Codimension-m Bifurcation Theorems Applicable to the Numerical Verification Methods

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.

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 Section 4. 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. Let 𝑋 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(ℝ^𝑚×𝑋,𝑌) have the following properties: 𝑔(Λ,𝑣)∈𝑌₁foranyΛ∈ℝ^𝑚,𝑣∈𝑋₁,𝑔_𝑢(Λ,𝑣)𝑋_𝑗⊂𝑌_𝑗foranyΛ∈ℝ^𝑚,𝑣∈𝑋₁𝑗=1,2,𝑔_𝑢𝑢Λ,𝑣₁𝑣₂𝑤∈𝑌₂foranyΛ∈ℝ^𝑚,𝑣₁,𝑣₂∈𝑋₁,𝑤∈𝑋₂. (1)(2)(3) We denote by 𝐞₁∶=(1,0,…,0)∈ℝ^𝑚 the first row vector of the identity matrix of order 𝑚. We define 𝐽∶ℝ^𝑚×𝑋→ℝ^𝑚×𝑌 by 𝐽Λ𝑣𝑤∶=𝑙𝑤−𝐞₁𝑔()𝑔Λ,𝑣_𝑢(Λ,𝑣)𝑤∈ℝ^𝑚×𝑌₁×𝑌₂for(Λ,𝑣,𝑤)∈ℝ^𝑚×𝑋₁×𝑋₂. (4) Here, 𝑙∈ℒ(𝑋,ℝ^𝑚), and we assume that 𝑙𝑣=0 for any 𝑣∈𝑋₁. We define projections 𝑝 and 𝑃 by 𝑝∶𝑟₁,𝑟₂,…,𝑟_𝑚∈ℝ^𝑚↦𝑟₁𝑟∈ℝ,𝑃∶₁,𝑟₂,…,𝑟_𝑚∈ℝ^𝑚↦𝑟₂,…,𝑟_𝑚∈ℝ^𝑚−1forthecase𝑚≥2. (5)

In what follows, we always set formally ℝ^𝑚−1×𝑌∶=𝑌 for the case: 𝑚=1. We define 𝐺∶ℝ^𝑚×𝑋→ℝ^𝑚−1×𝑌 by (),𝐺Λ𝑢𝑔()().𝐺∶=𝑔𝑚=1∶=𝑃𝑙𝑢Λ,𝑢𝑚≥2 (6)

We set 𝑍∶=(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) ∈ℝ^𝑚×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 operator 𝑔_𝑢(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 ℝ^𝑚×𝑋, 𝑎∈ℝ₊, 𝜁∈𝐶¹((−𝑎,𝑎),ℝ^𝑚), 𝜂∈𝐶¹((−𝑎,𝑎),𝑍), an open neighborhood 𝑉 of 0 in ℝ^𝑚 and 𝑞∈𝐶²(Λ₀+𝑉,𝑋₁) such that 𝜁(0)=Λ₀, 𝜂(0)=0, 𝑞(Λ₀)=𝑣₀ and 𝐺⁻¹(0)(Λ;(Λ∪𝜁(𝛼)∩𝑊=Λ,𝑞)Λ,𝑞)∈𝑊,𝛼𝑤₀+𝛼𝜂(𝛼)+𝑞(𝜁(𝛼));|𝛼|<𝑎. (7)

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: 𝑦̇=𝑓𝜆,𝑦,𝑦,𝑓𝜆,𝑦∈ℝ^𝑑. (O) 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 𝜎₀𝑓_𝑦(𝜆₀,𝑦₀) (so, by the implicit function theorem, the matrix 𝜎₀𝑓_𝑦(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 𝜎₀𝑓_𝑦(𝜆₀,𝑦₀) for 𝑘∈ℤ−{−1,1}.

It is difficult to check rigorously by numerical methods the dynamic 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 Section 2, 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. Let ℋ and 𝒦 be Banach spaces. (1)We denote by ℝ 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 by 𝐞₁∶=(1,0,…,0)∈ℝ^𝑚 the first row vector of the identity matrix 𝐸_𝑚=(𝛿_𝑖𝑗) 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 𝐵_ℋ(𝑢;𝑟)∶={𝑣∈ℋ;‖𝑣−𝑢‖<𝑟}. The subscript will be often omitted if no possible confusion arises.(4)We denote by ℒ(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)Let 𝐴∶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 set 𝑉⊂ℋ, we denote by 𝐶^𝑛(𝑉,𝒦) the space of 𝑛-times continuously differentiable functions from 𝑉 to 𝒦.(7)Let Ω be a domain of ℝ^𝑛. We denote Lebesgue spaces by 𝐿^𝑝(Ω)(1≤𝑝≤∞) with the norms: ‖𝑢‖_𝐿^𝑝_(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 ‖𝑢‖_𝐻1(Ω)∶=‖𝑢‖²_𝐿2(Ω)+^𝑛_𝑘=1𝜕𝑢𝜕𝑥_𝑘²_𝐿2(Ω)^1/2. (8)(8)Let 𝑁∈ℕ and 𝑍_𝑘(𝑘=1,2,…,𝑁) be norm spaces. We denote their direct product space 𝑍 usually by 𝑍=𝑍₁×⋯×𝑍_𝑁 and also by 𝑍=𝑍₁⊕⋯⊕𝑍_𝑁 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∶=(∑^𝑁_𝑘=1‖𝑢_𝑘‖²_𝑍𝑘)^1/2 for 𝑢=(𝑢₁,…,𝑢_𝑁)∈𝑍₁×⋯×𝑍_𝑁.

