z1 = σ1z1 + δ1z2z3 + β1z1z4 + [κ11|z1|2 + κ12(|z2|2 + |z3|2)]z1

+[µ11|z4|2 + µ12(|z5|2 + |z6|2)]z1 + ν1z1z5z6 + ξ1z2z3z4 + η1(z2z3z6 + z2z3z5),

z4 = σ2z4 + δ2z5z6 + β2z21 + [κ21|z1|2 + κ22(|z2|2 + |z3|2)]z4

+[µ21|z4|2 + µ22(|z5|2 + |z6|2)]z4 + ν2z1z2z3 + ξ2(z23 z5 + z2

2 z6).

54

不変部分空間

I1 = C(0,0,0,1,0,0)

I2 = C{(1,0,0,0,0,0), (0,0,0,1,0,0)}

I3 = C{(0,0,0,1,0,0), (0,0,0,0,1,0), (0,0,0,0,0,1)}

I4 = C{(1,0,0,0,0,0), (0,0,0,1,0,0), (0,0,0,0,1,0), (0,0,0,0,0,1)}

55

Busse, F. H. 1989 Fundamentals of thermal convection. In Mantle convection (ed. W. R. Peltier),pp. 23–95. London, UK: Gordon and Breach.

Busse, F. H. & Or, A. C. 1986 Subharmonic and asymmetric convection rolls. ZAMP 37, 608–623.(doi:10.1007/BF00945433)

Busse, F. H. & Sommermann, G. 1996 Double-layer convection: a brief review and some recentexperimental results. In Advances in multi-fluid flows (eds Y. Y. Renardy, A. V. Coward, D. T.Papageorgiou & S.-M. Sun), pp. 33–41. Philadelphia, PA: SIAM.

Buzano, E. & Golubitsky, M. 1983 Bifurcation on the hexagonal lattice and the planar Benardproblem. Phil. Trans. R. Soc. A 308, 617–667. (doi:10.1098/rsta.1983.0018)

Buzano, E. & Russo, A. 1987 Bifurcation problems with O(2)4Z2 symmetry and the buckling of acylindrical shell. Ann. Mat. Pura Appl. (IV) 146, 217–262. (doi:10.1007/BF01762366)

Carr, J. 1981 Applications of centre Manifold theory. Berlin, Germany: Springer.Chen, M. M. & Whitehead, J. A. 1968 Evolution of two-dimensional periodic Rayleigh cells of

arbitrary wavenumbers. J. Fluid Mech. 31, 1–15. (doi:10.1017/S0022112068000017)Clune, T. & Knobloch, E. 1994 Pattern selection in three-dimensional magnetoconvection. Physica

D 74, 151–176. (doi:10.1016/0167-2789(94)90031-0)Dangelmayr, G. 1986 Steady state mode interactions in presence of O(2) symmetry. Dyn. Stab.

Syst. 1, 159–185.Daumont, I., Kassner, K., Misbach, C. & Valance, A. 1997 Cellular self-propulsion of two-

dimensional dissipative structures and spatial-period tripling Hopf bifurcation. Phys. Rev. E 55,6902–6906. (doi:10.1103/PhysRevE.55.6902)

Dawes, J. H. P. 2000 The 1 :!!!2

pHopf/steady-state mode interaction in three-dimensional magneto

convection. Physica D 139, 109–136. (doi:10.1016/S0167-2789(99)00210-9)Dawes, J. H. P. 2001 Hopf bifurcation on a square superlattice. Nonlinearity 14, 491–511. (doi:10.

1088/0951-7715/14/3/304)Dionne, B., Silber, M. & Skeldon, A. C. 1997 Stability results for steady, spatially-periodic

planforms. Nonlinearity 10, 321–353. (doi:10.1088/0951-7715/10/2/002)Fujimura, K. 1997 Centre manifold reduction and the Stuart–Landau equation for fluid motions.

