# Normal Forms of Hopf Bifurcation for a Reaction-Diffusion System Subject to Neumann Boundary Condition

Authors: Cun-Hua Zhang; Xiang-Ping Yan
Journal: Journal of Applied Mathematics
 (2015)

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## Abstract

A reaction-diffusion system coupled by two equations subject to homogeneous Neumann boundary condition on one-dimensional spatial domain(0,lπ) with l>0 is considered. According to the normal form method and the center manifold theorem for reaction-diffusion equations, the explicit formulas determining the properties of Hopf bifurcation of spatially homogeneous and nonhomogeneous periodic solutions of system near the constant steady state (0,0) are obtained.


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## Body

## 1. Introduction

As an important dynamic bifurcation phenomenon in dynamical systems, Hopf bifurcation of periodic solutions has attracted great interest of many authors in the last several decades [1–8]. In general, the study of Hopf bifurcation includes the existence and the properties such as the direction of bifurcation and the stability of bifurcating periodic solutions. In application, however, it is more difficult to determine the properties of Hopf bifurcation than to find the existence of a Hopf bifurcation. An approach applied to determine the properties of Hopf bifurcation is to derive the projected equation of original equations on the associated center manifold, that is, the so-called normal form. Then one may explore the local dynamical behaviors of a higher dimensional or even infinitely dimensional dynamical system near a certain nonhyperbolic steady state according to the normal form obtained. The normal form of Hopf bifurcation in ordinary differential equations (ODEs) with or without delays has been established well [1, 3, 5] since in this case the equilibrium is always constant and there are also no effects of spatial diffusion.Under some certain conditions, the reaction-diffusion equations under the homogeneous Neumann boundary condition may have the constant steady state and thus one can study the Hopf bifurcation of system at this constant steady state. Compared with the ODEs, it is more difficult to derive the normal form of Hopf bifurcation for reaction-diffusion equations at the constant steady state. Although Hassard et al. [3] established the method computing the normal form of Hopf bifurcation in reaction-diffusion equations with the homogeneous Neumann boundary condition and also considered the Hopf bifurcation of spatially homogeneous periodic solutions in Brusselator system, using the same method, Jin et al. [9] and Ruan [10] as well as Yi et al. [11, 12] considered the Hopf bifurcation of spatially homogeneous periodic solutions for Gierer-Meinhardt system and CIMA reaction, respectively. There are few results regarding Hopf bifurcation of spatially nonhomogeneous periodic solutions for spatially homogeneous reaction-diffusion equations [7].Based on the reason mentioned above, in this paper we consider the normal form of Hopf bifurcation of reaction-diffusion equations at the constant steady state following the idea in [3]. In order to have a clearer structure, we are concerned with the following general reaction-diffusion system coupled by two equations defined on one-dimensional spatial domain (0,lπ) with l>0 and subject to Neumann boundary conditions; that is,(1)ut=d1uxx+f1λ,u,v,x∈0,lπ,t>0,vt=d2vxx+f2λ,u,v,x∈0,lπ,t>0,ux0,t=vx0,t=uxlπ,t=vxlπ,t=0,t>0,ux,0=u0x,vx,0=v0x,x∈0,lπ,in which d1,d2>0 are the diffusion coefficients, λ∈R is the parameter, and f1,f2:R×R2→R are Cr  (r≥5) functions with fk(λ,0,0)=0  (k=1,2) for any λ∈R. Although Yi et al. [7] described the algorithm determining the properties of Hopf bifurcation of spatially homogeneous and nonhomogeneous periodic solutions for (1) at (0,0) and also considered the Hopf bifurcation of a diffusive predator-prey system with Holling type-II functional response and subject to the homogeneous Neumann boundary condition, they did not give the normal form of Hopf bifurcation of spatially homogeneous and nonhomogeneous periodic solutions of the general reaction-diffusion system (1) at (0,0).This paper is organized as follows. In the next section, following the abstract method according to [3], we describe the algorithm determining the properties of Hopf bifurcation of spatially homogeneous and nonhomogeneous periodic solutions for system (1) at the constant steady state (0,0). In Section 3, the explicit formulas determining the properties of Hopf bifurcation of spatially homogeneous periodic solutions for system (1) at (0,0) are obtained. The explicit formulas determining the properties of Hopf bifurcation of spatially nonhomogeneous periodic solutions for (1) at (0,0) are also derived in Section 4.


