chossat[21] has shown that breaking the symmetry O(2) down to SO(2) through theaddition of small terms that break reflection symmetry generically destroys the heteroclinic cycles and replacesthem by a quasiperiodic orbit characterized by two small frequencies, one associated with the broken heteroclinicconnection and one with a slow drift along the group orbit of translations. Ashwin et al.[22] showed that thisperturbation must be dispersive: if reflection symmetry is broken by adding a constant throughflow the cycle willpersist. Others have examined the effects of breaking O(2) down to D4 [23] and O(2) down to D2 [24]. In allcases, the cycle is destroyed, although some structurally unstable heteroclinic orbits may be generated at the sametime.

In the present paper, we return to the problem investigated by Chossat, and consider the generic effects ofbreaking O(2) symmetry down to SO(2). We confirm Chossat’s conclusion, and show that new types of dy-namics are introduced into the system above and beyond the quasiperiodic orbit whose existence he estab-lished. Specifically, we find that secondary Takens–Bogdanov bifurcations, so-called T-points, and degener-ate heteroclinic cycles are responsible for the appearance of new types of global bifurcations, and with themof complex dynamics. We explore the effects of increasing the symmetry-breaking, and determine the typesof transitions the system undergoes as the reflection symmetry is increasingly broken. We envisage severalapplications of this type of analysis, both to problems involving throughflow, and more interestingly to vonKarman swirling flow between exactly counter-rotating parallel disks. This system has O(2) symmetry gen-erated by proper rotations together with an inversion through the origin, and as shown by Nore et al.[18]possesses structurally stable heteroclinic cycles in appropriate parameter regimes. The O(2) symmetry is bro-ken once the two disks (counter)-rotate at slightly different rates. The system then has a preferred directionof rotation and the bifurcation from Couette flow is to precessing states, called rotating waves, instead of thesteady states that are present when the disks counter-rotate exactly, and the structurally stable heteroclinic cy-cle is absent. It is this observation that motivates the theory described below; a comparison with the numer-ical solutions of the three-dimensional partial differential equations governing this system is presented else-where.

320 J. Porter, E. Knobloch / Physica D 201 (2005) 318–344

2. Normal form equations

In this section, we discuss the normal form equations that describe dynamics in the 1:2 mode interaction, firstfor the O(2)-symmetric case, and then with broken reflection symmetry. In either case, we consider an unfolding ofthe mode-interaction point where physical variables may be expanded as:

ψ(x, t) = z1(t)eikx + z2(t)e2ikx + c.c.+ · · · . (1)

Evolution equations for the complex amplitudesz1 andz2 are equivariant with respect to translations:

Tφ : (z1, z2) → (eiφz1,e2iφz2), 0 ≤ φ < 2π, (2)

and, in the O(2)-symmetric case, reflection:

R : (z1, z2) → (z1, z2). (3)

2.1. O(2)-symmetric equations

With O(2) symmetry represented byTφ andR, the cubic order normal form equations describing dynamics nearthe mode-interaction point take the form[11]:

z1 = µ1z1 + αz1z2 + z1(d11|z1|2 + d12|z2|2), (4a)

z2 = µ2z2 + βz21 + z2(d21|z1|2 + d22|z2|2). (4b)

Due to the reflection symmetryR, the coefficients in Eqs.(4), which can be computed via a center manifoldreduction from the governing equations, are real. Furthermore, providedα andβ are nonzero, the amplitudesz1 andz2 can be rescaled to setα = 1, β = ±1. The caseβ = −1 is of much greater interest and is the one that arises inhydrodynamics[25].

Eqs.(4) can be reduced to a three-dimensional system by factoring out the continuous symmetryTφ. This is doneby writingz1 = r1eiφ1,z2 = r2eiφ2, and introducing theTφ-invariant phaseθ = φ2 − 2φ1, to give the “polar” system:

r1 = µ1r1 + αr1r2 cosθ + r1(d11r21 + d12r

22), (5a)

r2 = µ2r2 + βr21 cosθ + r2(d21r21 + d22r

22), (5b)

θ = − 1

r2(βr21 + 2αr22) sinθ. (5c)

The evolution of the individual phases is slaved to that ofr1, r2, andθ:

φ1 = αr2 sinθ, (6a)

φ2 = −βr21

r2sinθ. (6b)

The system represented by Eqs.(4), or equivalently, Eqs.(5), has been studied extensively (see[11,13,6,12,20])and is known to contain a wealth of interesting dynamics. The nontrivial steady states of the system(5) come inthree varieties.Pure modes(P) withr1 = 0 bifurcate from the trivial state (O) whenµ2 = 0. The eigenvalues of P in system(4)

are 0, corresponding to spatial translations,−2µ2, corresponding to amplitude perturbations, and

σ± = µ1 − µ2d12

d22±√

− µ2

d22, (7)

describing the growth of perturbations away fromz1 = 0.


Table 1The group orbits of the five simple solutions of Eqs.(4)

O(2)-symmetric case SO(2)-symmetric case

Solution (z1, z2) Fixed by Solution

P Tφ(0, r2) Tπ, TφRT−φ PMM Tφ(r1, r2) TφRT−φ TWTW Tφ0+ωt(r1, r2eiθ) TWSW Tφ(r1(t), r2(t)) TφRT−φ MTWMTW Tφ0+φ(t)(r1(t), r2(t)eiθ(t)) MTW

The states P, MM and SW come in one-parameter families, generated byTφ. The TW break reflection symmetry and drift at a steady rateω

along the group orbit, while the MTW have two frequencies, one associated with an oscillation in (r1(t), r2(t), θ(t)) and the other with the driftφ(t). The final column indicates the fate of these solutions in the SO(2)-symmetric case: MM become TW, and SW become MTW.

Mixed modes(MM) emerge along the lineµ1 = 0 and connect to the branch of pure modes whenσ+ = 0 orσ− = 0. The MM satisfy

cosθ = ±1, r21 = −µ1 ± αr2 + d12r22

d11,

βµ1 ± (αβ + d21µ1 − d11µ2)r2 + (αd21 + βd12)r22 ± (d12d21 − d11d22)r32 = 0.(8)

Traveling waves(TW) are created whenαβ < 0 inR symmetry-breaking bifurcations from MM, and are char-acterized by steady drift along the group orbit generated byTφ. In the reduced system(5), they appear as steadystates satisfying

r22 = β(2µ1 + µ2)

d, r21 = −2α

βr22, cosθ = µ2(2αd11 − βd12) + µ1(βd22 − 2αd21)

αd√β(2µ1 + µ2)/d

, (9)

whered ≡ 2α(2d11 + d21) − β(2d12 + d22). The corresponding expression for cosθ in [12,20]has an incorrect signwhend < 0.

Hopf bifurcations on MM produce standing waves (SW), while Hopf bifurcations on TW lead to modulatedtraveling waves (MTW). The symmetry properties of the five simple states, P, MM, TW, SW, and MTW, aresummarized inTable 1. The final column of the table indicates what happens when theR symmetry is broken: MMand SW become slowly drifting TW and MTW, respectively.

Remarkably, Eqs.(4) also contain several types of heteroclinic cycles. Whenαβ < 0, there are open sets ofparameters where the system(4) exhibits structurally stable connections betweenπ-translates on the circle of puremodes[12]. These solutions, which may be attracting, are a robust feature of the 1:2 interaction, and have beenobserved in a variety of systems[14,15,17,18]. In addition to the structurally stable cycles connecting pure modes,more complicated cycles involving P, MM, SW, and O exist slightly further from the origin in the (µ1, µ2) plane[20]. The dynamics associated with these cycles can be chaotic (of Shil’nikov type) and are organized by a sequenceof transitions among five distinct heteroclinic cycles, one of which is structurally stable (see[20] for details).

