modelling instability of abc ﬂow using a mode interaction...

Physica D 215 (2006) 62–79www.elsevier.com/locate/physd

Modelling instability of ABC flow using a mode interaction between steadyand Hopf bifurcations with rotational symmetries of the cube

Olga Podviginab,c,d,∗, Peter Ashwina, David Hawkera

a Mathematics Research Institute, School of Engineering, Computer Science and Mathematics, University of Exeter, Exeter EX4 4QE, UKb Observatoire de la Cote d’Azur, CNRS UMR 6529, BP 4229, 06304 Nice Cedex 4, France

c International Institute of Earthquake Prediction Theory and Mathematical Geophysics, 79 bldg. 2, Warshavskoe Ave., 113556 Moscow, Russian Federationd Laboratory of General Aerodynamics, Institute of Mechanics, Lomonosov Moscow State University, 1, Michurinsky Ave., 119899 Moscow, Russian Federation

Received 2 March 2005; received in revised form 11 January 2006; accepted 17 January 2006Available online 3 March 2006

Communicated by C.K.R.T. Jones

Abstract

We consider hydrodynamic stability of symmetric ABC flow with A = B = C = 1, regarded as a steady state of the three-dimensionalNavier–Stokes equation with appropriate forcing. Numerical investigations have shown that its first instability on increasing the Reynolds numberis a Hopf bifurcation at R = 13.044, and that at this bifurcation the second dominant eigenvalue is real. Motivated by this, we study genericinteraction of steady-state and Hopf bifurcations in systems with rotational symmetry of the cube O with an eight-dimensional normal form.

The generic branching and bifurcation behavior of the third-order truncated normal form is investigated. This is used to analyse a range ofthe bifurcations for particular values of the coefficients, obtained by center manifold reduction from the hydrodynamic system. The normal formsystem shows a sequence of bifurcations and attractors that closely follows the sequence observed for the original hydrodynamic system up toabout R = 13.91. This includes a torus breakdown to a chaotic attractor and a crisis of attractors that leads to a change in symmetry.

We discuss numerical simulations of the hydrodynamic system for larger values of R. Finally, we present evidence that the system hasrobust heteroclinic cycles between fully symmetric ABC flow and six steady states with broken symmetry for a range of parameter values nearR = 15.c© 2006 Elsevier B.V. All rights reserved.

Keywords: Mode interaction; Hopf bifurcation with symmetry; Center manifold reduction; Torus bifurcation; Heteroclinic cycle; Navier–Stokes equation

1. Introduction

An ABC flow,

uABC = (A sin x3 + C cos x2, B sin x1 + A cos x3,

C sin x2 + B cos x1) (1)

(here A, B and C are arbitrary constants) is an exact solution ofthe force-free Euler equation.

Arnold [3] demonstrated that a steady solution of a force-free Euler equation can have chaotic streamlines only if

∗ Corresponding author at: International Institute of Earthquake PredictionTheory and Mathematical Geophysics, 79 bldg. 2, Warshavskoe Ave., 113556Moscow, Russian Federation. Tel.: +7 095 9775195; fax: +7 095 3107032.

E-mail address: [email protected] (O. Podvigina).

0167-2789/$ - see front matter c© 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.physd.2006.01.010

it possesses the so-called Beltrami property. He proposedthe ABC flows (1) as simple examples of such flows. TheLagrangian structure of ABC flows has been extensivelyinvestigated, and indeed for ABC 6= 0 the flow does possesschaotic streamlines [20,13]. This feature is essential for fastmagnetic field generation in conducting fluids, and thus ABCflows are commonly employed in fast dynamo studies [7,1,32] (see a review in [8]). For similar reasons, ABC flows arealso employed for the study of the development of turbulence[22,24,23]. Magnetic field reversals were observed in an ABC-force-driven fully nonlinear MHD system [25].

For any Reynolds number R, uABC is a steady solution ofthe Navier–Stokes equation

∂v∂t

= v × (∇ × v)− ∇ p +1R1v + f, (2)

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O. Podvigina et al. / Physica D 215 (2006) 62–79 63

subject to the incompressibility condition

∇ · v = 0 (3)

and the force

f =1R

uABC (4)

(p is the pressure). We only consider solutions that are periodicin x1, x2, x3:

v(x1, x2, x3) = v(x1 + 2kπ, x2 + 2lπ, x3 + 2mπ)

p(x1, x2, x3) = p(x1 + 2kπ, x2 + 2lπ, x3 + 2mπ)(5)

for any (k, l,m) ∈ Z3. By Ω we denote the periodicity cube.The flow has a discrete symmetry group which is the largest

possible when A = B = C [4,13] making ABC-forcedhydrodynamic and MHD systems amenable from the point ofview of applying equivariant bifurcation theory. Generically, (1)has a symmetry group isomorphic to D2. If two coefficients areequal, it is isomorphic to D4. For A = B = C the group,denoted by H, has 24 elements and it is isomorphic to therotation group of a cube, O [4,13]. We consider the latter case,

A = B = C = 1. (6)

The group O = S4 of permutations of four elements hasfive real irreducible representations W0, . . . ,W4; two of theseare on R3, one on R2 and two on R. The irreducible complexrepresentations consist of two copies of real representations.In the three-dimensional representation W0, the group O actsby rotations only. The other three-dimensional representationis the set of symmetries of a regular tetrahedron acting bypermuting its vertices; note that this includes reflections and hasno rotations of order four (but it does have a rotation–reflectionof order four).

Numerical results on bifurcations of time-evolving solutionsof the Navier–Stokes Eqs. (1)–(6) for 0 < R ≤ 50 werepresented in [22,23]. It can be proved that for R < 0.5 theflow (1) and (6) is a unique steady state of (1)–(6), and it isstable. Numerical computations show that the flow is stable forR ≤ 13.044 [15,22] and that it is a unique attractor of thishydrodynamic system for R ≤ 7.8 [22]. The trivial steady state– the 1:1:1 ABC flow (1) and (6) – becomes unstable in a Hopfbifurcation at R = Rc, where Rc = 13.044 [23].

The action of the group H on the center eigenspace atthis bifurcation is isomorphic to the standard representationof the group O (see [24]). After ABC flow becomes unstable,a complex sequence of bifurcations takes place [5]. TheHopf bifurcation gives rise to eight periodic orbits, afterwhich there is a second Hopf bifurcation resulting in atransition of these orbits into tori. There follows a complexsequence of bifurcations including a period doubling, a thirdHopf bifurcation, a transition to chaos and the merging ofeight chaotic attractors into a single one, which possesseson average all symmetries of the system. The analyticalstudy of Hopf bifurcation with the cubic symmetry [5]

explains only the first bifurcation – we observe in thesix-dimensional system obtained by the center manifoldreduction the appearance of eight periodic orbits, each with theZ3 symmetry group, the same as in the Navier–Stokes equation.The subsequent bifurcations are different in the two systems,though with the addition of S1 symmetry breaking termswe could reproduce some of the features of the breakdownto chaos. In [26] an eight-dimensional reduced system wasconstructed for the hydrodynamic system (1)–(6) using atruncated center manifold, involving the eigenspace associatedwith the eigenvalue with the next smallest real part. Thiseigenvalue is real and the action of the symmetry group on theeigenspace is isomorphic to the two-dimensional representationof the group.

Center manifold reduction is often applied to study bifur-cations in hydrodynamic systems, e.g. for the Taylor–Couettesystem [9] or convection in a plane layer [29,11]. Interaction ofseveral center eigenmodes is also often considered [2,28,12];however center eigenvalues with zero real parts are usually as-sumed; to the best of our knowledge, [26] is the first exam-ple of application of the center manifold theorem with centereigenvalues that are not on the imaginary axis. The study [26]of this eight-dimensional system showed that the bifurcationsmimicked those of the hydrodynamic system for a larger rangeof R than in [5], and this justifies the use of the modified centermanifold theorem to construct a low order truncation of the sys-tem, even though one cannot rigorously control the higher orderterms away from the bifurcation. An alternative interpretationof our approach is to say that we are exploring the neighbour-hood of a generic mode interaction between Hopf and steadymodes for a system with symmetry O, even though we have nonatural choice for a second parameter to vary that would allowus to locate the mode interaction as an ‘organizing center’ forthe dynamics.

