# Codimension-m Bifurcation Theorems Applicable to the Numerical Verification Methods

Authors: Tadashi Kawanago
Journal: Advances in Numerical Analysis
 (2013)

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## Abstract

We establish codimension-m bifurcation theorems applicable to the numerical verification methods. They are generalization of codimension-1 bifurcation theorems established by (Kawanago, 2004). As a numerical example, we treat Hopf bifurcation, which is codimension-2 bifurcation.


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## Body

## 1. Introduction

By the recent growth of the computer power, we can observe numerically bifurcation phenomena of solutions without difficulty for a lot of differential equations and systems. It is in general difficult, however, to analyze rigorously such phenomena by the use of pure analytical methods. Actually, it seems impossible at least at present to analyze by the use of pure analytical methods the Hopf bifurcation phenomena in the Brusselator model treated in Section4. We need some computer-assisted analysis to treat it. We now have various excellent bifurcation theorems from the theoretical point of view. It needs in general, however, some particular devices to apply them to a given concrete dynamical system since we are usually not able to check some conditions in such theorems directly by numerical methods.Another important approach to computer-assisted analysis for bifurcation problems is to establish new bifurcation theorems applicable directly to numerical verification methods. This approach is our theme in this paper. It is useful from the practical and applied mathematical point of view.  In [1], the author established some codimension-1 bifurcation theorems applicable directly to numerical verification methods. Using a symmetry-breaking bifurcation theorem [1, Theorem 3.1] and the numerical verification methods, we proved the existence of a ℤ2-symmetry-breaking bifurcation point  for a nonlinear forced vibration system described by a wave equation in [2], and Nakao et al. verified some symmetry-breaking bifurcation points for two-dimensional Rayleigh-Bénard heat convection system in [3, 4].In this paper, we establish codimension-m bifurcation theorems applicable to the numerical verification methods. They are generalization of codimension-1 bifurcation theorems mentioned above. In Section 4, we apply our new theorem to Hopf bifurcation, which is codimension-2 bifurcation.Here, we present our main theorem. LetX and Y be real Banach spaces. Let X1 and X2 be closed subspaces of X and let Y1 and Y2 be closed subspaces of Y. We assume that X=X1⊕X2 and Y=Y1⊕Y2. Here, ⊕ means the direct sum. Let m∈ℕ and g∈C2(ℝm×X,Y) have the following properties:
(1)g(Λ,v)∈Y1foranyΛ∈ℝm,v∈X1,(2)gu(Λ,v)Xj⊂YjforanyΛ∈ℝm,v∈X1(j=1,2),(3)guu(Λ,v1)v2w∈Y2foranyΛ∈ℝm,v1,v2∈X1,w∈X2.
We denote by e1:=(1,0,…,0)∈ℝm the first row vector of the identity matrix of order m. We define  J:ℝm×X→ℝm×Y by
(4)J(Λvw)∶=(lw-e1g(Λ,v)gu(Λ,v)w)∈ℝm×Y1×Y2for(Λ,v,w)∈ℝm×X1×X2.
Here, l∈ℒ(X,ℝm), and we assume that lv=0 for any v∈X1. We define projections p and P by
(5)p:(r1,r2,…,rm)∈ℝm↦r1∈ℝ,P:(r1,r2,…,rm)∈ℝm↦(r2,…,rm)∈ℝm-1forthecasem≥2.In what follows, we always set formallyℝm-1×Y∶=Y for the case: m=1. We define G:ℝm×X→ℝm-1×Y by
(6)G∶=g(m=1),G(Λu)∶=(Plug(Λ,u))(m≥2).We setZ:=(pl)-1(0)={u∈X;plu=0} and ℝ+:=(0,∞). Our main theorem is the following.Theorem 1.
In addition to the assumptions above, we assume that(Λ0,v0,w0)∈ℝm×X1×X2 satisfies the following  (H1) and (H2):(H1)
A point(Λ,v,w)=(Λ0,v0,w0) is an isolated solution of the extended system J(Λ,v,w)=0.(H2)
The linear operatorgu(Λ0,v0)∣X1:X1→Y1 is bijective.
Then, the point(Λ0,v0) is a bifurcation point of the equation G(Λ,v)=0. Exactly, there exist an open neighborhood W of (Λ0,v0) in ℝm×X, a∈ℝ+, ζ∈C1((-a,a),ℝm), η∈C1((-a,a),Z), an open neighborhood V of 0 in ℝm and q∈C2(Λ0+V,X1) such that ζ(0)=Λ0, η(0)=0, q(Λ0)=v0 and
(7)G-1(0)∩W={(Λ,q(Λ));(Λ,q(Λ))∈W}∪{(ζ(α),αw0+αη(α)+q(ζ(α)));|α|<a}.Roughly speaking, the well-known pitchfork bifurcation theorem [5, Theorem 1.7] by Crandall and Rabinowitz is equivalent to Theorem 1  with m=1, X1={0}, and Y1={0} (see Section 2). We immediately obtain a ℤ2-symmetry breaking bifurcation theorem [1, Theorem 3.1] by setting m=1 in Theorem 1  and by choosing the symmetric subspace of X as X1 and the anti-symmetric subspace as X2. We can apply Theorem 1  to Hopf bifurcation by setting m=2 and choosing an appropriate space of periodic functions as X, the subspace of steady functions as X1, and the complementary subspace of X1 as X2 (see Section 4 below).Finally, we illustrate by Hopf bifurcation  as an example that known bifurcation theorems are not in general applicable directly to the numerical verification methods.  We consider the next autonomous ordinary differential equation:(O)y˙=f(λ,y),y,f(λ,y)∈ℝd.
The classical Hopf bifurcation theorem reads as follows.Theorem 2.
The point(λ,y)=(λ0,y0) is a Hopf bifurcation point of (O), that is, a branch of periodic solutions of (O) bifurcates at the point (λ,y)=(λ0,y0) from a branch of steady solutions of (O) with the initial period 2πσ0, provided that the following conditions (C1)–(C4) are satisfied:(C1)
f(λ0,y0)=0,(C2)
±i are the simple eigenvalues of σ0fy(λ0,y0) (so, by the implicit function theorem, the matrix σ0fy(λ,y(λ)) has a pair of complex conjugate of eigenvalues μ(λ), μ(λ)¯ with μ(λ0)=i),(C3)
(Transversality condition of eigenvalues)Reμ′(λ0)≠0,(C4)
ik is not an eigenvalue of σ0fy(λ0,y0) for k∈ℤ-{-1,1}.It is difficult to check rigorously by numerical methods thedynamic condition (C3), and we need a particular device for it. See Remark 15 for more detailed information. On the other hand, the condition (C3) is implied by (H1) and (H2) in  Theorem 1, which are  static conditions, that is, regularity conditions for linear operators.The paper is organized as follows. In Section2, we mainly establish a basic bifurcation theorem, which is simpler than  Theorem 1. In Section 3, we prove  Theorem 1. In Section 4, we present a numerical example and treat Hopf bifurcation as an application of  Theorem 1  with m=2 and  our numerical verification method. We present some final remarks in Section 5.Notation.
Letℋ and 𝒦 be Banach spaces.   (1)
We denote byℝ the set of all real numbers, by ℚ the set of all rational numbers, by ℤ the set of all integers, and by ℂ the set of all complex numbers. We write ℝ+∶=(0,∞).(2)
We denote bye1∶=(1,0,…,0)∈ℝm the first row vector of the identity matrix Em=(δij) of order m.(3)
We denote by∥·∥ℋ the norm of ℋ. If ℋ is a Hilbert space, then ∥·∥ℋ stands for the norm induced by the inner product of ℋ. For u∈ℋ and r∈(0,∞), we write Bℋ(u;r)∶={v∈ℋ;∥v-u∥<r}. The subscript will be often omitted if no possible confusion arises.(4)
We denote byℒ(ℋ,𝒦) the space of bounded linear operators from ℋ to 𝒦, and ℒ(ℋ) stands for ℒ(ℋ,ℋ). For T∈ℒ(ℋ,𝒦), ∥T∥ℋ→𝒦 denotes the operator norm of T. The subscript will be omitted if no possible confusion arises.(5)
LetA:ℋ→𝒦 be an operator. We denote by 𝒟(A) the domain of A and by ℛ(A) the range of A. We define 𝒩(A)∶={v∈𝒟(A);Av=0}.(6)
For an open setV⊂ℋ, we denote by Cn(V,𝒦) the space of n-times continuously differentiable functions from V to 𝒦.(7)
LetΩ be a domain of ℝn. We denote Lebesgue spaces by Lp(Ω)(1≤p≤∞) with the norms: ∥u∥Lp(Ω)=(∫Ω‍|u(x)|pdx)1/pfor1≤p<∞,∥u∥L∞(Ω)=ess.sup{|u(x)|;x∈Ω}. We define the norm of the Sobolev space H1(Ω) by
(8)∥u∥H1(Ω)∶=(∥u∥L2(Ω)2+∑k=1n‍∥∂u∂xk∥L2(Ω)2)1/2.(8)
LetN∈ℕ and Zk(k=1,2,…,N) be norm spaces.  We denote their direct product space Z usually by Z=Z1×⋯×ZN and also by Z=Z1⊕⋯⊕ZN in Section 4.5 below. Actually, the direct product space is interchangeably called the direct sum space (see, e.g., [6, Section II.1]). We define ∥u∥Z:=(∑k=1N∥uk∥Zk2‍)1/2 for u=(u1,…,uN)∈Z1×⋯×ZN.