2.2. SO(2)-symmetric equations

When the reflection symmetryR is broken, the coefficients in the normal form equations are no longer forced tobe real and hence can be expected to acquire (small) imaginary parts. In the following, we measure the magnitudeof the symmetry-breaking by a (small) parameterε, and replace Eqs.(4) by:

z1 = (µ1 + iεω1)z1 + αz1z2 + z1(d11|z1|2 + d12|z2|2), (10a)

z2 = (µ2 + iεω2)z2 + βz21 + z2(d21|z1|2 + d22|z2|2), (10b)


where

(α, β, d11, d12, d21, d22) = (αr, βr, d11, d12, d21, d22) + iε(αi, βi, c11, c12, c21, c22). (11)

These equations are the subject of the remainder of this paper.To study these equations, we may once again factor out the continuous symmetryTφ by working with the real

amplitudesr1 andr2 and the relative phaseθ = φ2 − 2φ1 used to derive Eqs.(5). We letα ≡ |α|eiϕ1 andβ ≡ |β|eiϕ2

and rescale (r1, r2) using r1 → r1|αβ|−1/2, r2 → r2/|α|. After absorbing factors of 1/|αβ| and 1/|α|2 into thedefinitions ofd11, c11, d21, c21, andd12, c12, d22, c22, respectively, we obtain a three-dimensional polar systemanalogous to Eqs.(5):

r1 = µ1r1 + r1r2 cos(θ + ϕ1) + r1(d11r21 + d12r

22), (12a)

r2 = µ2r2 + r21 cos(θ − ϕ2) + r2(d21r21 + d22r

22), (12b)

θ = ν − r21

r2sin(θ − ϕ2) − 2r2 sin(θ + ϕ1) + c1r21 + c2r22, (12c)

whereν = ε(ω2 − 2ω1), c1 = ε(c21 − 2c11), c2 = ε(c22 − 2c12), and the individual phases (cf. Eqs.(6)) evolveaccording to:

φ1 = εω1 + r2 sin(θ + ϕ1) + εc11r21 + εc12r

22, (13a)

φ2 = εω2 − r21

r2sin(θ − ϕ2) + εc21r

21 + εc22r

22. (13b)

The main difference between Eqs.(12) and(5) is in the behavior ofθ: in Eq. (12c) the evolution ofθ is biased,without theθ → −θ symmetry of Eq.(5c).

The polar system(12) works well for most purposes, but fails whenr2 = 0. This is a natural consequence ofamplitude-phase coordinates, and is related to the discontinuity of the phase variableφ2 at the origin. Since most ofthe heteroclinic cycles we will consider, as well as other important solutions, cross the planer2 = 0, we use belowan alternative set of coordinates that avoids this (artificial) singularity. This issue does not arise atr1 = 0, since thisplane is invariant and trajectories cannot cross it. We define (x, y) = r2(cos(θ + ϕ1), sin(θ + ϕ1)) (cf. [26,27]) andobtain the equivalent system[28]:

r1 = µ1r1 + r1x+ r1(d11r21 + d12(x

2 + y2)), (14a)

x = µ2x− νy + 2y2 + r21 cosϕ + x(d21r21 + d22(x

2 + y2)) − y(c1r21 + c2(x2 + y2)), (14b)

y = µ2y + νx− 2xy + r21 sinϕ + y(d21r21 + d22(x

2 + y2)) + x(c1r21 + c2(x2 + y2)), (14c)

whereϕ = ϕ1 + ϕ2. Note that the phase differenceϕ1 − ϕ2 does not appear in these equations because it can beeliminated by a rotation, for example,z2 → eiϕ/2z2. The caseϕ = 0, which corresponds toαβ > 0 in Eqs.(4), is oflittle interest since it contains no TW, SW, MTW, or heteroclinic cycles whenε = 0 [11,12]. Consequently, in thefollowing we assume thatϕ = π +O(ε), i.e., that by breaking the reflection symmetryRwe perturb the interestingcaseαβ < 0 of Eqs.(4).

Within Eqs. (14), O(2) symmetry (more precisely, the reflection symmetryR : y → −y, sinceTφ has beenfactored out) is restored whenν = c1 = c2 = sinϕ = 0, i.e., it is not necessary to setε = 0. In this special case,the breaking of the reflection symmetry corresponds to looking at a steady-state mode interaction from a movingreference frame, a fact that is responsible for a number of special properties of the resulting system (cf.[21,22]).Thus, while withε = 0 a structurally stable P→ P heteroclinic cycle in Eqs.(14) represents, in Eqs.(4), an entirecircle of heteroclinic cycles connecting diametrically opposite pairs on the circle of steady P states, the same cycle


with ν = c1 = c2 = sinϕ = 0 but ε = 0 represents, in Eqs.(10), a homoclinicconnection from a rotating wave(i.e., a drifting P state) back to itself. Indeed, the transformation

z1 → e−if (z1,z2,t)z1, z2 → e−2if (z1,z2,t)z2, (15)

wheref is theTφ-invariant real-valued function:

f = ωt + a|z1|2 + b|z2|2 + · · · , (16)

changes Eqs.(4) to Eqs.(10), but withαi = βi = 0,ω2 = 2ω1 = 2ω, c21 = 2c11 = 4aµ1, andc22 = 2c12 = 4bµ2.In the following we exclude this special case.

2.3. General effects of symmetry-breaking

We now summarize the qualitative effects of breaking the reflection symmetryR. Here, and in the remainder ofthis paper, we use the notation Ws(X) and Wu(X) to refer to the stable and unstable manifolds, respectively, of afixed point or limit cycle X.

2.3.1. Generic solutions driftThis is the most obvious consequence of breaking reflection symmetry[29,30]. When 0< ε� 1, steady solutions

and standing waves drift slowly, forming traveling waves and modulated traveling waves, respectively. Thesesolutions result from the destruction of the invariant plane sinθ = 0 in Eqs.(5) and are characterized by an O(ε)frequency (see Eqs.(13)). In contrast, the left- and right-traveling waves that are present whenε = 0, αβ < 1typically have an O(1) frequency. When 0< ε� 1, the frequency of these waves remains O(1) but the waves are nolonger related by reflection symmetry. The loss of reflection symmetry thus splits the TW branch into two branchesof traveling waves with slightly different frequencies and amplitudes. Of course, the effect of nonzeroε is felt moststrongly when the TW have low frequency, i.e., near theε = 0 parity-breaking bifurcation from the circle of steadymixed states that produces them. Indeed, the loss of reflection symmetry “unfolds” this bifurcation into an imperfectpitchfork bifurcation, as described in[31,32]and depicted inFig. 1. One branch of TW is monotonic and displaysno bifurcations while the other, corresponding to waves traveling in the opposite direction, undergoes a saddle–nodebifurcation.

2.3.2. Primary bifurcations are Hopf bifurcationsWhen 0< ε� 1, the primary bifurcations atµ1 = 0 andµ2 = 0 become Hopf bifurcations[29,30] and lead

to branches of traveling waves. In the following we refer to these as TW and P, respectively, i.e., the TW is adrifting mixed mode (z1, z2) while P now represents the drifting pure modes (0, z2). In the reduced system(14) Palso emerges (atµ2 = 0) as a limit cycle. Ifνc2 < 1 this limit cycle is destroyed, asr2 increases, in a saddle–node

Fig. 1. Sketch illustrating the effect of brokenR symmetry on the transition from MM to TW. Stable (unstable) solutions are shown as solid(dashed) lines. (a) Withε = 0, TW are produced in anR symmetry-breaking bifurcation atµ = µ∗. (b) Whenε = 0, this symmetry-breakingbifurcation is replaced by a saddle–node bifurcation (see[31]) atµ = µ∗ +O(ε2/3) and there are two distinct branches of TW, correspondingto left- and right-traveling waves.


homoclinic bifurcation (also referred to as a saddle–node infinite period or “SNIPER” bifurcation) that takes placewhen 2r2 = |ν + c2r22|. A pair of fixed points

P± : (r1, x, y) =0,±

√r22 − (ν + c2r22)2

4,ν + c2r22

2

, r22 = − µ2

d22, (17)

replaces the limit cycle over the range

2 − νc2 − 2√

1 − νc2 < c22r22 < 2 − νc2 + 2√

1 − νc2, (18)

providedc2 = 0. If c2 = 0 these fixed points exist for allr22 > ν2/4. In this regime, mixed states (slowly drifting

TW) are produced byTπ symmetry-breaking bifurcations when one of the eigenvalues (cf. Eq.(7))

σ± = µ1 − µ2d12

d22±√

− µ2

d22− (ν − µ2c2/d22)2

4(19)

crosses zero. The conditionsσ± = 0 define steady-state bifurcations and thus do not introduce a second frequency.Outside of this regime, secondary bifurcations from P to mixed states are torus bifurcations that do generate a secondfrequency, and produce MTW.

