Physics $D (1982) 1-42 North-Holland Publishing Company

A PICTURE OF THE GLOBAL BIFURCATION DIAGRAM IN ECOLOGICAL INTERACTING AND DIFFUSING SYSTEMS

H. FUJII Department of Computer Sciences, Kyoto $angyo University. Kyoto 603. Japan

M. MIMURA Departmen~ of Mathematics, Hiroshima University, Hiroshima 730, Japan

and

Y. NISHIURA Department o[ Computer Sciences. K yoto Sangyo University, K yolo 603, Japan

Received 3 September 1981 Revised 13 January 1982

This paper concerns global phenomena of pattern formation in stationary reaction-diffusion equations, poss- essing Turing's diffusion-induced instability, which appear typically in mathematical biology. Glob~ bifurcation diagrams with respect to two dMusion parameters are presented by integrating two complementary' approaches - analytical and numerical, Simple and double bifurcation analysis using group theoretic methods for t~.e compact Lie Ilroup D,. results from singular perturbations when one diffusion constant is sufficiently ,-~,all, and a global existence theorem on primary bifurcated branches when the other diffusion is sufficiently large, are the main analytical results. A new numerical method is used to trace all bifurcating branch s. We examine interrelations between those local and semi-global results under the light of global pictures obtained by numerical studies. A varieq~ of interesting and new diagrams near double eigenvalues are observed. Coexistence of multiple stable stationary states and global extension and deformation of local double structures are one of the main conclusions.

1. Introduction

Plenty of attentio~l has been focused on reac- tion-diffusion equations 0 a 2 o'7 u = d, Fx u + f(u. v).

0 a 2 a-~ V = d2~-~ t~ + g(U, V).

(!.1)

subject to the zero flux boundary conditious

OU OV 0-x = Ox = 0 on the boundary, (!.2)

which often serve as nonlinear models of various phenomena with interaction and diffusion in, e.g., biochemistry, developmental biology, plasma physics, population dynamics and other fields. The interestitlg problem is the

phenomenon of pattern formation obse.'ved in steady-state solutions of (1. I)-41.2), i.e., of

d 2 d, d-~ u +/(u, v) = O.

d 2 d2~v +g(u, v)=O,

(1.3)

with the boundary conditions (I.2). The main tools for the study of pattern for-

mation are, on the one hand, the local bifur- cation analysis at simple or multiple eigen- values, and on the other hand, regular and/or singular perturbations when some physical parameters are controlled externally. (See. e.g.. Fife [4] and its bibliography.) Such research has extensively clarified questions on the onset ana existence of non-uniform patterns, their stabil- ity, existence of large amplitude patterns (e.g..

0167-278918210000-4}0001502.75 © 1982 North-Holland

2 H, Fujii el aLIA picture of the global bi[urcatlon diagram

14|, li3] and 121]), global existence of some bifurcated non-uniform patterns [26, 2";, 22], and s o on.

Even with such knowledge, it seems still open and difficult to have a complete picture of global parametric dependency of non-uniform stationary patterns. For instance, an attractive phenomenon of co-existence of multiple stable patterns in the system {1.1)-(I.2) has been noticed numerically. See. e.g. [3] and [17]. However, mathematical understanding of such a phenomenen from a global point of view remains incomplete and difficult.

As an approach to such a problem, one may resort to numeri:al methods, which may consist in solving (i. I)-(I.2) as an initial-boundary value problem, and letting t 1' +~: to obtain a sta- tionary pattern. A disadvantage of this approach is, of course, that only stable stationary states can be actually captured.

Another approach, which raay be free from such a difficulty, is to solve directly the sta- tionary problem (!.3)--(1.2) by some numerical techniques such as the shooting method [17] or the Galerkin methad [3]. Recently, a new ,':umerical algorithm along these lines has been developed by the first author [6]. This algorithm is ba.,ed on the principle that any symmetry- breaking singularities can be masked if the groul.,-theoretical decomposition is assumed on the solution space. One can get all bifurcation points, even secondary or tertiary ones in this algorithm. Thus, at least in principle, all bifur- cating branches of stationary solutions can be traced numerically, when some parameters are :'aried.

The aim of this paper is. by integrating two complementary approaches - analytical and numerical, t~ present a picture of global depev.- dency of one-dimensional non-uniform patterns on sor~e parameters. More specifically, we study the global diagram of bifurcating branches in R! ~ X, where \ is a suitable solution space. Fiery, we consider the diffusion constants td,, d.,)E/R! as external parameters to be con- trolled. There is no particular reason, in prin-

ciple, to limit such a choice of the parameters. We would emphasize, however, that this choice has an advantage that solutions can be analy- tically studied in the limits dt ~ +0, or d2 'F +~o.

Most works on pattern formation fall within one of three catagories. The first one is due to Turing's idea of diffusion-induced instabifity [34], which is mathematically realized through simple or multiple bifurcation analysis [28, 18]. The significance of double criticalities in under- standing secondary bifurcations has been well recognized (for instance, see [1]), and for Dirichlet boundary problems, a number of works on local double critical analysis have been published [3, 12, 16, 30]. However, it should be emphasized that, due to the presence of a high symmetry group (i.e., essentially, D®- symmetry, see, section 2.2), Neumann boundary problems as in (1.3)-(i.2) may have far more complicated local structures near double eigen- values than Dirichlet problems. We shall show a number of interesting and new (namely, not yet observed) diagrams. Also, the question of how the local structure near a bifurcation point affects the global bifurcation picture is, up to present, not ansxvered, and this is certainly one of our main interests here.

The second one concerns singular pertur- bation results in the limit dt J, 0. Fife [4] for Dirichlet problems, and our previous study [21] for (I.2)-(1.3), have shown the existence of large amplitude non-uniform patterns. It is, however, mathematically not known in general what the relation is between these large am- plitude singular perturba0on solutions and the small amplitude bifurcating ones. The third work has treated regular perturbation in the limit d~ t ~ 0 [13]. In this sit-ation, some recent work of one of the co-authors states that some bifurcating branches continue to exist globally with respect to dl. connecting to the singular perturbation solutions [22, 23].

Interrelations between these results may form the core of our motivation. We examine these local and semi-global results in the light of the global picture obtained through numerical stu

H. Fujii et aLIA picture oJ e the global bifurcation diagram 3

dies. We hope that this study may give an insight into the mechanism of various interes- ting phenomena in stationary reaction-diffusion equations.

We state here assumptions on t h e non- linearities J andg in (1,1), which are essentially derived from biological problems:

(A.1) f and g are sufficiently smoot~h functions of U - (u , v).

(A.2) T.~er, -~xists a unique constant state 0 - (~, ~:-~ ". J o f f ( a , ~) = g ( ~ , e ) = 0.

(A.3) At U = O, [ . > 0 , f . + g ~ < 0 and . f ,g~- f~g. > 0 .

The last assumption comes from the requirement of the occurrence of bifurcations in Turing's sense. Namely, (A.3) implies that 0 = (iL ~) is a stable critical point of

d d "~ u = f ( u , v), ~-~ v = g(u, v). ( ! .4)

One finds that (A.3) is divided up into two cases: One is the case where at U = 0.

(A.3.1) [. >0, /~ <0, g. >0 , g~ <0,

.f. + gv < 0 and/,go - /~g. > 0,

in which are included prey-predator models by Segei and Jackson [32], Okubo [~,4], Mimura [19], morphogenetic models by Gierer and Meinhardt [10], and substrate-inhibition models by Seelig [311. The other is where at U - U,

f. >0,fv >0 ,g . <0,g~ <0,

f. ~- g, < 0 and [.g~ - f,,g. > 0,

The precise form of (A.4) will be given in section 3.

The prey-predator model [19], the r,or- phogenetic model with saturation (see, [16~ in [10]) and the substrate-inhibition model [20]

• Q f , satisfy t h e assumption (A.4), A s a spectfi model which satisfies (A,I)(A,4), we give ~r ecological interacting and diffusing syster, i~ [19]:

0 0 2 0-7 u = d, ~ - ~ . + fo(=)u - k.v,

a a 2 O-t v = d, ~-~ v - go(v)v + kuv,

( t .5 )

where the functional forms of f0 and go are drawn in fig. 1. I. This model will be used in our numerical study.

From an ecological viewpoint, the problem is with questions concerning patch formation of two interacting species, ',¢ltich are dispersing randomly in a popu'tation reservoir with reflecting walls (a lake ~,r an islandt). Whether they form non-uniform distributions (=patchi- ness), and how the shape of the patterns depends on some ecological parameters, are the basic interests.

The outline of this paper Js the following: In section 2, we study local bifurcation structures near double criticalities. Our approach seems new in that we regard the one-dimensional Neumann boundary problem (1.2)(I.3) as a section uf a periodic boundary problem defined on a circle, the latter of which possesses the

in which the "Brusselator" by Prigogine and Lefever [25] is included. When equations of type (1.3) satisfy (A.3.1), they are called activa- tor-inhibitor systems.

In a part of our discussions, we shall make a further assumption on the global shape of the nonlinearity [. Tkis assumption is used to con- struct large amplitude ~atterns in section 3.

(A.4) The zero level curve of J(u, v)= 0 is S- shaped in (u, v)-space.

ktl - q o l v )

Fig. I.I. fo(u) ~ (119)(3S + 16u - u~), go(v) = 1 + (215)u.

4 H. Fujii el aLIA piclure of/he global bifurcatio~ diagram

symmetry group D , - a compact Lie group of plane rotations and a reflection. Then, using the above approach with the aid of group theoreti- :al methods, we classify the bifurcation equa- tions at simple and double critical points on D~- and D,-symmetric branches. (D,: a discrete subgroup of D=. which sends a regular n-poly- gon into itself.) We observe a variety of new bifurcation diagrams depending on symmetry groups of eigenfunctions.

We may complement here the above intro- duction by explaining why and how the group theoretical methods work for the analysis of the structure of the solution set near a singularity. The Lyapounov-Schmidt p~ocedure reduces th~is analysis to the study of the zeroes of a system of scalar equations, called the bifur- cciion equations. In general, the bifurcation equations can be very complicated. However, if the original equations governing the physical system is independent of the observer, the equation~ are expected to be symmetric under a certain group action, and they are called co~'ariant. In fact. our pro~,lem can be identified x~ith a D~-covarianl problem defined on a circle. Due to this group covariance, the bifurcation equations can be reduced to one of several for,ns. For the p:oblem under consideration, it can actual'v be determined which form appears at which singularities. This reduction is based on th,~, principle that the group covariance of the original equations in inherited to the bifurcation equations, and which comes from the methods of group representation theory (see, D.H. Sat- finger [29]).

