112 Landau Table 1 Poiseuille Rayleigh-Benard Rayleigh-Benard Figure 1; 3 19 Poiseuille $R_{c}^{-1}-R^{-1}$ $ z ^{2}$ 3 $\epsilon^{2}=r_{\mathrm{c}}^{

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1 Rayleigh-Benard (Kaoru Fujimura) Department of Appiied Mathematics and Physics Tottori University 1 Euclid Rayleigh-B\ enard Marangoni ( ) 3 Boussinesq 1 Rayleigh-Benard Boussinesq 2 Rayleigh-Benard Marangoni [2] [11] Boussinesq 1 Rayleigh-Benard 1 5 $[4][12]$ 5 Knobloch [13] Knobloch 2 Rayleigh-B6nard Boussinesq 1 Rayleigh-Benard Stuart-Landau Stuart-Landau $O(\epsilon^{7})$ Stuart-Landau $\frac{dz}{dt}=\epsilon^{2}(\sigma+\epsilon^{2}\sigma^{(1\}}+\epsilon^{4}\sigma^{(2)}+\cdots)z+(\lambda_{1}+\epsilon^{2}\lambda_{1}^{(1)}+\epsilon^{4}\lambda_{1}^{(2)}+\cdots) z ^{2}z$ $+(\lambda_{2}+\epsilon^{2}\lambda_{2}^{\{1)}+\cdots) z ^{4}z+(\lambda_{3}+\cdots) z ^{6}z+\cdots$

2 112 Landau Table 1 Poiseuille Rayleigh-Benard Rayleigh-Benard Figure 1; 3 19 Poiseuille $R_{c}^{-1}-R^{-1}$ $ z ^{2}$ 3 $\epsilon^{2}=r_{\mathrm{c}}^{-1}-r^{-1}$ ( $R$ Rc)/R $R$ $O(\epsilon^{2})$ Reynolds Rayleigh $\frac{dz}{dt}=\epsilon^{2}\sigma z+\lambda_{1} z ^{2}z+\lambda_{2} z ^{4}z+\lambda_{3} z ^{6}z+\cdots$ Poiseuille Stuart-Landau Herbert $[10]$ Table Figl Stua$\mathrm{r}\mathrm{t}$-Landau Poiseuille Rayleigh-Benard Stuart-Landau 4 Prandfl Landau Table 1 Fig 2 $P<02515$ Poiseuille Landau $P\geq 02515$ $P=02515$ 2 Landau

3 113 Figure Rayleigh-Benard $P=0025$ $(R-R_{c})/R_{\mathrm{c}}$ $ z ^{2}$ $P\geq 071$ Landau Fig 2 Prandtl Stuart-Landau Prandtl Poiseuille $\Gamma_{h}=\mathrm{D}_{6}\dotplus \mathrm{t}^{2}$ 6 $\overline{z}_{2}\overline{z}_{3}$ $z_{1}$ $z_{2}$ $z_{3}$ 1