2. Basic Bifurcation Theorems

Theorem 3 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}.

Let 𝑋 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 ℝ^𝑚. Let 𝑓∈𝐶¹(𝑉×𝑈,𝑌) be a map such that 𝑓(Λ,0)=0foranyΛ=Λ₁,…,Λ_𝑚∈𝑉, (9) and the partial Fréchet derivative 𝑓_Λ𝑘𝑢 exists and is continuous for 𝑘=1,…,𝑚. Here, for any fixed (Λ_⋆,𝑢_⋆)∈𝑉×𝑈, 𝑓_Λ𝑘𝑢(Λ_⋆,𝑢_⋆) represents the Fréchet derivative of the function 𝑢∈𝑈↦(𝜕𝑓/𝜕Λ_𝑘)(Λ_⋆,𝑢)∈𝑌 at 𝑢=𝑢_⋆. We denote 𝑓_Λ𝑢∶=(𝑓_Λ1𝑢,…,𝑓_Λ𝑚𝑢) for simplicity. We define 𝐻∶𝑉×𝑋→ℝ^𝑚×𝑌 by 𝐻Λ𝑢∶=𝑙𝑢−𝑒₁𝑓_𝑢()𝑢.Λ,0 (10) Here, 𝑙∈ℒ(𝑋,ℝ^𝑚). In what follows, we often use the same notations as in Section 1. We define 𝐹∶𝑉×𝑈→ℝ^𝑚−1×𝑌 by (),𝐹Λ𝑢𝑓()().𝐹∶=𝑓𝑚=1∶=𝑃𝑙𝑢Λ,𝑢𝑚≥2 (11) We set 𝑍∶=(𝑝𝑙)⁻¹(0)={𝑢∈𝑋;𝑝𝑙𝑢=0}.

Theorem 3. In addition to the assumptions above, we assume that(H) there exists 𝑢₀∈𝑈 such that (Λ,𝑢)=(0,𝑢₀) is an isolated solution of the extended system 𝐻(Λ,𝑢)=0.
Then, there exist an open neighborhood 𝑊 of (0,0) in ℝ^𝑚×𝑋, 𝑎∈ℝ₊ and continuous functions 𝜁∶(−𝑎,𝑎)→ℝ^𝑚, 𝜂∶(−𝑎,𝑎)→𝑍 such that 𝜁(0)=0, 𝜂(0)=0 such that 𝐹⁻¹(0));()}∪𝜁(𝛼)∩𝑊={(Λ,0Λ,0∈𝑊,𝛼𝑢₀+𝛼𝜂(𝛼);𝛼∈(−𝑎,𝑎). (12) Moreover, if 𝑓∈𝐶^𝑘+1 (𝑘∈ℕ), then 𝜁,𝜂∈𝐶^𝑘.

For simplicity, we write 𝐷𝐻⁰∶=𝐷𝐻(0,𝑢₀), 𝑓⁰_𝑢∶=𝑓_𝑢(0,0), 𝐹⁰_𝑢∶=𝐹_𝑢(0,0), 𝐹⁰_Λ𝑢∶=𝐹_Λ𝑢(0,0), and so on. We have 𝐷𝐻⁰Λ𝑢=𝑙𝑢Λ⋅𝑓⁰_Λ𝑢𝑢₀+𝑓⁰_𝑢𝑢=𝑝𝑙𝑢Λ⋅𝐹⁰_Λ𝑢𝑢₀+𝐹⁰_𝑢𝑢. (13)

Proof of Theorem 3. The proof is similar to that of  [1, Theorem 2.1] and [5, Theorem 1.7]. We obtain from (H) that 𝑝𝑙𝑢₀=1 and 𝐹⁰_𝑢𝑢₀=0. Let 𝑢 be any element of 𝒩(𝐹⁰_𝑢). We set 𝑐̂∶=𝑝𝑙𝑢. Then, we have 𝐷𝐻⁰(0,𝑢−𝑐𝑢̂₀)=0. Since 𝐷𝐻⁰ is one to one, 𝑢=𝑐𝑢̂₀. So we obtain 𝒩(𝐹⁰_𝑢)=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→ℝ^𝑚×𝑌 by ℎ()ℎ𝛼,Λ,𝑢∶=𝑙𝑢₀(𝛼,Λ,𝑢)withℎ₀()1𝛼,Λ,𝑢∶=𝛼𝑓Λ,𝛼𝑢₀(),𝑓+𝛼𝑢𝛼≠0_𝑢(Λ,0)𝑢₀+𝑢(𝛼=0). (14) By 𝐻(0,𝑢₀)=0, we have ℎ(0,0,0)=(0,𝑓⁰_𝑢𝑢₀)=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)→ℝ^𝑚, 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 Theorem 3 as a generalized version of [5, Theorem 1.7].