The notion of the group covariance is to be distinguished from the case in which certain ,~,lttti,,ns of interest are symmetric, called in- ~,~ri,mr. The group invariance of bifurcating ~,qutions is certainly important from the view- ~ in t of pattern (or, patch) formation of sta- ri,,nary solutions. The covariance of the prob- I,~m alloxvs the above reduction, while the in- ~:~riance of solutions comes from studying the ~ymmetry invariance of eigenfunctions at the

singularity under some subgroup of the full covariant group. This can be done with the aid of the "standard decomposition" of the whole solution space ~, and consequently of the ker- nel space with respect to the covariant group at the singularity (see, H. Fujii and M. Yamaguti [7]).

Section 3 is devoted to a summary of those analytical resuP r about two different pertur- bation problems (in the limits dl '¢ 1 and d~t~ 1) which have been obtained in our previous works [21, 23]. The first result is that there exist non-constant solutions of (1.3)-(1.2), (u,v) (x;dt. d:) for sufficiently small dt and d~ ~, and incidently justify the relation

lim lim (u. v)(x; dr, d,) dl ,[ O d2 T +~=

= lira lira (u, v)(x; Jr, d2). d-, 'r +~= d I ~ 0

Secondly, in the case d~ t ~ !, we obtain a global existence theorem ~th respect to the parameter dtER~.

Section 4 focuses on the numerica! study of the model problem (t.5) and (1.2) in order to find the global bifurcation diagram when (din, d9 are unrestricted in R~. We draw a picture of the global diagram which displays clearly the inter- relation between bifurcating solutions and per- turbed solutions in section 3. We observe in the diagram a number of interesting phenomena of bifurcating branches, including appearance of limit points, of secondary and tertiary bifur- cation points, disappearance of bifurcated branches, global existence of bifurcated bran- ches and so on.

Among these, co-existence of several stable patterns as a consequence of successive reco- very of stability is the most interesting phenomenon. This seems one of the first cases in which not only the existence but aise the mathematical mechanism of such a phenomenon have been convinced, although some numerical evidence has been known by other authors [3, 17]. A biological or physical consequence of this

tl. Fujil et aL/A picture of lhe global bifurcation diagram 5

result seems important, since this means that several biological or physical states are ad- missible in the nature. It is to be noted that a similar phenomenon has been experimentally observed in fluid mechanics. For instance, concerning the R a y l e ~ B 6 n a r d convection problem, J.P. Gollub points out in his survey [35] that "One important discovery is the observation that there are a number of stable convection patterns in cells with small aspect radio, depending on the past history or initial conditions of the system". It should be remar.. ked here that this may pose another mathema- tical and unsolved question that which pattern among those stable patterns can be realized from w~ich initial state.

Finally, section 5 is devoted to discussions about results in sections 2, 3 and 4. and to some remarks on the results obtained in this paper and on open questions about our subject.

List of notation

[I] D~; a compact Lie group of plane rotations r0(0<~0<2~r) and a reflection s(s:= I), namely, D~ ~ {ro, sro}o..o~:.,.

[2] D. ; a subgroup of order 7n, generated by the

rotation r=r2~, and s, n.amely, D , ~ {,k. srk}~:~o.

We shall have also subgroups C= and C. of D®, which consist only of rotations. See, e,g,

[3] X = H ~ I ) ( a real Hilbert space) =closure of {cos(n~rx/L)}::o in the

Sobolev space H2(1). [4] ~ = H~ff) (a complex Hilbert space)

= closure of {exp(incrx/L)}±,®,o in H2(f), where [ is a circle with length 2L.

X and ,~ are equipped with the usual Sobolev inner product, e.g., for any u, v E f~,

(v,, v)~ = f {AuA~ + Vu • Vt~ + u~} dx. I F !

[5] Y = L2(1), a real Hilbert space. [6] 5" = L2(f), a complex Hilbert space, the

complexification of Y. Y and f" are equipped with the usual

L'-inner Froduct. [7] X = (X)" and f~ -- (~):, where

2

(U. V)~ = Y~ (uj, ~'3~.

it,." any U = (ut , u.,) and V -- (vt , v2)E X.

[81 v = ( Y ) : and Y = (~)2.

2. Lock. structures near simple and double critical point, °

In this section, we deal with (I.3,-(I.2) under the assumptions (A. I)-(A.3). We rev,rite this problem by setting U = 0 + V, where V = (v,, v2):

d2 D(d) -dF v +

(P) ]d wm~-

Idx " '"' - dx

B • V + N ( V ) = O.

• ~ a . . , , f ~ v ,

X E I m (0, L ) . (2.1)

[d~ 0 ] and B = (b,j) (i, j = 1,2) is the Jacobian matrix of F at U = 0 : where D(d)= 0 d,

g,,Jwo (2.3)

where the elements of B satisfy (A.3).

6 H. Fajii el aLIA picture ol the global bifarcalion diagram

2. I. Primary bifurcation curves

By the assumption (A.3), the constant state U = 0 (namely, V = 0) is stable to small spatially homogeneous perturbations. It is also stable te spatially non-homogeneous perturbations when d = (d~, d,.) E R~ are sufficiently large.

The set of d ~ R~ where V = 0 loses its stability is obtained by solving the associated linear eigenvalue problem:

.T~d)~" = A(d)q ~, x E I,

d XP'(0) = ~_. ~(L) = 0.

,.ix

(2.4)

(2.5)

where • = (~,, ~:), and

d: .~(d) = D(d) d-~ + B. (2.6)

The set of d = (d~. d,.)E/R~ where (2.4)-42.5) has a zero eigenvalue ;~(d)= 0 corresponds to primary and time-independent bifurcation points where the trivial solution V = 0 loses stability. In fact, this set consists of an infinite number of hyperbolic curves {F,}~., in R2+, where

{ i ' r ]} r,, = (d,, d,.) d2 = h.(dD =- - ' ~ L3"n~d,_ b,, + b:: ~n = I, 2 . . . . ) (2.7)

and 3' -= (w/L) (see [18]). r . ( n ~> I) is called the nth primary bifurcation curve (with respect to V = 6), see fig. 2.1. The points r,,,. = r . N r, ,(n, m >~ I, n > m) where two primary bifurcation curves intersect is a

double critical point.

We note that the F, 's are all similar in the sense that the relation ll,(dt)= (i/n:)h,(n:d,)(n >~ !) holds.

It is immed,~ately seen from (A.3) that there exist no points d ER~ such that ,k(d) takes pure imaginary values. Accordingly, there are no primary bifurcations of Hopf type in the system (P).

2.2. CiassiSication of bifurcation equations and group symmetry o.f bifurcated branches

The aim of this subsection is to derive bifurcation equations at simple and double critical points on D~ and D,-symmetric branches, arQd to classify them according to symmetry groups of kernel spaces at those bifurcation points. We fo![ )w the group-theoretic methods due to D.H. Sattinger [29] and H. Fujii and M. Yamaguti [7]. However, we need to modify their arguments so as to apply them to our Neumann boundary value problem (P).

First, we note two points which are important to our discussion; as in [7], we use the notion of the standard decomposition of the whole solution space ~, which is the key in the analysis of group structure of kernels at critical points. We then label each bifurcating branch by its symmetry group. Next, we need to ider, tify the Neumann boundary value problem (P) with a section of the periodic boundary value problem (I 5) defined on the circle I = [ - L , L]:

d" ~!~ D(d) ~ ~r + B~? + N(',?) = 0, x E L (2.8)

~. ~ ~ ,,U,,, ~ . ~ of tl,, LCobal t,tf,,,~.~o, aosmm 7

d 2

unstable

!I i

t I

I I /

I

I I

• I

" i

....... t ............

0

stable

d~

Fig. 2.1. Primary bifJrcation curves in (d. d2)-space

An advantage of this identification is, as we shall see, that the virtual problem (f~) is covariant under the group action D~- a property which is not expected of the original problem (P). We note that (P) is covariant under only the reflection at x " L/2. If we denote by $ and ~ the set of solutions of (P) and (~), respectively, it is clear that S C ~, where we identify V E X with its even extension to I by reflection at x = 0. Moreover. we shall see that $ = ~ N 5(:° where 5(: is the closed subspace of 5( such that 5(: = {V E 5( ] V ( - x ) = V(x), x E [). "thus, our strategy is to look for all bifurcating solutions

of (f~, and then take the section $ of ~. We recast (P) as an operator equation

(P)' H(d, V) = O, (2.9)

where H :R+ 2 x Dora(H)--, Y (where 0• Dom(H)C X), and (~) as

(~)' ~(d. '~) = 0, ( 2 . t 0 )

where 1]' :R2+ × Dom(Jq)-., ~' (0E Dom(/q')C ~). The mappings/./ and l~' are real, that is, H(d, V )= H(d. V) and ~(d. ~') -" l~(d. '~) for all d E R~, V E~ Dora(H) and ~' E Dom(/~).

S H. i ~ i et aLIA I~rlm~ of the ~ol~l b/fright/on d~mm

Let G denote a compact continuous group, and T:G--* GL(V) a linear continuous representation of G on ~. Assuming that T(g)¢~ C ¢~ for all g E G, T is also a cont inuous representation of G on X.

Then, we say that H is ,'or, arian! under G if

T(g~fi~d. 9~ =/4(d, T(g)V), for all g E G and ~? E Dom(/~). (2.11)

It is easy to see that /~ is covariant under the compact Lie group D~, where T : D , , ~ G L ( Y ) is

defined by

(T(r+)gxx) = X:'(x + 0/~), (2A2)

(T~s~gR,r,~ = V( -x ) , where y = ,rlL.

In fact. this T defines a unitary representation on Y, and on ¢~ since the inner-product of X is stable under T(g) for all g E D~.

Let G be a subgroup of D~, and denote by dg the Haar measure of G. For example, if G = D®, dR :: d8:4, and if G is finite with n(G) < + ~ the order of G, then dg= lln(G). We now introduce the ~,llowing

~'fmifion " X÷ C :,~ is defined by ... i. The maximal G-symmetric subspace "c;

¢~.; = P:I;¢~. (2.13)

whece

T(g)flg (2.14) pC.

is the projection operator of ¢~ onto ~a. We define also the projection P_~ = I - P+~, to have the decomposition ~, = ~G @ ¢~c;. By (2.14) we have that

T(,e~" = (', for all g E G and ~7 ~ ~ . (2.15)

Namely. every, element of qc; is invariant under the group action G We c,,,,n see easily that

q D.= {span[l]}:, and ~o.. ={span[cos(~,nlx)]i=o}, for n = 1 ,2 , . . .

We note that since Dt --- {e, s}, (s: = e), f(~' is the space of reflection symmetric functions. For notational simplicity, we may use the symbols:

< , : ~ , D , and P ] - PD,=!(I +_ T(s)).

We have obviously the relations

q , - D " C q , ? " C X : = \ ( n , k = l , 2 . . . . ), (2.16)

~ here we always identify V ~ X with its even extension to L We next note that (15) ' is equivalently written as a system of equat,ons:

def ~P~" V:IZI(d.(9., (" )) =P'._H(d, 9 .+ V_)=O,

~vhe~-e

(2.17)

~7". = P : 9 ~ ~: . The D,-covariance 9f /4 yields that /~(d, (V+, 0))~ V+~ for all V+E)C~ (see

It. i:~lU ~ eLlA ptct~r, o! tl~ ~lobai~blllu~i~tO~dlatlram 9

[7]). We may thus make the following identitication of the o p e r a t o r / 4 : R+ z x ~+'-* ~,~:

~/(~, v) = (p~I)(~, (v, 0)), v ~ ~ .