2.3.3. The relative phaseθ can unlockIn the O(2)-symmetric case the planesθ = 0 andθ = π are invariant (see Eq.(5c)), a consequence of the

reflection symmetryR. As a result, the relative phaseθ is bounded and can neither rotate nor change sign. WithSO(2) symmetry these constraints are lifted. The two-frequency MTW solutions may be “locked” on average sothatθ oscillates about a constant value, as they are in the O(2)-symmetric case, or they may be “unlocked” in whichcaseθ evolves in an unbounded fashion. A locked MTW is illustrated inFig. 2, whereθ(t), φ1(t), andφ2(t) areshown over the same interval of time. For this solution,θ(t) oscillates periodically about a constant value and, onaverage,φ2 = 2φ1. Thus,z1 andz2 drift together. However, on the same MTW branch at a slightly smaller valueof µ2, we find an “unlocked” MTW. For this solution,z1 andz2 no longer move together, even in an average sense(seeFig. 3). We wish to be clear that this transition isnota bifurcation. In the reduced system(14), where the MTWare represented by a limit cycle, nothing special happens betweenµ2 = 0.6 (Fig. 2) andµ2 = 0.5 (Fig. 3). It issimply that on one side of this transition the projection of the MTW onto thez2 plane repeatedly encircles the origin(seeFig. 4b), while on the other side (Fig. 4a) it approaches the origin but fails to enclose it. In both cases, thez2mode spends significant periods of time drifting very slowly (withφ2 nearly constant), while its amplituder2 firstgrows and then decreases. Asr2 passes through its minimum value,z2 undergoes a period of rapid translation, withφ2 > 0 in the locked case (Fig. 2) andφ2 < 0 in the unlocked case (Fig. 3).

Fig. 2. A locked MTW solution of Eqs.(12) and(13) with µ1 = −0.3, µ2 = 0.6, d11 = −0.4, d12 = 1.6, d21 = −6, d22 = −0.5, ε = 0.1,ω1 = 0.3, ω2 = −0.4, ϕ1 = 0, ϕ2 = π, andc11 = c12 = c21 = c22 = 0. Observe thatθ(t) oscillates between 0.092 and 3.04 and, on average,φ2 = 2φ1.


Fig. 3. An unlocked MTW solution of Eqs.(12) and(13). All parameters are as inFig. 2 except thatµ2 = 0.5. Observe thatθ(t) < 0 for allt < t0 and thatz1 andz2 now rotate in opposite directions.

The transition from “locked” behavior to “unlocked” behavior discussed above contrasts with the scenariodescribed in[33], also for a system with SO(2) symmetry, where a similar transition takes place via a global(heteroclinic) bifurcation. In our case, this transition is associated with the singularity in Eq.(12c)whenr2 = 0 butis not a genuine bifurcation; in contrast, in[33] the relevant phase difference does not encounter such a singularity.

2.3.4. Waves can reverse their direction of propagationAlong with unlocked MTW, the breaking of reflection symmetryR allows for more complicated types of

traveling solutions in which one (or both) of the constituent modes repeatedly reverses its direction of rotation. Thisphenomenon is related to the so-called direction-reversing traveling waves[34,35]although here the symmetry isbroken and the wave does exhibit net drift (cf.[36]). In Fig. 5, we show a chaotic direction-reversing wave of thistype, with thez2 mode rotating in stick–slip fashion, but always in the same sense (cf.Fig. 3), while thez1 modeundergoes similar periods of rapid motion, but switches its direction of rotation chaotically. This type of chaoticwave bears a striking resemblance to solutions found in[33] for a two-mode Galerkin truncation of the complexGinzburg–Landau equation, and was found in[28] as well.

2.3.5. Structurally stable connections are destroyed, but new possibilities emergeIn the O(2)-symmetric case there are structurally stable connections, in an open region of parameter space,

between pure modes and theirπ-translates. These connections occur in reflection-invariant subspaces which, in theε = 0 version of Eqs.(14), are identified with they = 0 plane. In this plane, there are two pure mode solutionsP± : (r1, x, y) = (0,±√−µ2/d22,0) (cf. Eq.(17)). Each of these represents the entire circle of P states in the fullsystem(4), but with differentorientationswith respect toz1 perturbations (recall thatφ1 is undefined whenr1 = 0).Of these, P− has the least unstable eigenvector in thez1 plane while P+ has the most unstable eigenvector; P± maybe transformed into P∓ simply by “re-orienting” the perturbationz1 (i.e., changingφ1 by π/2), or by rotating thephase ofz2 byπ (with fixedz1 orientation), or by some combination of these actions. The structurally stable P→ P

Fig. 4. (a) The locked MTW solution shown inFig. 2 near the origin, and (b) the corresponding trajectory of the unlocked MTW solution inFig. 3, both projected onto the complexz2 plane. In both cases the loops precess in the direction indicated by the arrows in the first quadrant,but in (b) the loop encircles the origin (the unlocked case) while in (a) (the locked case) it does not.


Fig. 5. Solution of Eq.(14) where thez1 mode reverses its direction of propagation in a chaotic fashion, according to whether the (nearly)semi-circular part of the trajectory traverses they > 0 or they < 0 portion of the attractor. The parameters are as inFig. 2but withµ1 = −0.33andµ2 = 0.27.

cycles involve both of these phase changes. In these coordinates, the heteroclinic connection from a given P stateto itsπ-translate occurs (whenε = 0) in they = 0 plane of Eqs.(14); this connection is labeledΓ1 in Fig. 6a andjoins P+ to P−. The cycle is completed by “re-orienting” the trajectory at the pure mode from the stable manifoldalong which it entered to the unstable manifold so that it can leave again. This rotation can be either up (y > 0) ordown (y < 0), and occurs along (half of) the circlex2 + y2 = −µ2/d22 in Eqs.(14); these two choices are labeledΓ±

2 in Fig. 6a. The rotation connects P− to P+, so the whole process can repeat.When the reflection symmetryR is broken, heteroclinic connections that rely on the invariance of they = 0

plane are in general destroyed. This does not mean that they never occur, just that they are no longer robust. Theconnection P+ → P−, for example, is now codimension-one (seeFig. 6b) since it requires the intersection of theone-dimensional manifold Wu(P+) with the two-dimensional manifold Ws(P−) in three-dimensional space (ratherthan the two-dimensional planey = 0). Similarly, structurally stable connections in they = 0 plane from P+ or P− toan MM state, which play a part in the more complicated heteroclinic cycles of[20], are reduced to codimension-onephenomena whenε = 0.

Although the structurally heteroclinic connections in they = 0 plane are in general destroyed by the loss ofreflection symmetry, new types of structurally unstable homoclinic and heteroclinic connections are now permitted.Besides the simple codimension-one P− → P+ connection depicted inFig. 6b, more complicated “N-heteroclinics”become possible. Such solutions, like the 2-heteroclinic shown inFig. 6c, occur when the unstable manifold of P+passes through the neighborhood of P− several times before connecting up. Because Wu(P+) and Ws(P−) are

Fig. 6. (a) Schematic representation in the (x, y) plane of a structurally stable P→ P heteroclinic cycle withε = 0. The primary connectionΓ1

is in they = 0 plane.Γ±2 represent a re-orientation. (b) A heteroclinic cycle can still occur withε = 0 but it is no longer structurally stable.

(c) Multi-pulse heteroclinic cycles, which are impossible whenε = 0, can now occur as well. These “N-heteroclinics” make N visits to theneighborhood of P− before connecting up.


confined to they = 0 plane whenε = 0, N-heteroclinics only become possible forε = 0. In a similar fashion,symmetry-breaking allows for certain types of homoclinic connections that cannot occur withε = 0. For example,the MM involved in the heteroclinic cycles of[20] has a two-dimensional stable manifold Ws(MM) in the y = 0plane while Wu(MM), which is one-dimensional, is transverse to this plane. Without the involvement of anotherlimit set with y = 0 there is no way for Wu(MM) to re-enter the plane and connect back to MM (cf.[37]). Thus,heteroclinic connections to MM are possible whenε = 0 but nothomoclinicconnections. Whenε = 0 we expectto see MM homoclinics as well.

3. Detailed numerical examples

In this section, we illustrate the effects of breaking the reflection symmetryR with several detailed numericalexamples. These examples employ Eqs.(14) with ϕ = π, d11 = −0.4, d12 = 1.6, d21 = −6.0 andd22 = −0.5,allowing us to investigate the effect of symmetry-breaking not only on the P mode heteroclinic cycles but alsothe more complicated cycles present further away from (µ1, µ2) = (0,0) [20]. For simplicity, we setc1 = c2 = 0;small changes in these parameters and inϕ = π have no qualitative effect on the bifurcation sets described below.In all cases, we study the three-dimensional system(14) and so refer to torus bifurcations in the original system(10) as Hopf bifurcations. Thus, for example, the bifurcation from a drifting mixed mode (i.e., a TW) to MTW isreferred to as a Hopf bifurcation. This terminology should lead to no confusion since no locking occurs due to theunbroken translation invariance of the system. All our calculations are performed with the numerical continuationsoftware XPPAUT[38] and are presented in the (µ1, µ2) plane. We treatν ≡ −ε as the (sole) symmetry-breakingparameter and explore the effects of increasing this parameter, focusing on qualitative changes in the bifurcationsets as the symmetry O(2) is increasingly broken. This approach is particularly helpful for applications in whichone does not know a priori whether the symmetry-breaking effects present are small or large.