Proposition 4. The condition (H) in Theorem 3 is equivalent to the following (i) and (ii):(i)dim𝒩(𝐹⁰_𝑢)=1  and codimℛ(𝐹⁰_𝑢)=𝑚, (ii)∃𝑢₀∈𝑈  such that𝑝𝑙𝑢₀=1,𝒩𝐹⁰_𝑢=span𝑢₀,span𝐹⁰_Λ𝑢𝑢₀⊕ℛ𝐹⁰_𝑢=ℝ^𝑚−1×𝑌. (15)(16)Here, we denote span(𝐹⁰_Λ𝑢𝑢₀)∶={Λ⋅𝐹⁰_Λ𝑢𝑢₀;Λ∈ℝ^𝑚}.

Proof. First, we assume (H). By the proof of Theorem 3, we have dim𝒩(𝐹⁰_𝑢)=1 and (15). Since 𝐷𝐻⁰ is onto, we have span(𝐹⁰_Λ𝑢𝑢₀)+ℛ(𝐹⁰_𝑢)=ℝ^𝑚−1×𝑌. Let 𝑝∈span(𝐹⁰_Λ𝑢𝑢₀) and 𝑞∈𝑅(𝐹⁰_𝑢) satisfy 𝑝+𝑞=0. In order to prove (16), it suffices to show 𝑝=𝑞=0. We choose Λ∈ℝ^𝑚 and 𝑢∈𝑋 such that 𝑝=Λ⋅𝐹⁰_Λ𝑢𝑢₀ and 𝑞=𝐹⁰_𝑢𝑢. We set 𝑐̃∶=𝑝𝑙𝑢. Then, 𝐷𝐻⁰(Λ,𝑢−𝑐𝑢̃₀)=0. We have Λ=0 and 𝑢−𝑐𝑢̃₀=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 {𝐹⁰_Λ𝑘𝑢𝑢₀}^𝑚_𝑘=1 is linearly independent. Let Λ∈ℝ^𝑚 satisfy Λ⋅𝐹⁰_Λ𝑢𝑢₀=0. Then, we have 𝐷𝐻⁰(Λ,0)=0, which implies Λ=0. Thus, {𝐹⁰_Λ𝑘𝑢𝑢₀}^𝑚_𝑘=1 is linearly independent. So, dimspan(𝐹⁰_Λ𝑢𝑢₀)=𝑚. By this and (16), we have codimℛ(𝐹⁰_𝑢)=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)∈ℝ^𝑚×𝑋 satisfy 𝐷𝐻⁰(Λ,𝑢)=0. Then, by (13) we have 𝑝𝑙𝑢=0 and Λ⋅𝐹⁰_Λ𝑢𝑢₀+𝐹⁰_𝑢𝑢=0. We obtain from (16) that Λ⋅𝐹⁰_Λ𝑢𝑢₀=0 and 𝐹⁰_𝑢𝑢=0. We have Λ=0 since (i) and (16) imply span(𝐹⁰_Λ𝑢𝑢₀)=𝑚. By 𝑝𝑙𝑢=0 and (15), we have 𝑢=0. Hence, 𝐷𝐻⁰ is one to one. Finally, let (𝝀,𝑦) be an element of ℝ^𝑚×𝑌. By (16), there exist Λ∈ℝ^𝑚 and 𝑢∈𝑋 such that Λ⋅𝐹⁰_Λ𝑢𝑢₀+𝐹⁰_𝑢𝑢=(𝑃𝝀,𝑦). We set 𝑐̃∶=𝑝(𝝀−𝑙𝑢). Then, 𝐷𝐻⁰(Λ,𝑢+𝑐𝑢̃₀)=(𝝀,𝑦). Thus, 𝐷𝐻⁰ is onto. The proof is complete.

3. Proof of Theorem 1

We set 𝐷𝐽⁰∶=𝐷𝐽(Λ₀,𝑣₀,𝑤₀). We use the same notations as used in Section 2. We set 𝑔⁰_𝑢∶=𝑔_𝑢(Λ₀,𝑣₀), 𝑔⁰_Λ∶=𝑔_Λ(Λ₀,𝑣₀), 𝑔⁰_Λ𝑢∶=𝑔_Λ𝑢(Λ₀,𝑣₀), and 𝑔⁰_𝑢𝑢∶=𝑔_𝑢𝑢(Λ₀,𝑣₀). Then, we have 𝐷𝐽⁰Λ𝑣𝑤=𝑙𝑤Λ⋅𝑔⁰_Λ+𝑔⁰_𝑢𝑣Λ⋅𝑔⁰_Λ𝑢𝑤₀+𝑔⁰_𝑢𝑢𝑣𝑤₀+𝑔⁰_𝑢𝑤. (17)

