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Degenerate Bifurcations in the Taylor-Couette Problem
Lecture Notes in Num. Appl. Anal., 10, 121-128 (1989) Recent Topics in Nonlinear PDE IV, Kyoto, 1988
Degenerate Bifurcations in the Taylor-Couette Problem HISASHIOKAM O T O Department of Pure and Applied Sciences, University of Tokyo, Meguro-ku, Tokyo 153, Japan
$1. Introduction. The purpose of this paper is to explain certain bifurcation equations which describe newly discovered phenomena in the Taylor-Couette problem of the fluid motion between two concentric cylinders. The equations to be considered here are: (1.1)
EX + a + a t 2 + bz2 + CX’Z) + ( p + ez2>tz= 0, z( 6A
+ &x2 + 132)f x2 = 0)
and (1.2)
{
+ ax2 + b 2 ) f XZ = 0, z(6A + d z 2 + 2122 + t 2 2 ) + ( p + dZ’)Z2 = 0, x( €A + CY
where A, x and z are real variables, E , 6, a,b, c, d , e, B , 8 and 2 are real constants. We explain in the subsequent sections how the solutions to (1.1,2) fit the bifurcation diagram given in Tavener and Cliffe [7] in which Taylor vortices of new type bifurcating from the Couette flow are computed numerically. T h e equations (1.1,2) are derived from a certain degeneration of the equations given in Fujii, Mimura and Nishiura [l]. The equation (1.1) are considered in Fujii, Nishiura and Hosono [2], but (1.2) seems to be new. Although they considers in [1,2] a reaction diffusion system which has nothing t o do with the Taylor-Couette problem, the local structure of the bifurcation is of the same category. This is because the ofthogonal group O(2) acts on both problem.. . In this paper we announce a results in [5,6] in which we employed the singularity theoretic approach by Golubitsky and Schaeffer ( [3,4] ) to see systematically the structure of the equations. In 52 we state a precise statement of the Taylor-Couette problem. In 33 the relation between (1.1,2) and the Taylor-Couette problem is given. $ 4 is a final section for discussions. 121
Hisashi OKAMOTO
122
$2. The Taylor-Couette problem. I n this section we recall sonie newly d e w l o p p e d analysis for the nlc>chanisnlof vortex uuinber exchangt, in [7].The problem considered by thein is t o determine a fluid velocity field ( u , u, w ) and the pressure 11 which satisfy t h e fc,llowing stationary Navipr-StokPs equation (2.1-4):
(2.1 1
(2.2)
(2.3)
1 8 --(ru) r dr
R 111 += 0. 82
UJ a r e r , 8 , -~cumR is the Reyncilds number. In this paper, as in
w h e w the cylindrical coordinates are adopted. u , poririit, respectively and
"71. wc. w i i s i d r r cm1y
18,
velocity fields wliich are i n d e p r d e r i t o f t l w a z i m u tal cuudiiiate 8 and t h e t iiiir t. We haw, s u i t a b l y nondinirnsiunalized the qunntitirs so that ( 2 . 1 4 ) a r ~ satisfied in
The follow iiig boundary conciitic)ns are imposed : (2.5 j
(U,.L',.U?)
