# Hopf Bifurcation with the Spatial Average of an Activator in a Radially Symmetric Free Boundary Problem

Authors: YoonMee Ham
Journal: Mathematical Problems in Engineering
 (2010)

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## Abstract

An interface problem derived by a bistable reaction-diffusion system with the spatial average of an activator is studied on ann-dimensional ball. We analyze the existence of the
radially symmetric solutions and the occurrence of Hopf bifurcation as a parameter varies in two and
three-dimensional spaces.


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## Body

## 1. Introduction

The study of interfacial patterns is important in several areas of biology, chemistry, physics, and other fields [1–4]. Internal layers (or free boundary), which separate two stable bulk states by a sharp transition near interfaces, are often observed in bistable reaction-diffusion equations when the reaction rate is faster than the diffusion effect. We consider a reaction-diffusion system with a sufficiently small positive constant ε [5, 6](1.1)σεut=ε2∇2u+f(u,v),vt=D∇2v+g(u,v)t>0,x∈Ω,0=∂u∂ν=∂v∂ν,whereε, σ, and D are positive constants, Ω={x∈ℝn:|x|<R} is the ball in n-dimensional space, and ν stands for the unit outward normal on the boundary ∂Ω. The nonlinear functions are(1.2)f(u,v)=-u+H(u-a0)-v+(1-μ)〈u〉,g(u,v)=μu-v,where0<μ<1 and 〈u〉 denotes the spatial average, describing a global feedback effect, namely, (1.3)〈u〉=1|Ω|∫ΩudΩ=1|Ω|∫Ωu(x,t)dx.The system (1.1) with (1.2) is a model for flow discharges proposed by et al. [7], in which u is interpreted by the current density and v by the voltage drop across the gas gap [7, 8] in a gas-discharge system. This system also exhibits a codimension-two Turing Hopf bifurcation [9], where the conditions of a spatial Turing instability [10] with a certain wavelength ε and a temporal Hopf bifurcation with a certain frequency 1/εσ are met simultaneously. Equation (1.1) determines the dynamics of an internal layer, and equation (1.1) together with (1.2) represents a basic model of globally coupled bistable medium which is relevant for current density dynamics in large area bistable semiconductor systems [11–14]. The internal layer has a physical reason as the current filament has a sharp profile with a narrow transition layer connecting flat on- and off-states.Whenε in (1.1) is sufficiently small for the case of without the spatial average, the singular limit analysis is applied to show the existence and the stability of localized radially symmetric equilibrium solutions [15, 16]. In one-dimensional space for the case of without the spatial average, such equilibrium solutions should undergo certain instabilities, and the loss of stability resulting from a Hopf bifurcation produces a kind of periodic oscillation in the location of the internal layers [2, 17–19]. As the parameter