Proof of Theorem 1. By (H2) and the implicit function theorem, there exist an open neighborhood 𝑉 of 0 in ℝ^𝑚 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 𝑓(Λ,𝑢)∶=𝑔Λ₀+Λ,𝑢+𝑞Λ₀+Λ. (18) Then, we have (9), 𝑓⁰_𝑢=𝑔⁰_𝑢, and 𝑓⁰_Λ𝑢𝑤₀=𝑔⁰_Λ𝑢𝑤₀+𝑔⁰_𝑢𝑢𝐱₀𝑤₀with𝐱₀∶=𝑞^′Λ₀∈𝑋^𝑚₁. (19) It follows from (9) and (H2) that 𝐱₀∈𝑋^𝑚₁ is uniquely determined by the equation: 𝑓_Λ(0,0)=𝑔⁰_Λ+𝑔⁰_𝑢𝐱₀=0. (20) We define 𝐻∶𝑉×𝑋→ℝ^𝑚×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∶ℝ^𝑚×𝑋→ℝ^𝑚×𝑌 is bijective. Let (Λ,𝑢)∈ℝ^𝑚×𝑋 satisfy 𝐷𝐻⁰(Λ,𝑢)=0. It follows from (2), (3), (13), and (19) that 𝑔𝑙𝑢=0,_0𝑢𝑢=𝑓⁰_𝑢𝑢=−Λ⋅𝑓⁰_Λ𝑢𝑤₀=−Λ⋅𝑔⁰_Λ𝑢𝑤₀+𝑔⁰_𝑢𝑢𝐱₀𝑤₀∈𝑌₂. (21) 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 ℝ^𝑚×𝑌. Let 𝑦=𝑦₁+𝑦₂ with 𝑦₁∈𝑌₁ and 𝑦₂∈𝑌₂. Since 𝐷𝐽⁰ is onto, there exists (Λ,𝑣,̃𝑤)∈ℝ^𝑚×𝑋₁×𝑋₂ such that 𝐷𝐽⁰(Λ,𝑣,̃𝑤)=(𝝀,0,𝑦₂). By (H2), (17), (19), and (20), we have 𝝀=𝑙𝑤, 𝑣̃=Λ⋅𝐱₀, and 𝑦₂=Λ⋅(𝑔⁰_Λ𝑢𝑤₀+𝑔⁰_𝑢𝑢𝐱₀𝑤₀)+𝑔⁰_𝑢𝑤=Λ⋅𝑓⁰_Λ𝑢𝑤₀+𝑓⁰_𝑢𝑤. We set 𝑢∶=(𝑔⁰_𝑢|_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 Theorem 1 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: 𝜕𝑢₁−𝜕𝜕𝑡²𝑢₁𝜕𝑥²=𝐴+𝜂cos𝑥−(𝐵+1)𝑢₁+𝑢²₁𝑢₂+𝜆𝑢₃−𝑢₁for(𝑥,𝑡)∈(0,𝜋)×[0,∞),𝜕𝑢₂−𝜕𝜕𝑡²𝑢₂𝜕𝑥²=𝐵𝑢₁−𝑢²₁𝑢₂+𝜆𝜌𝑢₄−𝑢₂for(𝑥,𝑡)∈(0,𝜋)×[0,∞),𝜕𝑢₃−𝜕𝜕𝑡²𝑢₃𝜕𝑥²=𝐴+𝜂cos𝑥−(𝐵+1)𝑢₃+𝑢²₃𝑢₄+𝜆𝑢₁−𝑢₃for(𝑥,𝑡)∈(0,𝜋)×[0,∞),𝜕𝑢₄−𝜕𝜕𝑡²𝑢₄𝜕𝑥²=𝐵𝑢₃−𝑢²₃𝑢₄+𝜆𝜌𝑢₂−𝑢₄for(𝑥,𝑡)∈(0,𝜋)×[0,∞),𝜕𝑢_𝑗()=𝜕𝑥0,𝑡𝜕𝑢_𝑗()[)𝜕𝑥𝜋,𝑡=0for𝑡∈0,∞,𝑗=1,2,3,4. (B) 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 set 𝑢=(𝑢₁,𝑢₂,𝑢₃,𝑢₄). We denote the equations of (B) by 𝑢_𝑡−𝑢_𝑥𝑥=ℎ(𝜆,𝑢). We also write 𝑢_𝑡=𝜑(𝜆,𝑢) with 𝜑(𝜆,𝑢)∶=𝑢_𝑥𝑥+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: 𝑢_𝑡=𝜎𝜑(𝜆,𝑢). (22)