4 $\mathrm{t}^{2}$ $\mathrm{c}$ 114 $\Gamma_{h}$ $(z_{1} z_{2} z_{3})$ $(z_{1} z_{2} z_{3})arrow(\overline{z}_{1}\overline{z}_{2}\overline{z}_{3})$ $\mathrm{d}_{6}\{$ $\mathrm{d}_{3}\{\begin{array}{l}\mathrm{r}_{2\pi/3}(z_{1}z_{2}z_{3})arrow(z_{2}z_{3}z_{1})\sigma_{v}\cdot(z_{1}z_{2}z_{3})arrow(z_{1}z_{3}z_{2})\end{array}$ $(s t)\cdot z=(\mathrm{e}^{is}z_{1} \mathrm{e}^{-\iota^{-}(s+t\}}z_{2} \mathrm{e}^{it}z_{3})$ 1 (R) (H) 2 $\mathrm{r}\mathrm{a}$ 2 (T) ( ) $\Gamma_{h}$ $\dot{z}_{1}=z_{1}\mathcal{h}(\lambda u_{1} \sigma_{1} \sigma_{2} \sigma_{3} q)+\overline{z}_{2}\overline{z}_{3}\mathcal{p}(\lambda u_{1} \sigma_{1_{7}}\sigma_{2} \sigma_{3} q)$ $=h_{1}(\lambda \sigma_{1} \sigma_{2} \sigma_{3)}q)+u_{1}h_{3}(\lambda_{?}\sigma_{1} \sigma_{2} \sigma_{3} q)+u_{1}^{2}h_{5}(\lambda$ $\sigma_{1}$ $\sigma_{2}$ $\sigma_{3}$ $q)$ $\mathcal{p}=p_{2}(\lambda \sigma_{1} \sigma_{2} \sigma_{3_{7}}q)+u_{1}p_{4}(\lambda \sigma_{1} \sigma_{2} \sigma_{3} q)+u_{1}^{2}p_{6}(\lambda \sigma_{1} \sigma_{2} \sigma_{3} q)$ $u_{j}= z_{j} ^{2}$ $\sigma_{1}=u_{1}+u_{2}+u_{3}$ $\sigma_{2}=u_{1}u_{2}+u_{2}u_{3}+u_{3}u_{1}$ $\sigma_{3}=u_{1}u_{2}u_{3}$ $q=z_{1}z_{2}z_{3}+\overline{z}_{1}\overline{z}_{2}\overline{z}_{3}$ $p_{2}p_{4}p_{6}$ [4] $h_{1}$ $h_{3}$ $h_{5}$ $(z_{1} z_{2} z_{3})=(x 00)$ $x\in \mathrm{r}$ $0=H(x)$ $z_{\mathrm{j}}(t)=r_{j}(t)\mathrm{e}^{i\theta_{j}(t)}$ $\theta_{1}+\theta_{2}+\theta_{3}=\theta(t)$ $z_{1}=z_{2}=z_{3}$ $0=H(r \cos \mathrm{o}-)+r\cos\theta\cdot P(r \cos\theta)$ $0=\sin\Theta\cdot P(r \cos \mathrm{o}-)$ $\cos \mathrm{o}-=\pm 1$ $x\in \mathrm{r}$ $(z_{1} z_{2} z_{3})=(x x x)$ $+$ up-hexagons $l$-hexagons $\mathrm{h}_{+}$ down-hexagons $g$-hexagons $\mathrm{h}_{-}$ $0=\mathcal{H}(r;\cos\Theta=\pm 1)\pm r P(r;\cos\Theta=\pm 1)$ Sch\"ulter Lortz and Busse PDE $p_{2}$ 0 [19] $\cos\theta=\pm 1$ $q$ 4 $(O(4))$ $\cos\ominus\neq\pm 1$ $(z_{1} z_{2} z_{3})=(z z z)z\in \mathrm{c}$ $0=7\mathrm{f}(r \cos\theta)$ $0=\mathcal{P}(r \cos \mathrm{o}-)$ $r$ $\Theta(\neq n\pi)$ 5