From (2.18) and P'_ / ] (d , ( V , 0 ) -" 0 for all V ~ X~, w e see that 5 = ~ N ~ .

(2.18)

= ~ ~ . (2.19) tt,~l

The decomposition (2.19) is unique, and the projections P ~ " " -o . X --* X~ (~t -- I, 2, . . . . <I(G)) are given by

P° f , , - - n~ x~(g) T(g) dg, ~ = 1 ,2 , . . . , q(G). (2.2O) G

Here, we denote by % n~, and X~ the complete set of irreducible representations of G, the dimension of T~, and the character of ~-~, respectively; q(G) i s equal to the number of c0njugacy classes of G when G is a finite group, and q(G) --- + ~ when G is infinite.

The subspaces ~G ~, (~t - 1, 2 , . . . , q(G)) are characterized by

//it - o "o ~.a) -- %, and n~, ×~ ffi ~ X,.a, where (T ! ~G dim (~.~) ffi (2.21)

a s l

for all a ffi 1, 2 , . . . , a,,, and I~ -- 1, 2, . . . . q(G). This decompositiou is not unique but the multiplicity a , ~< + = is unique.

The subspace X~ is the one which corresponds to the identity representation ~'l(g)ffi x , (g)= 1 (Vg E G). Thus, from eqs. (2.14) and (2.20), it is immediately seen that

~+G X~ and ~_G :~2 o ~ ( ~ . ~ ' G = = • ' × q<O~- (2.22)

Suppose G = D®. The group D® has two one-dimensional irreducible representations ~-=+ and T=_:

l"®±(rs) = 1 (0 ~< 8 < 2~r) and 1"®±(s) = - 1, (2.23)

and a sequence of two-dimensional representations ~-f(! -- 1, 2 , . . . ) :

0 [exp(ilO/~/) exp(- i l0 /v)] (2.24) • ~(ro) = L 0

0] for ! -- 1, 2 , . . . Thus, the decomposition (2.19) gives exactly the Fourier decomposition of X:

whcre

X ~ = {span[l]}2' ~ = ~' (2.26)

~ - = {span[exp(-iTix)]} 2, ! = 1, 2 , . . .

l0 14. Fajii t,l eLlA piclMre of lht ~4obal bi/aA"alioa di~nram

S u p p o s e next that G = D. with n < + ~ . Recal l that D. is the g r o u p of o rde r 2n : !

D,, = {r~.. ~r t~"-~

whe re r, - r~., #, ~ 2~r/n. D , has the one -d imens iona l r ep resen ta t ions 1-,_. and r~.~2~*_ ( w h e n n is even):

T,-(r~) = 1, r , . ( s ) - ±1 ,

fo r k = 0. 1 . . . . . n - I. and a sequence of two-d imens iona l h i 2 - i w h e n n is even . and i . ~ --- (n - I)/2 w h e n n is odd:

0 ~(r"~= [exp(iloOd') exp(-ilOd,y)]

and

[0 Thus . the s tandard d e c o m p o s i t i o n of X b e c o m e s

- D,, , " t m e J t - D " x . . ~ ~ . _ ~ -o. -~.

~ ® ~ ® x: <. = odd),

where

t,: = {spanlcos(3,njx,~lT~0}:.

r.,.., = {span[sin(3,njx)]i~,}".

k,.,~.,. = {span[cos(3,n (,i + ~.)x)]j=0}, - D ~ I :c "~ \ , ~ _ = {span[si~(~/n(j + .Ox)]~=1}'.

and

where

and

X~-" {span[cos(3,(nj + l)x).cos(3,(n(! , 1 ) - l)x)]j=0},

o n e s

(2,27)

(2,28)

~, I -- 1, 2 . . . . .1..~, where Imp, =

(2.29)

(n = even) ,

(2.30)

(2.31)

'~I~" = {span[sir.,(~,(nj + l)x), s in(~(n(j -,- 1 ) - :)x)]~t} 2,

for I = 1, 2 . . . . . ima..

The not ion of the s y m m e t r y group of a subspace plays a role in the d iscuss ion of the s y m m e t r y of b i furca ted branches. For a subspace Z C X, we say that ~3[Z] is the symmetry group ol Z if

~d[Z]%f{g E D= I T(g)V = V for all V E Z}. (2.32)

~r:~ ~t aLIA picture of the global bifurcation diagram

Fo~ instance, we have by (2,15)that q3[X °] = G, The following lemma is immediate:

II

Lemma 2.1.

(i) - t ~ . . -~ ~J[X~]=D®, ~J[X~]=C®, and

~ [ X ( ~ - ~ = C~=

= C,.

(n = even),

¢~[~D,] _ Cgcd[nJ], for I ---- l, 2 , . . . , gmax,

where god[n, i] denotes the greatest common devisor of n and L

(iii) ~3[X~;] - D,, and ~315@-1 = Ct.

(iv) ~[~D.] = D~d[~], and ~3[X~_"]- C~dt~,~.

(2.33)

(2.34)

(2.35)

(2.36)

Now, we turn to the bifurcation problem. Let G denote either D,(n < +oo) or D®. We say in the following that a branch is G.symmetric if it consists of points in the space R 2 × ~+G. Suppose ~hat (d ~, V c) E R~ x ~+o is a real G-symmetric solution of (P) and (consequently, of) (P). We suppose also that

dim X = r ~> 1, (2.37)

where

,Y = ker DvH(d% V ~) C X+. (2.38)

Let

f/" = ker D~/](d ~, V ~) C )(. (2.39)

By the general theory (see, [29, 7]), the kernel ~ is invariant under G for all V ¢ E "~+~ (due to the G( C D®) covariance o f / J ) , and ~ is a direct sum of subspaces corresponding to the decomposition (2.25) or (2.30). Thus, in view of the relation DvH(d ~, V ~) - P ~ D ~ / ] ( d ~, (V ", 0)) (see, (2.18)), we can see that ~ is decomposed as

N = .,';~? ~_) ( ~ ~ * ) ~ if G = D®, (2.40)

and

i ?v, ,DA ,I .+ (~ (,,,:,:~+ ~ (i=~.~i ~, l+), ff O = D.,

J V ' - - I

/ ° " v . . = odd, "'~'"+ (~ ',t=, /

n - - c v e n ,

(2.41)

where

Let s (r ~ s ~ 0) denote the dimension of , / ~ (it O = D®) or of . Z ~ (;O,)+ (if O = D,, n < +oo). l ,et ~ = ',~1, ~2,.. •, ~,; ~,+1,. • . , ~,)E R' be the amplitude vector in the bifurcation equations of (P)

t." H. ~ a aLIA ~ttm~ ot t ~ g4o~! Nf, m'ettm e ' t q n ~

al ~d'. V'~. v, hich are obtained through the L y a p o u n o v - S c h m i d t procedure . Let

Ii = ll~¢r; ~ ) : R : × R' -~ R" (2.42)

be I r e b i furca t ion equa t ions of (PL where

er = d - d'. (2 43)

F r o m g2.,8tl or (2.41), the d imens ion P of .:~" is g iven by

r ~ s + 2 { r - s ) + t (for s o m e t ~ 0 ) ,

,~ ~: or of ,~' , . ~),~ 602~-° We can. h o w e v e r , a s sume tha t t = 0 without loss w h e r e t - 0 is t h e d imens ion of -.o - o. :'.o

of general i ty in our s u b s e q u e n t discussions . The ¢orrc:~pLmding b i furca t ion equat ions o f (P) are thus

(1 = H~o.- ,~ . . . . . & - : , . , . . . . . : , , ~ ,, . . . . . ~ , ) " R" x R ~ x r~'-"~--~ c ' . . ~ , (2.44)

and ~he complex conjugate func t ions H i (j = s + 1 . . . . . r). The f u n d a m e n t a l relat ion we shall use i~ the f olk'v~ ing

Prop~,sition 2. I

i l~o': ~t = ~ ( ~ : ¢, . . . . . ¢, : ¢ , . , . . . . . ¢ , : .& ~, . . . . . .~,), (2.45)

w h e r e

& = ~ e : ~ { i = s + l . . . . . r ) .

Here. ~,e may emphas ize that the mapp ing i~! (o', z; ~.) is G(__C D~)-covariant, and h e n c e the fo rm of f l can be easily obtained by the general theory [28]. Thu% the re la , ion (2.45) gives an easy way of gett ing general fo rms of the b i furca t ion equat ions of (P), accord ing to the s y m m e t r y g roup O of the original branch

\Vc arc no~ ready to d i scuss the classif icat ion of b i furca t ion equa t ions . Let

I, I~ I~. n = I..~ . . . . (2.46)

Then. ld'.OIE |'~× ~.D. is a simpic bifurcation point of (P) from the trivial branch. In view of the

decompos i t ion (2.40/ and (2.35), the e igen func t ion ~I' belongs to the D , - s y m m e t r i c space X ~ ( C ~:+~"). N:~_n..q~. in lhis case

.V=.~'~" . ~i th dim .'~'= I.

Pr~,posit~o~ 2.2. Tl~e bi furca t ion equat ion at (d', 0),~-r', × x~.'- t akes the form

|l ~ o,. t:) = ~[-(a,o-, - a,o-,) + 18~ "~ + (h.o.t.)], (2.47)

~ here ~h o ~.~ denotes higher order te rms o([_tr, I,*D and .£ ~ R.

Pr~ of. Since I'I~(o-, :, 5) is D=-covariant , it fo l lows that

fl~(o'. :. ~) = : [ - ( a , c r , - a~¢~+ t3!zl" + (h.o.t.)]

~,~ [2q]. Prop. 2.1 shows the assert ion.

From prop, 2.2, F" is a set ~g (either super- or sub-c•ical bifurcation points, depending on the sign

of #.

Proposit ion 2.3. Any bifurcatec' branch from F" is D,-symmetric, namely, every point of the

bifurcated branch is an element of R2+ × ~.D,.

The proof is essentially the same as in Fuji!" and Yamaguti [7], using the D.-invariance of the eigenfunction • E X.~ C X~" and the D , ~ c o v a r i ~ e of the orthogona/complement to the kernel ~'.

We shall call this branch "the primary D.-branch". Wenote also that a D.-branch does not lose the D.-symmetry neither at a regular point nor a t a limit point; See [7].

Remark 2.1. A (secondary, ter t iary, . . . ) bifurcation problem on a D,-branch can be considered as a

D,-covariant problem, since

(~.(d; W)~f l (d; V, + W), V, ~ ~?',

is a D,-covariant mapping.

We shall next consider a local bifurcaticn l~roblem on a D,-branch which is not necezsarily a

primary one.