4.2. The Setting of Functional Spaces

Let 𝐷∶=(0,𝜋)×(0,2𝜋), 𝒴∶=𝐿²(𝐷), and 𝑌∶=𝒴⁴. We define a real Hilbert space 𝒳 by 𝒳∶=𝑢=^∞∞_{𝑘=−∞𝑛=0}𝑐_𝑘𝑛𝑒^𝑖𝑘𝑡cos𝑛𝑥∈𝒴;𝑐_𝑘𝑛∈ℂ,𝑐_−𝑘,𝑛=𝑐_𝑘𝑛forany𝑘∈ℤ,𝑛∈ℕ∪{0}and𝑢_𝑡−𝑢_𝑥𝑥=∈𝒴_{∞∞𝑘=−∞𝑛=0}𝑐_𝑘𝑛𝑒^𝑖𝑘𝑡cos𝑛𝑥∈𝒴;𝑐_𝑘𝑛∈ℂ,𝑐_−𝑘,𝑛=𝑐_𝑘𝑛forany𝑘∈ℤ,𝑛∈ℕ∪{0}and_{∞∞𝑘=0𝑛=0}𝑘²+𝑛⁴𝑐_𝑘𝑛²<∞, (23) with the inner product (𝑢,𝑣)_𝒳∶=𝑢_𝑡−𝑢_𝑥𝑥+12𝜋²₀^2𝜋𝑑𝑡^𝜋₀𝑣𝑢𝑑𝑥,_𝑡−𝑣_𝑥𝑥+12𝜋²₀^2𝜋𝑑𝑡^𝜋₀𝑣𝑑𝑥_𝒴 (24) for 𝑢,𝑣∈𝒳. Then, we easily verify that 𝒳 is continuously embedded in 𝐻¹(𝐷) and that ‖𝑢‖_𝒳=(𝑢,𝑢)_𝒳=𝜋2𝑐₀₀²+2^∞_𝑘=−∞𝑘²𝑐_𝑘0²+^∞∞_{𝑘=−∞𝑛=1}𝑘²+𝑛⁴𝑐_𝑘𝑛^21/2 (25) for 𝑢=∑^∞_𝑘=−∞∑^∞_𝑛=0𝑐_𝑘𝑛𝑒^𝑖𝑘𝑡cos𝑛𝑥∈𝒳. We set 𝑋∶=𝒳⁴. We denote by 𝒱 (resp., 𝒲) the subspace of 𝒳 (resp., 𝒴) consisting of steady functions, that is, 𝒱∶=^∞_𝑛=0𝑐_𝑛cos𝑛𝑥;𝑐_𝑛∈ℝand^∞_𝑛=0𝑛⁴𝑐²_𝑛<∞,𝒲∶=𝐿²(0,𝜋). (26) We set 𝑋₁∶=𝒱⁴, 𝑌₁∶=𝒲⁴, 𝑋₂∶=𝑋^⟂₁, and 𝑌₂∶=𝑌^⟂₁.

We set Λ∶=(𝜆,𝜎). We define 𝑔(Λ,𝑢)∶=𝑢_𝑡−𝜎𝜑(𝜆,𝑢) and 𝑙∶=(𝑙₁,𝑙₂)∶𝑋→ℝ² by 𝑙₁1𝑢∶=𝜋²𝑢₁,sin𝑡_𝒴,𝑙₂1𝑢∶=𝜋²𝑢₁,cos𝑡_𝒴for𝑢=𝑢₁,𝑢₂,𝑢₃,𝑢₄∈𝑋. (27) 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 𝐽̂∶ℝ²×𝑋→ℝ²×𝑌 by 𝐽̂Λ𝑣𝑤∶=𝑙𝑤−𝐞₁𝜑()𝑔𝜆,𝑣_𝑢(Λ,𝑣)𝑤∈ℝ²×𝑌₁×𝑌₂for(Λ,𝑣,𝑤)∈ℝ²×𝑋₁×𝑋₂. (28) 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 operator 𝜑_𝑢(𝜆₀,𝑣₀)∣_𝑋1∶𝑋₁→𝑌₁ is bijective.

4.3. A Verification Result

By our numerical verification method described in Section 4.5, we verified the existence of a Hopf bifurcation point (𝜆₀,𝑣₀) of (B) satisfying 𝜆₀−𝜆₀²+𝜎₀−𝜎₀²+𝑣₀−𝑣₀²_𝑋+𝑤₀−𝑤₀²_𝑋≤6.46×10⁻⁸². (29) Here, ((𝜆₀,𝜎₀),𝑣₀,𝑤₀) is an isolated solution of the extended system 𝐽(̂Λ,𝑣,𝑤)=0, and ((𝜆₀,𝜎₀),𝑣₀,𝑤₀) 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 𝜆₀∶=0.9541600594715 and 𝜎₀∶=0.4534602327315. The functions 𝑣₀ and 𝑤₀ have the forms 𝑣₀∶=𝐚+𝐛cos𝑥+𝐜cos2𝑥with𝐚,𝐛,𝐜∈ℚ⁴,𝑤₀∶=𝐩₁sin𝑡+𝐩₂cos𝑡+𝐪₁sin𝑡+𝐪₂+𝐫cos𝑡cos𝑥₁sin𝑡+𝐫₂cos𝑡cos2𝑥 (30) with 𝐩_𝑘,𝐪_𝑘,𝐫_𝑘∈ℚ⁴(𝑘=1,2). We have 𝑣_0𝑋∈(20.8,20.9),𝑤_0𝑋∈(),𝐽̂Λ4.25,4.26₀,𝑣₀,𝑤_0ℝ2×𝑌∈4.57×10⁻⁹,4.58×10⁻⁹. (31) Here, Λ₀∶=(𝜆₀,𝜎₀).