In this paper we investigate bifurcations of the reducedsystem of [26] in detail. Firstly, we consider a generic steady-state/Hopf mode interaction with symmetry O, where theaction of the group on the real subspace corresponds to thetwo-dimensional representation of the group and the actionon the complex subspace corresponds to a six-dimensionalrepresentation of the group. We apply methods and results fromgeneric equivariant bifurcation theory to this problem and findthat the branches we investigate are all determined by a third-order truncation; see [17,16,5].

Secondly, we consider a particular case of the third-ordernormal form obtained by a center manifold reduction ofthe hydrodynamic system (1)–(6). We analyse instability andsecondary bifurcation of uABC for R > Rc, and compare theresults with those for the low dimensional system. Finally,we give numerical evidence that the hydrodynamic problemhas robust heteroclinic attracting states for a range of Rnear R = 15. The appendices include details of some ofthe calculations and a discussion of how breaking symmetryto A = B 6= C affects the normal form structure andbranching.

64 O. Podvigina et al. / Physica D 215 (2006) 62–79

2. Normal form for steady-state/Hopf mode interaction

2.1. The action of the group of symmetries for forced ABC flow

We consider the interaction of a steady-state and a Hopfbifurcation relevant to primary instabilities of the ABC flow in(1)–(6) on increasing the Reynolds number. The first bifurcationis Hopf and the action of the symmetry group on the kernel isisomorphic to a C3 irreducible representation of the group. Theeigenspace associated with the second dominant eigenvaluecorresponds to an R2(= C) irreducible representation. Asusual, in the normal form we assume a phase shift symmetryS1 that arises due to the symmetries of periodic orbits and canbe enforced in the normal form at all orders by near-identitytransformations (although this is typically broken on the centermanifold by higher order terms; see [17]).

Hence the interaction we consider has a center manifold thatis parametrized locally by (z0, z1, z2, z3) ∈ C4. The steadybifurcation modes are denoted by z0 while the zi for i = 1, 2, 3correspond to variables parametrizing the Hopf bifurcationmodes (this corresponds to a complex irreducible representationof O). The action of the group O × S1 on the Hopf variables C3

is generated by the threefold rotation

ρ111 : (z1, z2, z3) 7→ (z2, z3, z1), (7)

the fourfold rotation

ρ100 : (z1, z2, z3) 7→ (z1, z3,−z2), (8)

and the normal form S1 phase shift symmetry

γθ : (z1, z2, z3) 7→ eiθ (z1, z2, z3).

Other elements of the group are denoted by

κ+

110 : (z1, z2, z3) 7→ (z2, z1,−z3),

κ−

110 : (z1, z2, z3) 7→ (−z2,−z1,−z3),

where κ+

110 = ρ2111ρ001ρ

2111, κ−

110 = ρ2001κ

+

110. We similarlydefine ρ010, ρ001 etc.

Note that O can be regarded as the group of elements inSO(3) that permute the coordinate axes. The symmetry ρ111 oforder three corresponds to rotation about a vertex of the cube,ρ100 is of order four and corresponds to rotation about the centerof a cube face by π/2, κ+

110 is of order two and correspondsto rotation about a line through mid-points of opposite edgesof the cube by π . We refer to the (conjugacy class of the)group generated by κ−

110γπ as Z2(e) and to the one generatedby ρ2

100γπ as Z2( f ). The symmetries γ act as temporal phaseshift symmetries that are present in the normal form, but arebroken in generic problems by high order terms. The invariantsubspaces and corresponding isotropy subgroups for this C3

representation of the group are described in [5].We denote the primitive cube root of unity by

ζ := exp(

2π i

3

)and generators of the group O act on the steady mode subspaceC1 as follows (the S1 normal form acts trivially on this mode):

ρ111 : z0 7→ e2π i/3z0, (9)

Table 1

The action of a selection of the group elements in O × S1 on C4

Element Action on (z0, z1, z2, z3)

ρ111 (ζ z0, z2, z3, z1)

ρ2111 (ζ 2z0, z3, z1, z2)

ρ100 (z0, z1, z3,−z2)

ρ010 (ζ z0,−z3, z2, z1)

ρ001 (ζ 2z0, z2,−z1, z3)

κ+

011 (z0,−z1, z3, z2)

κ+

101 (ζ 2z0, z3,−z2, z1)

κ+

110 (ζ z0, z2, z1,−z3)

κ−

011 (z0,−z1,−z3,−z2)

κ−

101 (ζ 2z0,−z3,−z2,−z1)

κ−

110 (ζ z0,−z2,−z1,−z3)

γπ (z0,−z1,−z2,−z3)

and

ρ100 : z0 7→ z0. (10)

Hence ρ2100 = I and the action of the group is isomorphic to

that of D3. Bifurcations in systems with this action of D3 werestudied in detail e.g. in [17] but to our knowledge this is thefirst attempt to study the interaction of these two modes on C4.Table 1 summarizes the group action on C4.

We refer the three-dimensional representation consideredabove as O and to the other three-dimensional one1 as O−.The difference between these two representations is in theaction of ρ100; for O− it is defined by ρ100 : (z1, z2, z3) 7→

(−z1,−z3, z2). There exists an automorphism of the groupO × S1 given by

ρ111 7→ ρ111, ρ100 7→ γπρ100, γθ 7→ γθ , (11)

relating the representations [5]. By using this automorphismone can apply results of Sections 2 and 3 to the interactionswhere O is replaced by O−.

2.2. The truncated normal form

Appendix A gives a derivation of the general normal formon C4 that commutes with the action of O × S1 as describedabove. If we write the linear part as

z0 = δz0

z1 = (λ+ iω)z1

z2 = (λ+ iω)z2

z3 = (λ+ iω)z3

where δ and λ± iω are eigenvalues with δ and λ close to zero,then the most general commuting vector field truncated at thirdorder is

1 We use notation consistent with [17]: O− denotes a subgroup of O(3),isomorphic to O but different from it.


Fig. 1. The isotropy lattice for the action of O × S1 on C4. For convenience this is given in terms of typical points in the fixed point subspaces for an isotropysubgroup in the relevant conjugacy class. The subgroups themselves can be found in Table 2.

Table 2

A list of representative isotropy subgroups for the action of O × S1 on C4

Name Typical point Generators Dim Conjugates

O × S1 (0, 0, 0, 0) ρ111, ρ100, γθ 0 1

D4 × S1 (x, 0, 0, 0) ρ100, κ+

011, γθ 1 3

D2 × S1 (w, 0, 0, 0) ρ2100, ρ

2010, γθ 2 1

D3 (0, z, z, z) ρ111, κ−

011γπ 2 4

Z3 (0, z, ζ z, ζ 2z) ρ111γ2π/3 2 8

Z4 (x, 0, z, i z) κ−

011γπ/2 3 6

D2 (x, 0, z, z) κ+

011, κ−

011γπ 3 6

D4 (x, z, 0, 0) ρ100, κ+

011γπ 3 3

Z2(d) (w, z, 0, 0) ρ2100 4 3

Z2(e) (x, z, w,w) κ−

011γπ 5 6

Z2( f ) (w, z, u, 0) ρ2001γπ 6 3

1 (w, u, v, z) 8 1

Other isotropy subgroups can be obtained by conjugation. For typical points, xis a real quantity while u, v, w and z are complex. The Z2(e) and Z2( f ) arenamed to conform with the notation in [5].

z0 = δz0 + B1z02+ B2(|z1|

2+ ζ 2

|z2|2+ ζ |z3|

2)

+B3(|z1|2+ |z2|

2+ |z3|

2)z0 + B4|z0|2z0

z1 = (λ+ iω)z1 + A1|z1|2z1 + A2(|z2|

2+ |z3|

2)z1

A3(z22 + z2

3)z1 + A4(z0 + z0)z1 + A5|z0|2z1

z2 = (λ+ iω)z2 + A1|z2|2z2 + A2(|z1|

2+ |z3|

2)z2

+A3(z21 + z2

3)z2 + A4(ζ z0 + ζ 2z0)z2 + A5|z0|2z2

z3 = (λ+ iω)z3 + A1|z3|2z3 + A2(|z1|

2+ |z2|

2)z3

+A3(z21 + z2

2)z3 + A4(ζ2z0 + ζ z0)z3 + A5|z0|

2z3,

(12)

where B j are real and A j are complex numbers ( j = 1, . . . , 4).For convenience we define

A1 = A1r + i A1i , etc.