5 $\mathrm{r}$ 0 $\mathrm{h}$ $\mathrm{t}$ g_{1}}{\partial x_{1}}$ g_{1}^{r}}{\partial x_{1}}+2\frac{\partial g_{1}^{r}}{\partial x_{2}}(1)$ $3 \frac{\partial g_{1}^{i}}{\partial y_{1}}(1)$ g_{1}^{r}}{\partial x_{1}}$ $-$ g_{1}^{r}}{\partial x_{2}}(2)$ g_{1}^{\mathrm{i}}}{\partial y_{1}}+2\frac{\partial g_{2}^{i}}{\partial y_{2}}(1)$ g_{2}^{r}}{\partial x_{2}}-\frac{\partial g_{2}^{t}}{\partial x_{3}}(1)$ $\lambda_{1}$ $\lambda_{2}$ $\lambda_{1}\lambda_{2}=\frac{\partial g_{1}^{r}}{\partial x_{1}}(\frac{\partial g_{2}^{r}}{\partial x_{2}}+\frac{\partial g_{2}^{r}}{\partial x_{3}})-2\frac{\partial g_{1}^{r}}{\partial x_{2}}\frac{\partial g_{2}^{r}}{\partial x_{1}}$ g_{1}^{r}}{\partial x_{1}}$ $\lambda_{1}\lambda_{2}=(\frac{\partial g_{1}^{r}}{\partial x_{1}}+2\frac{\partial g_{1}^{r}}{\partial x_{2}})(\frac{\partial g_{1}^{\mathrm{i}}}{\partial y_{1}}+2\frac{\partial g_{1}^{i}}{\partial y_{2}})-(\frac{\partial g_{1}^{r}}{\partial y_{1}}+2\frac{\partial g_{1}^{\tau}}{\partial y_{2}})(\frac{\partial g_{1}^{i}}{\partial x_{1}}+2\frac{\partial g_{1}^{i}}{\partial x_{2}})$ $\mathrm{z}_{2}$ 115 Table 2 Label I 1 2 Jacobi ( ) Eigenvalue (multiplicity) $(6)$ (1) g_{1}^{t}}{\partial x_{1}}(1)$ g_{2}^{r}}{\partial x_{2}}+\frac{\partial g_{2}^{r}}{\partial x_{3}}(2)$ g_{2}^{r}}{\partial x_{2}}-\frac{\partial g_{2}^{r}}{\partial x_{3}}(2)$ 0 (2) RA 0 (2) $\lambda_{1}+\lambda_{2}=\frac{\partial g_{1}^{r}}{\partial x_{1}}+\frac{\partial g_{2}^{\tau}}{\partial x_{2}}+\frac{\partial g_{2}^{r}}{\partial x_{3}}$ 0 (2) $- \frac{\partial g_{1}^{r}}{\partial x_{2}}+\frac{\partial g_{1}^{i}}{\partial y_{1}}-\frac{\partial g_{1}^{i}}{\partial y_{2}}(2)$ $\lambda_{1)}\lambda_{2}$ $\lambda_{1}+\lambda_{2}=\frac{\partial g_{1}^{r}}{\partial x_{1}}+2\frac{\partial g_{1}^{r}}{\partial x_{2}}+\frac{\partial g_{1}^{i}}{\partial y_{1}}+2\frac{\partial g_{1}^{i}}{\partial y_{2}}$ $(z_{1} z_{2} z_{3})=(x y y)$ $x$ $y\in \mathrm{r}$ 2 up-hexagons down-hexagons Jacobi $g_{j}^{i}$ Table 2 $g_{j}^{r}$ $g_{\mathrm{i}}$ $x_{j}={\rm Re} z_{j}$ $y_{j}={\rm Im} z_{j}$ 4 $z=1/2$ $zarrow 1-z$ w\rightarrow -w $\thetaarrow-\theta$ $(z_{1} z_{2} z_{3})$ $\sigma_{h}$ $(z_{1} z_{2)}z_{3})arrow(-z_{1} -z_{2} -z_{3})$ \Gamma h\tilde =D6\dotplus T2\oplus Z2 $\dot{z}_{1}---z_{1}\mathcal{l}(\lambda u_{1} \sigma_{1} \sigma_{2} \sigma_{3} q^{2})+\overline{z}_{2}\overline{z}_{3}q\mathcal{m}(\lambda u_{1} \sigma_{1} \sigma_{2} \sigma_{3} q^{2})$ $=l_{1}(\lambda \sigma_{1} \sigma_{2} \sigma_{3} q^{2})+u_{1}l_{3}(\lambda \sigma_{1} \sigma_{2} \sigma_{3} q^{2})+u_{1}^{2}l_{5}(\lambda \sigma_{1} \sigma_{2} \sigma_{3} q^{2})$ $\mathcal{m}=m_{5}(\lambda \sigma_{1} \sigma_{2} \sigma_{3} q^{2})+u_{1}m_{7}(\lambda \sigma_{1} \sigma_{2} \sigma_{3_{2}}q^{2})+u_{1}^{2}m_{9}(\lambda \sigma_{1} \sigma_{2} \sigma_{3} q^{2})$ [12]