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.

Let ℋ 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). Let 𝑢₀∈ℋ and 𝐴∶=𝐹^′(𝑢₀) be bijective. We define a map 𝐺∶ℋ→ℋ by 𝐺(𝑢)∶=𝑢−𝐴⁻¹𝐹(𝑢). (32) Let 𝑅>0 be a constant and 𝑏∶[0,𝑅]→[0,∞) be a non-decreasing function such that sup𝐺^′(𝑢);𝑢∈𝐵𝑢₀;𝑟≤𝑏(𝑟)𝑓𝑜𝑟𝑎𝑛𝑦𝑟∈[0,𝑅]. (33) Let 𝜀₀≥0 be a constant such that 𝐴⁻¹𝐹𝑢₀≤𝜀₀. (34) We assume that there exist constants 𝑟₀ and 𝑟₁ such that 0<𝑟₀≤𝑟₁≤𝑅, 𝜀₀+^𝑟0₀𝑏(𝑟)𝑑𝑟≤𝑟₀𝑎𝑛𝑑𝑏𝑟₁<1. (35) Then, the equation 𝐹(𝑢)=0 has an isolated solution 𝑢₀∈𝐵(𝑢₀;𝑟₀). Moreover, the solution of 𝐹(𝑢)=0 is unique in 𝐵(𝑢₀;𝑟₁).

Theorem 6 plays a central role in our numerical verification method.

Remark 7. Theorem 6 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 Theorem 6, 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).

Let ℋ₀ and ℋ_𝑐 be closed subspaces of ℋ such that ℋ=ℋ₀⊕ℋ_𝑐. Let 𝒦₀ and 𝒦_𝑐 be closed subspaces of 𝒦 such that 𝒦=𝒦₀⊕𝒦_𝑐. We denote by 𝑃_𝑗∈ℒ(ℋ) the projection defined by 𝑃_𝑗𝑢∶=𝑢_𝑗 for 𝑢=𝑢₀+𝑢_𝑐∈ℋ with 𝑢_𝑗∈ℋ_𝑗(𝑗=0,𝑐). We denote by 𝑄_𝑗∈ℒ(𝐾) the projection defined by 𝑄_𝑗𝑣∶=𝑣_𝑗 for 𝑣=𝑣₀+𝑣_𝑐∈𝒦 with 𝑣_𝑗∈𝒦_𝑗(𝑗=0,𝑐). Let 𝑇∈ℒ(ℋ,𝒦) and ℒ∶ℋ→𝒦 be a closed linear operator. We set 𝐴∶=𝐿+𝑇. We assume ℛ𝑃₀⊂𝒟(𝐿),𝐿𝑃₀=𝑄₀𝐿on𝒟(𝐿). (36) We define 𝐴₀∶=𝐿+𝑄₀𝑇𝑃₀∶ℋ→𝒦. We denote by 𝐼_ℋ (resp., 𝐼_𝒦) the identity operator on ℋ (resp., 𝒦) and simply by 𝐼 the operators 𝐼_ℋ and 𝐼_𝒦 when there is no ambiguity.

Proposition 8. In addition to the above assumptions, we assume that 𝐴₀ is bijective and that 𝑆∶=𝐴₀⁻¹𝑄_𝑐𝑇(𝐼−𝐴₀⁻¹𝑃₀𝑇)𝑃_𝑐∈ℒ(𝐻) satisfies ‖𝑆‖<1. Then, 𝐴 is bijective. Moreover, we have 𝑃₀𝐴⁻¹𝑄₀=𝐴₀⁻¹𝑄₀+𝑄₀𝑇𝑃_𝑐(𝐼+𝑆)⁻¹𝐴₀⁻¹𝑄_𝑐𝑇𝐴₀⁻¹𝑄₀∶=𝐴¹¹,𝑃₀𝐴⁻¹𝑄_𝑐=𝐴₀⁻¹𝑄₀𝑇𝑃_𝑐(𝐼+𝑆)⁻¹𝐴₀⁻¹𝑄_𝑐∶=𝐴¹²,𝑃_𝑐𝐴⁻¹𝑄₀=−(𝐼+𝑆)⁻¹𝐴₀⁻¹𝑄_𝑐𝑇𝑄₀𝐴₀⁻¹𝑄₀∶=𝐴²¹,𝑃_𝑐𝐴⁻¹𝑄_𝑐=(𝐼+𝑆)⁻¹𝐴₀⁻¹𝑄_𝑐∶=𝐴²². (37)

Proposition 8 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 Proposition 8 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 {𝜙_𝑛}^∞_𝑛=1 of ℋ such that 𝜙_𝑛 is an eigenfunction of 𝐿 for any 𝑛∈ℕ. Let ℋ_𝑛 and 𝒦_𝑛 be subspaces of ℋ and 𝒦, respectively, such that both ℋ_𝑛 and 𝒦_𝑛 are equal to span{𝜙_𝑗}^𝑛_𝑗=1 as sets. Let 𝑃_𝑛 (resp., 𝑄_𝑛) be the orthogonal projection on ℋ (resp., 𝒦) onto ℋ_𝑛 (resp., 𝒦_𝑛), (A3)‖𝑇(𝐼−𝑃_𝑛)‖_ℋ→𝒦→0 and ‖(𝐼−𝑄_𝑛)𝑇‖_ℋ→𝒦→0as𝑛→∞.