We are concerned with the case where both of the linear growthrates δ and λ are close to zero and all other coefficients in thenormal form are in a generic position.

3. Bifurcations of the normal form for a general set ofcoefficients

Table 2 lists the isotropy subgroups for the action on C4.The lattice of isotropy subgroups is shown in Fig. 1. The latticefor the action we study here contains, as a sublattice, the oneconsidered in [5] for the action of O × S1 on C3. However, thedimensions are different and we obtain several new isotropysubgroups. Note that some of the maximal isotropy subgroups(such as Z3) for the action on C3 remain maximal while others(such as D4) do not.

In this section we investigate the two-parameter modeinteraction on C4 (12) assuming that δ and λ are close to zeroand the other parameters Ai , Bi and ω are in a general position.Note that the periodic orbits we examine are all S1 orbits andso we can find them by searching for relative equilibria for theequation; this can be done by reduction of the S1 symmetry asoutlined in Appendix B.

We examine firstly the maximal isotropy branches, then theother branches from the interaction δ = λ = 0 and finally somesecondary branches that bifurcate from steady solutions of theform (x±, 0, 0, 0). There are many other secondary branchesand indeed tertiary branches that we do not investigate here.

3.1. Maximal isotropy branches

Standard equivariant branching results [17] imply that thereare generically branches of periodic solutions with maximalsymmetries Z3 and D3 bifurcating from λ = 0. There isalso a branch of equilibria with maximal symmetry D4 × S1

bifurcating from δ = 0. Table 3 gives the branching equationsto leading order.

3.2. Global bifurcations of (x, 0, 0, 0)

A branch of solutions (x, 0, 0, 0) emerges at δ = 0via a transcritical bifurcation (see Table 3(b)) that is locallyunaffected by the value of B4. The term with coefficient B4does however affect the global behavior of the branch, and onincluding this third-order term we obtain a bifurcation scenario


Table 3Branching conditions for the maximal isotropy branches of (a) periodic orbits and (b) equilibria that bifurcate at δ = λ = 0

Symmetry Typical point Branching equation Number

D3 (0, z, z, z) |z|2 = −λ(A1r + 2A2r + 2A3r )−1 4

Z3 (0, z, ζ z, ζ 2z) |z|2 = −λ(A1r + 2A2r − A3r )−1 6

(a) Branches of relative equilibria bifurcating at λ = 0 for all δ. The number indicates the number of distinct periodic orbits in the group orbit of this relativeequilibrium.

Symmetry Typical point Branching equation

D4 × S1 (x, 0, 0, 0) x = −δ/B1

(b) Branches of equilibria branching at δ = 0 for all λ.

Only leading order terms for the branching are given. In particular, the steady branch of D4 × S1 symmetric solutions bifurcates transcritically at δ = 0 while theB4 terms determine the global properties of the branch.

Table 4

Branches of periodic orbits that generically exist in any neighbourhood of the origin (with x = o(|z|2) given by z0 = Z ) as long as 0 < |λ| |δ|

Symmetry Typical point Branching equation Number (Eqn)

D4 (x, z, 0, 0) |z|2 = −λ(A1r − 2A4r B4/δ)−1 3 (15)

D2 (x, 0, z, z) |z|2 = −λ(A1r + A2r + A3r − A4r B4/δ)−1 3 (17)

Z4 (x, 0, z, i z) |z|2 = −λ(A1r + A2r + A3r − A4r B4/δ)−1 6 (17)

Observe that the branches D2 and Z4 have identical branching equations for the third-order normal form; this is broken if one considers for example fifth-orderterms in the normal form. The final column refers to the equation that gives the location of secondary bifurcations on that branch.

Fig. 2. Diagram showing the bifurcation of steady solutions (x±, 0, 0, 0) of(12) on varying δ for fixed B1 and B4. This shows the case B1 > 0 and B4 < 0.Note that there are three pairs of such branches given by (ρk x±, 0, 0, 0) fork = 0, 1, 2.

consistent with the one that we observe in the hydrodynamicsystem. This is similar to the method used for examining otherbifurcations where transcritical branches appear [30].

On including the higher order terms in (12) one canexplicitly solve for branches of steady solutions of the form(x±, 0, 0, 0) to obtain

x± = −B1

2B4

(1 ±

√1 −

4δB4

B21

). (13)

These branches are created at a saddle-node bifurcation at δ0and exist for δ such that

4δB4 ≤ 4δ0 B4 where δ0 =B2

1

4B4.

As expected, the x− branches bifurcate transcritically at theorigin at δ = 0. Fig. 2 illustrates these bifurcations within theinvariant subspace (z, 0, 0, 0).

3.3. Other solutions branching from λ = 0

It follows from the analysis in [5] that for all δ 6= 0 there is aHopf bifurcation at λ = 0 creating a large number of branches

of periodic solutions. Some of these are listed in Table 3 butthis is not a full list of all generic branches that exist. In orderto obtain the branches, we need to perform a center manifoldreduction onto a manifold given by z0 = Z(z1, z2, z3) valid for|λ| |δ|. By symmetry, this manifold must have the form, atquadratic order

Z(z1, z2, z3) = α(|z1|2+ ζ 2

|z2|2+ ζ |z3|

2),

with α real depending on δ. Substituting this into the Eq. (12)we can determine that in the case λ = 0, δ 6= 0 we have

α = −B2

δ

and so the Hopf bifurcation at λ = 0 is determined by thenormal form on C3

z1 = (λ+ iω)z1 + A1|z1|2z1 + A2(|z2|

2+ |z3|

2)z1

−A4 B2

δ(2|z1|

2− |z2|

2− |z3|

2)z1 + A3(z22 + z2

3)z1

z2 = (λ+ iω)z2 + A1|z2|2z2 + A2(|z1|

2+ |z3|

2)z2

−A4 B2

δ(2|z2|

2− |z1|

2− |z3|

2)z2 + A3(z21 + z2

3)z2

z3 = (λ+ iω)z3 + A1|z3|2z3 + A2(|z1|

2+ |z2|

2)z3

−A4 B2

δ(2|z3|

2− |z1|

2− |z2|

2)z3 + A3(z21 + z2

2)z3

(14)

The remaining branches are listed in Table 4. One can observethat the branches can connect up with those branching from(x±, 0, 0, 0) in certain circumstances.

There are also periodic and quasiperiodic solutionsin subspaces such as (x, z1, z2, z2) and (w, z1, z2, 0),corresponding to other submaximal symmetries. Branches withthese symmetries can branch directly from the origin if the


coefficients lie in some open set. Conditions on the coefficientsthat imply the existence of such solutions and analysis of theirstability can be found in [5].

3.4. Secondary bifurcations from (x±, 0, 0, 0)

The symmetry group of the steady states (x±, 0, 0, 0) isthe D4 generated by ρ100 and κ+

011 (see Table 2). In a systemwith a non-trivial group of symmetries, bifurcations of a steadystate can be classified according to the action of the symmetrygroup on the center eigenspace. The group D4 has the fiveirreducible representations listed in Table 5. In the system(12) the center eigenspace of (x, 0, 0, 0) is generically either(0, z, 0, 0) or (0, 0, z1, z2). For the O representation, the actionof D4 on the center eigenspaces corresponds to the U2 or U4irreducible representations; for O− – to the U1 or U4 irreduciblerepresentations.