## 2. Algorithm Determining the Properties of Hopf Bifurcation

In this section, we will describe the explicit algorithm determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions of system (1) at (0,0).Define the real-valued Sobolev spaceX by (2)X=u,v∈H20,lπ×H20,lπ∣ux=vx=0,x=0,lπ.In terms of X, the complex-valued Sobolev space XC is given by (3)XC=X⊕iX=x1+ix2,x1,x2∈X,and the inner product 〈·,·〉 on XC is defined by(4)U1,U2=∫0lπu¯1u2+v¯1v2dx,forU1=u1,v1∈XC,U2=u2,v2∈XC.LetA(λ)=f1u(λ,0,0), B(λ)=f1v(λ,0,0), C(λ)=f2u(λ,0,0), and D(λ)=f2v(λ,0,0) and define the linear operator L(λ) with the domain DL(λ)=XC by(5)Lλ=d1∂2∂x2+AλBλCλd2∂2∂x2+Dλ.Assume that, for some λ0∈R, the following condition holds: (H)
There exists a neighborhoodO of λ0 such that, for λ∈O, L(λ) has a pair of simple and continuously differentiable eigenvalues α(λ)±iω(λ) with α(λ0)=0, ω(λ0)=ω0>0, and α′λ0≠0. In addition, all other eigenvalues of L(λ) have nonzero real parts for λ∈O.Then from [3, 7] we know that system (1) undergoes a Hopf bifurcation at (0,0) when λ crosses through λ0.Define the second-order matrix sequenceLj(λ) by(6)Ljλ=Aλ-d1j2l2BλCλDλ-d1j2l2,j∈N0.Then the characteristic equation of Lj(λ) is(7)β2-βTjλ+Djλ=0,j∈N0,where(8)Tjλ=Aλ+Dλ-d1+d2j2l2,Djλ=d1d2j4l4-d1Dλ+d2Aλj2l2+AλDλ-BλCλ.The eigenvalues ofL(λ) can be determined by the eigenvalues of Lj(λ)  (j∈N0) and we have the following conclusion.Lemma 1.
Ifβ(λ)∈C is an eigenvalue of the operator L(λ), then there exists some n∈N0 such that β(λ) is the eigenvalue of Ln(λ) and vice versa.Proof.
It is well known that the eigenvalue problem(9)-φ′′=μφ,x∈0,lπ,φ′0=φ′lπ=0has eigenvalues j2/l2  (j=0,1,2,…) with eigenfunctions cosj/lx. Assume that β(λ)∈C is an eigenvalue of the operator L(λ) and the corresponding eigenfunction is (ϕ,ψ)∈XC; that is,(10)Lλϕψ=λϕψ.Notice that (ϕ,ψ)∈XC can be represented as (11)ϕψ=∑j=0∞ajbjcos⁡jlx,where aj,bj∈C  (j∈N0). Then (10) can be written into(12)∑j=0∞Ljλajbjcos⁡jlx=βλ∑j=0∞ajbjcos⁡jlx.
From the orthogonality of the function sequencecosj/lxj=0∞, one can get from (12) that, for each j∈N0,(13)Ljλajbj=βλajbj.Since (ϕ,ψ)∈XC is the eigenfunction of L(λ) corresponding to the eigenvalue β(λ), (ϕ,ψ)≠0 and so there must be some n∈N0 such that 0≠(an,bn)∈C×C. Therefore, β(λ) is the eigenvalue of the matrix Ln.
Ifβ(λ) is the eigenvalue of some matrix Ln, then there exists a nonzero vector (an,bn)∈C×C such that (13) holds. Let (14)ϕψ=anbncos⁡nlx.Then (ϕ,ψ)≠0 and(15)Lλϕψ=Lλanbncos⁡nlx=Lnanbncos⁡nlx=λanbncos⁡nlx=βλϕψ.This demonstrates that β(λ) is an eigenvalue of L(λ) and thus the proof is complete.Lemma1 shows that, under assumption (H), there is a unique n∈N0 such that ±iω0 are purely imaginary eigenvalues of Ln(λ0); that is, Tn(λ0)=0 and Dn(λ0)>0. Furthermore, it is easy to see that Tj(λ0)≠0 for any j≠n. Therefore, Lj(λ0)  (j≠n) has eigenvalues with zero real parts if and only if Dj(λ0)=0. Assume that β(λ)=α(λ)+iω(λ) is the eigenvalue of L(λ) for λ sufficiently approaching λ0. Then by the smoothness of fk  (k=1,2) we know that β(λ) is also the eigenvalue of Ln(λ); namely, β(λ) satisfies the following equation:(16)β2-βTnλ+Dnλ=0.Under the assumption (H), differentiating the above equation with respect to λ at λ0 yields(17)dαλ0dλ=12A′λ0+D′λ0.Based on the above discussion, condition (H) has the