= (0,l.O)
(2.7)
I n this circumstance, t h e Couettt. flciw
( r = 1)
Taylor-Couette Problem
123
B = q2/(17 - 1). satisfies all the requirements if A = 1/(1 - 7 i 2 ) , The problem is to study the solutions bifurcating from this Couette flow. Notice t h a t the condition (2.7) makes it difficult t o examine the result,s by a laboratory experiment but there is no clificulty in performing computer simulation. In fact, [7] presents a. penetrating description of t h e stability exchange of the solutions. Let us statmebriefly the results in [7]. T h e y fix the parameter 7 = 1.0/0.615 arid let t h e riondimensional height of the cylinder r vary. It is known t h a t the primary bifurca.tion branch consists of the two-cell Taylor vortices for small value of r and t h a t it consists of the four-cell Taylor vortices for larger value of r. The number of the vortices, however, is integer a.nd l- can vary continuously. 'Therefore, there exist flows of "mixed type" when r is in a intermadiatr range. T h e hifurcation diagrams in [7] describe qualitatively how this exchange from two-cell to four cell occurs. Although they consider the excha.nge mechanism between two-cell and six-cell or four-cell and six-cell, we consider only t,he exchange of two and four, which requires the least mathematical technique to theorize. 93. Degenerate bifurcation equations. In this .sectmionwe show how (1.1,2) are derived. Our starting point is t o observe a hidden symmetry in (2.1-7). Let us consider another problem to seek ( u * , I,!*,w*,p*) which < t < r, the boundary condition satisfies (2.1-4) in 1 < T < r(, (2.5.6) and t h e periodic boundary condition on z = (, Xote that t h e height of the cylinder is double. ) We call this problem [2r]and the original problem [I?]. This n e w problem [2r'] has an a.dvantage t,hat it is 0(2)-equivsriant in the following sense: Let us define an actmionof the orthogonal group O(2) by
-r
(3.2)
-
3'( ..( r, 2 ), Z)( T, z ) 7 v.4 f', z ) , p( r, r ) ) (~U(T,~'+C~),'L~(T,~+CY);LL~(T,Z+~),~~~'T,~
if y E O(2') is a rotation with angle (3.3)
-r, r.
ct,
Y(ZL(T,2),21(T,Z),7JI(7;z),1~r,L))
-
( ~ ( -rz ,) , v ( T , -z),- w ( T , - z ) , & T ,
+a))
-i))
if 3' E O ( 2 ) is a. reflection. T h e n it is ea.sily checked t h a t the problem [2r] is 0(2)-equivariant: in other words, the governing equation is covariant with the 0(2)-action (3.2,3). We recall t h a t tho larger the symmetry group is, the simpler the equation becomes. Therefore the introduction clf [2r]is an advantage, since [r]is cova.riant. with only 5 discrete group. T h e relation between [2r]and [r]is given b y
Hisashi OKAMOTO
124
PROPOSITION 3.1. If ( u * ,~ ~ * , w * , pis* a) solution t o [2r] and if ( u * ,c * , w',p') is invariant with respect to (3.3). then it satisfies [r] in 0 < z < r. Conversely, if ( u , v , IIJ, p ) satisfies [r] and if we extend it to ( u * , z*,w*,p* ) in such a way that u*,v*,p* are even extention of u. v , p , respectively, and ZL.? is an o d d extension of ZU, then ( u*,V ,P C ~ * . ~is* )a solution to [2r]. In short, finding solutions t o [r]is equivalent to finding "symmetric" solutions t o pr]. F r o m now we consider [2r].I n the remaining part of this section, we explain how the analysis in [5] is applied t o [2r]. T h e r e is a numerical evidence t h a t , at some value of r, say J?o , the linearlized operator of (2.1-4) with (2.5,s) and the periodic boundary condition has a four tlimonsional null space spanned by 91 94 which are of the following form:
-
g3 ( r, 2 ) = ( U4(T j c o (~4 $), \$( T ) C O S ( 4 +) , W,(,?-)sin(44), P4( TI() os ( 411)j )
,
= (, CT4( r)sin (4,$) I$(,r)sin (4.14) Mr4( r)cos (4.G) , P4(r.)sin (4.2))) In these expressions Uj(T ) , V,( T'),Wj(T ) , Pj( 7') are functions of T only and y4( r. t )
are determined through a certain ordinary differential equation ( see [6] ).