## 2. Basic Bifurcation Theorems

Theorem3  in this section is a generalized version of [1, Theorem 2.1] and can be regarded as a refined version of  Theorem 1  with X1={0} and Y1={0}.LetX and Y be  real Banach spaces and U be an open neighborhood of 0 in X. Let m∈ℕ and V be an open neighborhood of 0 in ℝm. Let f∈C1(V×U,Y) be a map such that
(9)f(Λ,0)=0foranyΛ=(Λ1,…,Λm)∈V,
and the  partial Fréchet derivative fΛku exists and is continuous for k=1,…,m.  Here, for any fixed (Λ⋆,u⋆)∈V×U, fΛku(Λ⋆,u⋆) represents the Fréchet derivative of the function u∈U↦(∂f/∂Λk)(Λ⋆,u)∈Y at u=u⋆.  We denote fΛu∶=(fΛ1u,…,fΛmu) for simplicity. We define H:V×X→ℝm×Y by
(10)H(Λu)∶=(lu-e1fu(Λ,0)u).
Here, l∈ℒ(X,ℝm). In what follows, we often use the same notations as in Section 1. We define F:V×U→ℝm-1×Y by
(11)F∶=f(m=1),F(Λu)∶=(Pluf(Λ,u))(m≥2).
We set Z∶=(pl)-1(0)={u∈X;plu=0}.Theorem 3.
In addition to the assumptions above, we assume that(H)
there existsu0∈U such that (Λ,u)=(0,u0) is an isolated solution of the extended system H(Λ,u)=0.
Then, there exist an open neighborhoodW of (0,0) in ℝm×X, a∈ℝ+ and continuous functions ζ:(-a,a)→ℝm, η:(-a,a)→Z such that ζ(0)=0, η(0)=0 such that
(12)F-1(0)∩W={(Λ,0);(Λ,0)∈W}∪{(ζ(α),αu0+αη(α));α∈(-a,a)}.
Moreover, if f∈Ck+1 (k∈ℕ), then ζ,η∈Ck.For simplicity, we writeDH0∶=DH(0,u0), fu0∶=fu(0,0), Fu0∶=Fu(0,0), FΛu0∶=FΛu(0,0), and so on. We have
(13)DH0(Λu)=(luΛ·fΛu0u0+fu0u)=(pluΛ·FΛu0u0+Fu0u).Proof of Theorem3.
The proof is similar to that of  [1, Theorem 2.1] and [5, Theorem 1.7]. We obtain from (H) that plu0=1 and Fu0u0=0. Let u^ be any element of 𝒩(Fu0). We set c^∶=plu^. Then, we have DH0(0,u^-c^u0)=0. Since DH0 is one to one, u^=c^u0. So we obtain 𝒩(Fu0)=span{u0}. We choose ε∈ℝ+ and an open neighborhood K of 0 in X such that α(u0+u)∈U for (α,u)∈(-ε,ε)×K. We define a map h:(-ε,ε)×V×K→ℝm×Y by
(14)h(α,Λ,u)∶=(luh0(α,Λ,u))withh0(α,Λ,u)∶={1αf(Λ,αu0+αu)(α≠0),fu(Λ,0)(u0+u)(α=0).
By H(0,u0)=0, we have h(0,0,0)=(0,fu0u0)=0. We verify that the partial derivative of h(α,Λ,u) with respect to (Λ,u) at (α,Λ,u)=(0,0,0)  coincides with DH0 and is bijective.  So, by applying the implicit function theorem  to h(α,Λ,u) at (α,Λ,u)=(0,0,0), there exist an open neighborhood W⊂V×K of (0,0), a∈(0,ε) and continuous functions ζ:(-a,a)→ℝm, η:(-a,a)→Z such that ζ(0)=0, η(0)=0, and h(α,ζ(α),η(α))=0 for any α∈(-a,a) and that h(α,Λ,u)=0 and (α,Λ,u)∈(-a,a)×W imply Λ=ζ(α) and u=η(α).   Thus, F-1(0)∩W includes the right-hand side of (12). From the same argument as in the proof of [5, Theorem 1.7], we can verify that (12) actually holds.In view of the next result, we may consider  Theorem3  as a generalized version of [5, Theorem 1.7].Proposition 4.
The condition (H) in  Theorem3  is equivalent to the following (i) and (ii):(i)
dim𝒩(Fu0)=1  andcodimℛ(Fu0)=m,(ii)
∃u0∈U  such that(15)plu0=1,𝒩(Fu0)=span{u0},(16)span(FΛu0u0)⊕ℛ(Fu0)=ℝm-1×Y.Here, we denote span(FΛu0u0):={Λ·FΛu0u0;Λ∈ℝm}.Proof.
First, we assume (H). By the proof of  Theorem3, we have dim𝒩(Fu0)=1 and (15). Since DH0 is onto, we have span(FΛu0u0)+ℛ(Fu0)=ℝm-1×Y.  Let p∈span(FΛu0u0) and q∈R(Fu0) satisfy p+q=0. In order to prove (16), it suffices to show p=q=0. We choose Λ~∈ℝm and u~∈X such that p=Λ~·FΛu0u0 and q=Fu0u~.  We set c~∶=plu~.  Then, DH0(Λ~,u~-c~u0)=0. We have Λ~=0 and u~-c~u0=0 since DH0 is an injective linear map. So, we obtain p=0, which implies q=0. Thus, (16) holds. Next, we show by the same discussion as above that {FΛku0u0}k=1m is linearly independent. Let Λ∈ℝm satisfy Λ·FΛu0u0=0. Then, we have DH0(Λ,0)=0, which implies Λ=0. Thus, {FΛku0u0}k=1m is linearly independent. So, dimspan(FΛu0u0)=m. By this and (16), we have codimℛ(Fu0)=m.  Thus, (i) and (ii) hold.
We show the inverse. We assume (i) and (ii). Clearly, (15) implies lu0=e1 and H(0,u0)=0. So, it suffices to show that DH0 is bijective. Let (Λ^,u^)∈ℝm×X satisfy DH0(Λ^,u^)=0. Then, by (13) we have plu^=0 and Λ^·FΛu0u0+Fu0u^=0. We obtain from (16) that Λ^·FΛu0u0=0 and Fu0u^=0. We have Λ^=0 since (i) and (16) imply span(FΛu0u0)=m. By plu^=0 and (15), we have u^=0. Hence, DH0 is one to one. Finally, let (λ,y) be an element of ℝm×Y. By (16), there exist Λ~∈ℝm and u~∈X such that Λ~·FΛu0u0+Fu0u~=(Pλ,y). We set c~∶=p(λ-lu~). Then, DH0(Λ~,u~+c~u0)=(λ,y). Thus, DH0 is onto. The proof is complete.