following equivalent form:(18)Tnλ0=0,Dnλ0>0,A′λ0+D′λ0≠0for somen∈N0,Djλ0≠0for anyj∈N0.Then we know that ω0=Dn(λ0) and B(λ0),C(λ0) cannot be equal to zero simultaneously when the hypothesis (H) is satisfied. Therefore, the eigenvector of Ln(λ0) corresponding to the eigenvalue iω0 can be chosen as(19)anbn=1iω0-Aλ0+d1n2/l2Bλ0and thus the eigenfunction of L(λ0) corresponding to the eigenvalue iω0 has the form(20)q=anbncos⁡nlx=1iω0-Aλ0+d1n2/l2Bλ0cos⁡nlx.Let the linear operatorL∗(λ0) with the domain DL∗λ0=XC be defined by (21)L∗λ0=d1∂2∂x2+Aλ0Cλ0Bλ0d2∂2∂x2+Dλ0.Then L∗(λ0) is the adjoint operator of the operator L(λ0) such that 〈U,L(λ0)V〉=〈L∗(λ0)U,V〉 with U,V∈XC. Similar to the choice of the eigenfunction q of the operator L(λ0) corresponding to the eigenvalue iω0, we can choose(22)q∗=an∗bn∗cos⁡nlx=ω0+iAλ0-d1n2/l22ω0∫0lπcos2⁡n/lxdx-iBλ02ω0∫0lπcos2⁡n/lxdxcos⁡nlxsuch that(23)L∗λ0q∗=-iω0q,q∗,q=1,q∗,q¯=0.Define XC and XS by XC={zq+z¯q¯∣z∈C} and XS={U∈X∣〈q∗,U〉=0}, respectively. Then X can be decomposed as the direct sum of XC and XS; that is, X=XC⊕XS. Thus, for any U=(u,v)∈X, there exists z∈C and w=(w1,w2)∈Xs such that(24)U=zq+z¯q¯+w,oru=zancos⁡nlx+z¯a¯ncos⁡nlx+w1,v=zbncos⁡nlx+z¯b¯ncos⁡nlx+w2.DefineF(λ,U) by(25)Fλ,U=f1λ,u,v-Aλu-Bλvf2λ,u,v-Cλu-Dλv.Then system (1) can be rewritten into the following abstract form:(26)dUdt=LλU+Fλ,U.When λ=λ0, system (26) is reduced to(27)dUdt=Lλ0U+F0U,where F0(U)=F(λ,U)λ=λ0. In terms of (23) and decomposition (24), system (27) can be transformed into the following system in (z,w) coordinates:(28)dzdt=iω0z+q∗,F0,dwdt=Lλ0w+Hz,z¯,w,where(29)Hz,z¯,w=F0-q∗,F0q-q¯∗,F0q¯,F0=F0zq+z¯q¯+w.ForX=(x1,x2), Y=(y1,y2), and Z=(z1,z2)∈XC, define the symmetric multilinear forms Q(X,Y) and C(X,Y,Z), respectively, by(30)QX,Y=∑k,j=12∂2f1λ0,ξ1,ξ2∂ξk∂ξjξ1=ξ2=0xkyj∑k,j=12∂2f2λ0,ξ1,ξ2∂ξk∂ξjξ1=ξ2=0xkyj,(31)CX,Y,Z=∑k,j,l=12∂3f1λ0,ξ1,ξ2∂ξk∂ξj∂ξlξ1=ξ2=0xkyjzl∑k,j,l=12∂3f2λ0,ξ1,ξ2∂ξk∂ξj∂ξlξ1=ξ2=0xkyjzl.Then, for U=(u,v)∈X, we have(32)F0U=12QU,U+16CU,U,U+OU4.For the simplicity of notations, we will use QXY and CXYZ to denote Q(X,Y) and C(X,Y,Z), respectively.Let(33)Hz,z¯,w=H202z2+H11zz¯+H022z¯2+Oz3+Ozw.Then from (29) and (32), one can get(34)H20=Qqq-q∗,Qqqq-q¯∗,Qqqq¯,H11=Qqq¯-q∗,Qqq¯q-q¯∗,Qqq¯q¯.From the center manifold theorem in [3], we can rewrite w in the form(35)w=w202z2+w11zz¯+w022z¯2+Oz3.The second equation of (28), (33), and (35) yields(36)w20=2iω0I-Lλ0-1H20,w11=-Lλ0-1H11.Substituting (35) into the first equation of (28) gives the equation of reaction-diffusion system (1) restricted on the center manifold at (λ0,0,0) as(37)dzdt=iω0z+∑2≤k+j≤3gkjk!j!zkz¯j+Oz4,where g20=q∗,Qqq, g11=q∗,Qqq¯, g02=q∗,Qq¯q¯, and(38)g21=2q∗,Qw11q+q∗,Qw20q+q∗,Cqqq¯.The dynamics of (28) can be determined by the dynamics of (37).In addition, it can be observed from [3] that when λ approaches sufficiently λ0, the Poincaré normal form of (26) has the form (39)z˙=αλ+iωλz+z∑j=1Mcjλzz¯j,where z is a complex variable, M≥1, and cj(λ) are complex-valued coefficients with(40)c1λ0=i2ω0g20g11-2g112-13g022+g212=i2ω0q∗,Qqqq∗,Qqq¯-2q∗,Qqq¯2-13q∗,Qq¯q¯2+q∗,Qw11q+12q∗,Qw20q+12q∗,Cqqq¯.The direction of Hopf bifurcation and the stability of the bifurcating periodic solutions of (1) at (λ0,0,0) can be determined by the sign of Re⁡c1(λ0) and we have the following conclusion.Theorem 2.