D varies, the stability of the spherically symmetric solutions and their symmetry-breaking bifurcations into layer solutions for the case of without the spatial average have been examined in [5, 6].In this paper, the free boundary problem of (1.1) with (1.2) for the case when ε=0 in two- and three-dimensional space will be studied. Suppose that there is only one (n-1)-dimensional hypersurface η(t) which is a single closed curve given in the domain Ω in such a way that Ω×(0,∞)=Ω+(t)∪η(t)∪Ω-(t), where Ω-(t)={(x,t)∈Ω×(0,∞):u(x,t)>a0} and Ω+(t)={(x,t)∈Ω×(0,∞):u(x,t)<a0}. When D=1 and ε=0 in (1.1), the spatial average 〈u〉 satisfies(1.4)μ〈u〉〈H(x-η)〉-〈v〉,〈H(x-η)〉=1|Ωn|∫ΩnH(x-η)dx=1-(nR)n,where|Ω2|=πR2 and |Ω3|=(4/3)πR3. The spatial average of v is a solution of 〈v〉′(t)=μ〈u〉-〈v〉=1-(η/R)n-2〈v〉. Equation of η(t) is given by (see [20, 21])(1.5)dη(t)dt⋅ν=C(vi),x∈η(t),whereν is the outward normal vector on η(t), vi is the value of v on the interface η(t), and C(v) is the velocity of the interface. The reaction terms (1.2) satisfy the bistable condition, that is, the nullclines of f(u,v)=0 and g(u,v)=0 must have three intersection points and the nullcline f(u,v)=0 is the triple-valued function of u which is called h+, h-, and h0. From [2, 20, 22], the trajectory with a unique value of C=C(v) exists which is given by C(v)=h+-2h0+h-. Furthermore, the velocity of the interface C(v) is a continuously differentiable function defined on an interval I:=(-a0,1-a0), and thus the velocity of the interface can be normalized by(1.6)C((v(η),η,〈v〉),t)=-1-2a0-2Vσ(V+a0)(1-a0-V),whereV=v(η)-((1-μ)/μ)(1-(η/R)n-〈v〉).An analysis of the dynamics of this process has been shown (see, e.g., [2, 5, 6, 15]) to lead a free boundary problem consisting of the initial-boundary value problem(1.7)vt=∇2v+g(h±,v),(x,t)∈Ω±(t),v(x,0)=v0(x),v(η(t)-0,t)=v(η(t)+0,t),ddνv(η(t)-0,t)=ddνv(η(t)+0,t),η′(t)=C((v(η),η,w),t),w′(t)=1-(ηR)n-2w,w(0)=w0,wherew=〈v〉.The organization of the paper is as follows. In Section2, a change of variables is given which regularizes problem (1.7) in such a way that results from the theory of nonlinear evolution equations can be applied. In this way, we obtain enough regularity of the solution for an analysis of the bifurcation. In Section 3, we show the existence of radially symmetric localized equilibrium solutions for (1.7) and obtain the linearization of problem (1.7). In the last section we show the existence of the periodic solutions and the bifurcation of the interface problem as a parameter σ varies in two and three dimensions.