We show that Proposition 8 works well if we take ℋ₀∶=ℋ_𝑛 and 𝒦₀∶=𝒦_𝑛 for sufficiently large 𝑛. Let 𝐴_𝑛∶=𝐿+𝑄_𝑛𝑇𝑃_𝑛∶ℋ→𝒦.

Proposition 9. Under the assumptions above we have the following. (i)If  𝐴  is bijective, then  𝐴_𝑛  is bijective for sufficiently large  𝑛∈ℕ, and we have𝐴_𝑛⁻¹⟶𝐴⁻¹inℒ(𝒦,ℋ)as𝑛⟶∞. (38)(ii)If 𝐴_𝑛  is bijective and𝐴_𝑛⁻¹𝑇−𝑄_𝑛𝑇𝑃_𝑛<1, (39)then 𝐴  is bijective with the estimate ‖𝐴⁻¹‖≤‖𝐴_𝑛⁻¹‖/(1−‖𝐴_𝑛⁻¹‖‖𝑇−𝑄_𝑛𝑇𝑃_𝑛‖). (iii)We have𝐴_𝑛⁻¹⟶𝐴⁻¹𝑎𝑠𝑛⟶∞. (40)Here, we formally define ‖𝐴_𝑛⁻¹‖∶=∞ (resp., ‖𝐴⁻¹‖∶=∞) when 𝐴_𝑛 (resp., 𝐴) is not bijective.

Proof. We set 𝐵_𝑛∶=𝑇−𝑄_𝑛𝑇𝑃_𝑛∈ℒ(ℋ,𝒦). Then, 𝐴−𝐴_𝑛=𝐵_𝑛 on 𝒟(𝐿). We obtain from (A3) that ‖𝐵_𝑛‖→0 as 𝑛→∞.
(i) Let 𝐴 be bijective. Then, ‖𝐴⁻¹‖‖𝐵_𝑛‖<1 for sufficiently large 𝑛. Therefore, by [2, Corollary 2.4.1], 𝐴_𝑛 is bijective for sufficiently large 𝑛, and we obtain (38).
(ii) We immediately obtain the desired conclusion from [2, Corollary 2.4.1].
(iii) If 𝐴 is bijective, (i) implies (40). So, we consider the case where 𝐴 is not bijective. We proceed by contradiction. We assume that lim_𝑛→∞‖𝐴_𝑛⁻¹‖≠∞. Then, liminf_𝑛→∞‖𝐴_𝑛⁻¹‖<∞. So, there exists a large number 𝑁∈ℕ such that ‖𝐴_𝑁⁻¹‖‖𝐵_𝑁‖<1. So, (ii) implies that 𝐴 is bijective. This contradicts to our assumption. Therefore, (40) holds.

We consider the case where the condition of  Proposition 8 is satisfied. Let 𝑢∈𝒦. The estimate (41) below is useful when we apply Theorem 6 to concrete problems. We denote by 𝐴_𝑛^𝑖𝑗 the operator 𝐴^𝑖𝑗 with 𝐴₀, 𝑃₀, 𝑃_𝑐, 𝑄₀, and 𝑄_𝑐 replaced by 𝐴_𝑛, 𝑃_𝑛, 𝐼−𝑃_𝑛, 𝑄_𝑛, and 𝐼−𝑄_𝑛, respectively (𝑖,𝑗 = 1, 2). Then, we easily verify that 𝐴⁻¹𝑢≤𝐴_𝑛¹¹𝑢+𝐴_𝑛¹²𝑢²+𝐴_𝑛²¹𝑢+𝐴_𝑛²²𝑢^21/2 (41) 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 by 𝑆_𝑛 the operator 𝑆 defined in Proposition 8 with 𝐴₀, 𝑃₀, 𝑃_𝑐, and 𝑄_𝑐 replaced by 𝐴_𝑛, 𝑃_𝑛, 𝐼−𝑃_𝑛 and 𝐼−𝑄_𝑛, respectively. Though (39) is a sufficient condition for the existence of 𝐴⁻¹, another sufficient condition ‖𝑆_𝑛‖<1 is in general more efficient. The reason is that ‖𝑇(𝐼−𝑃_𝑛)‖ converges to 0 (𝑛→∞) more rapidly in general than ‖(𝐼−𝑄_𝑛)𝑇‖.