6 116 1 \S 3 (R) (H) (PQ) (RT) 2 (T) $\mathrm{r}\mathrm{a}$ ( ) 2 4 $x\in \mathrm{r}$ $z_{j}=r_{j}(t)\mathrm{e}^{i\theta_{j}(t\rangle}$ $\theta_{1}+\theta_{2}+\theta_{3}=\theta(t)$ $(z_{1} z_{2} z_{3})=(0 x x)$ $z_{1}=z_{2}=z_{3}$ $0=\mathcal{L}(r \cos^{2}\theta)+2r^{4}\cos^{2}\theta\cdot M(r \cos^{2}\theta)$ $0=\cos\Theta\sin\Theta\cdot M(r \cos^{2}\theta)$ $\cos\theta=\pm 1$ $(z_{1} z_{2} z_{3})=(x x x)$ $x\in \mathrm{r}$ $q$ 2 $0=\mathcal{L}(r;\cos^{2}\ominus=1)+2r^{4}\mathcal{M}(r_{7}\cos^{2}\Theta=1)$ $\cos\ominus=\pm 1$ up-hexagons down-hexagons 4 $\cos \mathrm{o}-=0$ $(z_{1} z_{2} z_{3})=(\mathrm{i}x \mathrm{i}x \mathrm{i}x)$ $x\in \mathrm{r}$ $0=\mathcal{L}(r;\cos \mathrm{o}-=0)$ $2r^{4}\lambda \mathit{4}$ $\cos^{2}\theta=10$ 7 5 \S 3 $\cos\theta\neq\pm 1$ $(z_{1} z_{2} z_{3})=(z z z)$ $z\in \mathrm{c}$ $0=\mathcal{L}(r \cos^{2}\theta)$ $0=\mathcal{M}(r \cos^{2}\mathrm{o}-)$ $r$ $\Theta(\neq n\pi/2)$ 7 \S Rayleigh-B6nard $\mathrm{d}_{6}\dotplus \mathrm{t}^{2}\oplus \mathrm{z}_{2}$ $z=1/2$ Jacobi Table Boussinesq $\rho_{0}\frac{d\overline{v}^{*}}{dt^{*}}=-\nabla^{*}p^{*}-pg\mathrm{e}_{z}+\mu\triangle*\overline{v}^{*}$ $\frac{dt^{*}}{dt^{*}}=\kappa\triangle*t^{*}$ $\nabla^{*\prec}$ $v^{*}=0$

7 $\mathrm{r}$ 0 $\mathrm{h}$ 0 $\mathrm{r}\mathrm{a}\mathrm{r}\mathrm{t}$ $\mathrm{t}$ g_{1}^{r}}{\partial x_{1}}(6)$ g_{1}^{r}}{\partial x_{1}}(1)$ g_{2}^{r}}{\partial x_{2}}(4)$ g_{3}^{r}}{\partial x_{3}}(1)$ g_{1}^{r}}{\partial x_{2}}(2)$ $3 \frac{\partial g_{1}^{i}}{\partial y_{1}}(1)$ $\frac{2\frac{\partial g_{1}^{i}}{\partial\mu_{2}g_{1}}\partial}{\partial x_{1}}+\frac{1)\partial g_{2}^{r}}{\partial x_{2}}\frac{\partial g_{3}^{r}-}{\partial x_{2}}(1\frac{\partial g_{1}^{i}}{r\partial y_{1}(1}+(\frac{\partial g_{1}^{i}}{\partial y_{1}+}\frac{\partial g_{1}^{f}}{\partial y_{2}(1)}(2)$ $0(2)0(2) \frac{\frac{\partial g_{1}^{r}}{\partial g_{2}^{r}\partial x_{1}}}{\partial x_{2}}-\frac{\partial g_{3})}{\partial x_{2}})$ g_{1}^{r}}{\partial x_{1}}-\frac{\partial g_{1}^{r}}{\partial x_{2}}+\frac{\partial g_{1}^{i}}{\partial y_{1}}-\frac{\partial g_{1}^{i}}{\partial y_{2}}(2)$ $\lambda_{1}+\lambda_{2}=\frac{\partial g_{1}^{r}}{\partial x_{1}}+2\frac{\partial g_{1}^{r}}{\partial x_{2}}+\frac{\partial g_{1}^{i}}{\partial y_{1}}+2\frac{\partial g_{1}^{\dot{\mathrm{z}}}}{\partial y_{2}})$ g_{3}^{i}}{\partial y_{3}}(1)$ $\lambda_{1}\lambda_{2}=(\frac{\partial g_{1}^{r}}{\partial x_{1}}+2\frac{\partial g_{1}^{r}}{\partial x_{2}})(\frac{\partial g_{1}^{i}}{\partial y_{1}}+2\frac{\partial g_{1}^{i}}{\partial y_{2}})-(\frac{\partial g_{1}^{r}}{\partial y_{1}}+2\frac{\partial g_{1}^{f}}{\partial y_{2}})(\frac{\partial g_{1}^{i}}{\partial x_{1}}+2\frac{\partial g_{1}^{i}}{\partial x_{2}})$ $\mathcal{l}$ g_{1}^{i}}{\partial y_{1}}+2\frac{\partial g_{2}^{i}}{\partial y_{2}}(1)$ 117 Table 3 Label I Jacobi ( ) Eigenvalue (multiplicity) (1) PQ 0(2) g_{1}^{r}}{\partial x_{1}}+\frac{\partial g_{2}^{r}}{\partial x_{1}}(1)$ g_{1}^{2}}{\partial x_{1}}-\frac{\partial g_{2}^{r}}{\partial x_{1}}(1)$ (2) $)$ g_{1}^{r}}{\partial x_{1}}+2\frac{\partial g_{1}^{r}}{\partial x_{2}}(1)$ g_{1}^{r}}{\partial x_{1}}$ $( \frac{\partial g_{1}^{r}}{\partial x_{1}}(\frac{\partial g_{2}^{r}}{\partial x_{2}}+\frac{\partial g_{3}^{r}}{\partial x_{3}})-2\frac{\partial g_{1}^{r}}{\partial x_{2}}\frac{\partial g_{2}^{r}}{\partial x_{1}})/(\frac{\partial g_{1}^{r}}{\partial x_{1}}+\frac{\partial g_{2}^{r}}{\partial x_{2}}+\frac{\partial g_{3}^{r}}{\partial x_{2}})(1)$ 0 (2) $\lambda_{1} \lambda_{2}$ $\rho=\rho_{0}[1-\alpha(t^{*}-t_{t})]$ 1 $R= \frac{\rho_{0}g\alpha^{(1)}(t_{b}-t_{t})d^{3}}{\mu\kappa}$ $P= \frac{\nu}{\kappa}$ Rayleigh Prandtl $(u v w p \theta)^{t}=\psi$ PDE }{\partial t}\mathrm{s}\psi-\mathcal{l}(r)\psi=n(\psi \psi)$ $\mathcal{b}\psi=0$ at $z=01$ $\mathrm{s}$ $B$ $N$ 2 $z=01$ $z=0$ $z=1$ $z=01$ 3 $B\psi=0$ Prandtl $P$ Rayleigh Busse balioon thermal imprinting $R$ [6]