Proposition 2.4. Suppose (d ~, V~) E R+ 2 × :~ , (n < +oo) is a simple critical point of (P), in which

,,V" = {~} = -~'~'~z)+, or ~ : , for some I E (2, 3 . . . . . i,,~).

Then, 'the bifurcation equation is given by

Ilt(cr; ~) = ~ [ - ( a : r l - a~rz) + a~ + 0~2+ (h.o.t.)], (2.48)

in w~ich

c~ = 0, if nli:~ 3. (2.49)

Moreover, the bifurcated sub-branch from (d ¢, V~) is qd[Xl-gymmetric, where q3[X] is the sym- metry group of ,N" as defined in lemma 2.1. More precisely, the bifurcating branch has the symmetry

- D n

D~2, if • E ~'c~2)+, (2.50)

Fox" example, on a D6-branch, a sub-branch is either D~-, Till- or D~-sym~etric. Only the D2-sub-

brancll is trans~critical i.e., a # 0, from (2.49). '

Fir~a,,y, at a double critical point d¢~ F,,,.,(n > m ~ 1) o q the trivial branch V = 0. we have the

following

L e m m a 2.2. The bifurcation equations (H~, If,) at Fro,, are

Ilm(~; ~, "q) = ~_,,' A~.~(o)~*'n q, (2.51) P , q

, tl. ~ i ef a|.lA #et.tt, of t ~ #otto bqRrca~ ~ m m

l i d , r : ~. ~) -- ~" .,t~(~r)~% q, (2.52) I*,q

where the ::ummations X' and X" are taken for all integers p, q ;= 0 provided there are two non-negative integers r and s such that

p ~ r ~ O and q;~s~ ,O, (2.53)

and that

m ( 2 r - p - l ) + n ( 2 s - q ) = O , fo rX ' ,

n ( 2 r - p ) + n ( 2 s - q - i ) = 0 , forX" ,

(2.54)

(2.55)

respectively.

Proof. Since the D, -covar ian t mappings

I)~( t r : :l, :-, ~,, Y.:) : R: x C 4--. C,

I~l,,(o': : , z:, ~.~, ~...): R" x C4~C,

are real. it suff, ces to compu te I'Im and I'I, (i.e.. the other two mappings are ~I,, and f l , ) . (See [29].) The group act ion D= on :-t. :--. :t and $2 is given by

r , ! z : / = e ' ' z., t - , | e~i"~ ~,, •

L o: J e"" ,~_~..

and

iii _0o0 , 0 0 o / u , t 0 1 0 0 j L ~ 2 J

We expand l~I~ as

y ~ - r ' - . t ' Fl.,(c,: : , . . . ". 5=, L.)= ~ .a(",,..,. ( tr)z,z:ztz: . r.r'.;.$'

Then. the T,-covariance of l~lm yields the relat ion

? ~ , . , .~.~ t, gT J , . 1 2 2 . - I ~- 2

E ~,~P , i ( r~tr t Z , a l s s)lt* I Z 2 Z 1 Z 2 -

e ~ ; .... :, . . . . . ~i~ = !. fc.r all O E [0, 2,r).

flora which follows

r o t - - r ' - I ) * n(s - s') = O. (2.56)

+ •

Now, from prop. 2.1, we+ ha~:e t h a t

l ira(+; ~,, 6.) = ~1=(+; +% ~+, ~,, ~ = ~ A (.) '+~+'+" "'+" r , r ' ,s ,s ' t J + l +2 •

We let

p - r + r ' and q - - - s + s ' .

Since r, r ' , s and s' are all non-negative integers, R should hold that p ~> r ~ O and q ~ s ~ 0 . We have then the sunu~ation Z ~ with the condition (2.54) in view of (2.56). We can show (2.52), (2.53) and (2.55) similarly.

Proposition 2.4. From (2.51)-(2.55), it follows that (l) If n/m = integer = k ( + 2),

IIm

I [A~.o (~r) + A~.~>(tr)vl + A<3.~)(~)6 ' + A~(~r)v~ + (h.o.t.)], • (m)

[A~.o(~r)+ A~.o(~r)~ + + rA(m)~o.~ + A(m)lo, ~+2 ~ ' [ t,o~ ~ ~.ot ~ +A~.~)(er)~+(h.o.t.)] (k~ '4) ,

[ 'In (n) = Ao.m(o')'q + A~0)(zr)~ k + A~."~(zr)62T/+ A<0."~(o')lq 3 + (h.o.t.).

(k =2),

(k = 3),

(2.57) (2.58)

(2.59)

(2.60)

(2) If him = non-integer, then

l'I= = 6 . [A~.~)(o ") + A~.~)(cr)62 + A~.~)(o')~: + (h.o.t.)], (2,61)

= (n) A ( n ) t o . x e 2 If, 7/• [Aoj(cr) + 1,2~ Js + A<0."~(~r)~ ~ + (h.o.t.)]. (2.62)

(3) It is if and only if n/m = 2 that second order terms (as 6~, 6~) remains in Ilm or in H,. Also, when nlm ffi k = 2, Hm is odd in ~ and H, is even in 6,

2.3. Local structure of the set of bifurcating solutions at double eigc:+,~alues

Double critical cases occur at F~,,(I ~< m < n) where two primary bifurcation curves Fm and F, intersect in R+ 2. In this subsection we classify the local structure of bifurcating branches at double eigenvalues and give qualitative pictures of them. Especially, we study the behavior of secondary branches appearing near the double critical point. The resuRs in this subsection are valid only locally, however, in later sections they are the key to understanding the global stn,~cture of the set of solutions of (2.1)-(2.2) bifurcating from V = 0.

d I First we write down the general system of bifurcation equations at F,~, = ( ,~,, d~,,) (see prop.

2.4):

(B.I) I Im= ~{-(alo', - a2o'z) + Pi1*f + P30~ 2 + P210] + P12~} 2 + (h.o.t,)} -- 0,

(B.2) 11, - -(b~or~ - b~cr2)Tl + q~o~ k + q21~2vl + qmTl3+ (h.o.t,) -'- 0 (k >~ 2),

Here \Ip, and V(. (resp. 9*, and 9:) are the normalized eigenfunctions corresponding to the zero Ggenvalue of the linearized operator Z(r,,,) (resp. its adjoint .P(r,,;:. pij and qi: are tE,e cor- responding coefficients of the Taylor expansicns with respect to 5 and q of

where \; ir; the orthogonal complement to the kernel space in the Lyapounov-Schmidt positkn: ( . ) denotes the inner product in (L’(Z))‘. We note that by a direct calculation

decom-

(2.63

In lvhat follows we give a .:lazsifcation OS preimages of zeros of (II,, II,). It is noted that in our ,~nnl~sis. all the ccefficients pli :nd qii are assumed to be fixed constant, and d = (d,, dz) varying in fpz .lre the bifurcation parameters.

In view of prop 2.4. the type of (El) is classified into five cases, by the ratio k = n/m, in the i~~tl~w.ing table.

Vanishing coefficients Type _--- -

[1

2 p, = 0 jizi

i

even

otherwise PII = PZI = 0 W) integer

3 Pll =o (13)

k= odd

otherwise PI1 = P?I = 0 (IOD)

non integral PII=P2l=o,qk,=o (NO

t l, h i l l el al, lA ph~l.~ of the global bifurcation diallmm 17

Remark 2.2. In every case,

~ = 0 and b ,cr , -b~2-~ qo-,q:+o(]71~ ~)

is a l w a y s a non~vial~ branch of r (B), We ca~ .this b r a n c h t h e ~l-cusp.

Let us study ~ sU~cmre of b i turcat~ solutions of each type and draw qualitative pictur,~s of them in (~, '1, o-~-space for each fLxe~l ¢2.

Remark 2.3. In what follows, we draw pictures under the assumption that p3o<O, pn<O, and q03<O (we also assume p30plr(p~l)2/4> 0 in (13)-type), and we omit the ~l-CUSp (also ~-cusp in (NI)-type) when we draw a projected picture on (~, ~l)-plane.

I. ( I2)- type

The bifurcation equations become

f { - ( a , c r l - a2cr2) + Pn'q + P3o~ ~+ PnVl 2+ (h.o.t.)} -- 0, (2.64/

- (btoh - b2cr2)Vl + q~of2 + q21~2~ + qo3~1~ + (h.o.t.) - 0. (2.65)

Dividing (2.64) by ~, we obtain by the implicit function theorem

crl = {a2cr2 + pnlq + p~of 2 + pl2~12 + (h.o.t.))/al. (2.66)

Substituting (2.66) into (2.65) and using the genera!:zed Morse lemma with respect to (~, 71), we have for or2 # 0,

¢~la(o'2) - ~:;/~(c~9 = 1, (2.67)

where

= ¢ + o(1¢1~),

(~21 = (c~ll~nY + O(Ic,~l~),

1~(~0 = {(c~)~/2~,,q~o) + o(!~1~),

~ = 2b~pt~la~,

c = b~(a~/a~- b~lb~) (>0).

(2.67) ~'epresents a solution branc_t proiected on (~, ~)-space for fixed or~_. l~ince a(o-..) is positive for small ~r~, we can see tha2 there are two differem types according to sign(p~q~0) (= sign(l~q~o)).

(12-h)-type (puq~0> 0, fig. 2.21 In this case (2.67) is a hyperbolic curve. The solution branch where p , >0 , q ~ > 0 is drawn in fig. 2.2.

(12-e)qype (p~q~o<0, fig. 2.3) (2.67) is an elliptic curve for this case. The typical figure for pt~ >0 ,

q~o < 0 is shown in fig. 2.3.

ts H. Flajii et all^ picture oi t ~ ~ i o ~ 1 b i t . r e m i m t diagram "

(127h) Pl l > 0 , q20 > 0

o 1

?-

O 1

t i

O ,

0 . 2 > 0 0 2 = 0 °2 < 0

+ + / - \

Fig. 2.2. (12-h), p,, > O, qzo > O.

One finds lhat the sign of P.q.,o plays the key role in the classification. The fol lowing proposit ion shows tLat the sign of it is determined by the direction of b i furcat ion of the pr imary D,.-branch (i.e., the branch emerging f rom (0, 0, a:crzlr,:~ in (~, ~, o'0space) for o'2 > 0. We know that the direction is de te rmined by the sign of /3 where ~ i. ~ the coefficient of the bifurcation equat ion (2.47) which de te rmines whether the simple bifurcation is super- or sub-critical. By using this/3, we have

Proposition 2.5.

,.,~;~ tt'~_l,~,...,, ,,, . . . . . . . . . . . . ~ if ,~,,,~-a ,,-,y--"" if "-u,c pr imary D, ,-branch bifurcates . . . . . . . . . . /3(o'z) > suoc rmcauy tm.e., 0 for small or z > 0).

~ii) (12-e) occurs if and only if the pr imary D,..-branch bi furcates supercritically (i.e., ~ ( o ' ~ < 0 for small cr: > 0).