4.5. Derivation of Our Verification Result

Here, we describe the outline of derivation of our verification result in Section 4.3. We define closed subspaces of 𝑋₂: 𝑊₁∶=^∞_𝑛=0𝐚_𝑛cos𝑡+𝐛_𝑛sin𝑡cos𝑛𝑥∈𝑋;𝐚_𝑛,𝐛_𝑛∈ℝ⁴,𝑊₂∶=_|𝑘|^∞_≥2𝑛=0𝐜_𝑘𝑛𝑒^𝑖𝑘𝑡cos𝑛𝑥∈𝑋;𝐜_𝑘𝑛∈ℂ⁴. (42) We similarly define closed subspaces of 𝑌₂: 𝑍₁∶=^∞_𝑛=0𝐚_𝑛cos𝑡+𝐛_𝑛sin𝑡cos𝑛𝑥∈𝑌;𝐚_𝑛,𝐛_𝑛∈ℝ⁴,𝑍₂∶=_|𝑘|^∞_≥2𝑛=0𝐜_𝑘𝑛𝑒^𝑖𝑘𝑡cos𝑛𝑥∈𝑌;𝐜_𝑘𝑛∈ℂ⁴. (43) 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=𝑔⁰_𝑢∣_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 𝐽̂. Then, we can represent 𝐽̂ by the direct sum of maps: 𝐽̂=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 operator 𝐾∶𝑊₂→𝑍₂ is bijective.

By the above discussion and Theorem 1, 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 𝑢₀∶=(Λ₀,𝑣₀,𝑤₀), 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∶=‖⋅‖_𝐿^𝑝_(𝐷).

Proposition 11. We have ‖𝑢‖₆≤𝐶₆‖𝑢‖_𝓍𝑓𝑜𝑟𝑎𝑛𝑦𝑢∈𝒳, (44) 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 max_{[]𝑥∈𝑎,𝑏}𝑓(𝑥≤_𝑏𝑎)𝑓^′(𝑥+1)𝑑𝑥𝑏−𝑎^𝑏_𝑎𝑓(𝑥)𝑑𝑥forany𝑓∈𝐻¹(𝑎,𝑏). (45) We easily verify that max‖𝑢‖₂,𝑢_𝑥2,𝑢_𝑡2≤‖𝑢‖_𝓍forany𝑢∈𝒳. (46) We set 𝒳₀∶={∑^𝑀_𝑘=−𝑀∑^𝑁_𝑛=0𝑐_𝑘𝑛𝑒^𝑖𝑘𝑡cos𝑛𝑥∈𝒳;𝑐_𝑘𝑛∈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 ₀^2𝜋𝑑𝑡^𝜋₀𝑢²≤𝑑𝑥_02𝜋max_{[]𝑦∈0,𝜋}𝑢𝑡,𝑦𝑑𝑡^𝜋₀max_{[]𝜏∈0,2𝜋}|𝑢(≤1𝜏,𝑥)|𝑑𝑥𝜋‖𝑢‖₁+𝑢_𝑥11‖𝑢‖2𝜋¹+12𝑢_𝑡1. (47) Let 𝑣∈𝒳₀. We substitute 𝑢=𝑣^𝑗(𝑗=2,3) to (47). Then, we obtain from (46) and Schwarz inequality that ‖𝑣‖⁴₄≤1𝜋‖𝑣‖²₂+2𝑣𝑣_𝑥11‖𝑣‖2𝜋²²+𝑣𝑣_𝑡1≤1𝜋‖𝑣‖²₂+2‖𝑣‖₂𝑣_𝑥21‖𝑣‖2𝜋²²+‖𝑣‖₂𝑣_𝑡2≤121𝜋+2²‖𝑣‖⁴_𝒳,‖𝑣‖⁶₆≤1𝜋‖𝑣‖³₃+3𝑣²𝑣_𝑥11‖𝑣‖2𝜋³³+32𝑣²𝑣_𝑡1≤‖𝑣‖⁴₄1𝜋‖𝑣‖₂+3𝑣_𝑥21‖𝑣‖2𝜋²+32𝑣_𝑡2≤‖𝑣‖⁴₄⋅121𝜋+3²‖𝑣‖²_𝒳. (48)(49) 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 Theorem 1 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 Theorem 1 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.
Let 𝐴∶=𝜎₀𝑓_𝑦(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 Re𝜇^′𝜆₀𝑓=Re^0𝜆𝑦𝜓₀+𝑓⁰_𝑦𝑦𝑣_∗𝜓₀,𝜓_∗𝜓₀,𝜓_∗. (50) Here, 𝑓⁰_𝜆𝑦∶=𝑓_𝜆𝑦(𝜆₀,𝑦₀), 𝑓⁰_𝑦𝑦∶=𝑓_𝑦𝑦(𝜆₀,𝑦₀), and 𝑣_∗∈ℝ^𝑑 satisfies 𝑓_𝜆(𝜆₀,𝑦₀)+𝑓_𝑦(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.

Acknowledgments

The author would like to express his sincere gratitude to Professor Atsushi Yagi and Professor Takaaki Nishida for their encouragement. The author is grateful to the referees for constructive comments.