8 $\overline{z}_{1}$ $\cdot$ 118 $\psi$ $2\pi/3$ 2 2 Fourier 3 $\psi(\mathrm{x} t)=\sum_{m=-\infty}^{\infty}\sum_{n=-\infty}^{\infty}\sum_{j=1}^{\infty}a_{mn}^{(j)}(t)\phi_{mn}^{\langle j)}(z)\mathrm{e}^{\mathrm{i}mk_{\mathrm{c}}x}\mathrm{e}^{ink_{\mathrm{c}}(\frac{-x}{2}+-_{2}\mathrm{k}\rangle}\sqrt{3}$ $\dot{a}_{mn}^{(j)}=\sigma_{mn}^{(j)}(\mu)a_{mn}^{\{j\}}+\sum_{kl}\lambda_{klm-kn-l}^{(jpq)}a_{kl}^{\langle p)}a_{m-kn-l}^{(q)}$ $A_{mn}^{(j)}$ $\mathrm{f}\phi_{\mathrm{f}\mathrm{l}}^{\overline{\mathrm{o}}}\xi\ovalbox{\tt\small $A_{01}^{(1)}$ $A_{10}^{(1)}$ REJECT}$ $A_{-11}^{(1)}$ $A_{-10}^{(1\rangle}$ $A_{0-1}^{(1)}$ $A_{11}^{(1)}$ 6 $A_{mn}^{(j)}=h_{mn}^{(j)}(A_{10}^{(1)} A_{01}^{(1\rangle} A_{-1-1}^{(1)} A_{-10}^{(1)} A_{0-1}^{(1)} A_{1_{\mathrm{J}}1}^{(1)})$ $h_{mn}^{(j)}(\mathrm{o})=dh_{mn}^{(j)}(0)=0$ $A_{10}^{(1)}arrow z_{1}$ $A_{01}^{(1)}arrow z_{2}$ $A_{-1-1}^{(1\}}arrow z_{3}$ $A_{-10}^{(1)}arrow$ $\mathrm{f}\mathrm{f}\mathrm{i}\text{ }$ [5] g $\text{ }\phi_{ib}^{\overline{o}}\text{ }r_{\mathrm{f}}$ $A_{0-1}^{(1)}arrow\overline{z}_{2}$ $A_{11}^{(1)}arrow\overline{z}_{3}$ $O(5)$ $\dot{z}_{1}=z_{1}[h_{1\lambda}(0)\lambda+h_{1\sigma_{1}}(0)\sigma_{1}+h_{3}(0)u_{1}+h_{1q}(0)q$ $+ \frac{1}{2}h_{1\sigma_{1}\sigma_{1}}(0\rangle\sigma_{1}^{2}+h_{1\sigma_{2}}(0)\sigma_{2}+h_{3\sigma_{1}}(0)u_{1}\sigma_{1}+h_{5}(0)u_{1}^{2}]$ $+\overline{z}_{2}\overline{z}_{3}[p_{2}(\mathrm{o})+$ $p_{2\sigma_{1}}(0)\sigma_{1}+p_{4}(0)u_{1}+p_{2q}(0)q]$ F Taylor 5 [9] [16] [17] $z=01$ 3 6 Boussinesq 1 $\mathrm{s}$ $\mathcal{l}$ $\mathcal{b}\psi=0$ 61 $\mathrm{z}_{2}$ \S Figure 3 $(P=7)$ 3