2

Proof. We parametdze the primary bi furcat ion curve I',. near F,..~,. as F,,(o'z), w h e r e or2 = d2 d c ii - 2and F.,(O) = F~ ..,. Let ~,s deno te the linearized opera tor on F,, by .~(o'z) = .~(Pm(o'z)). By definition, £g(o'z) ::!

! I q t

i~ i i / ' ~ "

Pl l ~>0, q20 <~0

o

t 9

0 2 > 0 0" 2 = 0 0 2 < 0

I

Fig. 2.3. (12-e), p11>0, q ~ < 0 .

has such eig,'.~values ~m(0.2) and ~2m(0.2) that {;m(c~2) = 0 for any 0.,, and ~2~(0) = 0. The coefficient ~(0.2]j (0., ¥ 0) associated with the simple bifurcation problem at F~(0.2) is given by (see, e.g. [18, Theorem 2~

0(0.2) - (Nc(~m(0.2)) - 2Nq(~Vm(0.2), K (0.2)Q(0.2)Nq(~Pm(0.2), 'Vm(0.2))), ~V*(o'2)),

where N~ and Nc are the quadratic a'~d cubic parts of the nonlinear term N, respectively; Q(0.2) is the projection operator onto R(0.2)- range .~(0.2) for 0.2 ~ 0; K(0.2) is the generalized inverse restricted Io R(0.2) for 0.2 # 0. Since ~2m(0.2)-* O, as 0.2-* 0, K(0.2) becomes unbounded as 0.~ -* 0. Therefore, # diverges as 0"2--,0. The principal part of ~ is ~ven by

(0"2) == -- 2~2m(O'~-tP ttq2o

Thus, noting the relat~ ~n ~2m(0.;~ < (resp. >) 0 for 0.2 > (resp. <) 0, we have the conclusion.

Example 2.1. For the prey-p,-edator model (1.5), the graph of ~ along I'~ (parametrized by dr) is illustrated in fig. 2.4. Therefore we find that (12-h) occurs.

~.0 H. Fujii et al.lA picture of ¢he giobat bifurcation diagram

Fig. 2.4. The graph of p along F~.

Example 2.2. We consider the Brusselator model ([25]):

d 2 d~ d x---~. u + A - ( B + l)u + u~v = 0 .

v "~ B u - u : v = 0 .

(2.68)

T. Erneux and J. Hiernaux [2] have shown that under (I.2) the point ~.A, ~ ) = (2, 4.6) falls in the (12-b) case. Howe,, er, a complete classificatl.on is possible with respect to the parameters A and B. In fact, we have tha~ (I2-h) is realized if and only if (A, B) belongs to {(A, B) [ B < A:'+ 1, 1 < B < 512 or B > 912} and (I2-e) occurs if ~.nd only if (A, B) belongs to {(A, B) [ B < A ~ + I and 512 < B < 9/2} (see fig. 2.5). The graph of ~ for the point (A, B) = (3, 3) is, for example, obtained as in fig. 2.6. "['hen, from prop. 25. this point falls in the (I2-e) case, which supports our result.

, \ \

• ,~ ° ~ , s , . o , o . . . o , . ' . . ~ . . "~. . , •

. , , ; ~ . . . o . • ¢ t "

I . . ' . : .

. , ; . , , . : . . . . . . . , ? . : . . : . . . . . , /~ : ! : : : . @...: :.- :.".,'.:.-:'.,:.: f ~ ::i/.. ;i ". '~ " ~ ~:" ,:':-'. :'::-:::.~?,::

/ . ,.. - , , , - , , . \ " . . - - ~

[ 2 - n \ • \

\

. . . . . . . . . . . . . . . . . . . . . . . . . A

Fig. 2.5. Classification for the Brusselator model.

A f f i 3

B = 3

d I

21

Fig. 2.6. The graph of # along FI.

If. ( IE)- type The bifurcat;ng equations take the following form

4~{-(alcrl - a~crT) + p3e~ = + pl=~l = + (h.o.t.)} = O, (2.69)

- (b loq - b2~r2)'q + qko~ k + qu~2~ + qo3~ 3 + (h.o.t.) = O, (2.70)

where k(~4) is an even integer. We obtain from (2.69)

oq = {a=cr2 + p30~ 2 + pn~ 2 + (h.o.t.)}/al. (2.71)

S~tbstituting (?.71) into (2.70), we have

b,Co'211 - qk0~ k + (A~2+ B~2)~ + (h.o.t.) = 0, (2.72)

where A ffi blpso/a= - q2:, B = b t p 1 2 [ a t - qo3 and C = a2/a, - b2/b~ > 0. Wl'en ¢r2 = 0, applying the New- ton ~olygon method to the following equation for ~1, we find two different pictures according to the number of real roots of it.

(Bw = + A ~ ' ) ~ - qko~ k + ( h . o . t . ) = 0.

Note that the number of real roots of (2.73) branches other than the ~1-cusp when (r2 = 0.

qualitative pictures by using (2.72).

(2.73)

for small ~( ~ O) corresponds to that of nontrivial For the perturbed case ors # 0, we may draw the

(IE-e ' )- type (fig. 2.7). In this case (2.73) has only one real root (i.e., A B > 0) and we have a qualitative picture as in fig. 2.7 .

(IEoh')-type (fig. 2.8). If (2.73) has three different real roots (i.e. A B < 0), typical pictures are given in llg. ~..0.

Ill. (13)-type In this case, the bifurcation equations become

~ { - (a,cr, - a=~r=) + p ~ = - - P~lfq + Pn~ z + Ot.o.t.)} = 0,

- ( b l c q - b2o',.)T/+ q3o~ s + q=I~2~ + qo~3 + (h.o.t.) = 0.

(2.74)

(2.75)

~ H. Fajli ~ aLIA pielare ol t/ee g4o~l ~,nm~km a~allmm

(IE-.e') qk0 ~>0, A < 0 , B < O

o,

J j -

, f f 1 J !

0 2 > 0 o 2 = 0

S

o 2 < 0

. . . . . . .

f / \

Fig. 2.7. (IE-e'), qko>0, A<0, B <0.

In a similar fashion to (IE)-type, wc obtain from (2.74) and (2.75),

btCo.~'q - q ~ + (A~" + 2 1 ~ + B.q ~).~ + (h.o.t.) = 0. (2.76)

where D = b,p2~/2al (A, 8, and C are the same as before).

At o'2 = 0, in view of the Newton polygon method, it is knowa that three different pictures appear according te, the distribution of real roots of the following equation for ~,

(Be/': + 2D~j~ + A~j~)~ - q3oe + (h.o.t.) = 0. ' (2.77)

For the perturbed case o'2 ~ 0, it is suggested that there exist three different pictures corresponding to three cases at tr: = 0.

(12,-Sy.type (fig. 2.9), If (2.77) has only one real root for any fixed small 6, we have a qualitative diagram like fig. 2.9.

rl

¢

73

o!

,

/

0 2 ~> 0 0 2 = 0

11

qk0 > 0 , A > O, B < 0

o 1

i \

0 2 <, 0

t7

F \ / \

-...3

Fig. 2.8. (IE-h'), qk0 > 0, A > 0, B < 0.

/ f

(D-S')-type (fig. 2.10). In this case, (2.77) has three different real roots with the same sign. A typical figure is given by fig. 2.10.

(13.h'~).type (fig. 2.11). If (2.77) has three different real roots and which have not the same sign, we have qualitative pictures as in fig. 2. l 1.

In the above three cases, two secc,ndary bifurcations from the ~-cusp are both tmnscritica! due to the existence of the term pz~f=~l in (2.74). Note that (13) is the only type where transcrifical bifurcations appear, L-.d which can be understoo(! more deeply from the group theoreticalarguments as in (2.49).

IV. (lOD),type The bifurcation equations are

~{-(alerl - a2(r2) + p~t[ 2 + Pn~: + (h.o.t.)} = 0,

-(bl(rl - b2(r2)vl + qk04~ t + q214~1 + qo3~ 3+ (h.o.t.) ~: 0,

(2,78)

(2.79)

24 ~. Fuji/et aLIA picture of the global bifurcation d/agram

o 1

I

11" f

(I3-S)

I f

q30 > 0 , A > O , B < O , D > 0

o

0 2 ~>0

L

0 2 = 0

\

Fig. 2.9. (13-S), q~o > 0, A > 0, B < 0, D > 0.

o 2 <0

where k(>~.',~ • is an odd integer . After a similar calculat ion as be fo re , we obtain

bLCor2~ - qk0~ ~ + (A~: + By1:)1? + (h.o.t.) = 0.

In this case we kave two qual i ta t ively d i f ferent pictures acco rd ing to the n u m b e r of real roots of the fol lowing equati m:

(B-q" + Aff~ . - qko~ ~" + (h.o.t.) = O. (2.80)

~lOD-S)-type (fig. 2. t2). If (2.80) has only one real root for any fixed small 6 (i.e., A ~ > 0),, we can obtain a qualitative d iagram like f,g. 2.12.

(lOD-h")-type (fig. 2.13). If (2.80) has three di f ferent real roo ts (i.e., AB < 0 ) , typica l pictures are given by fig. 2.13.

H . i ~ i e! al,/A pictave oi the~obal b i f u r c a t i o n diagram 2S

( I 3 - S ' )

01 al I

t

1 i

, i

q30 ~> 0o A < 0, B<~0, D ~ . 0

o

0 2 > 0 02 - 0 0 2 < 0

i? I?

Fig. 2.10. (I3-S'1, q~0 > 0, A < 0, B < O, D < 0.

We note that two secondary bifurcations from the "q-cusp are both one-sided ones, which is different from (I3)-type (sec (2.49)).

V. (Nl)-type In this ~ase, we have for the bifurcation eqm.~ions

~{- (a t - i - a:o'z) + p ~ ' + pl=-q ~ + (h.o.t.)} : O, (2.81)

. r t L _ .. b 2 0 2 ~ T ~ e 2 ~ - -2 . z_ lh , . , t ~ i .~n (2.82~ 1]~- - l l .UlUl ~ ~, .L tl21qg , riO3, I , ~ , , .U .L . I I - - v . ._ . . . . .

We note that -q = O, a~o'l-a2o'? = p~06:+o(16} ~) (we call this branch the 6-cusp) is also a nontrivial branch of (2.81) and (2.82) in addition to the -q-cusp. Other n~ntrivial branches are obtained by solv;ng the following system of equations:

- ( a , o h - a.cr2) + p~o~2 + pI2"t12 + (h.o.t.) = O, (2.83)

- - ( b i O ' 1 - b2o '2) + q 2 , 6 2 + qo3-q 2-k (h.o,t.| = O.

26 If, Fajii et aLIA lPlclare oy fhe 8lobal bifamalion ~ q m m

(I3-h") q30>O'A>O' B<O, rO>O

01

I

11."

o I

9 o 2 >0 0 2 =0 02 < 0

i

r/

/ //

\ Fig. 2.11. (13-h'), q ~ > 0 , A > 0 , B < 0 , D > 0 .