Hence, periodic orbits with the following symmetries canbifurcate from the steady states (x, 0, 0, 0):

Symmetry Typical point Number

D4 (x, z, 0, 0) 3D2 (x, 0, z, z) 3Z4 (x, 0, z, i z) 6

Bifurcation of these branches requires that the (x, 0, 0, 0)solutions exist, and so they can only bifurcate in regionswhere 4δB4 ≤ B2

1 . Stability of these steady states inthe directions z1, z2, z3 is controlled by the respectiveeigenvalues:

Eigenvalue Eigenspace Multiplicity

λ+ iω + 2A4x± + A5(x±)2 (0, z, 0, 0) 2

λ+ iω − A4x± + A5(x±)2 (0, 0, z1, z2) 4

This implies that a standard Hopf bifurcation from(x±, 0, 0, 0) occurs whenever

λ−A4r B1

B4

(1 ±

√1 −

4δB4

B21

)+

A5r B21

4B24

(1 ±

√1 −

4δB4

B21

)2

= 0. (15)

One can locate these bifurcations by solving

λ2 B24 + δ2 A2

5r − 2λδA5r B4 + δ(

4B4 A24r − 2B1 A4r A5r

)+λ

(B2

1 A5r − 2B1 B4 A4r

)= 0. (16)

from which one can observe that there is a path of suchbifurcations in the (λ, δ) plane passing through the origin.There is a Hopf bifurcation from (x, 0, 0, 0) with D4 symmetry[31] whenever

λ+A4r B1

2B4

(1 ±

√1 −

4δB4

B21

)+

A5r B21

4B24

(1 ±

√1 −

4δB4

B21

)2

= 0. (17)

Table 5

Irreducible representations of the group D4 generated by ρ100 and κ+

011; x isreal and z complex

Name Dimension Action of generators

U0 1 ρ100(x) = x, κ+

011(x) = x

U1 1 ρ100(x) = −x, κ+

011(x) = x

U2 1 ρ100(x) = (x), κ+

011(x) = −x

U3 1 ρ100(x) = −(x), κ+

011(x) = −x

U4 2 ρ100(z) = i z, κ+

011(z) = z

Rewriting this in a similar way to (16) implies that

λ2 B24 + δ2 A2

5r − 2λδA5r B4 + δ(

B4 A24r + B1 A4r A5r

)+λ

(B2

1 A5r + B1 B4 A4r

)= 0. (18)

The bifurcations at (16) and in particular (18) give rise to manybranches, some of which we investigate for a specific choice ofcoefficients in Section 4.

4. Bifurcations for the reduced hydrodynamic system

4.1. Computation of the normal form coefficients

We now return to the hydrodynamic system (1)–(6) near thecritical value Rc = 13.044. The eight-dimensional subspacecorresponding to the eigenvalues with largest and second-largest real parts near the Hopf bifurcation of uABC reduces to asix-dimensional representation of O (corresponding to the Hopfvariables) and a two-dimensional representation of O (see alsoSection 5.2 and Fig. 7(a)). One can perform a reduction to acenter manifold whose third-order truncation is the normal form(12) and that is tangent to this subspace. The coefficients forthis normal form are obtained by the center manifold methoddescribed in [26]. Although there is no guarantee that thetruncated normal form dynamics will be a good approximationof the ‘true’ dynamics (except near the Hopf bifurcation) wewill see that it works very well for a range of R. As thebifurcation parameter we use

ε = 1/R − 1/Rc

so that the Hopf bifurcation takes place at ε = 0. Wepresent a brief summary of the method used to approximate thecoefficient values.

A center manifold is an invariant manifold tangent to thelinear space spanned by central eigenvectors of the linearizedsystem. The theorem [26], in which center eigenvalues withnon-vanishing real parts are allowed, states that a (local) centermanifold exists for finite-dimensional dynamical systems and itis Ck-smooth if

k|α±| < |β±|, (19)

where α+ and α− are upper and lower boundaries for real partsof center eigenvalues, β+ is a lower boundary for real parts ofunstable eigenvalues, β− is an upper boundary for real parts ofstable eigenvalues. If there are no eigenvalues with positive realparts, the center manifold is attracting. In [27] the theorem isproved for a class of infinite-dimensional systems, which satisfy


an additional hypothesis similar to the one introduced in [33],and it is shown that the Navier–Stokes equation (2) and (3) forthe boundary conditions (5) belongs to this class, provided (19)holds.

Fig. 7(a) shows the real parts of eigenvalues of thehydrodynamic system (1)–(6) linearized near uABC. We chooseα1 and α2 as center eigenvalues and perform the reduction atR = Rc, giving α2 ≈ −0.02667 and α3 ≈ −0.07666. Note that2|α2| < |α3| implying the existence of a C2-smooth manifold.The smoothness is sufficient for calculation of the coefficientsin (12), because second-order terms of the Taylor expansionof the center manifold are required for computation of thethird-order normal form coefficients [26]. Since the unstableeigenspace is empty, the center manifold is attracting.

Let L0 denote the linear part of (2) near uABC for R = Rc,Xc the eight-dimensional subspace spanned by eigenvectorsof L0 associated with eigenvalues with the two largest realparts, and Xh the L0-invariant subspace, complementary to Xcin the space of divergence-free vector fields. Let z and w becoordinates in Xc and Xh , respectively. We choose a coordinatesystem (z0, z1, z2, z3) ∈ C4 in Xc that corresponds to the actionof O given by (7)–(10) and set z j = x j + iy j . A mappingψ : Xc → Xh defines an invariant manifold of (2) if

4∑j=1

∂ψ

∂z jz j = w, (20)

where w = ψ(z), z j and w are projections of (2) into therespective subspaces.

A Taylor expansion of ψ(z) can be constructed bysubstituting the expansion into (20), equating expressions infront of equal powers of zl and solving successively theresulting equations of the form L0(ξ) + Aξ = b. Substitutingthe expansion into (2) and projecting into Xc gives a restrictionof the equation on the center manifold.

The original hydrodynamic system (1)–(6) can be reduced toan eight-dimensional center manifold near the Hopf bifurcationof uABC. Truncation of this center manifold system at thirdorder yields the reduced system, whose bifurcations in thevicinity of the trivial steady state uABC for 13.044 ≤ R ≤ 13.9(i.e. −0.0047 < ε ≤ 0) were studied numerically in [26].After a near-identity change of variables the system takes thesimplified form (12). In this paper we investigate the reducedsystem (12) with coefficients in Table 6 over a larger range of εthan in [26]. We examine bifurcations of steady states, periodicorbits and tori that are not necessarily attracting, both nearand far from the trivial steady state, using both analytical andnumerical techniques. In particular, we perform path-followingcomputations using XPPAUT on varying ε in the reducedsystem.

To check the quantitative and qualitative similarity ofattractors of the original and the normal form systems inSection 5, we compare xi (t) and yi (t) with Pxi (v(t)) andPyi (v(t)), respectively. Here xi (t) and yi (t) are components ofthe attracting solutions to (12), the Pxi ,yi are projections on therespective coordinate vectors in Xc and v(t) is a solution of thehydrodynamic system (1)–(6).

Table 6Coefficients calculated for the reduced normal form (12) of the hydrodynamicsystem (1)–(6) as functions of ε = 1/R − 1/Rc

Coefficient Value

B1 −0.1437870B2 −1.383707B3 1.248662B4 −0.03892806

A1 −1.369883 − 23.03124iA2 −0.07483223 + 10.80340iA3 −0.1102290 − 0.2098525iA4 0.02117706 + 0.4696667iA5 −0.3728020 − 0.5649899i

δ −0.02667258 − 0.7347284ελ −3.783456εω 0.2792756 − 1.296154ε

4.2. A summary of bifurcations for the reduced hydrodynamicsystem

A complicated sequence of bifurcations occurs on varyingε for the reduced normal form (12) with parameters fromthe hydrodynamic system as in Table 6. We only showa small selection of the branches and bifurcation points;there are many more that include branches of tori, perioddoublings, homoclinic bifurcations and solutions that have noremaining symmetries. A selection of the bifurcations andinterconnections between branches are shown in Fig. 3. Thesebranches were found using a mixture of the analysis in theprevious section (to locate the points A, B, H, I, P, T, U) andnumerical continuation from selected solutions using XPPAUT[14]. Fig. 3 is a schematic diagram that summarizes thebifurcations occurring and the interconnection of branches.

Recall that increasing R is equivalent to decreasing ε andR = Rc corresponds to ε = 0. A trivial steady state is a uniqueattractor for ε > 0.1444. The system becomes bistable at thesaddle-node bifurcation A. The trivial solution loses stability ondecreasing ε through B to give branches of periodic solutionswith Z3 symmetry. These attractors undergo a sequence ofbifurcations C–D–E–J changing at C from periodic solutionsto 2-tori, then at D to doubled tori and by a complicatedsequence of bifurcations to chaotic attractors that merge at E .