9 $\mathrm{h}$ $\mathrm{t}$ $\delta=0036$ 119 Figure 3 3 $P=7$ Table 2 0 $\delta>0$ \S \S 62 3 \S Fig 4 Figure 3 $\mathrm{h}_{+}$ Fig 4 $\mathrm{h}_{-}$ $\mathrm{h}_{+}$ 2 Figure 3 $\mathrm{r}\mathrm{a}$ 1 Fig 4 ( ) $\mathrm{h}_{+}$ 2 5 $\delta\simeq 02$ $\mathrm{h}_{-}$ $\mathrm{h}_{+}$ reentrant hexagons \S 7 $\mathrm{h}_{-}$ $\delta\simeq 04$ 62 Figure 5 Figure [12] Figure 3

10 120 Figure 4 5 $P=7$ Rayleigh-Benard Figure 5 5 Nishida Iida and Yoshihara $\mathrm{r}\mathrm{t}$ [15] PQ $\mathrm{r}\mathrm{a}$ ( ) 1 Fig 5 Figure 6 Fig 5 5 $\mathrm{r}\mathrm{t}$ $\mathrm{h}$ 1 $\mathrm{r}\mathrm{a}$ PQ 2 RA reentrant hexagons 7 Boussinesq [19] Boussinesq ltable 3 Nishida et al

11 $ $ 121 Figure 5 5 $P=7$ ( [2]) Assenheimer and Steinberg reentrant hexagons [1] $\delta=24$ Boussinesq $\mathrm{h}_{+}$ H- Dewel et al 1) 6 $\dot{z}_{1}=z_{1}[\sigma_{1}+\mu_{1} z_{1} ^{2}+\mu_{2}( z_{2} ^{2}+ z_{3} ^{2})+\mu_{3}z_{0}^{2}]+\delta_{1}z_{0}\overline{z}_{2}\overline{z}_{3}$ $\{$ $\dot{z}_{0}=z_{0}[\sigma_{0}+\mu_{3}( z_{1} ^{2}+ z_{2} ^{2}+ z_{3} ^{3})+\mu_{4}z_{0}^{2}]+\delta_{2}(\overline{z}_{1}\overline{z}_{2}\overline{z}_{3}+z_{1}z_{2}z_{3})$ 7 3 [S] 3 reentrant hexagons Marangoni Rayleigh-B\ enard $l\mathrm{h}$ Clever and Busse Boussiensq $P=7$ 2 Floquet Assenheimer and Steinberg reentrant hexagons Busse balloon [7] $\text{ }$ $[$ $\mathrm{s}\mathrm{f}_{6}$ by and Steinberg Boussinesq 4 $Q$ Boussinesq reentrant hexagons $Q$ $\text{ }f\ovalbox{\tt\small REJECT} \text{ }7R$ $[1\mathrm{S}]$ Madruga Riecke and Pesch Floquet [14] $\partial_{t}z_{1}=\xi^{2}(\mathrm{n}_{i}\cdot\nabla)^{2}z_{i}+\delta z_{i}-(\kappa+\delta\mu)\overline{z}_{2}\overline{z}_{3}-g_{1} z_{1} ^{2}z_{1}-g_{2}( z_{2} ^{2}+ z_{3} ^{2})z_{1}$ $Q\neq 0$ $\kappa$ 2 0 [19] $(\delta=0)$