## 3. Proof of  Theorem1

We setDJ0:=DJ(Λ0,v0,w0).  We use the same notations as used in Section 2. We set gu0:=gu(Λ0,v0), gΛ0:=gΛ(Λ0,v0), gΛu0:=gΛu(Λ0,v0), and guu0:=guu(Λ0,v0).  Then, we have
(17)DJ0(Λvw)=(lwΛ·gΛ0+gu0vΛ·gΛu0w0+guu0vw0+gu0w).Proof of Theorem1.
By (H2) and the implicit function theorem, there exist an open  neighborhoodV of 0 in ℝm and a map q=q(Λ)∈C2(Λ0+V,X1) such that q(Λ0)=v0 and g(Λ,q(Λ))=0 for any Λ∈Λ0+V. We define f:V×X→Y by
(18)f(Λ,u)∶=g(Λ0+Λ,u+q(Λ0+Λ)).
Then, we have (9), fu0=gu0, and
(19)fΛu0w0=gΛu0w0+guu0x0w0withx0∶=q′(Λ0)∈X1m.
It follows from (9) and (H2) that x0∈X1m is uniquely determined by the equation:
(20)(fΛ(0,0))=gΛ0+gu0x0=0.
We define H:V×X→ℝm×Y by (10), and set u0∶=w0. Then, we have (13). In view of  Theorem 3, the proof is complete if we show that the map DH0:ℝm×X→ℝm×Y is bijective. Let (Λ,u)∈ℝm×X satisfy DH0(Λ,u)=0. It follows from (2), (3), (13), and (19) that
(21)lu=0,gu0u=fu0u=-Λ·fΛu0w0=-Λ·(gΛu0w0+guu0x0w0)∈Y2.
So, by (2), (H2), (17), (20), and (21), we have u∈X2 and DJ0(Λ,Λ·x0,u)=0. We obtain (Λ,u)=0 since DJ0 is one to one. Hence, DH0 is one to one. Next, let (λ,y) be an element of ℝm×Y. Let y=y1+y2 with y1∈Y1 and y2∈Y2. Since DJ0 is onto, there exists (Λ~,v~,w~)∈ℝm×X1×X2 such that DJ0(Λ~,v~,w~)=(λ,0,y2). By (H2), (17), (19), and (20), we have λ=lw~, v~=Λ~·x0, and y2=Λ~·(gΛu0w0+guu0x0w0)+gu0w~=Λ~·fΛu0w0+fu0w~. We set u~:=(gu0|X1)-1y1∈X1. Then, by (19), we have DH0(Λ~,u~+w~)=(λ,y). Therefore, DH0 is onto. The proof is complete.


## 4. A Numerical Example

In this section, we present a numerical example. We treat Hopf bifurcation as an application of  Theorem1  with m=2. This section is organized as follows. In Section 4.1, we present a partial differential system we consider. In Section 4.2, we  rewrite our problem under appropriate setting of functional spaces. In Section 4.3, we describe our numerical verification result  on the existence of a Hopf bifurcation point. In Section 4.4, we present the principal abstract results in our numerical verification methods. In Section 4.5, we describe the outline of derivation of our verification result.


### 4.1. Brusselator Model

We consider the following Brusselator model:(B)∂u1∂t-∂2u1∂x2=(A+ηcosx)-(B+1)u1+u12u2+λ(u3-u1)for(x,t)∈(0,π)×[0,∞),∂u2∂t-∂2u2∂x2=Bu1-u12u2+λρ(u4-u2)for(x,t)∈(0,π)×[0,∞),∂u3∂t-∂2u3∂x2=(A+ηcosx)-(B+1)u3+u32u4+λ(u1-u3)for(x,t)∈(0,π)×[0,∞),∂u4∂t-∂2u4∂x2=Bu3-u32u4+λρ(u2-u4)for(x,t)∈(0,π)×[0,∞),∂uj∂x(0,t)=∂uj∂x(π,t)=0fort∈[0,∞),j=1,2,3,4.
Here, A, B, ρ, λ, and η are parameters of the problem. In what follows, we set A∶=2, B∶=5.9, ρ∶=0.1, and η∶=0.001 and consider λ as the bifurcation parameter.  By some numerical methods, Kubíček and Holodniok [7] found some period-doubling bifurcation points for an ordinary differential system related  closely to (B). On the other hand, we verified the existence of a Hopf bifurcation point for (B).We setu=(u1,u2,u3,u4). We denote the equations of  (B) by ut-uxx=h(λ,u). We also write ut=φ(λ,u) with φ(λ,u)∶=uxx+h(λ,u). Clearly, u is a 2πσ-periodic solution of (B) if and only if u is a 2π-periodic solution of the next equation:
(22)ut=σφ(λ,u).