## 2. Regularized System

We look for an existence problem of radially symmetric equilibrium solutions of (1.7) with |x|=r, where the center and the interface are located at the origin and r=η, respectively. The problem is given by(2.1)vt=∂2v∂r2+n-1r∂v∂r-(μ+1)v+μH(r-η(t))+(1-μ)(1-(η(t)R)n-w(t)),η′(t)=C((v(η),η,w),t),t>0,η(0)=η0,w′(t)=-2w(t)+1-(η(t)R)n,w(0)=w0,∂v∂rv(0,t)=0=∂v∂rv(R,t),0<r<R,t>0.As a first step we obtain more regularity for the solution by semigroup methods, consideringA:=-(∂2/∂r2)+((n-1)/r)(∂/∂r)+μ+1 as a densely defined operator A:D(A)⊂X→X, where D(A)={v∈H2,2((0,R)):(∂v/∂r)(0,t)=0=(∂v/∂r)(R,t)} and X:=L2((0,R)) with norm ∥·∥2.We defineg:[0,R]×[0,R]×ℂ→ℂ,(2.2)g(r,η,w):=A-1(μ(H(⋅-η))(r)+(1-μ)(1-(ηR)n-w))=μ∫ηRG(r,y)dy+1-μ1+μ(1-(ηR)n-w),andγ:[0,R]×ℂ→ℂ,(2.3)γ(η,w):=g(η,η,w).HereG:[0,R]2→ℝ is a Green's function of A satisfying the Neumann boundary conditions:  for n=2,(2.4)G(r,z)={zI0(r1+μ)(K0(z1+μ)+K1(R1+μ)I1(R1+μ)I0(z1+μ)),0<r<z,zI0(z1+μ)(K0(r1+μ)+K1(R1+μ)I1(R1+μ)I0(r1+μ)),z<r<R,and forn=3,(2.5)G(r,z)={1YRzsinh⁡(r1+μ)r1+μ(R1+μcosh⁡((R-z)1+μ)-sinh⁡((R-z)1+μ)),0<r<z1YRzsinh⁡(z1+μ)r1+μ(R1+μcosh⁡((R-r)1+μ)-sinh⁡((R-r)1+μ)),z<r<R,whereIi and Ki are modified Bessel functions (i=0,1) and YR is given by(2.6)YR=1+μ(R1+μcosh⁡(R1+μ)-sinh⁡(R1+μ)).Moreover,g(·,η,w)∈D(A) for all η and w, (∂g/∂η)(r,η,w)∈H1,∞((0,R)2×ℂ) and γ∈C∞([0,R]×ℂ).Applying the transformationu(t)(r)=v(r,t)-g(r,η(t),w(t)), then we obtain an equivalent abstract evolution equation of(2.1)(2.7)ddt(u,η,w)+Ã(u,η,w)=F(u,η,w),(u,η,w)(0)=(u0,η0,w0),whereÃ is a 3×3 matrix defined on D(Ã)=D(A)×(0,R)×ℂ and given by(2.8)(A02(1-μ)1+μ000002)The nonlinear forcing termF defined on the set W:={(u,η,w)∈C1([0,R])×(0,R)×ℂ:u(η)+γ(η,w)-((1-μ)/μ)(1-(η/R)n-w)∈I}⊂openC1([0,R])×(0,R)×ℂ as(2.9)F(u,η,w)=(f2(u,η,w)⋅(f1(η)+1-μ1+μnRnηn-1)+1-μ1+μ(1-(ηR)n)f2(u,η,w)1-(ηR)n),wheref1:(0,R)→X,f1(η)(r):=μG(r,η), and f2:W→ℂ,f2(u,η,w):=C(u(η),η,w). We define ξ(η):=μ∫ηRG(η,y)dyand then γ(η,w)=ξ(η)+((1-μ)/(1+μ))(1-(η/R)n-w). Let χ(u,η):=u(η)+ξ(η)-((1-μ)/(1+μ))(η/R)n+((1-μ)/μ)(η/R)n, then the velocity of η is written by(2.10)C(u(η),η,w)=C(χ(u,η),w)=-1σ1-2a0-2(χ(u,η)-((1-μ)/μ(1+μ))(1-w))(a0+χ(u,η)-((1-μ)/μ(1+μ))(1-w))(1-a0-(χ(u,η)-((1-μ)/μ(1+μ))(1-w))).Lemma 2.1.
The functionsf1:(0,R)→X, f2:W→ℂ and F:W→X×ℂ×ℝ are continuously differentiable with derivatives given by
(2.11)f1′(η)=μ∂G∂z(⋅,η),Df2(u,η,w)(û,η̂,ŵ)=Cχ(χ(u,η),w)(u′(η)η̂+û(η)+ξ′(η)η̂+1-μμ(1+μ)nηn-1Rnη̂)+Cw(χ(u,η),w)ŵ,DF(u,η,w)(û,η̂,ŵ)=f2(u,η,w)⋅(f1′(η)+1-μ1+μn(n-1)Rnηn-2,0,0)η̂+Df2(u,η,w)(û,η̂,ŵ)⋅(f1(η)+1-μ1+μnRnηn-1,1,0)+(-1-μ1+μnRnηn-1,0,-nRnηn-1)η̂,
where Cχ=∂C/∂χ and Cw=∂C/∂w.The well posedness of solutions was shown in [23] applying the semigroup theory using domains of fractional powers α∈(3/4,1] of A and Ã[24]. Moreover, they obtained that F:W∩D(Ãα)→X×ℂ×ℝ is a continuously differentiable function, where D(Ãα)=D(Aα)×(0,R)×ℂ.


## 3. Radially Symmetric Equilibrium Solutions and Linearization

The steady states are solutions of the following problem:(3.1)Au*=(μG(⋅,η*)+1-μ1+μnRn(η*)n-1)C(u*(η*),η*,w*)+1-μ1+μ(-2w*+1-(η*R)n),0=C(u*(η*)+γ(η*,w*)),0=-2w*+1-(η*R)n,u*′(0)=0=u*′(R)for(u*,η*,w*)∈D(Ã)∩W.Lemma 3.1.
Defineξ(η):=μ∫ηRG(η,y)dy<0. Then ξ′(η)<0, and ξ′(η)+μG(η,η)>0 for 0<η<R.Proof.