12 $\ovalbox{\tt\small REJECT}$ $\delta$ $\delta$ Figure 6 \pi \nearrow \nearrow \acute ff 5 $P=7$ $\text{ }$ ffi $\mathrm{h}\ovalbox{\tt\small REJECT}_{\mathrm{f}}^{\zeta 3}$ t 2 $\vdash^{*}\mathrm{f}\mathrm{g}5$ $\doteqdot\phi$ $\delta$ $\delta$ $\delta\mu$ $Q$ 2 1 $R$ $\alpha$ $Q=0$ 5 Boussinesq reentrant hexagons Clever and Busse $\delta\simeq 02$ \S 2 Stuart-Landau $\delta\simeq 02$ $P=7$ Nishida et al 2 reentrant hexagons 7 Busse and Clever t $Q=0$ $P=7$ 2 [3] reentrant $\text{ squares }$ Busse balloon Effl $\pi/2$ $\Gamma_{s}=\mathrm{D}_{6}\dotplus \mathrm{t}^{2}$ $\alpha$ reentrant hexagons $\alpha\simeq 23$ $\delta=0$ $\alpha\simeq 23$

13 $\mathrm{r}$ 0 $\mathrm{s}\mathrm{q}$ 0 g_{1}^{r}}{\partial x_{1}}(4)$ g_{1}^{r}}{\partial x_{1}}$ (1) g_{1}^{r}}{\partial x_{1}}+\frac{\partial g_{1}^{f}}{\partial x_{2}}$ g_{2}}{\partial x_{2}}(2)$ $\lambda_{1}+\lambda_{2}=\frac{\partial g_{1}^{r}}{\partial x_{1}}+\frac{\partial g_{1}^{r}}{\partial x_{2}}$ $\mathrm{t}^{2}$ $\sigma_{v}$ g_{1}^{r}}{\partial x_{1}}-\frac{\partial g_{1}^{r}}{\partial x_{2}}$ $\lambda_{1}\lambda_{2}=\frac{\partial g_{1}^{r}}{\partial x_{1}}\frac{\partial g_{2}^{r}}{\partial x_{2}}-\frac{\partial g_{1}^{f}}{\partial x_{2}}\frac{\partial g_{2}^{r}}{\partial x_{1}}$ 123 Table 4 Jacobi Label I Eigenvalue (multiplicity) (1) (2) RH 0(2) $\lambda_{1}$ $\lambda_{2}$ $z_{2}\overline{z}_{1}\overline{z}_{2}$ $z_{1}$ $z_{1}$ $z_{2}\in \mathrm{c}$ $\Gamma_{s}$ $\mathrm{r}_{\pi/2}$ $(z_{1} z_{2})arrow(z_{2}\overline{z}_{1})$ $\mathrm{d}_{4}\{$ $(z_{1} z_{2})arrow(z_{1}\overline{z}_{2})$ $(s t)\cdot z=(\mathrm{e}^{is}z_{1} \mathrm{e}^{it}z_{2})$ $\dot{z}_{1}=z_{1}[f_{1}(\lambda \tau_{1} \tau_{2})+ z_{1} ^{2}f_{3}(\lambda \tau_{1} \tau_{2})]$ $\tau_{1}= z_{1} ^{2}+ z_{2} ^{2}$ $\tau_{2}= z_{1} ^{2} z_{2} ^{2}$ $x\in \mathrm{r}$ (R) $( z_{1} z_{2} )=(x 0)$ (SQ) $( z_{1} z_{2} )=(x x)$ IW (RH) $( z_{1} z_{2} )=(x y)$ $x\in \mathrm{r}$ $y\in \mathrm{r}$ $x$ 2 $O(5)$ Table 4 5 $\delta$ reentrant squares $\delta\simeq 13$ Busse and Clever reentrant squares $\mathrm{t}$ ) 5 reentrant squares References [1] M Assenheimer and V Steinberg Observation of coexisting upfiow and downflow hexagons in Boussinesq Rayleigh-Benard convection Phys Rev Lett 76 (1996) 756 [2] $\mathrm{f}\mathrm{h}$ Busse The stability of finite amplitude cellular convection and its relation to an extremum principle JFluid Mech 30 (1967) 625

