大阪大学 基礎工学研究科 システム創成専攻博士後期過程２年次

奥田　孝志

Department of Systems Innovation,Graduate School of Engineering Science, Osaka University

D2 Takashi Okuda

Bifurcations for Turing Instability

without SO(2) symmetry

Czech and Slovak Twin Seminar in Applied Mathematics

First Slovak - Japan workshop on ComputationalMathematics

September 9 - 13, 2006, Kocovce chateuand

Czech-Japanese Seminar in Applied Mathematics 2006September 14 - 16, 2006,

Department of Mathematics, FNSPE CTU in Prague

Bifurcations for Turing Instability withoutSO(2) symmetry

Takashi OkudaDepartment of Systems Innovation,

Graduate school of Engineering Science, Osaka University

It is well known that Turing instability is the basic mechanism inthe pattern formation problems. We usually consider reaction-diffusionequations with natural boundary conditions, such as Neumann or pe-riodic boundary conditions. And the solutions to there problems auto-matically have SO(2) symmetry.

On the other hand, in [2], activator-inhibitor systems are consideredwith mixed boundary conditions which is not SO(2) symmetric. Namely,they analyze the system of two component reaction-diffusion equationswhich satisfy different boundary conditions, respectively.

One can also find a similar kind of study in the convection prob-lem. In fact, Mizushima-Nakamura[5] studied linearized stability of theRayleigh-Benard problem with partially nonslip boundary conditionswhich are also not SO(2) symmetric. They observed the repulsion of theeigenvalues, which means the separation of the neutral stability curvesfor different modes, by changing the nonslip parameter. In addition,Kato-Fujimura[4] studied Rayleigh-Benard convection with the bound-ary conditions which correspond to the one considered in [5]. Moreover,they obtained the global bifurcation diagram numerically, and they stud-ied local bifurcation structure by the multiple scale method, as well.

In this study, we consider the Swift-Hohenberg equation:

∂w

∂t=

{ν −

(1 +

∂2

∂x2

)2}w − w3, t > 0, x ∈ (0, L/2). (1)

with the following boundary conditions:

w(t, 0) = w(t, L/2) = 0,

δwx(t, 0) − wxx(t, 0) = 0 ,

δwx(t, L/2) + wxx(t, L/2) = 0 . (2)

Where w = w(t, x) is real valued function, ν, L > 0 and δ ≥ 0 areparameters.

We analyzed the linearized eigenvalue problem(Figure1), and we stud-ied numerically the global bifurcation structures(Figure2). Moreover, westudied local bifurcation structures of stationary solutions to (1) with(2) by analyzing the cubic normal forms.

0.4

0.2

0

L

403020100

nu

1.2

1

0.8

0.6

L

nu

1.2

35

1

0.8

30

0.6

0.4

25

0.2

02015105

図 1: Neutral stability curves drawn in (ν, L)-plane. [Left: They corre-spond to the critical curves for δ = 0], [Right: The critical curves drownbased on the numerical simulation when δ = 0.02. The m-th and n-thcurve are avoid crossing when m + n is even].

delta=0.0,nu=0.37.L

||w||

0. 5. 10. 15. 20. 25. 30. 35.

0.

1.

2.

3.

4.

5.

6.

7.

delta=0.05,nu=0.37.L

||w||

0. 5. 10. 15. 20. 25. 30. 35.

0.

1.

2.

3.

4.

5.

6.

7.

図 2: Bifurcation diagrams of (1) with (2) for ν = 0.37. The horizontalaxis and vertical axis are L and ||w||, respectively. [Left: δ = 0], [Right:δ = 0.05].

References

[1] J.Carr: Applications of Center Manifold Theory, Springer, 1981.

[2] R.Dillon, P.K.Maini, H.G.Othmer : Pattern formation ingeneralized Turing systems I. Stedy-state patterns in systems withmixed boundary conditions, J.Math.Biol.,32(1994),345-393.

[3] Y.Kabeya, H.Morishita and H.Ninomiya, Imperfect bifurca-tions arising from elliptic boundary value problems, Non. Anal.,48,2002, 663-684.

[4] Y. Kato and K. Fujimura: Folded Solution Branches inRayleigh-Benard Convection in the Presence of Avoided Crossingsof Neutral Stability Curves, J. Phys. Soc. Jap., vol.75, No.3, 2006.

[5] J.Mizushima and T. Nakamura: Repulsion of Eigenvalues in theRayleigh-Benard Problem, J. Phys. Soc. Jap., vol.71, No.3, 2002,pp.677-680.

[6] Yasumasa Nishiura: Far-from-Equilibrium Dynamics, AMS,2001.

[7] L.Tuckerman and D.Barkley: Bifurcation analysis of the Eck-haus instability, Physica 46 D, 1990,57-86.