### 4.2. The Setting of Functional Spaces

LetD∶=(0,π)×(0,2π), 𝒴∶=L2(D), and Y∶=𝒴4. We define a real Hilbert space 𝒳 by
(23)𝒳∶={u=∑k=-∞∞‍∑n=0∞‍ckneiktcosnx∈𝒴;ckn∈ℂ,c-k,n=ckn¯foranyk∈ℤ,n∈ℕ∪{0}andut-uxx∈𝒴∑n=0∞‍}={∑k=-∞∞‍∑n=0∞‍ckneiktcosnx∈𝒴;ckn∈ℂ,c-k,n=ckn¯foranyk∈ℤ,n∈ℕ∪{0}and∑k=0∞‍∑n=0∞‍(k2+n4)|ckn|2<∞},
with the inner product
(24)(u,v)𝒳∶=(ut-uxx+12π2∫02π‍dt∫0π‍udx,vt-vxx+12π2∫02π‍dt∫0π‍vdx)𝒴
for u,v∈𝒳. Then, we easily verify that 𝒳 is continuously embedded in H1(D) and that
(25)∥u∥𝒳=(u,u)𝒳=π{2|c00|2+2∑k=-∞∞‍k2|ck0|2+∑k=-∞∞‍∑n=1∞‍(k2+n4)|ckn|2}1/2
for u=∑k=-∞∞∑n=0∞ckneiktcosnx∈𝒳. We set X∶=𝒳4. We denote by 𝒱 (resp., 𝒲) the subspace of 𝒳 (resp., 𝒴) consisting of steady functions, that is,
(26)𝒱∶={∑n=0∞‍cncosnx;cn∈ℝand∑n=0∞‍n4cn2<∞},𝒲∶=L2(0,π).
We set X1:=𝒱4, Y1:=𝒲4, X2:=X1⊥, and Y2:=Y1⊥.We setΛ∶=(λ,σ). We define g(Λ,u)∶=ut-σφ(λ,u) and l∶=(l1,l2):X→ℝ2 by
(27)l1u∶=1π2(u1,sint)𝒴,l2u∶=1π2(u1,cost)𝒴foru=(u1,u2,u3,u4)∈X.
Namely, l1u (resp., l2u) equals to the Fourier coefficient  of sint (resp., cost)  for u1. Let J:ℝ2×X→ℝ2×Y be defined by (4) with m=2. We define J^:ℝ2×X→ℝ2×Y by
(28)J^(Λvw)∶=(lw-e1φ(λ,v)gu(Λ,v)w)∈ℝ2×Y1×Y2for(Λ,v,w)∈ℝ2×X1×X2.
Then, (H1) is equivalent to the following (K1) since σ0≠0 and φ(λ0,v0)=0:(K1)
A point(Λ,v,w)=(Λ0,v0,w0) is an isolated solution of the extended system J^(Λ,v,w)=0.Similarly, (H2) is equivalent to the following (K2):(K2)
The linear operatorφu(λ0,v0)∣X1:X1→Y1 is bijective.


### 4.3. A Verification Result

By our numerical verification method described in Section4.5, we  verified the existence of a Hopf bifurcation point (λ0,v0)  of (B) satisfying
(29)|λ0-λ~0|2+|σ0-σ~0|2+∥v0-v~0∥X2+∥w0-w~0∥X2≤(6.46×10-8)2.
Here, ((λ0,σ0),v0,w0) is an isolated solution of the extended system J^(Λ,v,w)=0, and ((λ~0,σ~0),v~0,w~0) is its numerical approximation. The initial period of bifurcating periodic solutions is 2πσ0. We have w0∈W1, which is defined by (42) below. We set λ~0∶=0.9541600594715 and σ~0∶=0.4534602327315. The functions v~0 and w~0 have the forms
(30)v~0∶=a+bcosx+ccos2xwitha,b,c∈ℚ4,w~0∶=(p1sint+p2cost)+(q1sint+q2cost)cosx+(r1sint+r2cost)cos2x
with pk,qk,rk∈ℚ4(k=1,2). We have
(31)∥v~0∥X∈(20.8,20.9),∥w~0∥X∈(4.25,4.26),∥J^(Λ~0,v~0,w~0)∥ℝ2×Y∈(4.57×10-9,4.58×10-9).
Here, Λ~0∶=(λ~0,σ~0).Remark 5.
In the process of deriving the above result, we often used computer arithmetic with double precision, without taking into account the effects of the round-off errors, though (31) holds rigorously. It is sufficient, however, for our purpose, which is to  check that Theorem 1  is applicable to the numerical verification method. We can make the above result completely rigorous if we always use the interval arithmetic or the computational method described in [2, Section 2.1]  in the process of deriving the conclusion.