Let, F = F( R I?; 'u,v, 'w, p ) be a nonlinear functional for [2r]realized i n sonie Banach space, a n d let P be a projection onto t h e 4-dimensional space spanned by 91, -,$4. T h e n the bifurcat,ion of Taylor vortices is governed in a neighborhood of ( R , rcl; 0, vo, 0 , ~ " ) by
--
where (3 is in a complement of the range of P. Since 0(2)-equivariance is reflected in this bifurcation equation, G' must, be of a special form given i n [1,5]. I n order t o explain the form in [1,5], we identify real (.r, y, z, w ) space with C 2 by [ = x iy, C = z zw. Then G niust b e of the following form: G = ( G 1 , G 2 ) ,
+
(3.4)
+
Taylor-Couette Problem
125
+
(3.5) G2 = f3c f4t2l where f,(j = 1, 2 , 3 , 4 ) are functions of R , r, ]
f i ( 0 . o ; 0,010,o) # 0, f 4 ( 0 , 0 ; 0,o. 01 0) # 0 which is considered in [I]. T h e degeneration which we mentioned at the beginning of the present paper is
(A )
(B)
f2(
0, O; O, 01 O l O)
= Ol
.f4(
Ol
O; O, O, O , 1 #
and
f4(0,0; O , o , 0,O) = 0. f i ( 0 , O ; 5,0,0,0> # 0, (C ) T h e case ( B ) is considered in [2] but ( C ) seems to be new. Since (3.4,5) is applicable to a number of problems, we think it is useful t o study (3.1.5) systematically. To this end, the machinaries by Golubitsky and Schaeffer [3,4]are easy to handle for application-oriented niathematicians. Below we suinmarize the results in [5], where a computations of normal forins for (3.4,5) via the method in [3,4] are given. In the case of ( A ) , the bifurcation equation is, when slightly perturbed, generically O( 2)-equivalent t o
where CY is a pert,urbation ( unfolding ) parameter. Not8e t h a t we only consider a n O( 2)-equivariant perturvation. By the O(2j-equivalence we mean t h a t the bifurcation equation is t,ransforrned t o one of this form by a suit,able coordinates change which preserves the O(2)- equivariance. Roughly, we can say that, (3.6) is a normal form in the ca.se of ( A ) . In the case of ( B ) , (3.7)
{
[(EX
+ CY + a l i j z + blC12 + cRe(E2<).) + ( P + el<12)f( = 0, ((SX + q# + B1CI2) f t2= 0 ,
is a norma.1 form. I n the case of (C), (3.8)
i
E( E X + a + a151’
((6X
+ hlCI’)
f
Tc = 0,
+ tilEz = U,
is a norrna.1 form. When we consider t h e problem [I?] , the solutions are invariant. with respect t o the reflection (3.3:). Therefore we obtain the bifurcation equations for [I’]by rest,rict,ing complex variables E, t.o real ones z,z . T h i s restriction produces (1.1,q from (3.7,13), respectively.
<
Hisashi OKAMOTO
116
$4. Discussions. We first reinark t h a t , when R is taken as a bifurcation parameter, there are additional splitting parameters r and q. Therrefore there is a good possibility t h a t the foHowing scenario holds true: In the (r,q ) plane there is a 1-dimensional variety where t.he critical Reynolds number of 2-cell Row and that, of 4-cell flow coincide. And on t h i s variety there a.re points at which ( B ) or ( C ) holds. If we assume t h a t these points exist and are not far from the values taken in [7], then i t is nat,ursl that, the figures in [7]m e captured by ( 1 . l i 2 ) . W e no\v show how the phenomana in [7]is explained by (1.1,2). We ckoose u and ,O suitably t o have zero setmaof (1.1). In Fig. 2-6, we give c l r a w i n e by a computer a.nd a X-Y plot,ter. These explain the bifurcation diagtanis in [7] for large I‘ ( Fig. 4.3 (D-I) ). From ( 1 . 2 ) we cibt.ain Fig. 1, which explains diagrams for smaller I? ( Fig. 4 . 3 ( : C )of [7] ). Althuugh there is a pitchpork bifurcation in Fig. 4 . 3 (A,B,C) of [7], this ca.iinot be explained by (1.2:). This ,however, niay be a consequence c,f inore globa.1 bifurcation. Except for this, our pictures fit, the diagrams i n [7] qtiite well. For the complete set of the bifurcation diagrams of ( I . l , ? ) . see [GI.
Acknowledgement. T h e author wishes t,o express his t h a n k s to Prof. H . Fujii a n d Prof. k’. Nishiura for showing [2] a.nd giving k i n d advices.