14 Reentrant $453$ {}^{t}\mathrm{o}\mathrm{n}$ the 124 [3] $\mathrm{f}\mathrm{h}$ $\mathrm{r}\mathrm{m}$ Busse and Clever Asymmetric squares as an attracting set in Rayleigh- B\ enard convection PhysRev Lett 81 (1998) 341 [4] E Buzano and M Golubitsky Bifurcation on the hexagonal lattice and the planar B\ enard problem Phil Trans RSoc Lond A 308 (1983) 617 [5] J Carr Applications of Centre Manifold Theory (Springer-Verlag1980) [6] $\mathrm{m}\mathrm{m}$ [7] $\mathrm{r}\mathrm{m}$ $\mathrm{j}\mathrm{a}$ Chen and Whitehead $\zeta$ Evolution of two-dimensional periodic Rayleigh convection cells of arbitrary wave-number JFluid Mech 31 (1968) 1 $\mathrm{f}\mathrm{h}$ Clever and Busse Hexagonal convection cells under conditions of vertical symmetry Phys Rev (1996) R2037 $\mathrm{e}53$ [8] G Dewel S M\ etens $\mathrm{m}\mathrm{f}$ Hilali and P Borckmans FhysRev Lett 74 (1995) 4647 [9] K Fujimura Centre manifold reduction and the Stuart-Landau equation for fluid $\mathrm{a}$ motion Proc R SocLond (1997) 181 [10] T Herbert Nonlinear stability of parallel flows by high-order amplitude expansions AIAA J 18 (1980) 243 [11] $\mathrm{e}\mathrm{l}$ Koschmieder B\ enard Cells and Tayfor Vortices (Cambridge 1993) $\mathrm{j}\mathrm{w}$ [12] M Golubitsky and Swift and E Knobloch Symmetries and pattern selection $\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}_{7}$ in Rayleigh-B\ enard Physica IOD (1984) 249 [13] E Knobloch ( $ \mathrm{p}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{n}$ selection in long-wavelength convection Physica $D41$ (1990) 450 [14] S Madruga H Riecke and W Pesch Reentrant Hexagons in non-boussinesq Convection (preprint) (2004) [15] T Nishida T Ikeda and H Yoshihara Pattern formation of heat convection problens in Mathematical Modeling and Nermerical Simulation in Continerum Mechanics I Babuska P G Ciarlet and T Miyoshi) Lecture Notes in Computational ($\mathrm{e}\mathrm{d}\mathrm{s}$ Sciences and Engineering 19 (Springer 2002) 209 [16] 1368(2004) [17] (2004) $\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{i}\mathrm{n}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{g}_{7}$ [18] A Roy and V hexagons in non-boussinesq Rayleigh-Benard convection effect of compressibility Phys Rev Lett 88 (2002) $ [19] A Schliiter D Lortz and F Busse convection J Fluid Mech 23 (1965) 129 stability of steady finite amplitude

$\hat{\grave{\grave{\lambda}}}$ $\grave{\neg}\backslash \backslash ^{}4$ $\approx \mathrm{t}\triangleleft\wedge$ $10^{4}$ $10^{\backslash }$ $4^{\math

$$\hat{\grave{\grave{\lambda}}}$ $\grave{\neg}\backslash \backslash ^{}4$ $\approx \mathrm{t}\triangleleft\wedge$ $10^{4}$ $10^{\backslash }$ $4^{\math$ $\mathrm{r}\mathrm{m}\mathrm{s}$ 1226 2001 76-85 76 1 (Mamoru Tanahashi) (Shiki Iwase) (Toru Ymagawa) (Toshio Miyauchi) Department of Mechanical and Aerospaoe Engineering Tokyo Institute of Technology