### 4.4. Principle of Our Numerical Verification Method

In this  subsection, we  present two important abstract results in our numerical verification method.Letℋ and 𝒦 be Banach spaces in what follows in this subsection. We assume F∈C1(ℋ,𝒦). We can immediately obtain the following result from [8, Theorem 3.2].Theorem 6 (the convergence theorem of simplified Newton method).
Letu~0∈ℋ and A:=F′(u~0) be bijective. We define a map G:ℋ→ℋ by
(32)G(u):=u-A-1F(u).
Let R>0 be a constant and b:[0,R]→[0,∞) be a non-decreasing function such that
(33)sup{∥G′(u)∥;u∈B(u~0;r)¯}≤b(r)foranyr∈[0,R].
Let ε0≥0 be a constant such that
(34)∥A-1F(u~0)∥≤ε0.
We assume that there exist constants r0 and r1 such that 0<r0≤r1≤R,
(35)ε0+∫0r0‍b(r)dr≤r0andb(r1)<1.
Then, the equation F(u)=0 has an isolated solution u0∈B(u~0;r0)¯. Moreover, the solution of F(u)=0 is unique in B(u~0;r1)¯.Theorem6  plays a central role in our numerical verification method.Remark 7.
Theorem6  is a corollary of [9, Theorem 1.1]. Though the latter is in general stronger than the former, the former is simpler. The former works well for our present problem since the  residue of the approximate solution is very small (see (31)).In applying  Theorem6, it is very important to find an upper bound of the norm of A-1.  Proposition 8  below is useful with respect to this point. We also need it in order to check the condition (H2).Letℋ0 and ℋc be closed subspaces of ℋ such that ℋ=ℋ0⊕ℋc. Let 𝒦0 and 𝒦c be closed subspaces of 𝒦 such that 𝒦=𝒦0⊕𝒦c. We denote by Pj∈ℒ(ℋ) the projection defined by Pju∶=uj for u=u0+uc∈ℋ with uj∈ℋj(j=0,c). We denote by Qj∈ℒ(K) the projection defined by Qjv∶=vj for v=v0+vc∈𝒦 with vj∈𝒦j(j=0,c). Let T∈ℒ(ℋ,𝒦) and ℒ:ℋ→𝒦 be a closed linear operator. We set A:=L+T. We assume
(36)ℛ(P0)⊂𝒟(L),LP0=Q0Lon𝒟(L).
We define A0:=L+Q0TP0:ℋ→𝒦. We denote by Iℋ (resp., I𝒦) the identity operator on ℋ (resp., 𝒦) and simply by I the operators Iℋ and I𝒦 when there is no ambiguity.Proposition 8.
In addition to the above assumptions, we assume thatA0 is bijective and that S∶=A0-1QcT(I-A0-1P0T)Pc∈ℒ(H) satisfies ∥S∥<1. Then, A is bijective. Moreover, we have
(37)P0A-1Q0=A0-1{Q0+Q0TPc(I+S)-1A0-1QcTA0-1Q0}(∶=A11),P0A-1Qc=A0-1Q0TPc(I+S)-1A0-1Qc(∶=A12),PcA-1Q0=-(I+S)-1A0-1QcTQ0A0-1Q0(∶=A21),PcA-1Qc=(I+S)-1A0-1Qc(∶=A22).Proposition8  is a generalization of [9, Proposition 2.1] and can be proved in the same way. We note that we can apply  Proposition 8  to  our problem  but cannot directly apply [9, Proposition 2.1].Finally, we describe how  Proposition8  works well for our problem.  In our present context, we can assume in addition the following (A1)–(A3):(A1)
ℋ and 𝒦 are Hilbert spaces, and ℋ is embedded continuously and densely in 𝒦,(A2)
there is a CONS{ϕn}n=1∞ of ℋ such that ϕn is an eigenfunction of L for any n∈ℕ.Let ℋn and 𝒦n be subspaces of ℋ and 𝒦, respectively, such that both ℋn and 𝒦n are  equal to span{ϕj}j=1n as sets. Let Pn (resp., Qn) be the orthogonal projection on ℋ (resp., 𝒦) onto ℋn (resp., 𝒦n),   (A3)
∥T(I-Pn)∥ℋ→𝒦→0 and ∥(I-Qn)T∥ℋ→𝒦→0asn→∞.We show that  Proposition8  works well if we take ℋ0∶=ℋn and 𝒦0∶=𝒦n for sufficiently large n. Let An∶=L+QnTPn:ℋ→𝒦.Proposition 9.
Under the assumptions above we have the following.(i)
If  A  is bijective, then  An  is bijective for sufficiently large  n∈ℕ, and we have(38)An-1⟶A-1inℒ(𝒦,ℋ)asn⟶∞.(ii)
IfAn  is bijective and(39)∥An-1∥∥T-QnTPn∥<1,thenA  is bijective with the estimate∥A-1∥≤∥An-1∥/(1-∥An-1∥∥T-QnTPn∥).(iii)
We have(40)∥An-1∥⟶∥A-1∥asn⟶∞.Here, we formally define ∥An-1∥∶=∞ (resp., ∥A-1∥∶=∞) when An (resp., A) is not bijective.Proof.
We setBn:=T-QnTPn∈ℒ(ℋ,𝒦). Then, A-An=Bn on 𝒟(L). We obtain from (A3) that ∥Bn∥→0 as n→∞.
(i) LetA be bijective. Then, ∥A-1∥∥Bn∥<1 for  sufficiently  large  n. Therefore, by [2, Corollary 2.4.1], An is bijective for sufficiently large n, and we obtain (38).
(ii) We immediately obtain the desired conclusion from [2, Corollary 2.4.1].
(iii) IfA is bijective, (i) implies (40). So, we consider the case  where A is not bijective. We proceed by contradiction. We assume that limn→∞∥An-1∥≠∞. Then, liminfn→∞∥An-1∥<∞. So, there exists a large number N∈ℕ such that ∥AN-1∥∥BN∥<1. So, (ii) implies that A is bijective. This contradicts to our assumption. Therefore, (40) holds.We consider the case where the condition of  Proposition8  is satisfied. Let u∈𝒦. The estimate (41) below is useful when we apply  Theorem 6  to concrete problems. We denote by Anij the operator Aij with A0, P0, Pc, Q0, and Qc replaced by An, Pn, I-Pn, Qn, and I-Qn, respectively (i,j = 1, 2). Then, we easily verify that
(41)∥A-1u∥≤{(∥An11u∥+∥An12u∥)2+(∥An21u∥+∥An22u∥)2}1/2
and that the right-hand side of (41) converges to ∥A-1u∥ as n→∞ in view of (A3). So, the  right-hand side of (41) is a sharp upper bounds of ∥A-1u∥ when n is sufficiently large.Remark 10.
We denote bySn the operator S defined in  Proposition 8  with A0, P0, Pc, and Qc replaced by An, Pn, I-Pn and I-Qn, respectively. Though (39) is a sufficient condition for the existence of A-1, another sufficient condition ∥Sn∥<1 is in general more efficient. The reason is that ∥T(I-Pn)∥ converges to 0 (n→∞) more  rapidly in general than ∥(I-Qn)T∥.


### 4.5. Derivation of Our Verification Result

Here, we describe the outline of derivation of our verification result in Section4.3. We define closed subspaces of X2:
(42)W1∶={∑n=0∞‍(ancost+bnsint)cosnx∈X;an,bn∈ℝ4},W2∶={∑|k|≥2‍∑n=0∞‍ckneiktcosnx∈X;ckn∈ℂ4}.
We similarly define closed subspaces of Y2:
(43)Z1∶={∑n=0∞‍(ancost+bnsint)cosnx∈Y;an,bn∈ℝ4},Z2∶={∑|k|≥2‍∑n=0∞‍ckneiktcosnx∈Y;ckn∈ℂ4}.
Clearly, X2=W1⊕W2, and Y2=Z1⊕Z2. We set J*:=J^∣ℝ2⊕X1⊕W1:ℝ2⊕X1⊕W1→ℝ2⊕Y1⊕Z1 and K:=J^∣W2=gu0∣W2:W2→Z2.  Here, we replace the symbol of direct product by the direct sum (see the notation  (8) just after Section 1), which helps us to describe the decomposition of J^.   Then, we can represent J^ by the direct sum of maps: J^=J*⊕K. Since w0∈W1, the condition (K1) is equivalent to the following (L1) and (L2):(L1)
A point(Λ,v,w)=(Λ0,v0,w0) is an isolated solution of the extended system J*(Λ,v,w)=0.(L2)
The linear operatorK:W2→Z2 is bijective.By the above discussion and  Theorem1,  it suffices to verify (L1), (L2), and (H2) in order to show that (λ,v)=(λ0,v0) is a Hopf bifurcation point of (B). We verified (L1) and (29) by applying  Theorem 6  with F:=J* and u~0∶=(Λ~0,v~0,w~0), and (L2), (H2) by  Proposition 8. We need the following embedding inequality to check the conditions in  Theorem 6  and  Proposition 9. For simplicity, we denote ∥·∥p:=∥·∥Lp(D).Proposition 11.
We have(44)∥u∥6≤C6∥u∥𝓍foranyu∈𝒳,
where C6:={(1/2)(1/π+2)(1/π+3)}1/3∈(1.566,1.567).Proof.
The proof is similar to that of [2, Proposition 2.3.1].
First, let-∞<a<b<∞. In a similar way to the proof of [2, Lemma 2.3.1(ii)], we easily verify that
(45)maxx∈[a,b]|f(x)|≤∫ab‍|f′(x)|dx+1b-a∫ab‍|f(x)|dxforanyf∈H1(a,b).
We easily verify that
(46)max(∥u∥2,∥ux∥2,∥ut∥2)≤∥u∥𝓍foranyu∈𝒳.
We set 𝒳0:={∑k=-MM∑n=0Nckneiktcosnx∈𝒳;ckn∈ℂ and M,N∈ℕ∪{0}}. It suffices to prove (44) for any u∈𝒳0 since 𝒳0 is dense in 𝒳. Let u∈𝒳0. It follows from (45) and [2, Lemma 2.3.1(ii)] that
(47)∫02π‍dt∫0π‍u2dx≤∫02π‍maxy∈[0,π]|u(t,y)|dt∫0π‍maxτ∈[0,2π]|u(τ,x)|dx≤(1π∥u∥1+∥ux∥1)(12π∥u∥1+12∥ut∥1).
Let v∈𝒳0. We substitute u=vj(j=2,3) to (47). Then, we obtain from (46) and Schwarz inequality that
(48)∥v∥44≤(1π∥v∥22+2∥vvx∥1)(12π∥v∥22+∥vvt∥1)≤(1π∥v∥22+2∥v∥2∥vx∥2)(12π∥v∥22+∥v∥2∥vt∥2)≤12(1π+2)2∥v∥𝒳4,(49)∥v∥66≤(1π∥v∥33+3∥v2vx∥1)(12π∥v∥33+32∥v2vt∥1)≤∥v∥44(1π∥v∥2+3∥vx∥2)(12π∥v∥2+32∥vt∥2)≤∥v∥44·12(1π+3)2∥v∥𝒳2.
By (48) and (49), we have (44) with u replaced by v. The proof is complete.Remark 12.
Though our numerical verification method is not difficult, the verification process consists of a lot of steps and is very complicated. So, we omit its details here. They are much similar to the  discussions in [2, 9]. In particular, the verification process is described fully in detail in [2].