Periodic solutions of type (0, z, ζ z, ζ 2z) are stable on B–Cand undergo a Hopf bifurcation at C to a stable torus on C–D.Fig. 4 shows some projections of attractors for this system onthe branch C–D–E . The doubling and chaotic solution shownin Fig. 4(e,f) cannot be modelled by the Hopf normal form usedin [5]. By contrast to the reduction considered in that paper,the availability of extra degrees of freedom due to the steadyinteraction allows us to model the dynamics of a hydrodynamicsystem similar to D–E without having to break the normal formsymmetry.

The reduced model also manages to model the furtherbreakdown of the attractor to a more chaotic state as shownin Fig. 5. Observe the symmetry increasing bifurcation at Ebetween ε = −0.00055 and ε = −0.00057 where severalattractors with average symmetry Z3 merge to a single attractor


Fig. 3. Bifurcation diagram showing a notional solution norm on varying ε for the reduced system (12) with parameters as in Table 6. Stable branches are shownby bold lines; note that there are two types of attractor in the region A–J . The branch B–C is a Z3 periodic attractor that undergoes a number of bifurcationsC–N–O–J . For values of ε lower than J only the stable solution created at the saddle node A appears as an attractor. The diagram shows only a small selection ofbranches and some secondary bifurcation points, of which most (F,G, N , O, Q, S) are Hopf bifurcations that lead to branches of tori. N.B. the diagram is not toscale either vertically or horizontally; we reproduce only the ordering in ε and the interconnections between branches.

with full symmetry on average. Similar fully symmetric chaoticattractors are observed down to a crisis of chaotic attractors [18]J at ε ∼ −0.005 beyond which only chaotic transients and theattractors (x+, 0, 0, 0) can be found.

Fig. 3 shows (as predicted in [5]) five different symmetryclasses of primary branches of periodic solutions from thetrivial solution with full symmetry, given by the five maximalisotropy subgroups for the Hopf bifurcation at B. Thereare also primary branches of equilibria that pass throughthe steady transcritical bifurcation at P . Secondary periodicbranches bifurcate from the steady branch (x, 0, 0, 0) atHopf bifurcations H, I, T,U . The bifurcations at H and Thave D4 symmetry and so produce several branches. Theother bifurcations shown produce a maze of interconnectingbranches, some of which appear as unstable structures in thechaotic attractor found near the point J . None of the latter areshown on the diagram.

5. Bifurcations of solutions to the hydrodynamic system

In this section we present numerical simulations of time-dependent solutions to (1)–(6) for 0 < R ≤ 25 (correspondingto ε > −0.03666). We compare these to the observed

bifurcations using the symmetry-based analysis in Sections 3.2and 4.2. We also summarize some results previously reportedin [24,5,26].

5.1. Symmetries and Fourier coefficients

Standard pseudospectral methods [6] were used to computesolutions to the Navier–Stokes equations. We represent aspatially periodic solution v(t) to the hydrodynamic system(1)–(6) as a Fourier series

v(t) =

∑k

vk(t)eikx . (21)

Differentiation was performed in the Fourier space, thenonlinear terms were evaluated in the physical space and fastFourier transformation was used to switch between the two.Computations were performed with the resolution 163 Fourierharmonics for R ≤ 14 and 323 for larger R. Integration in timewas done using the fourth-order Runge–Kutta scheme.

The symmetry group of the system and its action in physicalspace are described in [5]. The action of the group O on thespace of Fourier coefficients can be decomposed into a directsum of five isotypic components F = W0 ⊕ W1 ⊕ W2 ⊕

W3 ⊕ W4, corresponding to five irreducible representations


Fig. 4. Numerically obtained attractors for the reduced system (12) with parameters as in Table 6, (a,b) ε = −0.000409, (c,d) ε = −0.000490 and (e,f)ε = −0.000495 projected into the (x1, x2) and (x1, x0) planes, where zk = xk + iyk . The period-doubling bifurcation marked D in Fig. 3 occurs between(a,b) and (c,d). In all cases we show a projection of the trajectory onto y1 = 0. Observe that x0 is non-zero on these attracting orbits. For cases (a,b) and (c,d) thereare attracting periodic orbits that are related by a period doubling at ε = −0.0004577. The attractor in (e,f) is apparently chaotic.

of O. Relations between Fourier coefficients for vectors fromeach isotypic component are given (see [24]).

To visualize the temporal behavior of solutions, we considerprojections

(v(t) · qk) ≡

∫Ω

v(t) · qkdx3

for

q1 = (0, cos x1,− sin x1), q2 = (0, cos x1, sin x1).

q3 = (0, sin x1, cos x1), q4 = (cos x2, 0, sin x2),

q5 = (sin x3, cos x3, 0),

Note [24] that q1 ∈W0, q2 ∈W1 and qi ∈W2 for i = 1, 2, 3,where W0, W1 and W2 correspond to the O, O− and D3representations of O. Since isotypic components are mutuallyorthogonal, these projections can be used to isolate dynamicalbehavior corresponding to the respective isotypic components.

For a vector field v presented as (21) and a symmetry γdefine n(γ, v) = ‖v − γ (v)‖, with the norm ‖v‖ = max j,k |v

jk|

(in simulations, we identify v as having a symmetry γ if n(γ, v)is below the accuracy of computation). We define E(v) =12

∑k v2

k, the kinetic energy of the flow v.

5.2. Steady states

In addition to the trivial steady state uABC the system (1)–(6)has three branches of steady states possessing symmetry groupD4. The symmetry ρ111 cyclically permutes the branches. Threepairs of stable and unstable steady states, denoted by u+

i (R) andu−

i (R), i = 1, 2, 3, appear at R = 7.9 (ε = 0.0499) in a saddle-node bifurcation (see Fig. 6). At R = 190 the three u−

i (R)intersect with uABC via a transcritical bifurcation generic tosystems with the symmetry group D3 [24]. The saddle-node andtranscritical bifurcations appear in the truncated normal formas the bifurcations marked A and P respectively (see Fig. 3),though at parameter values that are not very close to these ofthe hydrodynamic system.

Fig. 7(a) shows the four largest real parts of eigenvalues ofthe linearization of (2) at uABC. The dominant eigenvalue α1(i.e. that with the largest real part) is complex and crosses theimaginary axis at Rc = 13.044 (ε = 0) giving rise to instabilityof uABC. The associated eigenspace belongs toW0. For R = Rcthe eigenvalue α2 with the second largest real part is real, andits associated eigenspace belongs to W2. The next eigenvalueα3 = −1/R is associated with the six-dimensional eigenspaceof space-periodic functions satisfying u = ∇ × u [22]. This


Fig. 5. For more negative values of ε for the reduced systems (12) with parameters as in Table 6 we obtain attractors (a,b) ε = −0.00055, (c,d) ε = −0.00057and (e,f) ε = −0.00070. These are shown as time series of x2 and projected in the (x2, x3) planes. There is a symmetry increasing crisis between (a,b) and (c,d)that results in an intermittent switching between neighbourhoods of the group orbit of the attractors shown in (a,b). The rate of switching increases on decreasing εfurther from these values. On continuing the time series in (c,d), it explores all symmetric images of (a,b) in an intermittent manner.

Fig. 6. The products (u+

1 (R),q3), (u−

1 (R),q3) and (uABC,q3) (vertical axis)as functions of the Reynolds number (horizontal axis). Stable and unstablestates are shown by solid and dashed lines, respectively.

eigenspace can be decomposed as a sum of three subspacesbelonging to theW2,W3 andW4 components. The eigenvalueα4, which has the second largest real part for R > 23, iscomplex and its associated eigenspace belongs toW1.

Fig. 7(b) shows real parts of eigenvalues of the linearizationof (2) at u+

i (R). The eigenspace associated with the complexeigenvalue β1, dominant for R < 22, has one complexdimension, it corresponds to the U1 representation of D4 (seeTable 5). The eigenvalue β2 is also complex and the actionof the group on the associated eigenspace is trivial (the U0representation). The eigenspace associated with the complexeigenvalue β3 belongs to the U2 component.

5.3. Bifurcations for R ≤ 13.9

The first bifurcation of the system – the emergence ofthree branches of steady states via a saddle-node bifurcationat R = 7.9 – is discussed in the previous subsection. Afterthis bifurcation the system has three steady attractors u+

i (R), inaddition to the attractor uABC.