Forn=2, the derivative of ξ is given by
(3.2)ξ′(η)=μη(-I0(η1+μ)K0(η1+μ)+I1(η1+μ)K1(η1+μ))-μη(K1(R1+μ)I1(R1+μ)(I02(η1+μ)+I12(η1+μ))).Let h(η)=I0(η1+μ)K0(η1+μ)-I1(η1+μ)K1(η1+μ). Then limη→0h(η)=∞, and h(R)>0. The derivative of h is
(3.3)h′(η)=-1+μ((I0(η1+μ)+I1(η1+μ))K1(η1+μ)+2I1(η1+μ)K0(η1+μ))≤-21+μ(I1(η1+μ))(K1(η1+μ)-K0(η1+μ))≤0.Thus h(η)>0, and thus ξ′(η)<0 for 0<η<R. Moreover,
(3.4)ξ′(η)+μG(η,η)=ηI1(η1+μ)I1(R1+μ)(I1(R1+μ)K1(η1+μ)-I1(η1+μ)K1(R1+μ)).
Since Ii are increasing functions and Ki are decreasing functions (i=0,1),
(3.5)I1(R1+μ)K1(η1+μ)-I1(η1+μ)K1(R1+μ)=(I1(R1+μ)-I1(η1+μ))K1(η1+μ)+(K1(η1+μ)-K1(R1+μ))I1(η1+μ)>0,
and thus ξ′(η)+μG(η,η)>0 for 0<η<R. For n=3, the derivative of ξ is
(3.6)ξ′(η)=μM2(R1+μ+1)[1-(1-2η1+μ+2η2(1+μ))e2η1+μ]-μM2(R1+μ-1)[e2R1+μ-(1+2η1+μ+2η2(1+μ))e2(R-η)1+μ],
where M2=2η2(1+μ)1+μ[R1+μ+1+(R1+μ-1)e2R1+μ]. Since e2η1+μ≥(1+2η1+μ+2η2(1+μ)), we have ξ′(η)≤-(μ/M2)(1+R1+μ)4η4(1+μ2)<0 for 0<η<R. Moreover, ξ′(η)+μG(η,η)>2(μ/M2)(R-η)1+μ[(η1+μ+1)+(η1+μ-1)e2η1+μ]>0 for 0<η<R.Theorem 3.2.
Suppose that (i)0<(1/2)-a0<(2μ-1)/2μ and ξ′(η)+(1-μ)/2μ(1+μ)(n/Rn)ηn-1<0 or (ii) 0<(2μ-1)/2μ<(1/2)-a0 and ξ′(η)+((1-μ)/2μ(1+μ))(n/Rn)ηn-1>0. Then the stationary problem of (2.7) has the only stationary solution (u*,η*,w*) for all σ≠0 with u*=0 and 2w*=1-(η*/R)n,η*∈(0,R). The linearization of F at (0,η*,w*) is
(3.7)DF(0,η*,w*)(û,η̂,ŵ)=(4σ(û(η*)+γη(η*,w*)η̂+γw(η*,w*)ŵ+1-μμ(nRn(η*)n-1η̂+ŵ))⋅(μG(⋅,η*)+1-μ1+μnRn(η*)n-1)-1-μ1+μ(2ŵ+nRn(η*)n-1η̂)4σ(û(η*)+γη(η*,w*)η̂+γw(η*,w*)ŵ+1-μμ(nRnη*n-1η̂+ŵ))-2ŵ-nRnη*n-1η̂).
The pair (0,η*,w*) corresponds to a unique steady state (v*,η*,w*) of (2.1) for σ≠0 with v*(r)=g(r,η*,w*).Proof.
From the system (3.1), η* and w* are solutions of the following equations:
(3.8)u*=0,C(0,η*,w*)=02w*=1-(η*R)n.We only check the existence of η* of (2.10) and (3.8), and thus we let
(3.9)Γ(η):=12-a0-χ(0,η)+1-μμ(1+μ)(1-1-(η/R)n2).Then
(3.10)Γ(R)=12-a0,Γ(0)=12-a0-ξ(0)+1-μ2μ(1+μ),Γ′(η)=-ξ′(η)-1-μ2μ(1+μ)nRnηn-1.
Since ξ′(η)<0 and ξ(0)=μ/(1+μ) for n=2,3,there is a unique η*∈(0,R) when Γ′(η)<0, Γ(0)>0, Γ(R)<0, or Γ′(η)>0, Γ(0)<0,Γ(R)>0.
The formula forDF(0,η*,w*) follows from Lemma 2.1, the relation Cχ(0,η*,w*)=4/σ, and Cw(0,η*,w*)=4(1-μ)/σμ(1+μ). The corresponding steady state (v*,η*,w*) for (2.1) is obtained using the transformation and Theorem 2.1 in [17].Definition 3.3.