Assume that condition (H) (or equivalently (18)) holds. Then system (1) undergoes a supercritical (or subcritical) Hopf bifurcation at (0,0) when λ=λ0 if (41)1α′λ0Re⁡c1λ0<0resp.>0.In addition, if all other eigenvalues of L(λ0) have negative real parts, then the bifurcating periodic solutions are stable (resp., unstable) when Re⁡c1(λ0)<0 (resp.,>0).


## 3. Spatially Homogeneous Hopf Bifurcation

From the description in the previous section we know that Hopf bifurcation of (1) at (λ0,0,0) is spatially homogeneous if condition (18) holds when n=0. In the present section, we compute Re⁡c1(λ0) in (40) in order to determine the direction of spatially homogeneous Hopf bifurcation and the stability of bifurcating periodic solutions of (1) at (λ0,0,0) following the algorithm described in this pervious section.Lemma 3.
If condition (18) is satisfied when n=0, then H20=H11=0.Proof.
From (20) and (22) one can see(42)a0=1,b0=iω0-Aλ0Bλ0,a0∗=ω0+iAλ02lπω0,b0∗=-iBλ02lπω0,where(43)ω0=Aλ0Dλ0-Bλ0Cλ0=-A2λ0-Bλ0Cλ0.
Let all the partial derivatives offk(λ,u,v)  (k=1,2) be evaluated at (λ0,0,0), and let ck0, dk0, and ek0  (k=1,2) be defined, respectively, by(44)ck0=fkuu+2fkuvb0+fkvvb02,dk0=fkuu+fkuvb¯0+b0+fkvvb02,ek0=fkuuu+fkuuv2b0+b¯0+fkuvv2b02+b02+fkvvvb02b0.Then from (30) and (31) we can get(45)Qqq=c10c20,Qqq¯=d10d20,Cqqq¯=e10e20.Therefore,(46)q∗,Qqq=∫0lπa¯0∗c10+b¯0∗c20=lπa¯0∗c10+b¯0∗c20,q¯∗,Qqq=∫0lπa0∗c10+b0∗c20=lπa0∗c10+b0∗c20,q∗,Qqq¯=∫0lπa¯0∗d10+b¯0∗d20=lπa¯0∗d10+b¯0∗d20,q¯∗,Qqq¯=∫0lπa0∗d10+b0∗d20=lπa0∗d10+b0∗d20.From (34) and (46), one can obtain(47)H20=c10-lπa0∗+a¯0∗c10+b0∗+b¯0∗c20c20-lπa¯0∗b0+a0∗b¯0c10+b¯0∗b0+b0∗b¯0c20,H11=d10-lπa0∗+a¯0∗d10+b0∗+b¯0∗d20d20-lπa¯0∗b0+a0∗b¯0d10+b¯0∗b0+b0∗b¯0d20.