## 4.1. Brusselator Model

We consider the following Brusselator model:(B)∂u1∂t-∂2u1∂x2=(A+ηcosx)-(B+1)u1+u12u2+λ(u3-u1)for(x,t)∈(0,π)×[0,∞),∂u2∂t-∂2u2∂x2=Bu1-u12u2+λρ(u4-u2)for(x,t)∈(0,π)×[0,∞),∂u3∂t-∂2u3∂x2=(A+ηcosx)-(B+1)u3+u32u4+λ(u1-u3)for(x,t)∈(0,π)×[0,∞),∂u4∂t-∂2u4∂x2=Bu3-u32u4+λρ(u2-u4)for(x,t)∈(0,π)×[0,∞),∂uj∂x(0,t)=∂uj∂x(π,t)=0fort∈[0,∞),j=1,2,3,4.
Here, A, B, ρ, λ, and η are parameters of the problem. In what follows, we set A∶=2, B∶=5.9, ρ∶=0.1, and η∶=0.001 and consider λ as the bifurcation parameter.  By some numerical methods, Kubíček and Holodniok [7] found some period-doubling bifurcation points for an ordinary differential system related  closely to (B). On the other hand, we verified the existence of a Hopf bifurcation point for (B).We setu=(u1,u2,u3,u4). We denote the equations of  (B) by ut-uxx=h(λ,u). We also write ut=φ(λ,u) with φ(λ,u)∶=uxx+h(λ,u). Clearly, u is a 2πσ-periodic solution of (B) if and only if u is a 2π-periodic solution of the next equation:
(22)ut=σφ(λ,u).


## 4.2. The Setting of Functional Spaces

LetD∶=(0,π)×(0,2π), 𝒴∶=L2(D), and Y∶=𝒴4. We define a real Hilbert space 𝒳 by
(23)𝒳∶={u=∑k=-∞∞‍∑n=0∞‍ckneiktcosnx∈𝒴;ckn∈ℂ,c-k,n=ckn¯foranyk∈ℤ,n∈ℕ∪{0}andut-uxx∈𝒴∑n=0∞‍}={∑k=-∞∞‍∑n=0∞‍ckneiktcosnx∈𝒴;ckn∈ℂ,c-k,n=ckn¯foranyk∈ℤ,n∈ℕ∪{0}and∑k=0∞‍∑n=0∞‍(k2+n4)|ckn|2<∞},
with the inner product
(24)(u,v)𝒳∶=(ut-uxx+12π2∫02π‍dt∫0π‍udx,vt-vxx+12π2∫02π‍dt∫0π‍vdx)𝒴
for u,v∈𝒳. Then, we easily verify that 𝒳 is continuously embedded in H1(D) and that
(25)∥u∥𝒳=(u,u)𝒳=π{2|c00|2+2∑k=-∞∞‍k2|ck0|2+∑k=-∞∞‍∑n=1∞‍(k2+n4)|ckn|2}1/2
for u=∑k=-∞∞∑n=0∞ckneiktcosnx∈𝒳. We set X∶=𝒳4. We denote by 𝒱 (resp., 𝒲) the subspace of 𝒳 (resp., 𝒴) consisting of steady functions, that is,
(26)𝒱∶={∑n=0∞‍cncosnx;cn∈ℝand∑n=0∞‍n4cn2<∞},𝒲∶=L2(0,π).
We set X1:=𝒱4, Y1:=𝒲4, X2:=X1⊥, and Y2:=Y1⊥.We setΛ∶=(λ,σ). We define g(Λ,u)∶=ut-σφ(λ,u) and l∶=(l1,l2):X→ℝ2 by
(27)l1u∶=1π2(u1,sint)𝒴,l2u∶=1π2(u1,cost)𝒴foru=(u1,u2,u3,u4)∈X.
Namely, l1u (resp., l2u) equals to the Fourier coefficient  of sint (resp., cost)  for u1. Let J:ℝ2×X→ℝ2×Y be defined by (4) with m=2. We define J^:ℝ2×X→ℝ2×Y by
(28)J^(Λvw)∶=(lw-e1φ(λ,v)gu(Λ,v)w)∈ℝ2×Y1×Y2for(Λ,v,w)∈ℝ2×X1×X2.
Then, (H1) is equivalent to the following (K1) since σ0≠0 and φ(λ0,v0)=0:(K1)
A point(Λ,v,w)=(Λ0,v0,w0) is an isolated solution of the extended system J^(Λ,v,w)=0.Similarly, (H2) is equivalent to the following (K2):(K2)
The linear operatorφu(λ0,v0)∣X1:X1→Y1 is bijective.


## 4.3. A Verification Result

By our numerical verification method described in Section4.5, we  verified the existence of a Hopf bifurcation point (λ0,v0)  of (B) satisfying
(29)|λ0-λ~0|2+|σ0-σ~0|2+∥v0-v~0∥X2+∥w0-w~0∥X2≤(6.46×10-8)2.
Here, ((λ0,σ0),v0,w0) is an isolated solution of the extended system J^(Λ,v,w)=0, and ((λ~0,σ~0),v~0,w~0) is its numerical approximation. The initial period of bifurcating periodic solutions is 2πσ0. We have w0∈W1, which is defined by (42) below. We set λ~0∶=0.9541600594715 and σ~0∶=0.4534602327315. The functions v~0 and w~0 have the forms
(30)v~0∶=a+bcosx+ccos2xwitha,b,c∈ℚ4,w~0∶=(p1sint+p2cost)+(q1sint+q2cost)cosx+(r1sint+r2cost)cos2x
with pk,qk,rk∈ℚ4(k=1,2). We have
(31)∥v~0∥X∈(20.8,20.9),∥w~0∥X∈(4.25,4.26),∥J^(Λ~0,v~0,w~0)∥ℝ2×Y∈(4.57×10-9,4.58×10-9).
Here, Λ~0∶=(λ~0,σ~0).Remark 5.
In the process of deriving the above result, we often used computer arithmetic with double precision, without taking into account the effects of the round-off errors, though (31) holds rigorously. It is sufficient, however, for our purpose, which is to  check that Theorem 1  is applicable to the numerical verification method. We can make the above result completely rigorous if we always use the interval arithmetic or the computational method described in [2, Section 2.1]  in the process of deriving the conclusion.