3.1. O(2)-symmetric case:ε = 0

Here, for the sake of comparison, we reproduce results from Case I of[20]. Fig. 7shows the important bifurcationsets in the (µ1, µ2) plane. Each bifurcation set is labeled by the corresponding bifurcation, symmetry-breaking (SB),heteroclinic (Het), or Hopf, and by the solution produced. When the SB bifurcation is anR symmetry-breakingbifurcation from a group orbit of states (MM or SW), we refer to it as aparity-breakingbifurcation. Note that wedistinguish between two branches of mixed modes according to the sign ofx; the MM+ branch bifurcates from P+and the MM− branch from P−. The two most interesting regions in the (µ1, µ2) plane are highlighted. Region 1contains the structurally stable P+ → P− heteroclinic cycles. Region 2 marks, in an approximate sense, the domainof the various heteroclinic cycles (see[20]) involving P±, MM−, SW, and O. The two regions are separated by theSB bifurcation that produces MM− from P−.

We now describe the bifurcations in the O(2)-symmetric case, beginning in the third quadrant of the (µ1, µ2)plane where only the trivial state O is stable, and proceeding aroundFig. 7in clockwise fashion. These are illustrated

in Fig. 8for a radius|µ| ≡√µ2

1 + µ22 = 0.5. The primary bifurcation to P± occurs whenµ2 = 0 and is supercritical

becaused22 < 0. Pure modes exist for allµ2 > 0 and are stable until a Tπ symmetry-breaking bifurcation that givesrise to a stable branch of MM+. This bifurcation occurs whenσ+ = 0 (see Eq.(7)). In the variables (r1, x, y), thisbifurcation manifests itself as a bifurcation of P+. P− is always unstable to P+ so when the eigenvalueσ− (see Eq.(7)) becomes positive at a secondTπ symmetry-breaking bifurcation this generates an unstable MM− branch.

In region 1, the SB bifurcation to MM− is preceded by the formation of a heteroclinic connection P+ → O inthey = 0 plane—note that since there is a two-dimensional surface,r1 = 0, x2 + y2 ≤ −µ2/d22, of structurallystable connections from O to P+, this bifurcation produces a continuum of heteroclinic cycles (seeFig. 9). Thebreaking of this codimension-one connection produces, in region 1, a structurally a stable P+ → P− cycle, as well


Fig. 7. Bifurcations sets in the (µ1, µ2) plane for the O(2)-symmetric case withϕ = π, d11 = −0.4,d12 = 1.6,d21 = −6.0,d22 = −0.5 (Case Iof [20]) over the range−1 ≤ µ1 ≤ 0.3,−0.2 ≤ µ2 ≤ 1. Bifurcations are labeled according to type: symmetry-breaking (SB), heteroclinic (Het),or Hopf, and also by the solution generated. Structurally stable P+ → P− heteroclinic cycles are found in region 1, while the more complicated

cycles involving P±, MM−, SW and O are present in region 2. The dashed circular path at|µ| ≡√µ2

1 + µ22 = 0.5 is used to generate the

bifurcation diagram ofFig. 8.

as an SW, both of which lie in they = 0 plane. On the boundary of region 2, where the P+ → O heteroclinicconnection formsafter the bifurcation to MM−, the P+ → P− heteroclinic cycle does not form (because P− is nowunstablewithin they = 0 plane), and only the SW is produced. In addition to the SW, we may also associate theappearance of astableMTW with this curve of heteroclinic bifurcations (at least in the region shown inFig. 7),although in some regimes this MTW emerges in a separate, albeit nearby, bifurcation. Indeed, for|µ| � 1.1, thisbifurcation is a parity-breaking bifurcation of the newly created SW (see[20]). The resulting MTW are unstablebut acquire stability almost immediately in a torus bifurcation. On the scale ofFig. 7, the location of these twobifurcations cannot be distinguished from the curve of heteroclinic connections P+ → O. In contrast, in the lowerpart of region 1, stable MTW arise directly through the loss of stability of the P+ → P− heteroclinic cycles. This

Fig. 8. Bifurcations diagram corresponding to the dashed circular path shown inFig. 7. Stable (unstable) solutions are denoted by thick (thin)lines. The vertical hash-marks on the P± branch indicate the presence of (unstable) P+ → P− heteroclinic cycles.


Fig. 9. (a) Sketch of the P+ → O heteroclinic connection. (b) A structurally stable P+ → P− cycle forms when this connection breaks and oneenters region 1 (seeFig. 7). An unstable SW forms at the same time.

parity-breaking bifurcation takes place along the lineµ2 = µ1d22/d12, defined byσ+ = −σ− [6]; the structurallystable heteroclinic cycles are attracting below this line.

Further along our clockwise path around the origin in the (µ1, µ2) plane, we find that the SW oscillationsdisappear in a Hopf bifurcation on the MM+ branch at which the MM+ lose stability. This bifurcation is subcriticalfor µ1 � − 0.2 and supercritical for−0.2 � µ1 ≤ 0. The unstable MM+ branch then undergoes a parity-breakingbifurcation to (unstable) TW before terminating on O atµ1 = 0,µ2 > 0. The TW are stabilized at a Hopf bifurcationthat destroys the (stable) MTW created earlier, and remain stable until their termination in a second parity-breakingbifurcation at which the TW transfer stability to the MM− branch. These states remain stable until they disappearin a primary bifurcation atµ1 = 0,µ2 < 0.

There are two important global bifurcations in region 2 that are not included inFig. 7 due to the difficulty offollowing them numerically. These organize the interesting heteroclinic dynamics treated in detail in[20], whichwe now briefly summarize. The phase-space portrait appropriate to region 2 is sketched in Fig. 9 of[20]. In thisregion, the eigenvaluesσ± (see Eq.(7)) of the P± states are both positive, indicating the presence oftwomixedstates MM±. Furthermore, the MM+ state is surrounded by a hyperbolic limit cycle that is unstable within they = 0 plane but stable transverse to it. The first of the two global bifurcations occurs when the two two-dimensionalmanifolds Wu(P−) and Ws(SW) intersect tangentially, resulting instructurally stableheteroclinic cycles of theform O → P− → SW → O, that persist, as one moves counter-clockwise through region 2, until the second globalbifurcation at which the one-dimensional manifold Wu(MM−) falls within Ws(SW). In fact, the latter bifurcationcreates four distinct heteroclinic cycles: O→ P− → P+ → MM− → SW → O, O→ P+ → MM− → SW → O,O → P− → MM− → SW → O, and SW→ MM− → SW, each of which plays a role in the behavior of the systemat nearby parameter values. The region between the two global bifurcations is filled with an infinite cascade of isolasrepresenting finite-period periodic solutions of the three-dimensional system (see Figs. 3 and 8 of[20]) with anever-increasing number of rotations around the SW limit cycle. The isolas possess characteristic regions reflectingeach of the above heteroclinic cycles and these accumulate on the associated cycle as the oscillation period becomesinfinite. Several regions contain (partial) cascades of saddle–node and period-doubling bifurcations that are in turnassociated with chaotic (Shil’nikov-type) dynamics.

3.2. Broken symmetry:ε = 0.01

We now show howFig. 7is modified when the reflection symmetryR is broken. Although the magnitude of thesymmetry-breaking in this section is small,ε = 0.01 (ν = −0.01), it has a marked effect on both global and localbifurcations, as illustrated inFig. 10. Many of the effects described below were anticipated in Section2.3. Notableis the splitting of several of the bifurcation sets, and the introduction of new codimension-two points such as theTakens–Bogdanov (TB) point, the interaction between a heteroclinic cycle connecting the pure modes P± and a


Fig. 10. Bifurcations sets withν = −0.01,ϕ = π, c1 = c2 = 0. The remaining parameters, and the range ofµ1 andµ2, are as inFig. 7. The TWhomoclinic (Hom) bifurcation and the new codimension-two bifurcations: Takens–Bogdanov (TB), heteroclinic-SB (Het-SB), and the T-point,are discussed in the text.

Fig. 11. Close-up of the bifurcations sets inFig. 10to the right of the origin taking 0< µ1 < 0.18,−0.06< µ2 < 0.15, and in the further zoomof the boxed region 0< µ1 < 0.01,−0.006< µ2 < 0.006. Note the cusp of SN bifurcations branching off the curve of SB bifurcations on P−,and the SN-Hopf point visible in the left figure. Forµ1 � 0.004, there are actually five curves that on this scale are impossible to distinguishfrom theµ1 axis: the Hopf bifurcation (µ2 = 0) producing the drifting pure mode state, the SN-homoclinic (µ2 = −ε2d22/4) introducing P±,the Hopf bifurcation on TW, the homoclinic (Hom) bifurcation on TW, and the SB bifurcation on P−, where this TW terminates.