When R increases above the Hopf bifurcation Rc = 13.044,uABC becomes unstable and eight periodic attractors of the typeZ3 (see Table 2 in [5]) with frequency f1 ≈ 0.043 emerge; theyremain attractors for R ≤ 13.09. In the interval 13.044 ≤ R ≤

13.13 (−0.0005 ≤ ε ≤ 0) in the vicinity of uABC the system has


Fig. 7. Real parts of eigenvalues with largest real parts of the hydrodynamic system (1)–(6) linearized near uABC (a) and u+

1 (R) (b). The horizontal axis shows theReynolds number.

eight symmetry-related attractors that undergo a complicatedsequence of bifurcations.

At R = 13.1 (ε = −0.00032) the behavior becomesquasi-periodic: a torus with a second frequency, f2 ≈ 0.0017,emerges at a Hopf bifurcation of the periodic orbit at R = R1(13.09 < R1 < 13.1). The next bifurcation at R = R2(13.1 < R2 < 13.11, ε ∼ −0.00035) is a torus doubling wherea frequency f2/2 emerges. Further on, there is a transition to3-frequency quasiperiodicity at R = R3 (13.117 < R3 <

13.1175). There are many more bifurcations of the attractorsin this region; for our simulations we found at R = 13.1175that the emerging frequency f3 ≈ 0.00011 is very close(but not exactly equal) to f2/18, and f3 varies with R muchfaster than f1 and f2. Attractors found in computations are:R = 13.1177 – a 3-torus; R = 13.118 – a 2-torus withthe main frequencies f1 and f2/14; R = 13.119 – a 2-toruswith the main frequencies f1 and f2/10; R = 13.1195 – a2-torus with the main frequencies f1 and f2/16; R = 13.12and 13.13 – chaotic, reminiscent of the former 3-torus whichbecomes unstable. In fact, at the small interval 13.1177 ≤ R ≤

13.1195 we found new attractor at each value of R considered,suggesting that the system may even lack structural stabilityon this interval. However one clearly needs to be very cautiousin interpreting the fine details of these observed bifurcations,due to the sensitivity to numerical inaccuracies and roundingeffects.

At R = R4 (13.13 < R4 < 13.14, ε ∼ −0.000513) theeight symmetry-related chaotic attractors merge into a singleattractor possessing on average all symmetries of the system –in the terms of [10] this is a symmetry increasing bifurcation.For 13.14 ≤ R ≤ 13.8 the system possesses a single chaoticattractor in the vicinity of the trivial steady state, and thishas on average all symmetries of the system. For R ≥ 13.85(ε ≤ −0.00446) sample trajectories, initially close to the trivialsteady state, are attracted by one of the u+

i (R), which are stableup to R = 13.91.

5.4. Comparison of hydrodynamic and reduced systemdynamics

All bifurcations found in the hydrodynamic system for R ≤

13.9 (ε ≥ −0.00472; except for the interval 13.1177 ≤

R ≤ 13.1195 where the system is structurally unstable) canbe identified in the sequence of the normal form bifurcations(Fig. 3), although at values of ε that get more inaccurate aswe get further from ε = 0. For example, the Hopf bifurcationlabelled C occurs at ε = −0.0003641 in the normal formsystem and at ε = −0.00032 for the hydrodynamic system.Moreover, there is a marked similarity of Poincare sections ofthe hydrodynamic (Fig. 8) and normal form attractors (Fig. 4)at similar points in the bifurcation sequence.

5.5. Instability of u+

i

For R slightly larger than 13.91 the steady states u+ becomeunstable and periodic orbits appear via a Hopf bifurcation. Thebranches (x±, 0, 0, 0) can undergo two distinct types of Hopfbifurcations (see Section 3.2) but no bifurcations of the x+

branch are found for the normal form (12) with coefficientslisted in Table 6. The observed bifurcations of u+

i (R) in thehydrodynamics system cannot be reproduced by the normalform (12) due to the difference in action of the symmetries: forthe (x±, 0, 0, 0) bifurcations, the center eigenspaces correspondeither to the U2 or U4 representations of D4, while for theinstability of u+

i (R) the group is acting on the eigenspace asin U1 representation.

One could perform the center manifold reduction near uABCfor an R in the interval 19 < R < 25, assuming that thecenter subspace is spanned by eigenvectors associated witheigenvalues α2 and α4. On this interval |α2| < 2|α3| and|α4| < 2|α3|, thus Theorem 3 in [26] guarantees the existence ofa C2-smooth center manifold. As mentioned in Section 5.2, theaction of the symmetry group O on the eigenspaces associatedwith α2 and α4 is isomorphic to the D3 and O− representations.

As found in Sections 2 and 3 (results of these Sections arealso applicable for the O− representation), there are two typesof Hopf bifurcations of u±

i (R) with eigenspaces correspondingto the U1 or U4 representations of D4. For the instability ofu+

i (R) the action of D4 on the center manifold is isomorphicto the U1 representation. Thus, the bifurcation of u+

i (R) can beexplained considering the steady-state/Hopf mode interaction,where the eigenspace associated with the complex eigenvaluecorresponds to the O− representation of the group O. Thesymmetry group of the emerging periodic orbit is D4 (see


Fig. 8. Projections of Poincare sections Py1 (v) = 0 for attractors of the hydrodynamic system at R = 13.1 (ε = −0.000328) (a,b), R = 13.117 (ε = −0.000427)(c,d) and R = 13.12 (ε = −0.000444) (e,f). On (a,c,e) the x-axis – Px1 (v), the y-axis – Px2 (v); on (b,d,f) the x-axis – Px1 (v), the y-axis – Px0 (v).

Table 2), (11) implying that the periodic solution lies withinthe fixed point subspace of D2.

The next observed bifurcation of the hydrodynamic systemon increasing R is a Hopf bifurcation of periodic orbits to tori(one for each u+

i ), the latter also have the point symmetry groupD2. These tori are attractors for 14.0 ≤ R ≤ 15.0.

5.6. Heteroclinic cycles

As pointed out above, for R > 13.91 attractors of thehydrodynamic system involve modes belonging to the O−

representation, which cannot be modelled by the normal form(12); hence we do not discuss these transitions in terms of ε. ForR ≥ 14.7 in addition to tori we find a range of attractors withintermittent dynamics that we believe are robust heterocliniccycles. In order to explain the bifurcations occurring at R ≥

14.7 we first consider a restriction of the hydrodynamic systemto an invariant subspace.

5.6.1. Dynamics in the ρ2100-fixed subspace

Denote by V100 the invariant subspace fixed by ρ2100, and by

V 1,+100 the invariant subspace fixed by ρ100. Both subspaces are

invariant subspaces for the hydrodynamic system and all of thesteady states discussed so far – uABC and u±

i for i = 1, 2, 3 –belong to V100. The states uABC and u±

1 belong to V +

100 if wechoose such numbering that ρ100u+

1 = u+

1 .

Consider the hydrodynamic system restricted to V100. ForR > 13.044 the unstable subspace of uABC is in V 1,+

100 . AfteruABC becomes unstable, a stable (within V100) periodic orbit inV 1,+

100 emerges. However, the orbit is destroyed at higher R in acollision with u−

1 (see line B–C–J in Fig. 3). As discussed in

Section 5.2, for R ≤ 19 u+

1 (R) is stable in the V 1,+100 subspace.

Thus, trajectories in V +

100 starting near uABC are attracted byu+

1 (R) and there is a two-dimensional set of trajectories from


u+

1 to uABC in this interval of R. In the normal form (12) we canalso see this family of trajectories.

Denote by V 2,+100 the κ+

011-symmetric subspace of V100, whichis also an invariant subspace for the hydrodynamic system. InV 2,+

100 uABC is stable for R < 23.1 and u+

1 (R) is a saddlefor R > 13.91. Thus, a robust connection from u+

1 (R) touABC is possible. We observe in the numerics that there is aconnection for R = 14.7 giving rise to what we believe is aheteroclinic cycle with a continuum of connecting orbits. SinceV 2,+

100 belongs to the O− component of the phase space, thispart of the cycle cannot be reproduced by the normal formconsidered in Section 4.