## 4.4. Principle of Our Numerical Verification Method

In this  subsection, we  present two important abstract results in our numerical verification method.Letℋ and 𝒦 be Banach spaces in what follows in this subsection. We assume F∈C1(ℋ,𝒦). We can immediately obtain the following result from [8, Theorem 3.2].Theorem 6 (the convergence theorem of simplified Newton method).
Letu~0∈ℋ and A:=F′(u~0) be bijective. We define a map G:ℋ→ℋ by
(32)G(u):=u-A-1F(u).
Let R>0 be a constant and b:[0,R]→[0,∞) be a non-decreasing function such that
(33)sup{∥G′(u)∥;u∈B(u~0;r)¯}≤b(r)foranyr∈[0,R].
Let ε0≥0 be a constant such that
(34)∥A-1F(u~0)∥≤ε0.
We assume that there exist constants r0 and r1 such that 0<r0≤r1≤R,
(35)ε0+∫0r0‍b(r)dr≤r0andb(r1)<1.
Then, the equation F(u)=0 has an isolated solution u0∈B(u~0;r0)¯. Moreover, the solution of F(u)=0 is unique in B(u~0;r1)¯.Theorem6  plays a central role in our numerical verification method.Remark 7.
Theorem6  is a corollary of [9, Theorem 1.1]. Though the latter is in general stronger than the former, the former is simpler. The former works well for our present problem since the  residue of the approximate solution is very small (see (31)).In applying  Theorem6, it is very important to find an upper bound of the norm of A-1.  Proposition 8  below is useful with respect to this point. We also need it in order to check the condition (H2).Letℋ0 and ℋc be closed subspaces of ℋ such that ℋ=ℋ0⊕ℋc. Let 𝒦0 and 𝒦c be closed subspaces of 𝒦 such that 𝒦=𝒦0⊕𝒦c. We denote by Pj∈ℒ(ℋ) the projection defined by Pju∶=uj for u=u0+uc∈ℋ with uj∈ℋj(j=0,c). We denote by Qj∈ℒ(K) the projection defined by Qjv∶=vj for v=v0+vc∈𝒦 with vj∈𝒦j(j=0,c). Let T∈ℒ(ℋ,𝒦) and ℒ:ℋ→𝒦 be a closed linear operator. We set A:=L+T. We assume
(36)ℛ(P0)⊂𝒟(L),LP0=Q0Lon𝒟(L).
We define A0:=L+Q0TP0:ℋ→𝒦. We denote by Iℋ (resp., I𝒦) the identity operator on ℋ (resp., 𝒦) and simply by I the operators Iℋ and I𝒦 when there is no ambiguity.Proposition 8.
In addition to the above assumptions, we assume thatA0 is bijective and that S∶=A0-1QcT(I-A0-1P0T)Pc∈ℒ(H) satisfies ∥S∥<1. Then, A is bijective. Moreover, we have
(37)P0A-1Q0=A0-1{Q0+Q0TPc(I+S)-1A0-1QcTA0-1Q0}(∶=A11),P0A-1Qc=A0-1Q0TPc(I+S)-1A0-1Qc(∶=A12),PcA-1Q0=-(I+S)-1A0-1QcTQ0A0-1Q0(∶=A21),PcA-1Qc=(I+S)-1A0-1Qc(∶=A22).Proposition8  is a generalization of [9, Proposition 2.1] and can be proved in the same way. We note that we can apply  Proposition 8  to  our problem  but cannot directly apply [9, Proposition 2.1].Finally, we describe how  Proposition8  works well for our problem.  In our present context, we can assume in addition the following (A1)–(A3):(A1)
ℋ and 𝒦 are Hilbert spaces, and ℋ is embedded continuously and densely in 𝒦,(A2)
there is a CONS{ϕn}n=1∞ of ℋ such that ϕn is an eigenfunction of L for any n∈ℕ.Let ℋn and 𝒦n be subspaces of ℋ and 𝒦, respectively, such that both ℋn and 𝒦n are  equal to span{ϕj}j=1n as sets. Let Pn (resp., Qn) be the orthogonal projection on ℋ (resp., 𝒦) onto ℋn (resp., 𝒦n),   (A3)
∥T(I-Pn)∥ℋ→𝒦→0 and ∥(I-Qn)T∥ℋ→𝒦→0asn→∞.We show that  Proposition8  works well if we take ℋ0∶=ℋn and 𝒦0∶=𝒦n for sufficiently large n. Let An∶=L+QnTPn:ℋ→𝒦.Proposition 9.
Under the assumptions above we have the following.(i)
If  A  is bijective, then  An  is bijective for sufficiently large  n∈ℕ, and we have(38)An-1⟶A-1inℒ(𝒦,ℋ)asn⟶∞.(ii)
IfAn  is bijective and(39)∥An-1∥∥T-QnTPn∥<1,thenA  is bijective with the estimate∥A-1∥≤∥An-1∥/(1-∥An-1∥∥T-QnTPn∥).(iii)
We have(40)∥An-1∥⟶∥A-1∥asn⟶∞.Here, we formally define ∥An-1∥∶=∞ (resp., ∥A-1∥∶=∞) when An (resp., A) is not bijective.Proof.
We setBn:=T-QnTPn∈ℒ(ℋ,𝒦). Then, A-An=Bn on 𝒟(L). We obtain from (A3) that ∥Bn∥→0 as n→∞.
(i) LetA be bijective. Then, ∥A-1∥∥Bn∥<1 for  sufficiently  large  n. Therefore, by [2, Corollary 2.4.1], An is bijective for sufficiently large n, and we obtain (38).
(ii) We immediately obtain the desired conclusion from [2, Corollary 2.4.1].
(iii) IfA is bijective, (i) implies (40). So, we consider the case  where A is not bijective. We proceed by contradiction. We assume that limn→∞∥An-1∥≠∞. Then, liminfn→∞∥An-1∥<∞. So, there exists a large number N∈ℕ such that ∥AN-1∥∥BN∥<1. So, (ii) implies that A is bijective. This contradicts to our assumption. Therefore, (40) holds.We consider the case where the condition of  Proposition8  is satisfied. Let u∈𝒦. The estimate (41) below is useful when we apply  Theorem 6  to concrete problems. We denote by Anij the operator Aij with A0, P0, Pc, Q0, and Qc replaced by An, Pn, I-Pn, Qn, and I-Qn, respectively (i,j = 1, 2). Then, we easily verify that
(41)∥A-1u∥≤{(∥An11u∥+∥An12u∥)2+(∥An21u∥+∥An22u∥)2}1/2
and that the right-hand side of (41) converges to ∥A-1u∥ as n→∞ in view of (A3). So, the  right-hand side of (41) is a sharp upper bounds of ∥A-1u∥ when n is sufficiently large.Remark 10.
We denote bySn the operator S defined in  Proposition 8  with A0, P0, Pc, and Qc replaced by An, Pn, I-Pn and I-Qn, respectively. Though (39) is a sufficient condition for the existence of A-1, another sufficient condition ∥Sn∥<1 is in general more efficient. The reason is that ∥T(I-Pn)∥ converges to 0 (n→∞) more  rapidly in general than ∥(I-Qn)T∥.