Fig. 12. Bifurcation diagram (cf.Fig. 8) corresponding to the dashed circular path shown inFig. 10. Stable (unstable) solutions are denoted bythick (thin) lines. The region of structurally stable P+ → P− cycles is replaced by a structurally unstable heteroclinic cycle (labeled Het). Twofamilies of MTW bifurcate in opposite directions from Het. One of these terminates on a TW branch at a Hopf bifurcation near the polar angle1.47, while the other (hardly visible) disappears in a homoclinic bifurcation (labeled Hom) just after the bifurcation from P− to TW.

wavelength-doubling bifurcation of the P− (Het-SB), and the appearance of a T-point, all of which organize thebehavior of the system in their vicinity. Additional codimension-two points are located close to the origin, and arevisible only in the nested enlargements shown inFig. 11. In particular,Fig. 11shows that the rightmost curve of Hopfbifurcations (near theµ2 axis) is in fact detached from the origin and now collides with a curve of saddle–node (SN)bifurcations of the drifting mixed modes TW. In contrast to the case we consider in Section3.4, this saddle–nodeHopf (SN-Hopf) interaction appears to be one of the “uninteresting” cases (i.e., without a heteroclinic connectionin the normal form; see[39]). The curve of Hopf bifurcations that emerges to the right of this SN-Hopf pointsoon turns around and heads toward the intersectionQ : (µ1, µ2) = −ε2(d12, d22)/4 of a saddle–node homoclinicbifurcation producing P± and a wavelength-doubling bifurcation. However, even on the scale of the largest zoomthese bifurcation sets, as well as those corresponding to secondary Hopf bifurcations and TW homoclinics, areimpossible to separate from the axisµ2 = 0. At Q, the single pure mode (r1, x, y) = (0,0,−ε/2) has two zeroeigenvalues (geometric and algebraic multiplicity two). This degeneracy is responsible for much of the observedbehavior near the origin; various unfoldings of this codimension-two degeneracy are examined in[40]. Fig. 11alsoshows that the set of TW homoclinic bifurcations that emerges from the “lower” TB point at a larger value ofµ1(andµ2 < 0) also heads toward this codimension-two point. Finally, the figure reveals the presence of a cusp of SNbifurcations involving the drifting mixed modes TW near the lower TB bifurcation. Evidently, the loss of symmetryhas a profound effect on the various higher codimension bifurcations, whether these occur locally near the originor further away from it. In the following, we describe some of the more robust aspects of this “unfolding” thatcharacterize its properties farther from the origin, with particular reference to the transitions occurring along theclockwise circular path|µ| = 0.5 indicated inFig. 10. The corresponding bifurcation diagram is shown inFig. 12for comparison withFig. 8.

3.2.1. Effect on local bifurcationsThe primary bifurcation atµ2 = 0 to P, as discussed in Section2.3, is now a Hopf bifurcation. The corresponding

limit cycle in Eqs.(14) persists untilµ2 = −ε2d22/4, where it is replaced by the pair of fixed points P± of Eqs.(17), which represent drifting waves that can entrain (phase-lock)z1 perturbations. The saddle–node homoclinicbifurcation that marks the birth of P± is indistinguishable fromµ2 = 0 in Fig. 10and is not labeled. The P± stateschange stability atσ± = 0, producing slowly drifting TW that resemble the MM± of the O(2)-symmetric case, i.e.,waves with twice the wavelength of the drifting P± states. In addition, in the O(2)-symmetric case, the MM stateswent through parity-breaking bifurcations to TW. Whenε = 0, these bifurcations are unfolded as already described(seeFig. 1), and hence are replaced inFig. 10by curves of SN bifurcations. Beyond the SN bifurcation, three TWbranches are present, two of which are related byR symmetry whenε = 0 and exhibit O(1) phase velocities whilethe third takes the place of the unperturbed MM branch and consists of waves drifting with an O(ε) phase velocity.


The two former branches are perturbations of the TW branch present whenε = 0 and exist, more-or-less, betweenthe two SN bifurcations on either side of theµ2 axis; in the middle of this region, these two TW undergoseparateHopf bifurcations to MTW (seeFig. 10). These MTW take the form of a traveling wave with superposed amplitudemodulation.

The SW states likewise become MTW, i.e., SW with an additional slow frequency. As a result, the parity-breakingbifurcations from SW to MTW that occur near the leftmost boundary of region 2 inFig. 7are also replaced by SNbifurcations (seeFig. 1). In Fig. 10, this new SN bifurcation, located along the left boundary of what were regions1 and 2 inFig. 7, generates two MTW, one of which approximates the SW solution of theε = 0 case, i.e., it has (inthe full system(10)) one fast frequency associated with amplitude modulations, and oneO(ε) frequency associatedwith a drift along the group orbit generated byTφ, while the other is a traveling wave with superposed amplitudemodulation.

3.2.2. Effect on global bifurcationsThe loss ofR symmetry is manifest most dramatically in the kind of global bifurcations seen in Eqs.(14). In

Fig. 7, for example, a P+ → O heteroclinic bifurcation produces SW and, in region 1, structurally stable P+ → P−heteroclinic cycles. Since Wu(P+) and Ws(O) are both one-dimensional their coalescence in a plane is a codimension-one phenomenon but becomes of codimension-two when they = 0 plane ceases to be invariant and these manifoldscan explore the full three-dimensional phase space. Consequently, we do not expect to see a P+ → O connectiononceε = 0. This is indeed what happens in region 1, i.e., below the curveσ− = 0. However, whenσ− > 0, thisconnection is replaced by a P− → O connection. This is possible because in that region Wu(P−) is two-dimensionaland a codimension-one intersection with Ws(O) does not require an invariant plane. In fact, this connection is theremnant of the P− → P+ → O connection that forms part of the surface of heteroclinic cycles in theε = 0 case(seeFig. 9); for smallε this connection still passes very close to P+.

Chossat[21] shows that the P+ → P− heteroclinic cycles, which exist throughout region 1 ofFig. 7, are in generaldestroyed and replaced by MTW onceε = 0. In fact,Fig. 10shows that a single curve of P+ → P− cycles doesremain. This heteroclinic bifurcation, which is now of codimension-one (seeFig. 6b), generates two MTW. For theparameters we use these MTW emerge on opposite sides of the bifurcation (Fig. 12). The original structurally stablecycle is thus approximated by MTW throughout its range of existence, consistent with the results of[21]. One ofthese MTW eventually terminates (typically after a SN bifurcation) at one of the two Hopf bifurcations to the rightof theµ2 axis that replace the Hopf bifurcation from TW to MTW whenε = 0. The other is ultimately annihilated,typically after one or more SN bifurcations, in either the Hopf bifurcation that replaces the Hopf bifurcation to SWwhenε = 0, or in a homoclinic bifurcation involving the TW created whenσ− = 0 (see below). We do not attemptto describe all possible scenarios for the various MTW branches, nor to calculate all of the SN bifurcation sets, asthis would add unnecessary clutter toFig. 10; the figure includes only those bifurcations which are essential for theoverall picture, such as the SN bifurcation of MTW that follows the lower boundary of what were regions 1 and 2whenε = 0.

Homoclinic bifurcations involving the mixed states are possible only forε = 0. In fact, these are an essentialpart of the symmetry-breaking scenario, appearing just after the bifurcation from P± to MM± (TW whenε = 0)that forms (part of) the boundary of region 1. InFig. 7 the SB bifurcation on P+ is isolated from region 1 andits P+ → P− cycles and, unlike in the case of P−, does not give rise to TW homoclinics whenε = 0. With otherparameters, however, both of the P± bifurcations help to define region 1 and the TW homoclinics that becomepossible are illustrated inFig. 13. In either case, these homoclinic orbits resemble theε = 0 heteroclinic cycle,approaching it more and more closely asε decreases. Indeed, it is straightforward to show that in general these TWhomoclinics lie anO(ε2) distance from the SB bifurcation that produces the TW.