Proc. R. Soc. A 453, 181–203. (doi:10.1098/rspa.1997.0011)Golubitsky, M. & Stewart, I. 2002 The symmetry perspective. Basel, Germany: Birkheuser Verlag.Golubitsky, M., Swift, J. W. & Knobloch, E. 1984 Symmetries and pattern selection in Rayleigh–

Benard convection. Physica D 10, 249–276. (doi:10.1016/0167-2789(84)90179-9)Golubitsky, M., Stewart, I. & Schaeffer, D. G. 1988 Singularities and groups in bifurcation theory.

Berlin, Germany: Springer.Judd, S. L. & Silber, M. 2000 Simple and superlattice Turing patterns in reaction–diffusion

systems: bifurcation, bistability, and parameter collapse. Physica D 136, 45–65. (doi:10.1016/S0167-2789(99)00154-2)

Manogg, G. & Metzener, P. 1994 Strong resonance in two-dimensional non-Boussinesq convection.Phys. Fluids 6, 2944–2955. (doi:10.1063/1.868121)

Mercader, I., Prat, J. & Knobloch, E. 2001 The 1 : 2 mode interaction in Rayleigh–Benardconvection with weakly broken midplane symmetry. Int. J. Bifurc. Chaos 11, 27–41. (doi:10.1142/S0218127401002006)

Mercader, I., Prat, J. & Knobloch, E. 2002a The 1 : 2 mode interaction in Rayleigh–Benardconvection with and without Boussinesq symmetry. Int. J. Bifurc. Chaos 12, 281–308. (doi:10.1142/S0218127402004401)

Mercader, I., Prat, J. & Knobloch, E. 2002b Robust heteroclinic cycles in two-dimensionalRayleigh–Benard convection without Boussinesq symmetry. Int. J. Bifurc. Chaos 12,2501–2522. (doi:10.1142/S0218127402006047)

Nore, C., Tuckerman, L. S., Daube, O. & Xin, S. 2003 The 1 : 2 mode interaction in exactlycounter-rotating von Karman swirling flow. J. Fluid Mech. 477, 51–88. (doi:10.1017/S0022112002003075)

Nore, C., Moisy, F. & Quartier, L. 2005 Experimental observation of near-heteroclinic cycles in thevon Karman swirling flow. Phys. Fluids 17, 064103. (doi:10.1063/1.1926827)

K. Fujimura152

Proc. R. Soc. A (2008)

Busse, F. H. 1989 Fundamentals of thermal convection. In Mantle convection (ed. W. R. Peltier),pp. 23–95. London, UK: Gordon and Breach.

Busse, F. H. & Or, A. C. 1986 Subharmonic and asymmetric convection rolls. ZAMP 37, 608–623.(doi:10.1007/BF00945433)

Busse, F. H. & Sommermann, G. 1996 Double-layer convection: a brief review and some recentexperimental results. In Advances in multi-fluid flows (eds Y. Y. Renardy, A. V. Coward, D. T.Papageorgiou & S.-M. Sun), pp. 33–41. Philadelphia, PA: SIAM.

Buzano, E. & Golubitsky, M. 1983 Bifurcation on the hexagonal lattice and the planar Benardproblem. Phil. Trans. R. Soc. A 308, 617–667. (doi:10.1098/rsta.1983.0018)

Buzano, E. & Russo, A. 1987 Bifurcation problems with O(2)4Z2 symmetry and the buckling of acylindrical shell. Ann. Mat. Pura Appl. (IV) 146, 217–262. (doi:10.1007/BF01762366)

Carr, J. 1981 Applications of centre Manifold theory. Berlin, Germany: Springer.Chen, M. M. & Whitehead, J. A. 1968 Evolution of two-dimensional periodic Rayleigh cells of

arbitrary wavenumbers. J. Fluid Mech. 31, 1–15. (doi:10.1017/S0022112068000017)Clune, T. & Knobloch, E. 1994 Pattern selection in three-dimensional magnetoconvection. Physica

D 74, 151–176. (doi:10.1016/0167-2789(94)90031-0)Dangelmayr, G. 1986 Steady state mode interactions in presence of O(2) symmetry. Dyn. Stab.