REFERENCES 1 . I< Fujii, b1 Mimura and Y. Niehiura, A ptclur-e of the g f O b 5 1 brfurration dtagram
und d i g u r r n g ryrfernr, Phgsica 5D (1982),1 4 2 . Fujii. Y. Nishiura an4 Y . Hosono, O n t h e r l r u r t u r e of m u l t i p l e r r i r t e n e e o j r l a b l e s i e l i o n a r y r o l u l i o n r in r y r f e r n s of r e o c f r o n - d i f l u r i o n e q u a f i o n r , Patterns snd W a v e s - Qualitative analysis of nonlinear differcnt,ial equations, e d s . T.Niehida. M . R4iinura and H . Fujii , North-Holland (1986),167-218. M . G ol i i hi tsk y and D . S h a e f f e r , Im.pcrjrrl bifirrcnlion in the p r c i e n c r o f #urnt n r f r y , C‘#>mni.\ l a t h Phya. 6 7 ( l 9 7 9 ) , 205-232. M (:;olut,itnky and D. Yihaeffer. “Yingiilaritivls and G r u u y o in Lifurcatiun theory. V I ? ~ 1.” 5pring.r Vrrlag. N e w York. 1985. ti Oknmi, to, 0 (‘ .? )-epu i v ar I o n i b , f i r rcof io z q ua 1 i o n 3 IU I t h tu.o 131 o d r I i n it- ra c l i u m , . ( preprint ) . H O k a n i o t o and 5 . Tavenvr, A d r g c n r r n f c 0 ( 2 ) - P q u r r ~ n r ~ r n nbf a f ~ t r c n l i o ne q u a t i o n rind i t s ( r p p l i r n l i o n l o t h e T a y l o r p r o b l c m , ( preprint ) . T a v r n e l - ari,l K . A . Cliffe. P r i m n r y f l o w e r c h a i i y a r n e c h v n i r m . 8 In T a y l o r l . . - o u e t t ~ f l o u . a p p l y i n g n u n - p u z b o u n d a r y c o n d i l t o n r , ( preprint ) . in c c o l o g i c a l tntrrncirng
?. H
3.
4.
=,. C
7.
-
l i r uusordr. O( L!-eqiiivariance. multiple bifurcatiun. Tavlor v o r t e x
Taylor-Couette Problem
\
m
128
Hisashi OKAMOTO
Degenerate Bifurcations in the Taylor-Couette Problem HISASHIOKAM O T O Department of Pure and Applied Sciences, University of Tokyo, Meguro-ku, Tokyo 153, Japan
$1. Introduction. The purpose of this paper is to explain certain bifurcation equations which describe newly discovered phenomena in the Taylor-Couette problem of the fluid motion between two concentric cylinders. The equations to be considered here are: (1.1)
EX + a + a t 2 + bz2 + CX’Z) + ( p + ez2>tz= 0, z( 6A
+ &x2 + 132)f x2 = 0)
and (1.2)
{
+ ax2 + b 2 ) f XZ = 0, z(6A + d z 2 + 2122 + t 2 2 ) + ( p + dZ’)Z2 = 0, x( €A + CY
where A, x and z are real variables, E , 6, a,b, c, d , e, B , 8 and 2 are real constants. We explain in the subsequent sections how the solutions to (1.1,2) fit the bifurcation diagram given in Tavener and Cliffe [7] in which Taylor vortices of new type bifurcating from the Couette flow are computed numerically. T h e equations (1.1,2) are derived from a certain degeneration of the equations given in Fujii, Mimura and Nishiura [l]. The equation (1.1) are considered in Fujii, Nishiura and Hosono [2], but (1.2) seems to be new. Although they considers in [1,2] a reaction diffusion system which has nothing t o do with the Taylor-Couette problem, the local structure of the bifurcation is of the same category. This is because the ofthogonal group O(2) acts on both problem.. . In this paper we announce a results in [5,6] in which we employed the singularity theoretic approach by Golubitsky and Schaeffer ( [3,4] ) to see systematically the structure of the equations. In 52 we state a precise statement of the Taylor-Couette problem. In 33 the relation between (1.1,2) and the Taylor-Couette problem is given. $ 4 is a final section for discussions. 121
Hisashi OKAMOTO
122
$2. The Taylor-Couette problem. I n this section we recall sonie newly d e w l o p p e d analysis for the nlc>chanisnlof vortex uuinber exchangt, in [7].The problem considered by thein is t o determine a fluid velocity field ( u , u, w ) and the pressure 11 which satisfy t h e fc,llowing stationary Navipr-StokPs equation (2.1-4):
(2.1 1
(2.2)
(2.3)
1 8 --(ru) r dr
R 111 += 0. 82
UJ a r e r , 8 , -~cumR is the Reyncilds number. In this paper, as in
w h e w the cylindrical coordinates are adopted. u , poririit, respectively and
"71. wc. w i i s i d r r cm1y
18,
velocity fields wliich are i n d e p r d e r i t o f t l w a z i m u tal cuudiiiate 8 and t h e t iiiir t. We haw, s u i t a b l y nondinirnsiunalized the qunntitirs so that ( 2 . 1 4 ) a r ~ satisfied in
The follow iiig boundary conciitic)ns are imposed : (2.5 j
(U,.L',.U?)