## 4.5. Derivation of Our Verification Result

Here, we describe the outline of derivation of our verification result in Section4.3. We define closed subspaces of X2:
(42)W1∶={∑n=0∞‍(ancost+bnsint)cosnx∈X;an,bn∈ℝ4},W2∶={∑|k|≥2‍∑n=0∞‍ckneiktcosnx∈X;ckn∈ℂ4}.
We similarly define closed subspaces of Y2:
(43)Z1∶={∑n=0∞‍(ancost+bnsint)cosnx∈Y;an,bn∈ℝ4},Z2∶={∑|k|≥2‍∑n=0∞‍ckneiktcosnx∈Y;ckn∈ℂ4}.
Clearly, X2=W1⊕W2, and Y2=Z1⊕Z2. We set J*:=J^∣ℝ2⊕X1⊕W1:ℝ2⊕X1⊕W1→ℝ2⊕Y1⊕Z1 and K:=J^∣W2=gu0∣W2:W2→Z2.  Here, we replace the symbol of direct product by the direct sum (see the notation  (8) just after Section 1), which helps us to describe the decomposition of J^.   Then, we can represent J^ by the direct sum of maps: J^=J*⊕K. Since w0∈W1, the condition (K1) is equivalent to the following (L1) and (L2):(L1)
A point(Λ,v,w)=(Λ0,v0,w0) is an isolated solution of the extended system J*(Λ,v,w)=0.(L2)
The linear operatorK:W2→Z2 is bijective.By the above discussion and  Theorem1,  it suffices to verify (L1), (L2), and (H2) in order to show that (λ,v)=(λ0,v0) is a Hopf bifurcation point of (B). We verified (L1) and (29) by applying  Theorem 6  with F:=J* and u~0∶=(Λ~0,v~0,w~0), and (L2), (H2) by  Proposition 8. We need the following embedding inequality to check the conditions in  Theorem 6  and  Proposition 9. For simplicity, we denote ∥·∥p:=∥·∥Lp(D).Proposition 11.
We have(44)∥u∥6≤C6∥u∥𝓍foranyu∈𝒳,
where C6:={(1/2)(1/π+2)(1/π+3)}1/3∈(1.566,1.567).Proof.
The proof is similar to that of [2, Proposition 2.3.1].
First, let-∞<a<b<∞. In a similar way to the proof of [2, Lemma 2.3.1(ii)], we easily verify that
(45)maxx∈[a,b]|f(x)|≤∫ab‍|f′(x)|dx+1b-a∫ab‍|f(x)|dxforanyf∈H1(a,b).
We easily verify that
(46)max(∥u∥2,∥ux∥2,∥ut∥2)≤∥u∥𝓍foranyu∈𝒳.
We set 𝒳0:={∑k=-MM∑n=0Nckneiktcosnx∈𝒳;ckn∈ℂ and M,N∈ℕ∪{0}}. It suffices to prove (44) for any u∈𝒳0 since 𝒳0 is dense in 𝒳. Let u∈𝒳0. It follows from (45) and [2, Lemma 2.3.1(ii)] that
(47)∫02π‍dt∫0π‍u2dx≤∫02π‍maxy∈[0,π]|u(t,y)|dt∫0π‍maxτ∈[0,2π]|u(τ,x)|dx≤(1π∥u∥1+∥ux∥1)(12π∥u∥1+12∥ut∥1).
Let v∈𝒳0. We substitute u=vj(j=2,3) to (47). Then, we obtain from (46) and Schwarz inequality that
(48)∥v∥44≤(1π∥v∥22+2∥vvx∥1)(12π∥v∥22+∥vvt∥1)≤(1π∥v∥22+2∥v∥2∥vx∥2)(12π∥v∥22+∥v∥2∥vt∥2)≤12(1π+2)2∥v∥𝒳4,(49)∥v∥66≤(1π∥v∥33+3∥v2vx∥1)(12π∥v∥33+32∥v2vt∥1)≤∥v∥44(1π∥v∥2+3∥vx∥2)(12π∥v∥2+32∥vt∥2)≤∥v∥44·12(1π+3)2∥v∥𝒳2.
By (48) and (49), we have (44) with u replaced by v. The proof is complete.Remark 12.
Though our numerical verification method is not difficult, the verification process consists of a lot of steps and is very complicated. So, we omit its details here. They are much similar to the  discussions in [2, 9]. In particular, the verification process is described fully in detail in [2].


## 5. Final Remarks

Remark 13.
Our main result  Theorem1  is new and useful from the practical and applied mathematical point of view. It does not mean, however, that it is new from the theoretical point of view, though there is no known result equivalent to our main result with m≥2 as far as the author knows. In view of  Proposition 4,  Theorem 1  is theoretically considered as a corollary of [5, Theorem 1.7] for the case m=1 and is generalization of it for the case m≥2. We refer to [10] by Hale and [11] by López-Gómez where some related bifurcation theorems were obtained. They are generalization of [5, Theorem 1.7] different from our main result. Actually, Theorem 1 implies Theorem 2, as shown in our work mentioned in Remark 14 below. On the other hand, the bifurcation theorems in [10, 11] do not imply Theorem 2.Remark 14.
In a near future work, we will study the equivalent relations between conditions in Theorem1 with m=2 and those in Theorem 2 and show that Theorem 1 with m=2 is stronger than the existence part of [12, Theorem 1.11].Remark 15.
Nishida et al. studied problems of stability and bifurcation of solutions for some fluid equations in [13, 14]. We can apply the same technique as in [13, 14] to check rigorously the condition (C3) in Theorem 2 by the numerical method.
LetA∶=σ0fy(λ0,y0), ψ0∈N(A-i)-{0}, and ψ*∈N(A*+i)-{0}. Then, we can obtain in the same way as in [13] and [14, Section 4] that
(50)Reμ′(λ0)=Re(fλy0ψ0+fyy0v*ψ0,ψ*)(ψ0,ψ*).
Here, fλy0:=fλy(λ0,y0), fyy0:=fyy(λ0,y0), and v*∈ℝd satisfies fλ(λ0,y0)+fy(λ0,y0)v*=0. We can check the condition (C3) since we can find by the numerical verification method an accurate approximate value of the right-hand side of (50) with a rigorous error bound.



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