Syst. 1, 159–185.Daumont, I., Kassner, K., Misbach, C. & Valance, A. 1997 Cellular self-propulsion of two-

dimensional dissipative structures and spatial-period tripling Hopf bifurcation. Phys. Rev. E 55,6902–6906. (doi:10.1103/PhysRevE.55.6902)

Dawes, J. H. P. 2000 The 1 :!!!2

pHopf/steady-state mode interaction in three-dimensional magneto

convection. Physica D 139, 109–136. (doi:10.1016/S0167-2789(99)00210-9)Dawes, J. H. P. 2001 Hopf bifurcation on a square superlattice. Nonlinearity 14, 491–511. (doi:10.

1088/0951-7715/14/3/304)Dionne, B., Silber, M. & Skeldon, A. C. 1997 Stability results for steady, spatially-periodic

planforms. Nonlinearity 10, 321–353. (doi:10.1088/0951-7715/10/2/002)Fujimura, K. 1997 Centre manifold reduction and the Stuart–Landau equation for fluid motions.

Proc. R. Soc. A 453, 181–203. (doi:10.1098/rspa.1997.0011)Golubitsky, M. & Stewart, I. 2002 The symmetry perspective. Basel, Germany: Birkheuser Verlag.Golubitsky, M., Swift, J. W. & Knobloch, E. 1984 Symmetries and pattern selection in Rayleigh–

Benard convection. Physica D 10, 249–276. (doi:10.1016/0167-2789(84)90179-9)Golubitsky, M., Stewart, I. & Schaeffer, D. G. 1988 Singularities and groups in bifurcation theory.

Berlin, Germany: Springer.Judd, S. L. & Silber, M. 2000 Simple and superlattice Turing patterns in reaction–diffusion

systems: bifurcation, bistability, and parameter collapse. Physica D 136, 45–65. (doi:10.1016/S0167-2789(99)00154-2)

Manogg, G. & Metzener, P. 1994 Strong resonance in two-dimensional non-Boussinesq convection.Phys. Fluids 6, 2944–2955. (doi:10.1063/1.868121)

Mercader, I., Prat, J. & Knobloch, E. 2001 The 1 : 2 mode interaction in Rayleigh–Benardconvection with weakly broken midplane symmetry. Int. J. Bifurc. Chaos 11, 27–41. (doi:10.1142/S0218127401002006)

Mercader, I., Prat, J. & Knobloch, E. 2002a The 1 : 2 mode interaction in Rayleigh–Benardconvection with and without Boussinesq symmetry. Int. J. Bifurc. Chaos 12, 281–308. (doi:10.1142/S0218127402004401)

Mercader, I., Prat, J. & Knobloch, E. 2002b Robust heteroclinic cycles in two-dimensionalRayleigh–Benard convection without Boussinesq symmetry. Int. J. Bifurc. Chaos 12,2501–2522. (doi:10.1142/S0218127402006047)

Nore, C., Tuckerman, L. S., Daube, O. & Xin, S. 2003 The 1 : 2 mode interaction in exactlycounter-rotating von Karman swirling flow. J. Fluid Mech. 477, 51–88. (doi:10.1017/S0022112002003075)

Nore, C., Moisy, F. & Quartier, L. 2005 Experimental observation of near-heteroclinic cycles in thevon Karman swirling flow. Phys. Fluids 17, 064103. (doi:10.1063/1.1926827)

K. Fujimura152

Proc. R. Soc. A (2008)

at the quadratic order.On the other hand, at the cubic order, the z1 interactswith z2and z3 interacts with z4 non-resonantly through their moduli. Therefore, couplingsbetween z1 and z3 and between z2 and z4 are much stronger than those between z1and z2 and between z3 and z4. We thus expect that, on a square lattice, two-dimensional patterns will be preferred to three-dimensional patterns. To concludethis, concrete analyses are needed.