We c~n see from (2.83) and the generalized Morse lemma that there are two different types according to the sign of A~B~, where A, and Bt are defined by

.A~ = alqo~- blp12 and Bi = blp3o- alq,.1. (2.84)

(NlH)- type (AIBt > 0, fig. 2.14). There are two secondary bifurcations of one-sided type for tr2 ~ 0 ,rod which are transferred from the ~-cusp to "q-cusp as ~r2 changes its sign. T)pical pictures are given by fig. 2.14.

( NIE)-type (Alat < 0, fig. 2.15). le this case, there appears a loop whicL contracts to one point when ~:r2 ~ 0. See fig. 2.15.

. ! ~ | ¢, aLlA: pic~un~ o f the Xlob~-~ation, ~ m 2"I

(ZOO-S} qkO > 0 ' A < O , B < 0

01

t

I i i

a 2 > 0 0 2 = 0 o 2 < 0

I T?

i

Fig. 2.12. (lOr).s), qhe>o, ,4 <0 , B < 0

2s ti, Fal~i ¢, aLIA pictam o! the I1to~1 l, i run'm~n

(IOD-h") qko>O, A > O , B < O

o I

T 01 o I

I

~12 ~ 0 o 2 = 0 o 2 < 0

\

Fig. 2.13. (IOD-h"), q~,o>0, A>O, B <0.

r/

. . . . i

|NIH)

01 O,

J

A 1 < 0 , B 1 < 0

o,

0 2 > 0 0 2 - - 0 a 2 < 0

7/

~ J

f ~

Fig. 2.14. (NIH), At<0, Bi--~0.

~0 N. Fujii et aLIA picture of the global bifurcation diagrQm

(NIE) A 1 >0o 8, < 0

n

0 2 > 0

01

a 2 = 0

Ol

o 2 ' < 0

Fig. 2,15. (NIE) , A ~ > 0 , BL<O.

H. Fajii ~ al:IA pictt, re o/the global I~ifu~ation diagram 31

In this section, under the assumetions (A.I), (A.2), (A.3.1), and (A.4), we consider the system

obtain from the second equatica of (1.3)

v,a =0. (3.1)

Here we restate the assumption (A.4) in a precise form:

(A.4.1) The zero-level curve of f(u, v) is S- shaped in (u, v)-spa¢e, i.e., when it is solved wi,'h respect to u, it consists of three different branches u - h~(v) defined on the open interval J~ (i ffi 1, 2, 3) respectively such that h,(v) < h3(v) < h,(v) holds in the inter- section t3 ?=l J* ( # 0). Moreover, there exist two intervals Jj = [c~, d~](i ffi 1, 2) such that .lj C 3"*, J~ N 3'2 # ~ and the partial derivative of f(u, v) with respect to u satisfies the following:

~ ] ( u , v ) < O at = hi(v) U

in J~(i'- 1,2) .

, 4 (A.4.2) GI(v)ECt(J~) and d-~G~(v)<0 in J~

(i = 1, 2), whe ~ O,(v) = S(h~(v), ~) (i = 1,2) .

For the proofs of the results in this section, see [21] and [23].

3.1. Largeness of the inhibitor's di~usion coenicient (di t "1{ 1)

First we derive a limit system of (1.3)-(1.2) as a ~ 0, where a = di ~. Suppose that the solutions of (1.3)-(I.2) remain bounded as a ~ 0, Then we

j . . . .

0

holds independently of a. Thus we are led to solve the following system:

d 2 dt ~ u + f (u , 6) = 0 (3.3)

L

f g(u, ~) dx -- O, o

(3.4)

d c l u(O) = ~ u ( L ) = ~},

dx uJ~ (3.5)

where v - £ is , constant function. We call (3.3)-(3.5) the shadow system for (1.3) and (1.2). In what follows we only consider monotone solutions for u, in other words, one-mode solu- tions of the shadow system, since other (mode) solutions can be constructed from one-mode solutions.

We will show that the structure of solu- tion branches of the shadow system is a good approximation to those of (1.3)-(1.2). Let us introduce some notation. f~ = R+ I × X ffi {(dl, U)[ dl E R+ I, U • X}. qg~ (resp. qg~) denotes the component in the closure of the set of non-constant solutions of (3.3)--(3.5) (resp. (1.3)-(1.2) with a =: d~ I being fixed) in ~, which contains (b , l% 0) (resp. (d~, U) with (d], a ' I) ~ FI). Seedl(W) denotes the set {w [ (dl, w) ~ W} for arbitrary subset W of ~.

It follows from (A.4.1) that f(u,£.~--0 has three zeros ul( f )< uc(~) < u,(6) fo r 6 = "~-i 3~.

32 H. Fajii e! al.lA pitt.re oy the global biIurcation diagram

Here we assume that

(A.51 J l~) = l f (u , O d u has a unique i

at zero

d = r* and 7-~J( r*) < 0 .

t i c

U n d e r ;he a s sumpt ions (A. l l , (A.2), (A.3.1), IA.4L and (A.5L we have a global result for the shadow sys tem.

Theorem 3.1. ~ is a locally one-parametr izabl , : smoo th curve in ~" wh ich bifurcates f rom (h,~/~.Cr) and goes into (O,(u*(x), t ,*)) and !O. tt,*(L - x ) . t'*)) as d, ,~ O, where

{thit, *), O ~ x < x*. u*(x) = u,(r*) , x* < x ~< L. (3.6)

aad -~* is uniquely d e t e r m i n e d by the relat ion

z

I g(u*(x), r*) dx = O. (3.7) t ~

d , , r e o v e r . there are no o the r solution b ranches of (3.3gq3.5~ which go into (O,(u*(x), v*)) and , O . ( u ' ( l . - x ) . t'*)).

~ t when dt ~ 0, we construct the singularly per- turbed solutions of (1.3)-(1.2) in the next sub- section.

3.2. Smal lness o[ the ac t iva tor ' s dif fusion

coe~fcient (d, ,~ 1)

In this subsection, we consider (1.3)--(1.2) in the case where d, = • 2 is a sufficiently small perameter. Thus, for this case, the singular per- turbation technique is applicable. For brevity of discussions, we concentrate on a ecological model (!.5). However, we nete that the analysis is valid for the general form (1.3) under (A.I), (A.2), (A.3.1), (A.4) and (A.5) (see [21])

We look for a solut ion of (1.5) and (I.2) in the form of

u(x; e) = uo(x) + e.u~(x) + . • . ,

v(x ; ~) = v0(x) + (v,(x) + . • .. (3.8)

We subs t i tu te (3.8) into (1.5) so that the first term in p o w e r s of • leads to

ii I(.;;oo> = 0, • (3.9)

I , dx----7, v0 + g(uo, Vo) =: 0 x E I,

subject to b o u n d a r y condi t ions

+~cm,~rk 3. I. The local a rcwise proper ty of Y,~ in h e ~-,bove theorem holds in generic sense (for the precise meaning, see [23]), and when it is )ccally arcwise it is c lear that ~'J~ has no secondarv bifurcations.

I " . |

< ~; ,~ a good approx ima t ion to q~,, when the solut ion space is res tr ic ted to ~'a = [& ~ ) × X. In fact we have

Fho,re . , 3.2. For any posi t ive ~ and ,-, the re e\i,,t,, , ,ome positive cons tan t c~ such that %'~, t, eX~n~, to the ¢ -ne ighbourhood of kq~ w h e n both b ranches are res t r ic ted to the space ~ , %r .:n~ ~, ~ i th 0~-~ ~<c~t.

tn order to study the asympto t i c behavior nf

d uo(0)= d d~ ~ vo(L) = 0. (3.10)

Solving the first of (3.9) wi th r e spec t to Uo, we have three di f ferent func t ions

{ h~(vo) = O, in JT = (0, +~ ) , un= h,(vo) , i n J ~ = ( 0 , m) ,

h4vo) , in J~¢ = (fo(O)/k, m),

where m = max.~ofo(u) /k and h l < h 3 < h 2 in J ~ f q J ~ f q J $ (see fig. 3.1). Then (3.9) is I -duced to a single equa t ion for vo:

d: d . -d~ x . vo+ gi(vo)=O, x E I, (3.11)

where g~(v)= g(hi(v), v). (3.11) can be handled more easi ly than (3.9). Next, by using two of {h+} (i = 1 ,2 ,3) , we c o n s t r u c t the fol low-

Y

hl(V 0 )

-k---

0 ,,,L , ,

Fig. 3.1. Three branches of f(Uo, Vo) = 0.

ing special solution of f(Uo, vo) = O:

[hi(v), in J~' A {v < O, u = h(v; ~) = th2(v), in J~ A {~ < v < m},

where ~ E .f~' A J~. Therefore h(v; ~) has a dis- continuity at v = 6 Then we obtain

d 2 d2~-~ Vo+ g(vo; ~) = 0, (3.12)

where g(Vo; ~) = g(h(vo; ~), Vo). Because of the discontinuity of g, we must define a weak solu- tion of (3.12) and (3.10) by

Vo E H I(l) d d ok) = (g(vo; ~), ck ), (3.13) vo,

for all ~b ~ H t(i),

where ( , ) is the inner p~roduct on L~(I). Fur- thermore, a weak solution of (3.9) and (3.10) is also defined by

(u0, v0) E L2(I) x HI(I J, f(uo, Vo) = 0, almost everywhere ia I,

d ,k) = (g(uo, vo), ( d2 VO, -d-

for all ~k E H~(I).

depending on d2 and g(v; 6) such that

'p.x n = no. no+ 1, . . . are solutions of (3.14), where u~ -- h(v3; ~,).

Proof. ([21]). First we consider (3.13) by using the s tandard plane analysis and obtain "J~(x). Then we can get u~(x) from the relation u - h(v; !~) (see fig. 3.2).

Remark 3.2. (A.4) implies that there exist two intervals ./d = [cl, dd(c l~)(i = I, 2) such that

d (i) ~ gl(v; O < 0, in 1,(i = I, 2),

(ii) g~(v; 6) < 0 < g2(v; 6) in (cl, d2),

where g.~(v; 6) = g(h~(t,), v).

Remark 3.3. The family of solutions (u~(x), v~(x)) obtained in prop. 3.1 are not all solu- tions of the reduced problem (3.14). In fact, let vo(x; ~)(s = I, II) be solutions of

( d ) d~ d : Vo, ~-~ cb = (g(v0; 60, ~), for ~ E H'(I),

where 61 ~ 6n. U dng vo(x:. 6~)(s = I, If) suitably, we can construct ~ solution Vo(X; 6~, 6u) other than two solutions vo(x; 6.,)(s = I, II)(see fig. 3.3).

Here we cor~sider the relation between d2 and the critical number no for fixed nonlinearity g(v; ~) in prop. 3. !. Therefore we consider again the problem (3.12).

(3.14)

/ We first show the existence ,ff solutions of (3.13).