Consider a general system x = f (x) with a non-trivialsymmetry group Γ . Suppose ξ j , j = 1, . . . ,m, are hyperbolicequilibria of the system such that dim W u(ξ j ) ≥ 1 and

W u(ξ j ) \ ξ j ⊂

⋃γ∈Γ

W s(γ ξ j+1) (22)

for all 1 ≤ j ≤ m (where it is denoted as ξm+1 = ξ1). Then theset of group orbits of unstable manifolds

X = W u(γ ξ j ), j = 1, . . . ,m, γ ∈ Γ

constitutes a heteroclinic cycle [21]. The cycle is structurallystable (i.e. persists under small Γ -invariant perturbations) if foreach j there exists a fixed-point subspace Pj of an isotropysubgroup of Γ such that W u(ξ j ) ⊂ Pj and ξ j+1 is a sink inPj . The cycle is asymptotically stable (i.e. trajectories startingnear the cycle are attracted to it) if the inequality

m∏j=1

min(c j , e j − t j ) >

m∏j=1

e j , (23)

holds, where e j , c j and t j are, respectively, expanding,contracting and transverse eigenvalues of the linearized systemat ξ j (for definitions of the eigenvalues, see [21]).

In our case there are two hyperbolic equilibria: uABC andu+

1 (m = 2). We cannot prove or check numerically that thecondition (22) is satisfied as the unstable manifolds are two-dimensional; indeed the unstable manifold of uABC presumablycontains a large number of possible invariant sets. However, itdoes appear to be the case that within V100 we have

W u(uABC) \ uABC ⊂ W s(u+

1 )

since in our simulations any trajectory starting in V100 nearuABC is attracted to a symmetric image of u+

1 . We find thatthis occurs similarly in the normal form for any ε < −0.005.Computations suggest that the condition W u(u+

1 ) \ u+

1 ⊂

W s(uABC) is satisfied for R close to R = 14.7 as well.Thus by the definition above uABC and u+

1 are connected bya heteroclinic cycle. These cycles are in effect more complexversions of robust heteroclinic cycles that manifest themselvesas bursting to the origin [19].

The cycle, should it exist, is structurally stable since we canchose P1 = V 1,+

100 and P2 = V 2,+100 . As follows from Section 5.2

and Fig. 7, the condition (23) for asymptotic stability of the

Fig. 9. Projection of trajectories on (q1,q2,q3) for R = 14.7 (a), R = 15.3(b) and R = 15.5 (c) for trajectories in the subspace V100. (v(t),q3) – x-axis, (v(t),q1) – y-axis, (v(t),q2) – z-axis. For (a,c) only one transitionuABC → u+

1 → uABC is displayed while for (b) several transitions u+

1 ↔ u−

1are displayed.

cycle takes the form

Re(α4) Re(β3) > Re(α1) Re(β1) (24)

which holds for R ≤ 17.7.Fig. 10(a) shows the observed cycles in simulations of the

hydrodynamic system. There appear to be intervals of constantenergy and rapid transitions between them, for R = 14.7and a randomly chosen initial condition in V100 in the vicinityof uABC. Fig. 10(b–d) show that the trajectories repeatedlycome very close to the invariant subspaces for periods of time,before moving away. In particular, the figure shows that after atransient the trajectory approaches points with symmetry ρ100to within numerical accuracy 10−14, for the transition from


Ftr

ig. 10. Graphs of E(v(t)) (a), n(ρ100, v(t)) (b), n(κ011, v(t)) (c) and n(ρ010, v(t)) (d) (vertical axes) as functions of time (horizontal axis) for R = 14.7 for aajectory in the V100 subspace.

Fig. 11. Graphs showing E(v(t)) (vertical axis) as a function of time (horizontal axis) for R = 15.1 (a) and R = 15.3 (b) for trajectories in the subspace V100.

uABC to u+

1 (R) (e.g. during 2000 < t < 4000). For thetransition the other way the trajectory approaches points withsymmetry κ001 (e.g. during 4000 < t < 8000), confirming thatthe heteroclinic trajectories occur within invariant subspacesfixed by these group elements. Fig. 9(a) also shows that onepart of the trajectory belongs to V 1,+

100 (horizontal plane) and

the remaining part to V 2,+100 (vertical plane). The steady states

that are connected are in the intersection of the two invariantsubspaces.

For 14.8 ≤ R ≤ 15.0 the behavior is similar to that forR = 14.7. For R in the interval 15.0 < R < 15.1, the

unstable set of u+

1 (R) collides with the attracting torus resultingin the demise of the torus and a more complex structure ofthe unstable set of u+

1 (R) (cf. the behavior of trajectories afterthey leave u+

1 (R) for R = 14.7 and R = 15.1 on Fig. 10(a)and 11(a)). For R = 15.2 and 15.3 the trajectory is shownfollowing heteroclinic connections between u+

1 and u−

1 . The

section u+

1 to u−

1 lies in V 2,+100 and the return section lies in

V ++

100 (see Fig. 9(b)). For 15.4 ≤ R ≤ 17 the behavior isagain similar to that for R = 14.7, the trajectory follows theuABC ↔ u+

1 heteroclinic cycle (see Fig. 9(c)). For larger R,intervals between successive visits of uABC become smaller and


Fig. 12. Graphs showing E(v(t)) (a) and n(ρ2100, v(t)) (b) (vertical axis) as functions of time (horizontal axis) for R = 14.7.

Fig. 13. Projections of Fourier modes of the velocity (v(t),q3) (solid line), (v(t),q4) (dashed line), and (v(t),q5) (dotted line) as functions of time (horizontalaxis) for (a) R = 16 and (b) R = 20.

the overall behavior is less regular. For R ≥ 18 a trajectory isattracted by steady states u+

i , i = 2 or 3, which are stable in thesubspace V001.

We suggest the following explanation for the observedbehavior. The heteroclinic cycle connecting uABC and u+

1 (R)ceases to exist at R ≈ 15.2 when a connection from u+

1 (R) tou−

1 (R) emerges. The connection disappears at R ≈ 15.4 withthe cycle restored. Graphs of eigenvalues displayed on Fig. 7imply that (24) holds for R ≤ 17.7, which is consistent withinstability of the cycle for R ≥ 18. For an R in the interval15.0 < R < 15.4 a heteroclinic cycle connecting u+

1 and u−

1can exist, but it is structurally unstable since the subspaces Piinvolved in the condition for structural stability do not exist (seeFig. 9(b)).

5.6.2. Attractors for larger RSimilarly to V100, one can define subspaces V010 and

V001. Since the subspaces are cyclically mapped by the ρ111symmetry, all the above results for V100 are applicable for theother two subspaces. Below we refer as V0 to one of the threesubspaces.

For R ≤ 15.9 the temporal behaviors of solutions withoutsymmetry restrictions and of those with the symmetry imposedare similar (cf. for instance Fig. 10(a) and 12(a)), a trajectory

does not escape for a long time from the subspace V100 (seeFig. 12(b)). There are three distinct attractors related by the ρ111

symmetry.

For R ≥ 16 the three attractors join into one and a trajectoryjumps between the former attractors. For R = 16 the jumpsare rare, after making several cycles along a V0-attractor, itswitches to another one, the transition happens near uABC (seeFig. 13(a)). For R = 20 a trajectory jumps to another formerattractor just after one iteration (see Fig. 13(b)), for R = 25 thecycles are separated by intervals of irregular behavior.

A heteroclinic cycle will still exist for the system inthe original phase space (i.e. without symmetry restrictions)because of the robust connections between uABC and u+

1 . Thiscycle includes the cycle considered in the previous subsectionand all its images under the action of symmetries ρ j

111, j = 1, 2.We believe that our arguments establishing the existence andstructural stability of the heteroclinic cycle can be adapted tothese cases, and that condition (24) remains the condition forasymptotic stability of the cycle for R ≤ 17.7. An interestingopen question is why for R ≤ 15.9 a trajectory follows just onesubcycle uABC ↔ u+

j while for larger R it jumps between allthree subcycles.


6. Conclusions

We have investigated the dynamics of a normal formfor generic interaction of Hopf and steady bifurcations withsymmetries O. This was used to successfully extend themodelling commenced in [5,26] of a very complicated sequenceof bifurcations and attractors that appear in the modelhydrodynamic system (1)–(6). As in [5,26] the assumption ofspatial periodicity of all solutions means that we have notconsidered any ‘long wavelength’ instabilities of these patterns;we merely note that these are certainly of significance for theReynolds numbers we consider.