In this paper, we have examined the bifurcation of solutions z in (3.4). Therelationship between the amplitude z and the disturbance j is given by (3.1) where

the eigenfunctions f1, ., f6 are normalized by hSf!j "mn; ~f

!j "mniZ1 as is described in

the first paragraph of §4. In figure 9, we show the temperature field of thedisturbance in a cross section of a roll-type structure. Figure 9a shows thetemperature field for kZkc, whereas figure 9b shows the one for kZ2kc. Figure 9cshows a cross section of the solution S4. For the solution S4, z1 and z4 are typically10K5 and 10K4, respectively, for rZ10K4 in case (iii). Figure 9c is thus due to thesuperposition of figure 9a,b after multiplying by 10K5 and 10K4, respectively.Consider a situation in which thermo-sensitive liquid crystal powder is added tothe working fluids. We may visualize a planform of convection pattern from aboveor below by applying a horizontal light sheet to the fluid layers. Since the planformdepend on the height as is clearly seen in figure 9c, planforms like figure 2 can beobserved when the height of the light sheet is appropriately adjusted.

The author expresses his sincere thanks to referees for their helpful and constructive comments onthe manuscript.

References

Andereck, C. D., Colovas, P. W. & Degen, M. M. 1996 Observations of time-dependent behavior inthe two-layer Rayleigh–Benard system. In Advances in multi-fluid flows (eds Y. Y. Renardy,A. V. Coward, D. T. Papageorgiou & S.-M. Sun), pp. 3–12. Philadelphia, PA: SIAM.

Armbruster, D. 1987 O(2)-symmetric bifurcation theory for convection rolls. Physica D 27,433–439. (doi:10.1016/0167-2789(87)90042-X)

Armbruster, D., Guckenheimer, J. & Holmes, P. 1988 Heteroclinic cycles and modulated travelingwaves in systems with O(2) symmetry. Physica D 29, 257–282. (doi:10.1016/0167-2789(88)90032-2)

Busse, F. H. 1967 The stability of finite amplitude cellular convection and its relation to anextremum principle. J. Fluid Mech. 30, 625–649. (doi:10.1017/S0022112067001661)

Busse, F. H. 1978 Nonlinear properties of thermal convection. Rep. Prog. Phys. 41, 1929–1967.(doi:10.1088/0034-4885/41/12/003)

1 + D –1

1

(a) (b) (c)

z

0xxx

Figure 9. Examples of the structure of convection in case (iii) for P1Z150:76 and P2Z7 with e1Z0and e2Z0:1. (a) Temperature field of roll structure with kZkc, (b) cross section of rolls S2 withkZ2kc and (c) cross section of the mixed roll structure formed by the solution S4 with z 1Z10K5

and z4Z10K4.

151The 1 : 2 resonance on a hexagonal lattice

Proc. R. Soc. A (2008)

I3 = C{(0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 1)}

Fujimura & Yamada 2008 Proc. R. Soc. Ser. A 464, 2721 ! 273.

正六角形格子上でのパターン形成

56

!"!

!!!!!!"

!"!!

!""

!!"

"!!

!!!

(a)

S2

S!3

S+3

S5

0 0 005 0 01 0 015

(b)

R1 ! R1c

R1c

S2

S!3

S+3

S++6

S!+6

S4

S5

S8

0. 0.005 0.01 0.015

I3 = C{(0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 1)}

P1 = P2 = 7, !1 = !2 = 0.1

57

S2

S4S4

TW1 TW1

(a)

0 1 20.5 1.5

+ + !

!!!

!!!!!!

! + +

! + +

+ + +

! + !

!! +

!/"

S2

S!3

S+3

S4S4

S+!6S!+

6S!!

6S++6

S8S8

S8

S11S11

S12 S12

S13

(b)

!/"

0 1 2

TW2

TW1

0.5 1.5++ !!++ !!

σ1 = ρ cos ϕ, σ2 = ρ sinϕ

ρ = 10−4

I2 = C{(1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0)}

P1 = P2 = 7, !1 = !2 = 0.158

周期軌道に対するニュートン反復法

次の問題を考える：

x = f(x, µ), x ∈ Rn, µ ∈ R, f : Rn × R→ Rn

x(t) = x(t + T ) for T � 1 であると仮定し，つぎのようにおく：

ξ =t

T↔ t ∈ [0, T ] and ξ ∈ [0,1]

τdx

dξ= f(x, µ) where τ =

1

T

x(ξ) = φξ(x(0),0; τ, µ)

与えられたパラメター µ に対し次を解く：

x(0) = φ1(x(0),0; τ, µ) for x(0) and τ

59

その代わりとして，

ξ =2t

T− 1 ↔ t ∈ [0, T ] and ξ ∈ [−1,1].