Suppose thata0 and μ satisfy 0<(1/2)-a0<(2μ-1)/2μ and ξ′(η)+((1-μ)/2μ(1+μ))(n/Rn)ηn-1<0. One defines (for 1≥α>3/4) the operator B that is a linear operator from D(Ãα) to D(Ã) as
(3.11)B:=σ4DF(0,η*,w*).
One then defines (0,η*,w*) to be a Hopf point for (2.7) if there exists an ϵ0>0 and a C1-curve
(3.12)(-ϵ0+τ*,τ*+ϵ0)⟼(λ(τ),ϕ(τ))∈C×D(Ã)C
(YC denotes the complexification of the real space Y) of eigendata for -Ã+τB such that (i)
(-Ã+τB)(ϕ(τ))=λ(τ)ϕ(τ),(-Ã+τB)(ϕ(τ)¯)=λ(τ)¯ϕ(τ)¯,(ii)
λ(τ*)=iβ with β>0,(iii)
Re(λ)≠0 for all λ in the spectrum of (-Ã+τ*B)∖{±iβ},(iv)
Reλ′(τ*)≠0 (transversality), where τ=4/σ.


## 4. Hopf Bifurcation Analysis

We will show that there is a Hopf bifurcation from the curveσ↦(0,η*,w*) of radially symmetric stationary solution. The linearized eigenvalue problem of (2.7) is(4.1)-Ã(u,η,w)+τB(u,η,w)=λI3(u,η,w),whereI3 is a 3 by 3 identity matrix. This is equivalent to(4.2)(A+λ)u=τ(u(η*)+γη(η*,w*)η+γw(η*,w*)w+1-μμ(w+nRn(η*)n-1η))⋅(μG(⋅,η*)+1-μ1+μnRn(η*)n-1)+1-μ1+μ(-2w-nRn(η*)n-1η),λη=τ(u(η*)+γη(η*,w*)η+γw(η*,w*)w+1-μμ(w+nRn(η*)n-1η)),λw=-2w-nRn(η*)n-1η.Our main theorem is stated as follows.Theorem 4.1.
Suppose thata0 and μ satisfy 0<(1/2)-a0<((2μ-1)/2μ) and ξ′(η)+((1-μ)/2μ(1+μ))(n/Rn)ηn-1<0, the problem (2.7), and (2.1), has a unique stationary solution (u*,η*,w*), where u*=0 and w*=(1/2)(1-(n/Rn)(η*)n-1), and (v*,η*,w*), respectively, for all τ>0. Then there exists a unique τ* such that the linearization -Ã+τ*B has a purely imaginary pair of eigenvalues β. The point (0,η*,w*,τ*) is then a Hopf point for (2.7), and there exists a C0-curve of nontrivial periodic orbits for (2.7) and (2.1), bifurcating from (0,η*,w*,τ*) and (v*,η*,w*,τ*), respectively.We will show the following three theorems that verify the above theorem. The next theorem shows that the steady state is the only Hopf point.Theorem 4.2.
Forτ*∈ℝ∖{0}, the operator -Ã+τ*B has a unique pair of purely imaginary eigenvalues {±iβ}. Then the point (0,η*,w*,τ*) satisfies the conditions (i), (ii), and (iii) in Definition 3.3.Proof.
In the sequel, we denotebn=(n/Rn)(η*)n-1, n=2,3. We assume without loss of generality that β>0 and Φ* is the (normalized) eigenfunction of -Ã+τ*B with eigenvalue iβ. We have to show that (Φ*,iβ) can be extended to a C1-curve τ↦(ϕ(τ),λ(τ)) of eigendata for -Ã+τB with Re(λ′(τ*))≠0. For this, let Φ*:=(ψ0,η0,w0)∈D(Ã). First, we note that if w0=0, then  η0=0 (vice versa) in the last equation of (4.2). We see that η0≠0 and w0≠0, for otherwise, by (4.2), (A+iβ)ψ0=iβ(μG(·,η*)η0+((1-μ)/(1+μ))bn+((1-μ)/(1+μ))w0)=0, which is not possible because A is symmetric. So without loss of generality, let η0=1. Define
(4.3)E:D(A)C×C×C×R⟶XC×C×C,E(u,w,λ,τ)≔((A+λ)u-τ(u(η*)+γη+γww+1-μμ(bn+w))(μG(⋅,η*)+1-μ1+μbn)+1-μ1+μ(2w+bn)λ-τ(u(η*)+γη+γww+1-μμ(bn+w))λw+2w+bn).