Proposition 3.1. Consider the problem (3.13) under the assumptions (A.I), (A.2), (A.3.1), and (A.4). Then there exists an integer n o > 0

It. Fu|li et aLIA picture of the slobat bifurcation diagram 33

Fig. 3.2.

~4 H. Fa~ii el aLIA piclune oi the global bilarcation diagram

VoiX: | ' L~

Fig. 3.3.

Theorem 3.3. Suppose that (A. 1), (A.2), (A.3.1), (A.4), and (A.5). Then, there exist eo and Wo such that for each 0 < rr = suplg.(ue, re)]< ere, (1.5) and (I.2) have an e- family of solutions (u(x; e), v(x; e)) of the form 0 .8 ) for O< ~ < eo satisfying

lira u(x; ~) = u0(x), almost everywhere on I, e10

lim v(x; e) = Vo(X), uniformly on I, El0

where (ue(x), vo('x)) is a solution of (3.9) and (3.10) obtained in prop. 3.1 when ~ = v*.

Lerr, raa 3. i. Let w(x) be a monotone increasing :olution of

d: w+g(Wo:~)=O ( 0 < x < L ) .

a__ c l w(O) = % w(L) = 0,

dx QX

for a t y fixed L. Then - w ( 0 ) and w(L) are both monotone increasing wilh respect to L.

This lemma implies that the maximum length f-= k(d,_) ~: exists for ~ome k >0 , which is obtained by considering

d ~ d: dx---~, w '-- .¢~.,0: ¢) (0 <~,.': </S) ,

d = :=- w(f.) = 0.

dx

w(/-_) = m.

Thus we have the following

Propositibn 3.2. If d, < d~(n) = L:/(nk):, there never exists any solution v0(x)E ,~o, when ~ is fixed.

As will be setn in section 4 and section 5, this result on non-existence is closely related to the global existence of the primary D,-branch.

We next consider the case where ~ is not zero but sufficient!/small.

Remark 3.4. Consider the system with ad- ditional parameter d2 = l / a :

~2 . u + I ( u , v ) = O,

(3.15) I d 2 a d x 2 v + g ( u ' v ) O.

As a direct consequence of the above proposi- tion, we find that there exist ao and eo such that for each fixed a ( 0 < a <ao) , an e-family of s,,lufions (u(x; ~, a), v(x; e, a)) exists for 0 <

< c0, and that for each fixed e(0 < • < e0), there exists a~ such that (.3.15) has no other solutions in a small neighbourhood (which is independent of a ) of solution (u(x;e , a) , v (x ;~ ,a ) ) in the CLtopology for any a with 0 < a < al.

3.3. Regular-singular perturbation in the case of

We can integrate the results of the previous two subsections in the case when a = d ~ 1, i.e., c~ coincides with singularly perturbed solutions for small d~.

First we can prove that for fixed e, (u(x; ~., a), v(x; ~, a)) in remark 3.4 constitutes a

Cauchy sequence in X as a ~ 0, and we denote its limit solution by (u*(x; e), 6*(e)) where 6*(e) is a constant function. Secondly, it follows from the uniqueness property of q~ for small • (see theorem 3.1) that (u*(x; e), 6*(e)) coincides with

li. ~ e~ aLIA picture o~ the global bi],,rcation d~agram J$

~ for small ~. In other words, ( u ( x ; ~ , a ) , v(x; ~, a)) converges to ~o ~ as a ~ 0. Thus com- bining theorem 3.2 and the local uniqueness property of !u(x; ~, a), v(x; ~., a)) (see remark

We' ,in

using t~e piecewise linear finite elements with a lumpi~q~ approximation to the nonlinear terms. For a justification and convergence studies, see [6]. We note that throughout this section, the

Theorem 3.~. (From bifurcation to singular per- turbat;on).

There exists a small positive constant a~ such that for 0 < a < as, ~ , crests globally with res- pect to d~ in the sense that SeCd,(~) ~ ~ for any d ,~(0,d~) , and coincides with the singularly perturbed solutions in theorem 3.3 when d~ is sufficiently small. Moreover~ the following rela- tion holds:

lim lim (u(x; ~, a), v(x; ~, a)) d~Oa~O

-- l im l im (u(x ; ~, a) , v~x; e, a)) a~O dl J,O

= ( u * ( x ) , v*).

4. Numerical results

We summarize the results of numerical com- putations that trace bifurcating branches of the system:

d 2 d~ ~ u + { f0(u) - v}u = O,

d 2 d2 ~-~ v - {g0(v)- u}v = O,

(4.1) in x E I ,- (0, L) ,

subject to homogeneous Neumann boundary conditions at x = 0 and x = L. In (4.1), f0 and go a r e

fo(U) = ~(35 + 16u - u 2) and go(v) = 1 + ~v.: i4.2)

In the following, L = 5 is assumed. The algorithm used ~ the static iteration

method together with the group-theoretic decomposition of the solution space. Detailed discussion of this technique is in Fujii [6]. The ~pace discretization of (4.1) has been done by

tical values is theoretically known [6]. We begin with a remark about the labeling of

bifurcating branches. Due to the D,-covariance of (1:3)-(1.2), each branch has its symmetry group D~ C D~. For example, the trivial branch 0 = (~, #) is D®-symmetric, while the primary bifurcating branch from a point on F~ is D,- symmetric (prop. 2.3). Secondary branches from a Dk-symmetric branch again have a subgroup Dk, ~Dk, ;ncluding the possibifity that k ' = 1 (namely, the case of no symmetry). When k is even, there exist two equivalent branches which have possibly different stability. Whenever necessary (a~d if possible), we distinguish them by the symbols D~ and D~, where D~ are the continuations ot V~(O = - + ~ + ~7(--0, ~ ~> O; Vk is the nolmaT.,zed eigenvector of (2.4)-(2.5) with d E F~, in section 2.1; V is the orthogonal ¢omplement to the kernel sp,~ce (~k) in the Lyapounov-Schmidt decomposition.

4.1. The primary D~- and D~-branches

In view of the result of local double ~ifur. cation allalysis at FI.2 = FIN F2 in section 2.3, we have the picture of the :~c~d structure n~,'ar 1"!.2 (fig. 4.1), which is of (I2-h) type. (See example 2.1). It may be emphasized that this picture gives no information about the destination of the Drbranch, eve~a in the vicinity of Ft.,.

However, it is observed that the secondary bifurcatioa at R ~ of the D~-branch recovers the stability of D~. The double criticality FI.~, is obviously responsible for this recovery.

Fig. 4.2 indicates the destination of the D~- ~,ranch at the cross-section d2 = 50. This figure

L-, N, F. j i i tit aLIA plelM~ o[ the S lo~ l b i /u~a l /on d i a spo ra

;?. (1) / ; i __-~'~2 o +, //

,~+ , / + / / D I

L . . , ,

I:Ig 4.1. L e g a l b i fu rca t ion d i a g r a m n e a r F i , . F~.,-~ (I.600. . . . | t |~ :~, "¢~ qR 9'. i~ t he firsl ~ecoverv oo in ! o f D~; _.,. is t he f i rs t

~ , , ' q o g p ~ i r l l o f Dr.}

f-f . . . . . . '. ~" ~!t~ \ .... "

,, : ~ k . . . , - - ----~- ' 2. / "

.,,.. ~ / < t

[-~g 4 " N u m e r i c a l result at a: = 50: he re . !', = 3.108, I'., = t -~0. R , : = ] ~25. N =- 1 .144= S',.

,h , ,u , some interesting and somewhat surpris- ing behavior of the D,-branch, which is a result of our numerical computations.

Starting from F,, the Dr-branch proceeds to the ieft until it arrives at Su. where it turns back and then is absorbed by the D;-branch ~t R~'.'. & is a limit point of D,. The D,-branch, ~hich is o."iginally unstable, recovers its stability at R ' . " - a symmetry-breaking secondary bifurca- tion point of D:. and continues to exist for d~ .-*. 1 ~vithout losing stability.

kke note that the numerical branch appears to c~iq to the limit d, ~0 at this cross-section d ¢, " . : . ¢ , . the accuracy of solutions in the range d-~ h {h- the mesh length) may not be guaran- teed by the theory of numerical analysis [61, ~incc the solution branch enters into the region ,,f ~ingu]ar perturbation theory.

Numerically obtained ~atterns of ( , , v) at the points specified on the diagram (fig. 4.2) are shown in fig. 4.3.

Fig 4.2 shows such pSenomena as: (1) Coexistence of two stable branches Dr and

D~ for the range d, E (&, n m ) 1 % 2 , - .

(2) Existence of a limit point & on the D r branch, and disappearance of Dr for dr < S i ,

(3) A 'k;,nd of hysteresis phenomena m~y be observed ff a situation is assumed where the coefficient dr may fluctuate in a neigh- borhood of S~ and Re, u

Considering the limit point S~ as a function of d.,, S; = &(d,.), we may be interested in the behavior of St(d,+) as d~. varies. It is found that &(d,.) is decreasing for large d2, and for sufficiently large d.+, for instance d., = 200, the point S| "runs through" the ~al l dn =0 , and disappears. See, the conceptual picture fig. 4.4. If S,(d,.) is continuous (hopefully), there is a threshold value d., = d~ such that

lira St(d,.) = 0. (4 .3 ) d , ,--, ,IS.

Numerically, this value lies around d"+.= 165 (though this value depends on the mesh length in the numerical computations).

Global existence of the Di-branch to the limit d, ~, 0, which is stable throughout, for the range d2>d',., is thus suggested by our numerical computations.

As d,. becomes smaller, the " loop" structure

la ~ . .I [ . . . . . . . . .

Fig, 4.3. (a) at A(dl ~2 .800) , (b) at Sl(dl ~ 1.144), (c) at I l l

R : _ ( & ~ 1.325), (d) at B(d, ~ 0.100).

//. F~tl e, aLIA pictk:~ of the glol~l bipJrcat|on diagram 3?

/ o÷ : , \ / -..... / 2 ~ ' 2 , + \ /

Y - s , . . - ~ T - - - - / f Y Y - 7 ' l / / °2 \ . . - - " -, / / " , ~ ' f - / ~ - -

, / ) 2,- / I

Fig. 4.4. Di-branch at d2 = 200.

of the Drb ranch is contracting. Below F1.2, both the starting and ending points transfer to the Di-branch through the exchange of branches at Ft,2. See figs. 4.1 and 4.5. The secondary branch from L~ ~_ is the one which has been called DI on the upper side of Fla,

Our numerical results seem to support the conjecture that the origin of this "loop" is a degenerate simple critical point Z which may lie on the D~-sheet. See, fig. 4.6.

"/-' D~ ' "7

Fig. 4.5. Dt-bnlnch moves on D:. below F~.2.

/

Fig. 4.6.

d 2

d:(l~

" ~ F2~3

d I Fig. 4.7.

Assuming the existence of such an origin Z - a degenerate critical point, it is placed some- where on the triangle formed by FL2, F2.3 ;tnd F,.3 in (dr, d~p lane .