The dynamics of the reduced normal form agrees wellwith several bifurcations found in the hydrodynamic systemup to R = 13.91 (ε = −0.004773). For R > 13.91 wefind more complex attractors in the hydrodynamic system thatinvolve modes not included in the normal form model; inparticular we have found evidence that near R = 14.7 thereare attractors consisting of robust heteroclinic cycles betweenthe ‘trivial’ steady solution uABC and symmetry broken steadysolutions. In principle, this could be modelled using a similarcenter manifold reduction, including one extra type of modesto produce a 14-dimensional system. The high dimension ofthe unstable manifold of the trivial solutions in this regionsuggests that a full analysis of these attracting cycles isprobably impossible, but in future we do hope to gain a betterunderstanding of the structure of cycles and their attractivity.We also hope to understand the more complex intermittentdynamics observed for example in Fig. 13.

Acknowledgements

Part of the research of OP was carried out during visits to theUniversity of Exeter, UK, in May–July 2002 and January–April2004. We are grateful to the Royal Society for supportof the visits. Some numerical results were obtained usingcomputational facilities provided by the program “SimulationsInteractives et Visualisation en Astronomie et Mecanique(SIVAM)” at Observatoire de la Cote d’Azur, France. The workof OP at the Observatoire de la Cote d’Azur was supported bythe French Ministry of Education. OP has been partly financedby the grant from the Russian Foundation for Basic Research04-05-64699. PA was partly funded by a Leverhulme ResearchFellowship during 2004. DH was supported by an EPSRC DTAstudentship.

Appendix A. Derivation of the normal form

We show (cf. [5]) that the normal form at the interaction ofthe bifurcations is (12) by considering a general polynomialvector field on C4 that commutes with the desired action ofO × S1 on C4. Namely, suppose that

z0 =

∑(m1,m2,l1,l2,l3,q1,q2,q3)∈J

C0m1,m2,l1,l2,l3,q1,q2,q3

× zm10 z0

m2 zl11 |z1|

2q1 zl22 |z2|

2q2 zl33 |z3|

2q3 ,

where J is the subset of (m1,m2, l1, l2, l3, q1, q2, q3) ∈ Z8

such that

m1 ≥ 0m2 ≥ 0l1 + l2 + l3 = 0l1, l2, l3 evenq1 ≥ max(−l1, 0)q2 ≥ max(−l2, 0)q3 ≥ max(−l3, 0)

and complex coefficients C0m,l,q satisfy

C0m,l,q = e2π i(m1−m2−1)/3C0

m,l2,l3,l1,q2,q3,q1

= e−2π i(m1−m2−1)/3C0m,l3,l1,l2,q3,q1,q2

,

C0m,l,q = C0

m,l1,l3,l2,q1,q3,q2.

Similarly, if we assume that

z1 =

∑(m1,m2,l1,l2,q1,q2,q3)∈I

C1m1,m2,l1,l2,q1,q2,q3

zl11 |z1|

2q1

× Wm1,m2,l1,l2,q2,q3

with

W = (zm10 z0

m2 zl22 z1−l1−l2

3 |z2|2q2 |z3|

2q3

+ zm20 z0

m1 zl23 z1−l1−l2

2 |z3|2q2 |z2|

2q3)

then I is the subset of (m1,m2, l1, l2, q1, q2, q3) ∈ Z7 such that

m1 ≥ 0m2 ≥ 0l1 oddl2 even

l2 ≥1 − l1

2q1 ≥ max(−l1, 0)q2 ≥ max(−l2, 0)q3 ≥ max(1 − l1 − l2, 0);

expressions for z2 and z3 can be obtained from the equations:

C1m,l,q = e2π i(m1−m2−1)/3C2

m,l3,l1,l2,q3,q1,q2

= e−2π i(m1−m2−1)/3C3m,l2,l1,l3,q2,q1,q3

.

Some of the terms can be eliminated by a near-identity changeof coordinates. Considering terms in the above sums up to order3 and performing the change of variables, one obtains the third-order normal form (12).

Appendix B. Reduction of the S1 normal form symmetry

In order to efficiently compute branches of solutions, bothanalytically and using XPPAUT [14], we reduce the system(12) by one dimension using the S1 symmetry. We consider themodified ODE on (v0, v1, v2, v3) ∈ C4

z0 = f0(z0, z1, z2, z3)

z1 = f1(z0, z1, z2, z3)+ iaz1

z2 = f2(z0, z1, z2, z3)+ iaz2

z3 = f3(z0, z1, z2, z3)+ iaz3,

(25)

where the original equations (12) are expressed as zi =

fi (z0, z1, z2, z3), and where we set


a =−Im( f1z1)

|z1|2 .

This equation is well defined for all z1 6= 0 and its solutionsare in correspondence with group orbits of those of (12) in thesense that

zk(t) 7→ zk(t) exp(iγ (t))

but moreover solutions of (25) preserve arg(z1). Hence we canset =(z1) = 0, assume R(z1) ≥ 0 and solve the reduced systemin seven instead of eight dimensions, subject to taking care nearthe singularity z1 = 0. In addition to this giving a dimensionreduction of the phase space, periodic orbits (resp. tori) that arerelative equilibria (resp. periodic orbits) in the original systembecome equilibria (resp. periodic orbits) in (25).

Appendix C. Symmetry breaking to ABC flow with A =

B 6= C

One can similarly consider a symmetry breaking from Oto D4 due to perturbation of the force to one that breaks thesymmetry;

A = C = 1, B = 1 + µ, (26)

where we assume µ to be small. This causes the symmetry tobreak from O × S1 to the subgroup D4 × S1 generated by ρ100,κ011 and γθ and results in the change of the linear part of (12) to

z0 = δ1z0 + δ2z0

z1 = (λ1 + iω1)z1

z2 = (λ2 + iω2)z2

z3 = (λ2 + iω2)z3.

(27)

One can use the techniques of [26] for computing the coeffi-cients in Table 6 to obtain coefficients of (27):

δ1 = δ + 0.009µ

δ2 = 0.174µ

λ1 = λ+ 0.107µ

ω1 = ω + 0.769µ

λ2 = λ+ 0.091µ

ω2 = ω + 0.573µ.

(28)

Hence the symmetry broken system can be approximated as

z0 = δ1z0 + δ2z0 + B1z02+ B2(|z1|

2+ ζ 2

|z2|2+ ζ |z3|

2)

+B3(|z1|2+ |z2|

2+ |z3|

2)z0 + B4|z0|2z0

z1 = (λ1 + iω1)z1 + A1|z1|2z1 + A2(|z2|

2+ |z3|

2)z1

A3(z22 + z2

3)z1 + A4(z0 + z0)z1 + A5|z0|2z1

z2 = (λ2 + iω2)z2 + A1|z2|2z2 + A2(|z1|

2+ |z3|

2)z2

+A3(z21 + z2

3)z2 + A4(ζ z0 + ζ 2z0)z2 + A5|z0|2z2

z3 = (λ2 + iω2)z3 + A1|z3|2z3 + A2(|z1|

2+ |z2|

2)z3

+A3(z21 + z2

2)z3 + A4(ζ2z0 + ζ z0)z3 + A5|z0|

2z3,

(29)

to the lowest order, with coefficients given by (28) and Table 6.

We have not analysed the system (29) in detail as theoriginal system is already so complicated; however we notethat setting µ non-zero will have the effect of splittingmany of the bifurcation points in Fig. 3 into a number ofbifurcation points. For example, the three branches of steadystates (x, 0, 0, 0) and its images under the ρ111 symmetry willsplit into two branches with the D2 symmetry group relatedby ρ100 symmetry and one branch with the D4 symmetrygroup. Thus, the saddle-node bifurcation P will split intotwo bifurcations. Similarly, the Hopf bifurcation B will splitinto a D4 Hopf and a symmetry-free Hopf; bifurcations suchas H and T will also split into several copies, one ofwhich will remain as a D4 Hopf and the others will splitinto two bifurcations that could in principle be examinednumerically.

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modelling instability of abc ﬂow using a mode interaction...

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