The symmetry-breaking has an even more dramatic effect on the various heteroclinic cycles in region 2, destroyingseveral important heteroclinic connections. In particular, the connections P+ → MM− and SW→ O, which werestructurally stable withε = 0, now require the tuning of an additional parameter (i.e., they become of codimension-one). The effect of finiteε on the heteroclinic cycles of region 2 is summarized inTable 2. Whenε = 0, only


Fig. 13. Sketch of the homoclinic bifurcations that arise near the boundary of region 1 whenε = 0: (a) just after the bifurcation from P−, as inFig. 10, and (b) after a bifurcation from P+. We have shown the “up” case, which utilizes the upper piece of Wu(P−) or Ws(P+), but the “down”case also occurs.

two of these cycles (the first and last entries) remain of codimension-one or less. Consequently, one may expectto encounter these two cycles when traversing region 2. However, the situation near the MTW limit cycle ismuch more subtle. Whenε = 0, the connections P− → SW → O and MM− → SW → MM− include the SWbynecessity, since a connection to the SW provides the only mechanism for re-entering the invarianty = 0 plane.This is no longer the case when the symmetryR is broken. In addition to the codimension-one cycles ofTable2 (replacing SW with MTW and MM− with TW), we can expect an infinite number of heteroclinic cycles of theform O → P− → (near MTW)→ O and TW→ (near MTW)→ TW, which merely visit the neighborhood of theMTW, executing a finite number of rotations, before leaving again. The situation is further complicated by thepossibility of using either the “up” or the “down” parts of Wu(P−) or Wu(TW) (see alsoFig. 6). The latter cycles arein fact the most prominent cycles one observes numerically in the parameter regime containing the infinite cascadeof isolas whenε = 0. In Fig. 14we show the period of the various periodic solutions of Eqs.(14) that are relatedto these isolas (cf. Fig. 3 of[20]).

3.2.3. New codimension-two pointsThe new homoclinic and heteroclinic bifurcations discussed above are organized by a series of new codimension-

two points. The simplest of these is the Takens–Bogdanov point, resulting from the intersection of a SN and a Hopfbifurcation with zero frequency[39]. Under appropriate conditions, which appear to be satisfied for our parameters,the TB bifurcation generates a curve of homoclinic connections. InFig. 10, the TB points serve as endpoints forthe TW homoclinics that emerge near the SB bifurcation from P− to TW. Note the (new) Hopf bifurcation (whichis rather difficult to see inFig. 10) that extends between the two TB points, and the TW homoclinic that appearsjust below the SB bifurcation on P− in the lower right part ofFig. 10(also difficult to see).

The point labeled Het-SB inFig. 10 marks the intersection of the P+ → P− heteroclinic cycle with the SBbifurcation on P− that destroys them. At this point there are two degenerate heteroclinic cycles, one “up” and

Table 2Codimension of the various heteroclinic cycles found in region 2 whenε = 0

Heteroclinic cycle ε = 0 codimension ε = 0 codimension

O → P− → SWε→ O 0 1

O → P− → P+ε→ MM− → SW

ε→ O 1 3

O → P+ε→ MM− → SW

ε→ O 1 3

O → P− → MM− → SWε→ O 1 2

SW → MM− → SW 1 1

A small ε indicates which connections are disrupted by the symmetry-breaking. Recall also that MM− and SW become TW and MTW whenε = 0.


Fig. 14. The period of several periodic solutions of Eqs.(14) with ε = 0.01 in the parameter regime containing an infinite cascade of isolaswhenε = 0 [20] as a function of the angle along a circle of radius|µ| = 1.5 around (µ1, µ2) = (0,0). The global bifurcations that replace theisolas whenε = 0.01 are of three basic types: O→ P− → (near MTW)→ O, TW → (near MTW)→ TW, and TW→ TW, labeled A, B, andC, respectively. The global bifurcation C occurs after a subcritical Hopf bifurcation that destroys the periodic orbit replacing SW, and involvesthe TW replacing MM+, in contrast to bifurcation B which involves the TW that replaces MM−. All the solutions shown are unstable.

one “down” (seeFig. 15a), involving the nonhyperbolic fixed point P−. Two corresponding curves of TW ho-moclinics (up and down) emerge on opposite sides of the Het-SB point, and tangent to the SB bifurcationcurve.

Fig. 10shows that the TW homoclinic bifurcation to the left of the Het-SB point stays very close to the curveof SB bifurcations that separates regions 1 and 2 inFig. 7 and then terminates in a T-point[41] at the bottom ofthe P− → O heteroclinic curve. The nature of this T-point is illustrated inFig. 15b. The heteroclinic connectionfrom TW to O is the most important feature; it is a codimension-two event because both Wu(TW) and Ws(O)are one-dimensional. The remaining two segments of the cycle are structurally stable: O→ P− takes place in theinvariantr1 = 0 plane (i.e., within Wu(O)), while the intersection of Wu(P−) and Ws(TW) is robust because bothmanifolds are two-dimensional. It is straightforward to show (seeAppendix A), using appropriate Poincare returnmaps, that the unfolding of this T-point contains both the TW homoclinic bifurcation and the P− → O heterocliniccycle, as suggested byFig. 10. Because the eigenvalues of O in ther1 = 0 plane are complex (see Eqs.(14)), thecurve of TW homoclinics, all of which pass through a neighborhood of O, takes the form of a logarithmic spiral (ina sufficiently small neighborhood of the T-point). However, whenε = 0.01, it is not possible to discern this spiralwithout substantial magnification ofFig. 10. Indeed, as seen next, much of the behavior just described becomesclearer whenε is increased, although some of the more detailed features are lost.

Fig. 15. Sketches of (a) the degenerate P+ → P− cycle(s) at the Het-SB point, and (b) the heteroclinic cycle TW→ O → P− → TW thatdefines the T-point inFig. 10.


Fig. 16. Bifurcations sets forν = −0.1, with the remaining coefficients as inFig. 10.


Whenε is increased by a factor of 10, the qualitative behavior remains largely the same although a few interestingtransitions are lost (seeFig. 16). However, others become clearer, and it is these that we focus on. Comparison withFig. 10reveals that the TW homoclinics have moved further away from the SB bifurcation on P−, although theyremain anchored at the Het-SB point. The two curves of torus bifurcations to MTW (to the right of theµ2 axis,labeled Hopf) coincide whenε = 0 but for ε = 0.1 are quite distinct: one makes its way toward the origin, asbefore, while the other, in contrast toFig. 10, veers sharply toward the TB point in the first quadrant where itterminates. Other than that, the main visible differences betweenFigs. 10 and 16are in the details of the SN andHopf bifurcations occurring in the lower right portion of the figure; we omit these and several additional globalbifurcations, but describe the latter for a yet larger value ofε in the next subsection.


Whenε = 0.5, with the remaining coefficients held fixed, one obtainsFig. 17. Note that, in contrast withFigs.10 and 16, the line of saddle–node homoclinic (SN-Hom) bifurcations that replaces the P mode limit cycle with P±is now easily distinguished from theµ1 axis. The codimension-two point where this bifurcation intersects the SBbifurcation to TW contains in its unfolding curves of P+ → P− heteroclinic cycles and of Hopf bifurcations[40],both of which are seen in the figure. The T-point, identified in bothFigs. 10 and 16, has shifted further from thecurve of SB bifurcations on P− where it originated (seeFig. 10), and now exhibits clear spiraling on both the TWhomoclinic curve and the P− → O heteroclinic curve that emerge from it. The (logarithmic) spiraling of the set ofP− → O connections indicates that the TW have acquired complex eigenvalues, in contrast to the case depicted inFig. 15b.


Fig. 17. Bifurcations sets in−1 ≤ µ1 ≤ 0.1, −0.1 ≤ µ2 ≤ 1.2 whenν = −0.5, with the remaining coefficients as inFig. 10.

Fig. 18. Close-up ofFig. 17focusing on the two codimension-two global bifurcations (the flip and SN-Hopf) and the codimension-one globalbifurcations connecting them. The unpaired curve (!) is special and is indicated inFig. 19as well.


Fig. 19. Close-up of the SN-Hopf point inFigs. 17 and 18. Two intertwined curves of TW homoclinics emerge from an exponentially thin regionnear the SN-Hopf point. One of these terminates on the TB point while the other (!) ends in the flip bifurcation near the Het-SB point. The (pairsof) curves below (!) are disconnected from each other; they begin and end at the flip bifurcation.