Notice from (42) that(48)a0∗+a¯0∗=b¯0∗b0+b0∗b¯0=1lπ,b0∗+b¯0∗=a¯0∗b0+a0∗b¯0=0.The conclusion follows by substituting (48) into (47).Lemma3 and (36) imply w20=w11=0 when n=0 in (18) and thus we have (49)g21=q∗,Cqqq¯=∫0lπa0∗e10+b0∗e20=lπa¯0∗e10+b¯0∗e20.It follows from (40) that(50)2Re⁡c1λ0=Re⁡iω0q∗,Qqqq∗,Qqq¯+q∗,Cqqq¯=Re⁡ilπa¯0∗c10+b¯0∗c20lπa¯0∗d10+b¯0∗d20ω0+lπa¯0∗e10+b¯0∗e20.We representA(λ0), B(λ0), and C(λ0) by A,B, and C, respectively, for the simplicity of notations and under assumption (18) with n=0, substituting b0 in (42) into (44) yields that, for k=1,2,(51)Re⁡ck0=fkuuB2-2fkuvAB+fkvv2A2+BCB2,Im⁡ck0=2ω0fkuvB-fkvvAB2,dk0=fkuuA-2fkuvB-fkvvCB,Re⁡ek0=fkuuuB2-3fkuuvAB+2fkuvvA2+fkvvvACB2,Im⁡ek0=ω0fkuuvB-2fkuvvA-fkvvvCB2.From (42), (50), and (51) one can derive (52)2Re⁡c1λ0=C0λ04ω02B2,where(53)C0λ0=f1uuA2+f2uu-2f1uvAB-f1vvAC-2f2uvB2-f2vvBCf1uuB+f1vvC+2f2uvB-2f2vvA+f1uuA-2f1uvB-f1vvCf1uuAB+2f1uvBC-f1vvAC+f2uuB2-2f2uvAB+f2vv2A2+BC-2A2+BC·f1uuu+f2uuvB2-2f1uuv+f2uvvAB-f2vvvBC.Thus we have the following result.Theorem 4.
Assume that condition (18) is satisfied when n=0 and C0(λ0) is defined by (53). Then the spatially homogeneous Hopf bifurcation of system (1) at (λ0,0,0) is supercritical (resp., subcritical) if(54)C0λ0A′λ0+D′λ0<0resp.>0.Moreover, if each eigenvalue of Lj(λ0) has negative real parts for all j∈N, then the above spatially homogeneous bifurcating periodic solutions are stable (resp., unstable) when (55)C0λ0<0resp.>0.


## 4. Spatially Nonhomogeneous Hopf Bifurcation

Notice that the spatially nonhomogeneous periodic solutions of (1) at (λ0,0,0) from Hopf bifurcation are unstable. Accordingly, in this section we will calculate Re⁡c1(λ0) in (40) in order to determine the direction of Hopf bifurcation of spatially nonhomogeneous periodic solutions of system (1) at (λ0,0,0). To this end, we always assume that n∈N in (18) throughout this section and still represent A(λ0), B(λ0), and C(λ0) by A,B, and C, respectively. Thus q∗ defined in (22) has the form(56)q∗=an∗bn∗cos⁡nlx=ω0+iA-d1n2/l2lπω0-iBlπω0cos⁡nlx,where(57)ω0=A-d1n2l2D-d2n2l2-BC=-A-d1n2l22-BC.Since whenn∈N, (58)∫0lπcos3⁡nlxdx=0,one can