= (0,l.O)
(2.7)
I n this circumstance, t h e Couettt. flciw
( r = 1)
Taylor-Couette Problem
123
B = q2/(17 - 1). satisfies all the requirements if A = 1/(1 - 7 i 2 ) , The problem is to study the solutions bifurcating from this Couette flow. Notice t h a t the condition (2.7) makes it difficult t o examine the result,s by a laboratory experiment but there is no clificulty in performing computer simulation. In fact, [7] presents a. penetrating description of t h e stability exchange of the solutions. Let us statmebriefly the results in [7]. T h e y fix the parameter 7 = 1.0/0.615 arid let t h e riondimensional height of the cylinder r vary. It is known t h a t the primary bifurca.tion branch consists of the two-cell Taylor vortices for small value of r and t h a t it consists of the four-cell Taylor vortices for larger value of r. The number of the vortices, however, is integer a.nd l- can vary continuously. 'Therefore, there exist flows of "mixed type" when r is in a intermadiatr range. T h e hifurcation diagrams in [7] describe qualitatively how this exchange from two-cell to four cell occurs. Although they consider the excha.nge mechanism between two-cell and six-cell or four-cell and six-cell, we consider only t,he exchange of two and four, which requires the least mathematical technique to theorize. 93. Degenerate bifurcation equations. In this .sectmionwe show how (1.1,2) are derived. Our starting point is t o observe a hidden symmetry in (2.1-7). Let us consider another problem to seek ( u * , I,!*,w*,p*) which < t < r, the boundary condition satisfies (2.1-4) in 1 < T < r(, (2.5.6) and t h e periodic boundary condition on z = (, Xote that t h e height of the cylinder is double. ) We call this problem [2r]and the original problem [I?]. This n e w problem [2r'] has an a.dvantage t,hat it is 0(2)-equivsriant in the following sense: Let us define an actmionof the orthogonal group O(2) by
-r
(3.2)
-
3'( ..( r, 2 ), Z)( T, z ) 7 v.4 f', z ) , p( r, r ) ) (~U(T,~'+C~),'L~(T,~+CY);LL~(T,Z+~),~~~'T,~
if y E O(2') is a rotation with angle (3.3)
-r, r.