The equation E(u,w,λ,τ)=0 is equivalent to λ being an eigenvalue of -Ã+τB with eigenfunction (u,1,w). By (4.2), we have E(ψ0,w0,iβ,τ*)=0 which is equivalent to
(4.4)(A+iβ)ψ0=iβ(μG(⋅,η*)+1-μ1+μ(bn+w0)),iβ=τ*(ψ0(η*)+γη+γww0+1-μμ(bn+w0)),iβw0=-2w0-bn.
To apply the implicit function theorem to E, we have to check that E is in C1 and that
(4.5)D(u,w,λ)E(ψ0,w0,iβ,τ*)∈L(D(A)C×C×C,XC×C×C)isanisomorphism.
In addition, the mapping
(4.6)D(u,w,λ)E(ψ0,w0,iβ,τ*)(û,ŵ,λ̂)=((A+iβ)û+λ̂ψ0-τ*(û(η*)+γwŵ+1-μμŵ)(μG(⋅,η*)+1-μ1+μbn)+21-μ1+μŵλ̂-τ*(û(η*)+γwŵ+1-μμŵ)λ̂w0+iβŵ+2ŵ)
is a compact perturbation of the mapping
(4.7)(û,ŵ,λ̂)⟼((A+iβ)û,ŵ,λ̂)
which is invertible. In order to verify (4.5), it suffices to show that the system
(4.8)D(u,w,λ)E(ψ0,w0,iβ,τ*)(û,ŵ,λ̂)=0,
which are
(4.9)(A+iβ)û+λ̂ψ0=τ*(û(η*)+γwŵ+1-μμŵ)(μG(⋅,η*)+1-μ1+μbn)-21-μ1+μŵ,λ̂=τ*(û(η*)+γwŵ+1-μμŵ),λ̂w0=-iβŵ-2ŵ,
necessarily implies that û=0,ŵ=0, and λ̂=0. We define ϕ:=ψ0-(μG(·,η*)+((1-μ)/(1+μ))bn+((1-μ)/(1+μ))w0), then the first equation of (4.9) is given by
(4.10)(A+iβ)û+λ̂ϕ=iβ1-μ1+μŵ.
Since (v,η,w,λ)=(ψ0,1,w,iβ) solves (4.2), ϕ is a solution to the equation
(4.11)(A+iβ)ϕ=-μδη*-(1-μ)(bn+w0),(4.12)iβτ*=ϕ(η*)+1-μμ(bn+w0),(4.13)(2+iβ)w0=-bn.
Multiply (4.11) by rn-1(n=2,3) and integrate, then
(4.14)∫0R(-ϕrr-n-1rϕr+(μ+1+iβ)ϕ)rn-1dr=∫0R(-μδη*-(1-μ)(bn+w0))rn-1dr
which implies that
(4.15)∫0R(∂∂r(-rn-1ϕr)+(μ+1+iβ)∫rn-1ϕ)dr=-μ(η*)n-1-(1-μ)(bn+w0)Rnn.
Therefore we obtain
(4.16)(μ+1+iβ)∫rn-1ϕ=-μ(η*)n-1-(1-μ)(bn+w0)Rnn.
Multiply (4.10) by rn-1(n=2,3) and (4.11) by rn-1ϕ. Now we integrate the resulting equation to obtain
(4.17)(μ+1+iβ)∫rn-1û=-λ̂∫rn-1ϕ+iβ1-μ1+μ⋅Rnnŵ.