obtain(59)q∗,Qqq=q∗,Qqq¯=q¯∗,Qqq=q¯∗,Qqq¯=0.Thus, in order to calculate Re⁡c1(λ0), it remains to compute(60)q∗,Qw11q,q∗,Qw20q¯,q∗,Cqqq¯.Let all the second- and third-order partial derivatives offk(λ,u,v)  (k=1,2) with respect to u and v be evaluated at (λ0,0,0) and let(61)ckn=fkuu+2fkuvbn+fkvvbn2,dkn=fkuu+fkuvb¯n+bn+fkvvbn2,ekn=fkuuu+fkuuv2bn+b¯n+fkuvv2bn2+bn2+fkvvvbn2bn,k=1,2.Then from (30) and (31), one can observe(62)Qqq=c1nc2ncos2⁡nlx=12c1nc2n1+cos⁡2nlx,Qqq¯=d1nd2ncos2⁡nlx=12d1nd2n1+cos⁡2nlx,Cqqq¯=e1ne2ncos3⁡nlx.In view of (34), (59), and (62), we have(63)H20=Qqq=12c1nc2n1+cos⁡2nlx,H11=Qqq¯=12d1nd2n1+cos⁡2nlx.Equalities (63) show that the calculation of [2iω0-L(λ0)]-1 and [L(λ0)]-1 will be restricted on the subspaces spanned by eigenmodes 1 and cos2n/lx.Let(64)α1=12d1d2-3d12n4l4-3d2-d1An2l2-3ω02,α2=6d1+d2n2ω0l2,α3=d12n4l4+d2-d1An2l2-3ω02,α4=-2d1+d2n2ω0l2.Then, under condition (18), one can derive(65)2iω0I-L2nλ0-1=1α1+iα2·2iω0+A+3d2-d1n2l2BC2iω0-A+4d1n2l2,2iω0I-L0λ0-1=1α3+iα3·2iω0+A-d1+d2n2l2BC2iω0-A.From (36) and (61), we have(66)w20=2iω0I-L2nλ0-12cos⁡2nlx+2iω0I-L0λ0-12c1nc2n=12α1+iα2·2iω0+A+3d2-d1n2l2c1n+Bc2nCc1n+2iω0-A+4d1n2l2c2n·cos⁡2nlx+12α3+iα4·2iω0+A-d1+d2n2l2c1n+Bc2nCc1n+2iω0-Ac2n.Similarly, we can get(67)w11=12α5A+3d2-d1n2l2d1n+Bd2nCd1n+4d1n2l2-Ad2n·cos⁡2nlx+12α6·A-d1+d2n2l2d1n+Bd2nCd1n-Ad2n,where(68)α5=12d1d2-3d12n4l4-3d2-d1An2l2+ω02,α6=d12n4l4+d2-d1An2l2+ω02.From (30) we have(69)Qw20q¯=f1uuξ+f1uvη+f1vvγf2uuξ+f2uvη+f2vvγcos⁡nlxcos⁡2nlx+f1uuτ+f1uvχ+f1vvζf2uuτ+f2uvχ+f2vvζcos⁡nlx,Qw11q=f1uuξ~+f1uvη~+f1vvγ~f2uuξ~+f2uvη~+f2vvγ~cos⁡nlxcos⁡2nlx+f1uuτ~+f1uvχ~+f1vvζ~f2uuτ~+f2uvχ~+f2vvζ~cos⁡nlx,with(70)ξ=2iω0+A+3d2-d1n2/l2c1n+Bc2n2α1+iα2,η=2iω0+A+3d2-d1n2/l2d1n2/l2-A-iω0+BC2Bα1+iα2·c1n+5d1n2/l2-2A+iω02α1+iα2c2n,γ=Cc1n+2iω0-A+4d1n2/l2c2nd1n2/l2-A-iω02Bα1+iα2,τ=2iω0+A-d1+d2n2/l2c1n+Bc2n2α3+iα4,χ=2iω0+A-d1+d2n2/l2d1n2/l2-A-iω0+BC2Bα3+iα4·c1n+d1n2/l2-2A+iω02α3+iα4c2n,ζ=Cc1n+2iω0-Ac2nd1n2/l2-A-iω02Bα3+iα4,ξ~=A+3d2-d1n2/l2d1n+Bd2n2α5,η~=A+3d2-d1n2/l2iω0-A+d1n2/l2+BC2Bα5d1n+iω0-2A+5d1n2/l22α5d2n,γ~=iω0-A+d1n2/l2Cd1n+4d1n2/l2-Ad2n2Bα5,τ~=A-d1+d2n2/l2d1n+Bd2n2α6,χ~=A-d1+d2n2/l2iω0-A+d1n2/l2+BC2Bα6d1n+iω0-2A+d1n2/l22α6d2n,ζ~=iω0-A+d1n2/l2Cd1n-Ad2n2Bα6.Notice