ct,
Y(ZL(T,2),21(T,Z),7JI(7;z),1~r,L))
-
( ~ ( -rz ,) , v ( T , -z),- w ( T , - z ) , & T ,
+a))
-i))
if 3' E O ( 2 ) is a. reflection. T h e n it is ea.sily checked t h a t the problem [2r] is 0(2)-equivariant: in other words, the governing equation is covariant with the 0(2)-action (3.2,3). We recall t h a t tho larger the symmetry group is, the simpler the equation becomes. Therefore the introduction clf [2r]is an advantage, since [r]is cova.riant. with only 5 discrete group. T h e relation between [2r]and [r]is given b y
Hisashi OKAMOTO
124
PROPOSITION 3.1. If ( u * ,~ ~ * , w * , pis* a) solution t o [2r] and if ( u * ,c * , w',p') is invariant with respect to (3.3). then it satisfies [r] in 0 < z < r. Conversely, if ( u , v , IIJ, p ) satisfies [r] and if we extend it to ( u * , z*,w*,p* ) in such a way that u*,v*,p* are even extention of u. v , p , respectively, and ZL.? is an o d d extension of ZU, then ( u*,V ,P C ~ * . ~is* )a solution to [2r]. In short, finding solutions t o [r]is equivalent to finding "symmetric" solutions t o pr]. F r o m now we consider [2r].I n the remaining part of this section, we explain how the analysis in [5] is applied t o [2r]. T h e r e is a numerical evidence t h a t , at some value of r, say J?o , the linearlized operator of (2.1-4) with (2.5,s) and the periodic boundary condition has a four tlimonsional null space spanned by 91 94 which are of the following form:
-
g3 ( r, 2 ) = ( U4(T j c o (~4 $), \$( T ) C O S ( 4 +) , W,(,?-)sin(44), P4( TI() os ( 411)j )
,
= (, CT4( r)sin (4,$) I$(,r)sin (4.14) Mr4( r)cos (4.G) , P4(r.)sin (4.2))) In these expressions Uj(T ) , V,( T'),Wj(T ) , Pj( 7') are functions of T only and y4( r. t )
are determined through a certain ordinary differential equation ( see [6] ).
Let, F = F( R I?; 'u,v, 'w, p ) be a nonlinear functional for [2r]realized i n sonie Banach space, a n d let P be a projection onto t h e 4-dimensional space spanned by 91, -,$4. T h e n the bifurcat,ion of Taylor vortices is governed in a neighborhood of ( R , rcl; 0, vo, 0 , ~ " ) by
--
where (3 is in a complement of the range of P. Since 0(2)-equivariance is reflected in this bifurcation equation, G' must, be of a special form given i n [1,5]. I n order t o explain the form in [1,5], we identify real (.r, y, z, w ) space with C 2 by [ = x iy, C = z zw. Then G niust b e of the following form: G = ( G 1 , G 2 ) ,
+
(3.4)
+
Taylor-Couette Problem
125
+
(3.5) G2 = f3c f4t2l where f,(j = 1, 2 , 3 , 4 ) are functions of R , r, ]
f i ( 0 . o ; 0,010,o) # 0, f 4 ( 0 , 0 ; 0,o. 01 0) # 0 which is considered in [I]. T h e degeneration which we mentioned at the beginning of the present paper is
(A )
(B)
f2(
0, O; O, 01 O l O)
= Ol
.f4(
Ol
O; O, O, O , 1 #
and
f4(0,0; O , o , 0,O) = 0. f i ( 0 , O ; 5,0,0,0> # 0, (C ) T h e case ( B ) is considered in [2] but ( C ) seems to be new. Since (3.4,5) is applicable to a number of problems, we think it is useful t o study (3.1.5) systematically. To this end, the machinaries by Golubitsky and Schaeffer [3,4]are easy to handle for application-oriented niathematicians. Below we suinmarize the results in [5], where a computations of normal forins for (3.4,5) via the method in [3,4] are given. In the case of ( A ) , the bifurcation equation is, when slightly perturbed, generically O( 2)-equivalent t o
where CY is a pert,urbation ( unfolding ) parameter. Not8e t h a t we only consider a n O( 2)-equivariant perturvation. By the O(2j-equivalence we mean t h a t the bifurcation equation is t,ransforrned t o one of this form by a suit,able coordinates change which preserves the O(2)- equivariance. Roughly, we can say that, (3.6) is a normal form in the ca.se of ( A ) . In the case of ( B ) , (3.7)
{
[(EX
+ CY + a l i j z + blC12 + cRe(E2<).) + ( P + el<12)f( = 0, ((SX + q# + B1CI2) f t2= 0 ,
is a norma.1 form. I n the case of (C), (3.8)
i
E( E X + a + a151’
((6X
+ hlCI’)
f
Tc = 0,
+ til
is a norrna.1 form. When we consider t h e problem [I?] , the solutions are invariant. with respect t o the reflection (3.3:). Therefore we obtain the bifurcation equations for [I’]by rest,rict,ing complex variables E, t.o real ones z,z . T h i s restriction produces (1.1,q from (3.7,13), respectively.