とおく，すると次を得る：

τdx

dξ= f(x, µ) where τ =

2

T

擬弧長：(δs)2 = (δx)2 + (δµ)2 + (δτ)2

x = x + x, µ = µ + µ, and τ = τ + τ

⇒

(τ + τ)d(x + x)

dξ= f(x+x, µ+µ) and x(−1)+x(−1) = x(1)+x(1).

(sn − sn−1)2 = (x + x− xn−1)

2 + (µ + µ− µn−1)2 + (τ + τ − τn−1)

2

60

線形化すると，

τdx

dξ+ τ

dx

dξ−

df

dx(x, µ)x−

∂f

∂µ(x, µ)µ = −τ

dx

dξ+ f(x, µ) + O(2)

2(x− xn−1)x + 2(µ− µn−1)µ + 2(τ − τm−1)T

= δs2 − (x− xn−1)

2 − (µ− µ−1)2 − (τ − τn−1)

2

and x(−1)− x(1) = −x(−1) + x(1).

x(ξ) と x(ξ) を次のようにチェビシェフ多項式に展開する：

xk(ξ) =�

j

A(k)j

Tj(ξ) and xk(ξ) =�

j

a(k)j

Tj(ξ)

さらに，任意の位相シフトが生じないように，適当な l に対して

xl(−1) = xl(1) = 0

を課す．

61

T2j−1(ξ) = T2j+1(ξ)− T1(ξ)

T2j(ξ) = T2j+2(ξ)−T �2j+2(1)T2(ξ)

T �2(ξ)

−(T2j+2(1)−T �2j+2(1)T2(1)

T �2(1)

)T0(ξ)

62

Traveling Waves TW2

zj(t) = rj(t) ei!j(t), j = 1, · · · , 6, !1 = "1 + "2 + "3, !2 = "4 + "5 + "6,

#1 = "4 ! 2"1, #2 = "5 ! 2"2, #3 = "6 ! 2"3

We require r2 = r3, r5 = r6, !1, !2, #1, #2, #3 : const.

"j " const. " "j(t) = "jt + $j , (j = 1, · · · , 6) for const. "j , $j .

Setting "1/k = c and % = x ! ct, we have

! = r1"1eik"+i#1 + r2"1e

ik(!"/2+"

3y/2)+i#2 + r2"1eik(!"/2!

"

3y/2)+i#3

+r4"4e2ik"+i(#1+$1) + r5"4e

ik(!"+"

3y)+i(2#2+$2) + r5"4eik(!"!

"

3y)+i(2#3+$2)

+c.c. + higher order terms.

TW2 lie on the group orbit &z for & = ("1t,!"1t/2) # T2.

63

Traveling Waves TW2

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Registered Charity No 207043

8 November 2008volume 464 . number 2099 . pages 2803–3088

Proc. R. Soc. A | vol. 464 no. 2099 pp. 2803–3088 | 8 N

ov 2008

journals.royalsociety.orgPublished in Great Britain by the Royal Society, 6–9 Carlton House Terrace, London SW1Y 5AG