The most obvious new feature ofFig. 17is the formation near the upper Takens–Bogdanov point inFig. 16ofa true SN-Hopf interaction, with a nonzero Hopf frequency at the interaction. This codimension-two singularityhas a three-dimensional normal form[39] containing a set of heteroclinic connections (between the two TW) in itsunfolding. This connection, however, relies on a normal form symmetry which is not a feature of the full problem.Indeed, as shown by Kirk[42,43], when appropriate symmetry-breaking corrections are taken into account, onefinds that the curve of heteroclinic cycles breaks apart, leaving two curves ofhomoclinicconnections, one for eachof the TW. These two curves oscillate with exponentially decreasing amplitude as they approach the SN-Hopf point,intertwining in a characteristic “out-of-phase” fashion[42–44]. One of these wiggly curves of TW homoclinicconnections has been included inFig. 17; it eventually terminates at the TB point. The fate of the other is shownin Fig. 18, a close-up ofFig. 17 that includes many additional global bifurcations, and inFig. 19, which showsa further close-up of the SN-Hopf region. The second curve of TW homoclinics (marked with! in both figures)veers away from the region of the SN-Hopf toward the Het-SB point. It cannot terminate here, however, but doesso close by, at a homoclinic flip bifurcation of the original “down” (single-loop) TW homoclinic (seeFig. 15a) thatemerges toward the right from the Het-SB point. The unfolding of this homoclinic flip bifurcation contains very

Fig. 20. Stable dynamics near the SN-Hopf bifurcation point inFig. 19: (a) a stable periodic orbit at (µ1, µ2) = (0.016501,0.79295); (b) anattracting torus at (µ1, µ2) = (0.0165,0.7931).


complicated dynamics, including “N-homoclinics” for any integer N and regions of chaotic shift-dynamics[45]. Aseries of these N-homoclinics, which make N visits to the neighborhood of the TW before reconnecting, is shownin Figs. 18 and 19. It appears likely that the N-homoclinics created in the homoclinic flip bifurcation connect tothe various homoclinics involved in closing the resonance tongues present near the SN-Hopf bifurcation[43]. Infact, all but one of the N-homoclinics begin and end at the flip bifurcation and only the curve (!) terminates in theSN-Hopf point.

Near the SN-Hopf point we find a variety of interesting dynamics including stable tori and periodic orbits typicalof the perturbed normal form[46]. Examples of these are shown inFig. 20. We have not attempted to locate thebranches of periodic orbits associateddirectlywith the TW homoclinic bifurcations that emerge from the SN-Hopfpoint even though both are of Shil’nikov type and the ratio of the real part of the complex eigenvalues of the TWto the real eigenvalue is approximately 0.35 in magnitude in both cases. This is thus the “interesting” case (see[47]) and one expects intervals of stable periodic orbits associated with one (but not both) of these TW homoclinicbifurcations.

4. Conclusion

In this paper we have explored the consequences of breaking reflection symmetry in the 1:2 spatial resonancewith O(2) symmetry. This bifurcation arises in a variety of applications, including Rayleigh–Benard convectionin a plane layer[6,17], Marangoni convection in a cylindrical container[9], cellular flame patterns on a circularporous plug burner[15], turbulent boundary layer breakdown[8], and the von Karman flow between two exactlycounter-rotating disks[18]. In each of these cases imperfections may be present that break the assumed O(2)symmetry, i.e., the symmetry generated by rotations and reflection of a circle, or translations and reflections alongthe real line, modulo an imposed spatial period. We have focused here on the simplest case, that of breaking thereflection symmetry, while preserving the rotation or translation invariance of the system, a situation that arises inconvection in the presence of weak throughflow, or of slow rotation of a cylindrical container. More interestingly,perhaps, this situation also arises in the von Karman flow between two counter-rotating disks when these do notcounter-rotate exactly. In all of these examples the breaking of the reflection symmetry introduces a preferreddirection. This, together with the rotation (translation) invariance, implies that the generic symmetry-breakingprimary bifurcation from the trivial state takes the form of a rotating (traveling) wave. As a result, steady states arenot expected and neither are the attracting structurally stable heteroclinic cycles that are characteristic of this modeinteraction.

We have seen that the loss of reflection symmetry destroys this heteroclinic cycle, and replaces it with a modulatedtraveling wave. At isolated parameter values structurally unstable heteroclinic connections are also expected. Whilethis result is in accord with earlier analyses[21], we have seen that much else happens besides. In particular, we haveseen that the loss of reflection symmetry is responsible for the introduction of three interesting codimension-twobifurcations into the dynamics. The behavior at two of these, the Het-SB bifurcation and the T-point, is summarizedin Fig. 15. Both of these bifurcations are associated with the disappearance of the heteroclinic cycle in the O(2)-symmetric system when the pure modes lose stability to mixed modes. The third is a (pair of) Takens–Bogdanovbifurcations on the branch of drifting mixed modes, referred to here as TW. When the forced symmetry-breakingis weak, these two sets of phenomena are unrelated, but as the forced symmetry-breaking increases in strengththey become interconnected in a remarkable way. In particular, we have identified near the Het-SB bifurcation ahomoclinic flip bifurcation; this bifurcation is responsible for the generation of much complex dynamics, althoughwe have focused here only on the appearance of the so-called N-homoclinics, i.e., infinite period orbits that starton a TW solution and return to its vicinity N times before reconnecting to it. We have traced the curves of theseN-homoclinics through parameter space, and found that they interact with a saddle–node Hopf bifurcation, therebyconnecting large regions of parameter space and two quite distinct codimension-two phenomena. This unexpectedobservation has led us to suggest a connection between the N-homoclinics and the homoclinics involved in the


Fig. 21. Close-up of relevant bifurcation sets fromFig. 16 with ε = 0.1 and several N-homoclinics. The SN-Hopf bifurcation has not yetemerged. The primary “down” TW homoclinic (Hom) terminates at the TB point while the remaining N-homoclinics form closed loops thatbegin and end at the flip bifurcation.

closure of the resonance tongues associated with the saddle–node Hopf bifurcation. The details of this connectionand the mechanism by which it comes into being are not entirely clear at this time, and constitute interesting topicsfor further research. We may, however, gain some insight by following the N-homoclinics ofFig. 18to smaller valuesof ε. In Fig. 21, computed forε = 0.1, we include several of these, along with the relevant bifurcation sets fromFig. 16. For this value ofε there is no saddle–node Hopf bifurcation and the “down” TW homoclinic that emergesfrom the Het-SB point connects directly to the TB point. In addition, there are closed loops of N-homoclinics thatextend toward the region of the TB point (and for largerε values, the saddle–node Hopf point) before turning backtoward the flip bifurcation. These “large” loops, which leave and return to the flip bifurcation along very differentpaths, should be contrasted with the “thin” loops ofFigs. 18 and 19present below the unpaired (!) homoclinic. Itseems that in increasingε from 0.1 to 0.5, the saddle–node Hopf bifurcation and its associated wiggly curves of TWhomoclinics manage to unzip and reconnect several loops of N-homoclinics, replacing one partner with another andtransforming large loops to thin ones. The unpaired (!) homoclinic marks the place (for a given value ofε) wherethis process stops.

In this connection, it is worth remarking that similar connections between diverse codimension-two bifurcationshave been observed in other strongly asymmetric systems. For example, Sneyd et al.[48] find a connection betweenbehavior near a T-point and near a saddle–node Hopf bifurcation across a large region of parameter space in theirstudy of calcium traveling waves in pancreatic acinar cells, while van der Heijden[49] investigates the “zipping” oftwo wiggly curves of periodic orbits created in two distinct Shil’nikov bifurcations. The latter process appears to bevery similar to the behavior revealed inFigs. 18 and 19. The 1:2 spatial resonance in Rayleigh–Benard convectionin a rotating cylinder[28] may also exhibit the type of behavior identified here.

The approach we have taken is motivated by applications. For this reason, we have focused on the effects ofgeneric low order terms that break the reflection symmetry while preserving the continuous symmetry. As in otherproblems of this type, this approach is expected to give correct results when the strength of these terms is small(ε� 1). However, no guarantee can be given that higher order terms will not produce new dynamics that is absentfrom our formulation. Likewise, we can make no guarantee that the behavior found at largerεwill be seen genericallyand that nothing else can occur. On the other hand, the behavior of even our simplest system is so complex that most


of the analysis has had to rely on delicate numerical computations. It is our view that for applications this approachprovides a good guide to the phenomena that may be expected as the reflection symmetry is increasingly broken,and we propose to compare our results with three-dimensional simulations of the von Karman vortex flow betweennearly counter-rotating disks in a future publication.

Acknowledgments

This work was supported by EPSRC grant GR/R52879/01. We are indebted to Caroline Nore and LauretteTuckerman for discussions, and to Vivien Kirk for comments on the manuscript.

Appendix A

In this appendix we give the details of the TW homoclinic and P− → O heteroclinic bifurcation sets in theneighborhood of the T-point (seeFigs. 10, 16 and 18). There are two cases depending on whether the TW hasreal eigenvalues (as inFigs. 10 and 16) or complex eigenvalues (as inFig. 18). The heteroclinic cycle defining theT-point and the Poincare sections we use to analyze it are sketched inFig. 22.