Multiply (4.10) and (4.11) by rn-1ϕ, and then eliminate the term λ̂(A+iβ)rϕ2. Integrating the resulting equation, we have(4.18)0=λ̂μ(η*)n-1ϕ(η*)+λ̂(1-μ)(bn+w0)∫rn-1ϕ+μ(η*)n-1û(η*)(μ+1+iβ)+(1-μ)(bn+w0)(μ+1+iβ)∫rn-1û+iβ1-μ1+μŵ(μ+1+iβ)∫rn-1ϕ=μ(η*)n-1(λ̂ϕ(η*)-iβ1-μ1+μŵ+(μ+1+iβ)(λ̂τ*-1-μμ(1+μ)ŵ))
by (4.16) and (4.17). Using (4.9), (4.12), and (4.13) in the above equation, then we obtain
(4.19)0=λ̂(μ+1+i2βτ*-(μG(η*,η*)+ξ′(η*))-1-μμbn+1-μμbn(2+iβ)2).
Suppose that λ̂≠0. Then the real and imaginary parts of (4.19) are given by
(4.20)0=μ+1τ*-(μG(η*,η*)+ξ′(η*))-(1-μ)bnμ⋅β4+9β2+12(4+β2)2,0=1τ*-1-μμ⋅2bn(4+β2)2.
From these equations, we have
(4.21)0=μG(η*,η*)+ξ′(η*)+(1-μ)bnμ⋅β4+9β2+10-2μ(4+β2)2.
This leads to a contradiction that the right hand side is positive for β>0 and 0<μ<1. Therefore, we should have λ̂=0. Thus ŵ=0 and û=0.Theorem 4.3.
Under the same condition as in Definition3.3, (0,η*,w*,τ*) satisfies the transversality condition. Hence this is a Hopf point for (2.7).Proof.
By implicit differentiation ofE(ψ0(τ),w(τ),λ(τ),τ)=0,
(4.22)D(u,w,λ)E(ψ0,w0,iβ,τ*)(ψ0′(τ*),w′(τ*),λ′(τ*))=((μG(η*,η*)+1-μ1+μbn)(ψ0(η*)+γη+γww0+1-μμbn+1-μμw′(τ*)))ψ0(η*)+γη+γww0+1-μμbn+1-μμw′(τ*))0).
This means that the functions ũ:=ψ0′(τ*),w̃:=w′(τ*), and λ̃:=λ′(τ*) satisfy the equations
(4.23)(A+iβ)ũ+λ̃ψ0-τ*(ũ(η*)+γww̃+1-μμw̃)(μG(η*,η*)+1-μ1+μbn)+21-μ1+μw̃=(ψ0(η*)+γη+γww0+1-μμbn+1-μμw0)(μG(η*,η*)+1-μ1+μbn),λ̃-τ*(ũ(η*)+γww̃+1-μμw̃)=ψ0(η*)+γη+γww0+1-μμ(bn+w0),λ̃w0+(2+iβ)w̃=0.
By letting ϕ:=ψ0-(μG(·,η)+((1-μ)/(1+μ))bn+((1-μ)/(1+μ))w0), then
(4.24)(A+iβ)ũ+λ̃ϕ=iβ1-μ1+μw̃.
From (4.9) and (4.23), we obtain
(4.25)λ̃τ*=ũ(η*)+(γw+1-μμ)w̃+iβτ*2.
We now multiply (4.24) by rn-1ϕ and (4.11) by rn-1ũ and then subtract these two equations, then we get
(4.26)λ̃rn-1ϕ2=μrn-1ũ(r)δη*+(1-μ)(bn+w0)rn-1ũ+iβ1-μ1+μw̃rn-1ϕ.
Comparing to (4.11) and then integrating, we have
(4.27)0=-μ(η*)n-1ϕ(η*)λ̃-λ̃(1-μ)(bn+w0)∫rn-1ϕ-μ(η*)n-1ũ(η*)(μ+1+iβ)-(1-μ)(bn+w0)(μ+1+iβ)∫rn-1ũ-iβ1-μ1+μw̃(μ+1+iβ)∫rn-1ϕ.
Using (4.16) and (4.17) in the above equation, then
(4.28)0=μ(η*)n-1(λ̃ϕ(η*)+(μ+1+iβ)ũ(η*)-iβ1-μ1+μw̃).
By substituting (4.12), (4.13), and (4.23), we have
(4.29)(μ+1+iβ)iβτ*2=λ̃(μ+1+2iβτ*-μG(η*,η*)-ξ′(η*)-1-μμbn+1-μμbn(2+iβ)2).