<
Hisashi OKAMOTO
116
$4. Discussions. We first reinark t h a t , when R is taken as a bifurcation parameter, there are additional splitting parameters r and q. Therrefore there is a good possibility t h a t the foHowing scenario holds true: In the (r,q ) plane there is a 1-dimensional variety where t.he critical Reynolds number of 2-cell Row and that, of 4-cell flow coincide. And on t h i s variety there a.re points at which ( B ) or ( C ) holds. If we assume t h a t these points exist and are not far from the values taken in [7], then i t is nat,ursl that, the figures in [7]m e captured by ( 1 . l i 2 ) . W e no\v show how the phenomana in [7]is explained by (1.1,2). We ckoose u and ,O suitably t o have zero setmaof (1.1). In Fig. 2-6, we give c l r a w i n e by a computer a.nd a X-Y plot,ter. These explain the bifurcation diagtanis in [7] for large I‘ ( Fig. 4.3 (D-I) ). From ( 1 . 2 ) we cibt.ain Fig. 1, which explains diagrams for smaller I? ( Fig. 4 . 3 ( : C )of [7] ). Althuugh there is a pitchpork bifurcation in Fig. 4 . 3 (A,B,C) of [7], this ca.iinot be explained by (1.2:). This ,however, niay be a consequence c,f inore globa.1 bifurcation. Except for this, our pictures fit, the diagrams i n [7] qtiite well. For the complete set of the bifurcation diagrams of ( I . l , ? ) . see [GI.
Acknowledgement. T h e author wishes t,o express his t h a n k s to Prof. H . Fujii a n d Prof. k’. Nishiura for showing [2] a.nd giving k i n d advices.
REFERENCES 1 . I< Fujii, b1 Mimura and Y. Niehiura, A ptclur-e of the g f O b 5 1 brfurration dtagram
und d i g u r r n g ryrfernr, Phgsica 5D (1982),1 4 2 . Fujii. Y. Nishiura an4 Y . Hosono, O n t h e r l r u r t u r e of m u l t i p l e r r i r t e n e e o j r l a b l e s i e l i o n a r y r o l u l i o n r in r y r f e r n s of r e o c f r o n - d i f l u r i o n e q u a f i o n r , Patterns snd W a v e s - Qualitative analysis of nonlinear differcnt,ial equations, e d s . T.Niehida. M . R4iinura and H . Fujii , North-Holland (1986),167-218. M . G ol i i hi tsk y and D . S h a e f f e r , Im.pcrjrrl bifirrcnlion in the p r c i e n c r o f #urnt n r f r y , C‘#>mni.\ l a t h Phya. 6 7 ( l 9 7 9 ) , 205-232. M (:;olut,itnky and D. Yihaeffer. “Yingiilaritivls and G r u u y o in Lifurcatiun theory. V I ? ~ 1.” 5pring.r Vrrlag. N e w York. 1985. ti Oknmi, to, 0 (‘ .? )-epu i v ar I o n i b , f i r rcof io z q ua 1 i o n 3 IU I t h tu.o 131 o d r I i n it- ra c l i u m , . ( preprint ) . H O k a n i o t o and 5 . Tavenvr, A d r g c n r r n f c 0 ( 2 ) - P q u r r ~ n r ~ r n nbf a f ~ t r c n l i o ne q u a t i o n rind i t s ( r p p l i r n l i o n l o t h e T a y l o r p r o b l c m , ( preprint ) . T a v r n e l - ari,l K . A . Cliffe. P r i m n r y f l o w e r c h a i i y a r n e c h v n i r m . 8 In T a y l o r l . . - o u e t t ~ f l o u . a p p l y i n g n u n - p u z b o u n d a r y c o n d i l t o n r , ( preprint ) . in c c o l o g i c a l tntrrncirng
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Hisashi OKAMOTO