It is convenient to use normal form coordinates centered at each of the fixed points O, P−, and TW. Near O, forexample, we may write

r1 = µ1r1, ρ = µ2ρ, φ = ν, (20)

whereµ1 < 0,µ2 > 0, andρ2 = x2 + y2. We use the Poincare sections

#aO : {(r1, ρ, φ)|r1 = ε, ρ ≤ ε}, #b

O : {(r1, ρ, φ)|ρ = ε, r1 ≤ ε}, (21)

and find the local map from#aO to#b

O to be

TO : (ε, ρ, φ) →(ε

∣∣∣ρε

∣∣∣−µ1/µ2, ε, φ −

(ν

µ2

)log

∣∣∣ρε

∣∣∣) . (22)

Fig. 22. Sketch of the heteroclinic cycle at the T-point (“up” case) and the various Poincare sections defined below. In case (a) the TW has purelyreal eigenvalues, as it does forε� 1, while in (b) there is a complex conjugate pair of eigenvalues, as withε = 0.5.


Near P− we useξ = √−µ2/d22 − r2 and write

r1 = σ−r1, ξ = σ2ξ, ˙y = σ1y, (23)

whereσ− > 0, σ2 = −2µ2 < 0, andσ1 = σ+ − σ− > 0 (see Eq.(19)). With the Poincare sections

#aP− : {(r1, ξ, y)|r1 ≤ ε, ξ = ε, |y| ≤ ε}, #b

P− : {(r1, ξ, y)|r1 = ε, ξ ≤ ε, |y| ≤ ε}, (24)

we find the local map from#aP− to#b

P− :

TP− : (r1, ε, y) →(ε, ε

∣∣∣ r1ε

∣∣∣−σ2/σ−, y

∣∣∣∣ εr1∣∣∣∣σ1/σ−

). (25)

Note that the map TP− assumes that trajectories cross#bP− ; in general this need not be the case since Wu(P−) is

two-dimensional and contains a curve in ther1 = 0 plane. In order for TP− to be relevant for heteroclinic connectionsfrom P− to TW, we must restrict to (r1, y) ∈ #a

P− such that ˆy|ε/r1|σ1/σ− < ε.The behavior near TW depends on the nature of the eigenvalues. If all three are real, we may write

u = γ1u, v = γ2v, ˙y = γ3y, (26)

whereγ1 < 0, γ2 < 0, andγ3 > 0, and use the Poincare sections

#a,rTW : {(u, v, y)|u = ε, |v| ≤ ε, |y| ≤ ε}, #

b,rTW : {(u, v, y)||u| ≤ ε, |v| ≤ ε, y = ε}. (27)

This leads to the local map:

TrTW : (ε, v, y) →

(ε

∣∣∣∣ yε∣∣∣∣−γ1/γ3

, v

∣∣∣∣ yε∣∣∣∣−γ2/γ3

, ε

). (28)

On the other hand, ifγ1 andγ2 collide and leave the real axis, we may write

R = γR, ϕ = ω, ˙y = γ3y, (29)

whereu = R cosϕ, v = R sinϕ, andγ < 0. Appropriate Poincare sections are then

#a,cTW : {(R, ϕ, y)|R = ε, |y| ≤ ε}, #

b,cTW : {(u, v, y)|

√u2 + v2 ≤ ε, y = ε}, (30)

and the local map from#a,cTW to#b,c

TW is given by:

TcTW : (ε, ϕ, y) →

(ε

∣∣∣∣ yε∣∣∣∣−γ/γ3

cos

(ϕ − ω

γ3log

∣∣∣∣ yε∣∣∣∣), ε

∣∣∣∣ yε∣∣∣∣−γ/γ3

sin

(ϕ − ω

γ3log

∣∣∣∣ yε∣∣∣∣), ε

). (31)

Note that by taking ˜y = ε in#b,rTW and#b,c

TW, we focus on the “up” case, i.e., a T-point formed by the coalescence ofthe upper part of Wu(TW) and Ws(O). This is indeed what we observe numerically with the parameters of Section3, but the “down” case, in principle, may also occur.

Appropriate global maps are obtained by linearizing about the heteroclinic cycle at the T-point, thus prompt-ing us to define the following intersections: Ws(P−) ∩#b

O = (0, ε, φ0), Wu(P−) ∩ Ws(TW) ∩#bP− = (ε,0, y0),

Wu(P−) ∩ Ws(TW) ∩#a,rTW = (ε, v0,0) and, when TW has complex eigenvalues, Wu(P−) ∩ Ws(TW) ∩#a,c

TW =(ε, ϕ0,0). While the real parametersφ0, y0, v0 or ϕ0, defined above, will change as one moves away from theT-point, this dependence is of little consequence. The primaryunfoldingcan be associated with the intersectionWu(TW) ∩#a

O = (ε, *, ϑ), i.e., with the two parameters* andϑ (which themselves can be thought of as functions


of µ1 andµ2); at the T-point we have* = 0. Using the above definitions, and taking into account the invariance ofr1 = 0, we define

TO→P− : #bO → #a

P− : (r1, ε, φ) → (a1r1, ε, a2r1 + a3(φ − φ0)), (32)

TrP−→TW : #b

P− → #a,rTW : (ε, ξ, y) → (ε, v0 + b1ξ + b2(y − y0), b3ξ + b4(y − y0)), (33)

TcP−→TW : #b

P− → #a,cTW : (ε, ξ, y) → (ε, ϕ0 + b1ξ + b2(y − y0), b3ξ + b4(y − y0)), (34)

TTW→O : #b,rTW or#b,c

TW → #aO : (u, v, ε) → (ε, η1 + c1u+ c2v, η2 + c3u+ c4v), (35)

wherea1, . . . , a3, b1, . . . , b4, c1, . . . , c4, η1, η2 are real constants witha1 > 0. Note that in the map TTW→O wehave used a Cartesian representation, (ρ cosφ, ρ sinφ), on#a

O in order to avoid difficulties in theρ → 0 limit;in the following we think of (η1, η2) = (* cosϑ, * sinϑ), which specify Wu(TW) ∩#a

O, as alternative unfoldingparameters.

To locate the set of TW homoclinics, we begin on Wu(TW) ∩#aO = (ε, *, ϑ) and require that this point is mapped

by TrP−→TW ◦ TP− ◦ TO→P− ◦ TO or Tc

P−→TW ◦ TP− ◦ TO→P− ◦ TO to Ws(TW). This leads, independent of whetherTW has complex eigenvalues, to the condition (˜y = 0):

b4y0 = A1*µ1σ2/(µ2σ−) + A2*

µ1σ1/(µ2σ−)[A3*

−µ1/µ2 + a3

(ϑ − φ0 −

(ν

µ2

)log*

)], (36)

where A1 = b3ε(a1εµ1/µ2)−σ2/σ− , A2 = b3ε(a1ε

µ1/µ2)−σ1/σ− , A3 = a2ε1+µ1/µ2, and φ0 = φ0 − (ν/µ2) logε.

Since*µ1σ1/(µ2σ−) diverges as*→ 0. Eq.(36)can only be satisfied if

ϑ − φ0 =(ν

µ2

)log*−

(A3

a3

)*−µ1/µ2, (37)

which defines a logarithmic spiral in the*→ 0 limit (see[41,50]).The set of P− → O heteroclinics can be located, when TW has purely real eigenvalues, by starting with Wu(P−) ∩

#bP− = (ε,0, y) and insisting that some point on this line, which is parameterized by−ε < y < ε, is mapped by

TTW→O ◦ TrTW ◦ Tr

P−→TW to Ws(O), i.e., thatρ = 0. This requirement leads to the equations:

(η1η2

)= B1

(c1c3

)|y|−γ1/γ3 + B2

(c2c4

)(v0 + b2y)|y|−γ2/γ3, (38)

whereB1 = −ε|ε/b4|γ1/γ3, B2 = −|ε/b4|γ2/γ3, andy = y − y0. In our case|γ2| > |γ1| so the first term dominatesfor smally but, regardless, Eq.(38) describes a half-line in the (η1, η2) plane, parameterized byy (cf. [41]). In the“up” case, which we focus on here, the relevant values satisfyb4y > 0 (see Eqs.(33) and (34)).

If Ws(TW) is associated with complex eigenvalues then, similarly, a P− → O heteroclinic connection requiresthat TTW→O ◦ Tc

TW ◦ TcP−→TW(ε,0, y) ∈ Ws(O) for some−ε < y < ε. In this case, the necessary conditions are:

(η1η2

)= B1|y|−γ1/γ3

(c1 c2c3 c4

)cos

(ϕ0 + b2y −

(ω

γ3

)log |y|

)

sin

(ϕ0 + b2y −

(ω

γ3

)log |y|

) , (39)

where ϕ0 = ϕ0 + (ω/γ3) log |ε/b4| and againy = y − y0. Eq. (39) describes a (linearly deformed) logarithmicspiral for smally (see[50]).


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