that, for n∈N,(71)∫0lπcos2⁡nlxdx=lπ2,∫0lπcos⁡2nlxcos2⁡nlxdx=lπ4.It follows from (69) that(72)q∗,Qw20q¯=lπ4a¯n∗f1uuξ+f1uvη+f1vvγ+b¯n∗f2uuξ+f2uvη+f2vvγ+lπ2a¯n∗f1uuτ+f1uvχ+f1vvζ+b¯n∗f2uuτ+f2uvχ+f2vvζ,q∗,Qw11q=lπ4a¯n∗f1uuξ~+f1uvη~+f1vvγ~+b¯n∗f2uuξ~+f2uvη~+f2vvγ~+lπ2a¯n∗f1uuτ~+f1uvχ~+f1vvζ~+b¯n∗f2uuτ~+f2uvχ~+f2vvζ~.Substituting bn=iω0-A(λ0)+d1n2/l2/B(λ0) into (61) gives(73)Re⁡ckn=fkuuB2-2fkuvA-d1n2/l2B+fkvv2A-d1n2/l22+BCB2,Im⁡ckn=2ω0fkuvB-fkvvA-d1n2/l2B2,dkn=fkuuA-d1n2/l2-2fkuvB-fkvvCB,Re⁡ekn=fkuuuB2-3fkuuvA-d1n2/l2B+2fkuvvA-d1n2/l22+fkvvvA-d1n2/l2CB2,Im⁡ekn=ω0fkuuvB-2fkuvvA-d1n2/l2-fkvvvCB2,k=1,2.Then from (70) and (73), we have(74)Re⁡ξ=A+3d2-d1n2/l2Re⁡c1n+BRe⁡c2n-2ω0Im⁡c1nα1+A+3d2-d1n2/l2Im⁡c1n+BIm⁡c2n+2ω0Re⁡c1nα22α12+α22,Im⁡ξ=A+3d2-d1n2/l2Im⁡c1n+BIm⁡c2n+2ω0Re⁡c1nα1-A+3d2-d1n2/l2Re⁡c1n+BRe⁡c2n-2ω0Im⁡c1nα22α12+α22,Re⁡η=A+3d2-d1n2/l2d1n2/l2-A+BC+2ω02Re⁡c1n-3ω0d1-d2n2/l2-AIm⁡c1n+B5d1n2/l2-2ARe⁡c2n-Bω0Im⁡c2n2Bα12+α22·α1+A+3d2-d1n2/l2d1n2/l2-A+BC+2ω02Im⁡c1n+3ω0d1-d2n2/l2-ARe⁡c1n+B5d1n2/l2-2AIm⁡c2n+Bω0Re⁡c2n2Bα12+α22·α2,Im⁡η=A+3d2-d1n2/l2d1n2/l2-A+BC+2ω02Im⁡c1n+3ω0d1-d2n2/l2-ARe⁡c1n+B5d1n2/l2-2AIm⁡c2n+Bω0Re⁡c2n2Bα12+α22·α1-A+3d2-d1n2/l2d1n2/l2-A+BC+2ω02Re⁡c1n-3ω0d1-d2n2/l2-AIm⁡c1n+B5d1n2/l2-2ARe⁡c2n-Bω0Im⁡c2n2Bα12+α22·α2,Re⁡γ=d1n2/l2-ACRe⁡c1n-A-4d1n2/l2Re⁡c2n-2ω0Im⁡c2n+ω0CIm⁡c1n-A-4d1n2/l2Im⁡c2n+2ω0Re⁡c2n2Bα12+α22α1+d1n2/l2-ACIm⁡c1n-A-4d1n2/l2Im⁡c2n+2ω0Re⁡c2n-ω0CRe⁡c1n-A-4d1n2/l2Re⁡c2n-2ω0Im⁡c2n2Bα12+α22α2,Im⁡γ=d1n2/l2-ACIm⁡c1n-A-4d1n2/l2Im⁡c2n+2ω0Re⁡c2n-ω0CRe⁡c1n-A-4d1n2/l2Re⁡c2n-2ω0Im⁡c2n2Bα12+α22α1-d1n2/l2-ACRe⁡c1n-A-4d1n2/l2Re⁡c2n-2ω0Im⁡c2n+ω0CIm⁡c1n-A-4d1n2/l2Im⁡c2n+2ω0Re⁡c2n2Bα12+α22α2,Re⁡τ=A+d1+d2n2/l2Re⁡c1n+BRe⁡c2n-2ω0Im⁡c1nα3+A+d1+d2n2/l2Im⁡c1n+BIm⁡c2n+2ω0Re⁡c1nα42α32+α42,Im⁡τ=A+d1+d2n2/l2Im⁡c1n+BIm⁡c2n+2ω0Re⁡c1nα3-A+d1+d2n2/l2Re⁡c1n+BRe⁡c2n-2ω0Im⁡c1nα42α32+α42,Re⁡χ=A-d1+d2n2/l2d1n2/l2-A+BC+2ω02Re⁡c1n-ω03d1+d2n2/l2-3AIm⁡c1n+Bd1n2/l2-2ARe⁡c2n-Bω0Im⁡c2n2Bα32+α42·α3+A-d1+d2n2/l2d1n2/l2-A+BC+2ω02Im⁡c1n+ω03d1+d2n2/l2-3ARe⁡c1n+Bd1n2/l2-2AIm⁡c2n+Bω0Re⁡c2n2Bα32+α42·α4,Im⁡χ=A-d1+d2n2/l2d1n2/l2-A+BC+2ω02Im⁡c1n+ω03d1+d2n2/l2-3ARe⁡c1n+Bd1n2/l2-2AIm⁡c2n+Bω0Re⁡c2n2Bα32+α42·α3-A-d