The real part of λ̃ is given by
(4.30)βτ*((μ+1)2+β2)Reλ̃=|λ̃|2((μ+1)Q-βP),
where
(4.31)P=μ+1τ*-μG(η*,η*)-ξ′(η*)-1-μμbn+(1-μ)bnμ⋅4-β2(4+β2)2,Q=2βτ*-1-μμ⋅4bnβ(4+β2)2,(μ+1)Q-βP=(μ+1)βτ*+β(μG(η*,η*)+ξ′(η*))+(1-μ)bnβμ(4+β2)2(β4+9β2+8-4μ).
Since (μ+1)Q-βP>0 for 0<μ<1, we have Reλ̃>0. Therefore, Reλ′(τ*)>0 for β>0 and for 0<μ<1, and thus by the Hopf-bifurcation theorem in [17], there exists a family of periodic solutions which bifurcates from the stationary solution as τ passes τ*.The next theorem shows that a critical Hopf pointτ* exists uniquely.Theorem 4.4.
Under the same condition as in Definition3.3, there exists a unique, purely imaginary eigenvalue λ=iβ of (4.2) with β>0 for a unique critical point τ*>0 in order for (0,η*,w*,τ*) to be a Hopf point.Proof.
We only need to show that the function(u,β,τ)↦E(u,w,iβ,τ) has a unique zero with β>0 and τ>0. This means solving the system (4.2) with λ=iβ, η0=1, and ψ0=ϕ+μG(·,η*)+((1-μ)/(1+μ))bn+((1-μ)/(1+μ))w0,
(4.32)(A+iβ)ϕ=-μδη*-(1-μ)(bn+w0),iβτ*=ϕ(η*)+μG(η*,η*)+ξ′(η*)+1-μμ(bn+w0),(2+iβ)w0=-bn.
The second equation becomes
(4.33)iβτ*=-μGβ(η*,η*)+μG(η*,η*)+ξ′(η*)-1μ+1+iβ(1-μ)(bn+w0)+1-μμ(bn+w0),
where Gβ is a Green's function of the differential operator A+iβ. The real and imaginary parts of this above equation are given by
(4.34)βτ*=-μIm⁡Gβ(η*,η*)+(1-μ)bnβμ(4+β2)+(1-μ)bnβ(1-μ+β2)(4+β2)((1+μ)2+β2)=-μIm⁡Gβ(η*,η*)+(1-μ)bnβμ(4+β2)((1+μ)2+β2)((1+μ)β2+3μ+1),0=-μReGβ(μ*,μ*)+μG(η*,η*)+ξ′(η*)+(1-μ)bn(2+β2)μ(4+β2)-(1-μ)bn((μ+1)(2+β2)+β2)((μ+1)2+β2)(4+β2)=-μReGβ(μ*,μ*)+μG(η*,η*)+ξ′(η*)+(1-μ)bnβμ(4+β2)((1+μ)2+β2)×(β4+3β2+2(1+μ)).
Since ImGβ(η*,η*)<0 in [17, Lemma 12] and μ>0, there is a unique τ in the first equation if it does guarantee the existence of β. Now, we let
(4.35)T(β):=-μReGβ(μ*,μ*)+μG(η*,η*)+ξ′(η*)+(1-μ)bnβμ(4+β2)((1+μ)2+β2)×(β4+3β2+2(1+μ)),
then T(∞)=μG(η*,η*)+ξ′(η*)>0 and T(0)=ξ′(η*)+(1-μ)bn/2μ(1+μ)<0 by assumption. If we show that T′(β)<0, then the existence of β is proved:
(4.36)T′(β)=-μ(ReGβ(η*,η*))′+2(1-μ)bnβμ(4+β2)2((1+μ)2+β2)2×((μ2+2μ+2)β4+4(1+μ)(1+2μ)β2+2(1+μ)(1+4μ-μ2)).
Since 1+4μ-μ2>0 for 0<μ<1 and (ReGβ(η*,η*))′<0 in [17, Lemma 12], we have T′(β)>0 for all β>0.There is a unique pure imaginary eigenvalueβ>0 and the critical point τ* of (2.1) and thus there exists a family of periodic solutions which bifurcates from the stationary solution as τ passes τ* under the condition of Theorem 4.4. Thus we also found the relationship between μ and a0 for which Hopf bifurcation occurs for the problem (2.1).



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*Source: 529869-2010-11-29.xml*