8 November 2008

ISSN 1364-5021

volume 464

number 2099

pages 2803–3088Invited reply. Reply to Gagliardini’s comment on ‘Creep and recrystallization of large polycrystalline masses’ by Faria and co-authors 2803S. H. Faria, K. Hutter & G. M. KremerSingle-molecule magnetic tweezer tests on DNA: bounds on topoisomerase relaxation 2811J. M. T. ThompsonNon-parametric estimation of residual moments and covariance 2831D. Evans & A. J. JonesMultiple coiling of an elastic sheet in a tube 2847V. Romero, T. A. Witten & E. CerdaThe method of separation of variables for solving equations describing molecular-motor-assisted transport of intracellular particles in a dendrite or axon 2867A. V. Kuznetsov & A. A. AvramenkoThe liquid blister test 2887J. Chopin, D. Vella & A. BoudaoudTwo-degree-of-freedom vortex-induced vibration of a pivoted cylinder below critical mass ratio 2907C. M. Leong & T. WeiIntrinsic and measured statistics of discrete stochastic populations 2929O. E. French, K. I. Hopcraft, E. Jakeman & T. J. ShepherdTristability of thin orthotropic shells with uniform initial curvature 2949S. Vidoli & C. MauriniOn radial crack and half-penny crack induced by Vickers indentation 2967Y. Tang, A. Yonezu, N. Ogasawara, N. Chiba & X. ChenRapidly dissolving dense bodies in an inviscid fluid 2985I. EamesNonlinear Euler buckling 3003A. Goriely, R. Vandiver & M. DestradeDouble cavity flow past a wedge 3021Y. A. Antipov & V. V. SilvestrovThe effect of surface tension on localized free-surface oscillations about surface-piercing bodies 3039R. Harter, M. J. Simon & I. D. AbrahamsNatural selection for least action 3055V. R. I. Kaila & A. AnnilaStrength of electromagnetic, acoustic and Schrödinger reflections 3071S. Mokhov & B. Ya. ZeldovichOn a connection between the polylogarithm function and the Bass diffusion model 3081P. Jodrá

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64

Oscillatory Solution in C6

We require r1 = r2 = r3, r4 = r5 = r6, !1 = !2 = !3 = 0, and "1 = 0

! #j = 0

The equations governing the oscillatory solution OS: two-dimensional for r1, r4

r1 = [$1 +%1r4 + &1r1 +('11 +2'12)r21 +(µ11 +2µ12 +(1)r

24 +(2)1 + *1)r1r4]r1,

r4 = $2r4 + %2r21 + &2r

24 + ('21 + 2'22 + 2*2)r

21r4 + (µ21 + 2µ22)r

34 + (2r

31.

65

Phil.T rans.R.Soc. 340 (1992) pp.95 ! 130

O(2) の下での近ヘテロクリニック軌道の周期

66

|z1|

|z2|

|z3|

|z4|

|z5|

|z6|0 1t (!10!5)

|z1|

|z2|

|z3|

|z4|

|z5|

|z6|9 10t (!10!5)

(z1, z2, z3, z4, z5, z6) = (10−4, 0, 0, 10−3i, 0, 0)

I2 = C{(1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0)}67

10 20 30 40 50 60 70

10000

20000

30000

40000

50000

60000

70000|z1|

|z2|

|z3|

|z4|

|z5|

|z6|0 1t (!10!5)

Single

Double

Quadruple

I4 = C{(1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 1)}

P1 = 150.76, P2 = 7, !1 = 0, !2 = 0.1

４次元不変部分空間の近ヘテロクリニックサイクル

68

|z1|

|z2|

|z3|

|z4|

|z5|

|z6|0 1t (!10!5)

|z1|

|z2|

|z3|

|z4|

|z5|

|z6|0 1t (!10!5)

δn = 0 δn = O(10−24)

I2 = C{(1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0)}

P1 = 150.76, P2 = 7, !1 = 0, !2 = 0.1

(104, !2, !3, 10!3i, !5, !6)

２，４次元不変部分空間の近ヘテロクリニックサイクル

69

x1 x4

x5

x1

6

4

1

3

6

4

2

5

2 3

1

|z1|

|z4|

|z5|

|z6|

t

12 3 4 56R

5

1

12

33

H

! = 10!3

P1 = 150.76, P2 = 7, !1 = 0, !2 = 0.1

Re z5

Re z1Re z4

４次元不変部分空間の近ヘテロクリニックサイクル

70

|z1|

|z4|

|z5|

|z6|

(a)

0 1 2t (!10!5)

|z1|

|z2|

|z3|

|z4|

|z5|

|z6|

(b)

0 3 6t (!10!5)

I2 → I4

δn = O(10−24) δn = O(10−8)

P1 = 150.76, P2 = 7, !1 = 0, !2 = 0.1

I2 ! I4 ! C6

近ヘテロクリニックサイクルの崩壊

71

Re z1

Re z4

|z5|

HC(I2), HC(I4), and OS(C6)72