1+d2n2/l2d1n2/l2-A+BC+2ω02Re⁡c1n-ω03d1+d2n2/l2-3AIm⁡c1n+Bd1n2/l2-2ARe⁡c2n-Bω0Im⁡c2n2Bα32+α42·α4,Re⁡ζ=d1n2/l2-ACRe⁡c1n-ARe⁡c2n-2ω0Im⁡c2n+ω0CIm⁡c1n-AIm⁡c2n+2ω0Re⁡c2n2Bα32+α42α3+d1n2/l2-ACIm⁡c1n-AIm⁡c2n+2ω0Re⁡c2n-ω0CRe⁡c1n-ARe⁡c2n-2ω0Im⁡c2n2Bα32+α42α4,Im⁡ζ=d1n2/l2-ACIm⁡c1n-AIm⁡c2n+2ω0Re⁡c2n-ω0CRe⁡c1n-ARe⁡c2n-2ω0Im⁡c2n2Bα32+α42α3-d1n2/l2-ACRe⁡c1n-ARe⁡c2n-2ω0Im⁡c2n+ω0CIm⁡c1n-AIm⁡c2n+2ω0Re⁡c2n2Bα32+α42α4,Re⁡η~=A+3d2-d1n2/l2d1n2/l2-A+BCd1n+B5d1n2/l2-2Ad2n2Bα5,Re⁡η~=ω0A+3d2-d1n2/l2d1n+Bd2n2α5,Re⁡γ~=d1n2/l2-ACd1n+4d1n2/l2-Ad2n2Bα5,Im⁡γ~=ω0Cd1n+4d1n2/l2-Ad2n2Bα5,Re⁡χ~=A-d1+d2n2/l2d1n2/l2-A+BCd1n+Bd1n2/l2-2Ad2n2Bα6,Im⁡χ~=ω0A-d1+d2n2/l2d1n+Bd2n2α6Bλ0,Re⁡ζ~=Cd1n-Ad2nd1n2/l2-A2Bα6,Im⁡ζ~=ω0Cd1n-Ad2n2Bα6.Sincelπa¯n∗=1-iA-d1n2/l2/ω0 and lπb¯n∗=-iB/ω0, one can get(75)Re⁡q∗,Qw20q¯=14f1uuRe⁡ξ+2Re⁡τ+f1uvRe⁡η+2Re⁡χ+f1vvRe⁡γ+2Re⁡ζ+A-d1n2/l24ω0f1uuIm⁡ξ+2Im⁡τ+f1uvIm⁡η+2Im⁡χ+f1vvIm⁡γ+2Im⁡ζ+B4ω0f2uuIm⁡ξ+2Im⁡τ+f2uvIm⁡η+2Im⁡χ+f2vvIm⁡γ+2Im⁡ζ,Re⁡q∗,Qw11q=14f1uuξ~+2τ~+f1uvRe⁡η~+2Re⁡χ~+f1vvRe⁡γ~+2Re⁡ζ~+A-d1n2/l24ω0f1uuξ~+2τ~+f1uvRe⁡η~+2Re⁡χ~+f1vvRe⁡γ~+2Re⁡ζ~+B4ω0f2uvIm⁡η~+2Im⁡χ~+f2vvIm⁡γ~+2Im⁡ζ~.In addition, it follows from∫0lπ‍cos4n/lxdx=3lπ/8 and (62) that (76)q∗,Cqqq¯=3lπ8a¯n∗e1n+b¯n∗e2n.Therefore,(77)Re⁡q∗,Cqqq¯=38B2f1uuuB2-3f1uuvA-d1n2l2B+2f1uvvA-d1n2l22+f1vvvA-d1n2l2C+A-d1n2l2·f1uuvB-2f1uvvA-d1n2l2-f1vvvC+Bf2uuvB-2f2uvvA-d1n2l2-f2vvvC.Now, by (40), we have(78)Re⁡c1λ0=Re⁡q∗,Qw11q+12Re⁡q∗,Qw20q¯+12·Re⁡q∗,Cqqq¯=18f1uuRe⁡ξ+2Re⁡τ+2ξ~+4τ~+f1uvRe⁡η+2Re⁡χ+2Re⁡η~+4Re⁡χ~+f1vvRe⁡γ+2Re⁡ζ+2Re⁡γ~+4Re⁡ζ~+A-d1n2/l28ω0f1uuIm⁡ξ+2Im⁡τ+2ξ~+4τ~+f1uvIm⁡η+2Im⁡χ+2Re⁡η~+4Re⁡χ~+f1vvIm⁡γ+2Im⁡ζ+2Re⁡γ~+4Re⁡ζ~+B8ω0f2uuIm⁡ξ+2Im⁡τ+f2uvIm⁡ηI+2Im⁡χ+2Im⁡η~+4Im⁡χ~+f2vvIm⁡γ+Im⁡ζ+2Im⁡γ~+4Im⁡ζ~+316B2f1uuuB2-3f1uuvA-d1n2l2B+2f1uvvA-d1n2l22+f1vvvA-d1n2l2C+A-d1n2l2·f1uuvB-2f1uvvA-d1n2l2-f1vvvC+Bf2uuvB-2f2uvvA-d1n2l2-f2vvvC.Thus we have the following result.Theorem 5.
Assume that condition (18) holds for n∈N. Then the spatially nonhomogeneous Hopf bifurcation of system (1) at (λ0,0,0) is supercritical (resp., subcritical) if (79)Re⁡c1λ0A′λ0+D′λ0<0resp.>0;here Re⁡c1(λ0) is given by (78).



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*Source: 657307-2015-08-03.xml*