In summary, t h : region of (d~, de)~ R~ where the Dr-sheet stably exists is illustrated in fig. 4.7. Fig. 4.8 shows the profile of the D : h e e t .

J ' ~ , / , /

Fig. 4.8. Profile of the Drsheet. Note: At Q, we observe a bifurcation of fourth order contact, i.e., 13 in (2.47) vanishes.

38 H. b"ujfi et al.l A picture of the global bifurcation diagram

4.2. Successive recover), and loss of stability

The origin of first recoveries of stability of the D~, and D; (k-> 3)-branches car~ be traced back to the double critical points 1" ~,_, and Ft.~ (k ~2), respectively. Conceptually, all secon- dary branches from the first recovery points R ~ on the D~-branches enter into the D~+~- branches, provided they do not reach the point d, = 0 before this entering. We show in fig. 4.9 a possible schematic picture of the primary bran- ches and those secondary branches which bifurcate from the first recovery points, in the case that F~.Es, are of the type: Ft..,:I2-h, F~.~: 13- S. F,.4:lE-e'. I'~::IOD-S, FL~:IE-e', and so on. We may emphasize here that this picture is not numerical: rather, this is a local theoretical pic- ture, in the sense that fig. 4.9 is valid in a small tubular neighborhood of the curve F~xO(~ R xx.

We show in fig. 4.10 a result of numerical computations, which trace the behavior of the secondary branch from the first recov,;ry point R~ ~ of the D~-branch. We observe that the

" ~ . ~ . I / , - /

Fig. 4.10.

secondary branch from D3 extends to the left, turns back then, and finally enters into the D~- branch. It exibits clearly that the secondary bifurcation from D~ is super-critical, while that of D~ is transcritical. This picture completely coincides with the schematic diagram in fig. 4.9. This suggests to us that the local structure near double singularities extends to hold (at least, "semi"-) globally with some deformations of branches.

It seems that, so long as numerical obser- vations are concerned, the second, third . . . . and ( k - 1)th recoveries of stability occur on each

[. b

/

/ /

D 4

I 5

/ / /

D 3

; 4 ]'3 12 ['I

D j

D 1

i / /' l / / , / /

i / //

./ ,/ ;///

~ ' l , b

1,5

1 , 4

/.

l , /

Fig. 4.9.

Er. Fu|ii el aLIA picture of the global bifurcation diagram 39

D~-branch, and hence every D~-branch has its own range in (dt, d2)ER+ 2 space, where D~ recovers its total stability. As a consequence, several s tab le parameter (dr, also observed by Hersc~owi tz -Kaufman arid Erneux [11] fo r the Bmsselator,

It is clear that some o f the origins of this phenomenon can be reduced to the double cri- tical structure at 1"~ for some k ~, 1 and J ~ 2 (see, section 2.3), as in the case of first reco- veries. However, complete understanding of such structures is not yet available and may be a subject of future studies.

Anot,er aspect of the global branching prob- lem is the "fold-up" principle. That is, if U(x; d) is a solution of (P) at d - (dr, dz), then R"(U)(x) is a solution of (P) at din', for n = 1,2, 3 . . . . . Here,

R"(U)(x) = f U ( n ( x - i / n ) ; d ) , if i is even,

I U(n(I/n - (x - i/n)); d), if i is odd,

for i l n ~ x ~ (i + l ) l n , ( i - O, 1 , . . . , n - I). As a consequence, if the Drbranch forms a

"loop" structure on a cross-section at a fixed d2o with d z ~ d2 ~ d], as we have observed (d z = the value of d2 at Z), then the D,-branches (k ;, 2) should also have a "loop" similar to that of Dr, which is contracted with the ratio I/k 2, on the cross-section dl t~ = ddk 2, We have for each Dt a pot-like profile as in fig. 4.8, which is similar to that of DI and at the bottom, there may exist a degenerate simple singularity.

This, however, does not mean that all the pot-like profiles have the same stability. On the contrary, this generally forces Dk-branches to S _ - _ ~ L - - - * - - - - t - - L ' I . J A - . ~ . . . . . . . l - t t L _ V ~ * . ~ ' L _ _ _

IOSC LilC:IE l l t a U l l l l L y , I p o r © x a m p z c , t H e t J 2 - o r a n -

ches should enter into the D,-branch (by the fold-up principle), as the D,-branch does into D2.

A group-theoretical consideration suggests that this entering occurs at the second recovery point R~ 2) of the D4-branch. Thus, near R~ a the bifurcating branch has either one or two posi- tive eigenvalues depending on whether it is sub-

or super-critical, while the Drbranch has al- ready recovered its total stability due to the first recovery. Consequently, the Dz-branch cannot enter into the D4-branch before it loses its stability at least once onthe w a y t o D4i

S. i)bemslon

The purpose of the di.~cussion in this section is to integrate the analytical arid numerical results in sections 2--4 into a unified view about the global bifurcation structure for our interac- ting and diffusing systems.

On the global existence of the Dr-branch. Reflect- ing on the results from three different approaches to the global question-namely, the regular-sin- gular perturbation result, theorem 3.4, the exis- tence of a singularly perturbed e-fami!v of solu- tions, i.e., theorem 3.3, and the numerical trac- ing of the bifurcating D1-branch, we are ira- mediately aware of the fact that there appear three critical values which are different in nature, corresponding to the three different ap- proaches.

The regular-singular perturbation result states that for sufficiently large d2, there exists a global Dr-branch (with respect to dl), which connects to the e-family of singular perturbation solu- tions. This suggests to us that there is a m~imal value d~(1)( ~ a [ I) such that for d2> d~'(1), the Dr-branch exists globally. We may call this d~(1) the first critical value. 3econdly, the numerical tracing of the D1-branch yields the second criti- cal value d~(1)(-d~ in section 4) which is defined as the point where the limit point S,(d2)

_ , e l e " u L _ ~ , . _ ~ a - - - - A L . . . . . , 1

oz t h e . . . . . ' w a , z 1 sm~©s m© d " O. u l - o r a n t ; n

Prop. 3.2 yields another critical value d~(t), under which value of d2, there exists no sin- gularly perturbed solutions as at ~ 0.

A very interesting question to ask is: what is

the relationship between :hem. Firstly, it seems reasonable to assume that the

critical value d~'(1) implies d~(1) itself. At the

40 H. Fu]ii e! aLIA piclure of th¢ global bifurcation diagram

upper part of d~(1), there exists a Dr-branch which hits the wall d~ = 0, and at the lower part of d~(l), the Dl-branch does not reach the wall. These two approaches give a completely coin- cident result, and support each other

This conjecture leads us to an insight why the D~-branch disappears as d~ J, 0. In fact. this is due to the existence of the limit point $1, the origin of which seems to be a degenerate sin- gularity Z that appears on the D~,-sheet.

Secondly, in order to know the relation be- tween d~(l) and the existence limit d~,(l) of singular perturbation solutions, we have cal- culated numerically an approximate value of d~(l) (see, prop. 3.5). It turns out that d$(1)~ 165 = d~(l)! It remains, o f course, a possibility that this coincidence is only accidental. However. we are inclined to conjecture that two critical values may coincide: d~(l) = d~(I).

= = d : ( I ) , There is a keystone d~(I) d~(l) ~' which interrelates the three approaches.

It is observed in our numerical computations (see, e.g., fig. 4.4) that the Dr-branch hits the wall without any loss of stability. If this is the case. the singularly perturbed ~-family of solu- tions are stable. It may be emphasized that, however, this does not exclude the possibility that as d~ ~ 0, some eigenvalues of the D~-branch might asymptotically tend to zero. It seems that a further study is necessary to clarify the situa- tion.

Another remark is that due to the fold-up principle, every D~-branch possesses a pot-like profile similar to that of D~ (see, fig. 4.8) Con- sequently, for d:>d~(k)= d~(i)lk~'=d~(k), the D~-branch hits the wall d~ = 0. However, their stability problem is open at present.

Finally, ~ , note that it is revealed from the abo,'e d~scussion that when one of the diffusion coefficients d~ is sufficiently large, the global behavior of solutions is "c lose" to that of scalar problems. As is shown in 12] for nonlinear Sturm-IAcuville problems, every bifurcating branch wP h preserving the nodal property exists globall~ ,o the limit when the diffusion

coefficient tends to zero. On the contrary, if d~ is not large, the figures in section 4 dramatically show the complexity and crucial difference of giobal behavior of branches, as compared with that of scalar problems.

6. Concluding remarks

As the figures in section 4 show, the significant role of local double critical structure on the global bifurcation picture seems clear: the double bifurcation point Fro.,, h i m -- integer, is always an origin of some exchange (i.e., recovery or losing) of stability of bifurcating branches. In fact, what we have observed numerically may be summarized in a few words: global extension and deformation of local dou- ble structures.

We have in this paper a variety of new bifur- cation diagrams near I',,.,, as a consequence of the introduction of the group D~. We note here that in our analysis, all the coefficients in the bifurcation equations are held fixed, and the two diffusion constants d =(d l , d2) are varied in R~ as bifurcation parameters. However , the bifur- cation problem at F,,, may have much larger codimension than ! or 2 in the sense of sin- gularity theory. Thus, what we have obtained are two-dimensional sections of complete bifurcation pictures in some parameter spaces of dimension ~>2. A complete analysis of such D~-covariant bifurcation problems, which are derived naturally from Neumann boundary- value problems, seems not to be done. Such a work would be done with the aid of the sin- gularity theory, for instance, by Golubitsky and Schaeffer [8, 9]. However, this requires the extension of the contact equivalence relation to the case of multi-bifurcation parameters under the presen(,e of high symmetries. We feel that this is an important subject for future research.

We remark that the existence of a degenerate, simp[e critical point Z on each primary bifur- cated sheet, say on D~ (as is pointed out in

H. FuJii et aLIA picture of the global bifurcation diagram 41

section 4), can be understood as some unfolding of the bifurcation problem 12, discussed in sec- tion 2.3. Such a result will be reported else- where.

In our specific model (1,5)-(1.2), we did not observe any transition from a stationary solu- tion to a time-dependent one as a result of successive bifurcations. However, it is noted that this is not a general situation in reaction- diffusion systems under (A.I)--(A.4). In fact, appearance of such a Hopf bifurcation is found in other systems [1 I].

As is observed in [11], the phenomenon of successive recovery of stability exists also in e.g., the Brusselator. This suggests that there may lie an essentially similar global structure behind nonlinear interacting--diffusing systems, other than our specific model, and even with the Dirichlet boundary conditions.

Though several problems are left open, we do believe and hope that our study might help in constructing nonlinear interaction-diffusion models, and in future studies of global bifur- cation problems of nonlinear reaction-diffusion equations.

Acknowledgements

The authors wish to thank Prof. Y. Hosono of Kyoto Sangyo University for his discussions and hearty help in the course of preparation of this paper.

Thay are also indebted to the Referees for the kind reading of the manuscript and for sugges- ting a number of improvements.

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