Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2010, Article ID 529869, 18 pages
doi:10.1155/2010/529869
Research Article
Hopf Bifurcation with the Spatial
Average of an Activator in a Radially Symmetric
Free Boundary Problem
YoonMee Ham
Department of Mathematics, Kyonggi University, Suwon 443-760, Republic of Korea
Correspondence should be addressed to YoonMee Ham, ymham@kyonggi.ac.kr
Received 5 August 2010; Accepted 21 October 2010
Academic Editor: Oded Gottlieb
Copyright q 2010 YoonMee Ham. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
An interface problem derived by a bistable reaction-diffusion system with the spatial average
of an activator is studied on an n-dimensional ball. We analyze the existence of the radially
symmetric solutions and the occurrence of Hopf bifurcation as a parameter varies in two and threedimensional spaces.
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 reactiondiffusion system with a sufficiently small positive constant ε 5, 6
σεut ε2 ∇2 u fu, v,
vt D∇2 v gu, v t > 0, x ∈ Ω,
0
1.1
∂u ∂v
,
∂ν ∂ν
where ε, 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.
2
Mathematical Problems in Engineering
The nonlinear functions are
fu, v −u Hu − a0 − v 1 − μ u,
gu, v μu − v,
1.2
where 0 < μ < 1 and u denotes the spatial average, describing a global feedback effect,
namely,
u 1
|Ω|
Ω
u dΩ 1
|Ω|
Ω
ux, tdx.
1.3
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.
When ε 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 − 1dimensional hypersurface ηt which is a single closed curve given in the domain Ω in such
a way that Ω × 0, ∞ Ω t ∪ ηt ∪ Ω− t, where Ω− t {x, t ∈ Ω × 0, ∞ : ux, t > a0 }
and Ω t {x, t ∈ Ω × 0, ∞ : ux, t < a0 }. When D 1 and ε 0 in 1.1, the spatial
average u satisfies
μu H x − η − v,
H x−η 1
|Ωn |
n n
H x − η dx 1 −
,
R
Ωn
1.4
where |Ω2 | πR2 and |Ω3 | 4/3πR3 . The spatial average of v is a solution of v t μu − v 1 − η/Rn − 2v. Equation of ηt is given by see 20, 21
dηt
· ν Cvi ,
dt
x ∈ ηt,
1.5
Mathematical Problems in Engineering
3
where ν is the outward normal vector on ηt, vi is the value of v on the interface ηt, and
Cv is the velocity of the interface. The reaction terms 1.2 satisfy the bistable condition,
that is, the nullclines of fu, v 0 and gu, v 0 must have three intersection points and
the nullcline fu, 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 Cv exists which is given by Cv h − 2h0 h− . Furthermore, the velocity of the interface Cv 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 − 2a0 − 2V
,
C v η , η, v , t − σ V a0 1 − a0 − V 1.6
where V vη − 1 − μ/μ1 − η/Rn − 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
vt ∇2 v g h± , v ,
x, t ∈ Ω± t,
vx, 0 v0 x,
v ηt − 0, t v ηt 0, t ,
d d v ηt − 0, t v ηt 0, t ,
dν
dν
η t C v η , η, w , t ,
w t 1 −
η n
R
− 2w,
1.7
w0 w0 ,
where w v.
The organization of the paper is as follows. In Section 2, 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.
4
Mathematical Problems in Engineering
The problem is given by
vt ηt n
∂2 v n − 1 ∂v −
μ
1
v
μH
r
−
ηt
1
−
μ
1
−
−
wt
,
r ∂r
R
∂r 2
η t C v η , η, w , t , t > 0, η0 η0 ,
w t −2wt 1 −
ηt
R
∂v
∂v
v0, t 0 vR, t,
∂r
∂r
2.1
n
,
w0 w0 ,
0 < r < R, t > 0.
As a first step we obtain more regularity for the solution by semigroup methods,
considering A : −∂2 /∂r 2 n − 1/r∂/∂r μ 1 as a densely defined operator
A : DA ⊂ X → X, where DA {v ∈ H 2,2 0, R : ∂v/∂r0, t 0 ∂v/∂rR, t} and
X : L2 0, R with norm · 2 .
We define g : 0, R × 0, R × C → C,
η n
g r, η, w : A
−w
μ H · − η r 1 − μ 1 −
R
−1
μ
R
η
2.2
η n
1−μ
1−
G r, y dy −w ,
1μ
R
and γ : 0, R × C → C,
γ η, w : g η, η, w .
2.3
Here G : 0, R2 → R is a Green’s function of A satisfying the Neumann boundary conditions:
for n 2,
⎧
K1 R 1 μ ⎪
⎪
⎪
zI0 r 1 μ K0 z 1 μ I0 z 1 μ ,
⎪
⎪
I1 R 1 μ
⎨
Gr, z ⎪
⎪
⎪
K1 R 1 μ ⎪
⎪
I0 r 1 μ ,
⎩zI0 z 1 μ K0 r 1 μ I1 R 1 μ
0 < r < z,
z < r < R,
2.4
Mathematical Problems in Engineering
5
and for n 3,
⎧
1 sinh r 1 μ ⎪
⎪
R 1 μ cosh R − z 1 μ − sinh R − z 1 μ ,
z
⎪
⎪
⎪
Y
r 1μ
R
⎪
⎪
⎪
⎪
⎨
0<r<z
Gr, z ⎪
⎪
1 sinh z 1 μ ⎪
⎪
z
R 1 μ cosh R − r 1 μ − sinh R − r 1 μ ,
⎪
⎪
⎪
Y
r 1μ
⎪
⎪ R
⎩
z < r < R,
2.5
where Ii and Ki are modified Bessel functions i 0, 1 and YR is given by
YR 1 μ R 1 μ cosh R 1 μ − sinh R 1 μ
.
2.6
Moreover, g·, η, w ∈ DA for all η and w, ∂g/∂ηr, η, w ∈ H 1,∞ 0, R2 × C and γ ∈
C∞ 0, R × C.
Applying the transformation utr vr, t − gr, ηt, wt, then we obtain an
equivalent abstract evolution equation of2.1
d
u, η, w F u, η, w ,
u, η, w A
dt
u, η, w 0 u0 , η0 , w0 ,
2.7
is a 3 × 3 matrix defined on DA
DA × 0, R × C and given by
where A
⎞
⎛
2 1−μ
A
0
⎜
1μ ⎟
⎟
⎜
⎟
⎜
⎟
⎜0 0
0
⎠
⎝
0 0
2
2.8
The nonlinear forcing term F defined on the set W : {u, η, w ∈ C1 0, R × 0, R × C :
uη γη, w − 1 − μ/μ1 − η/Rn − w ∈ I} ⊂open C1 0, R × 0, R × C as
⎛ η n ⎞
1 − μ n n−1
1−μ
1−
⎟
⎜f2 u, η, w · f1 η 1 μ Rn η
1μ
R
⎟
⎜
⎟
⎜
⎟,
⎜
F u, η, w ⎜
f2 u, η, w
⎟
⎟
⎜
⎠
⎝
η n
1−
R
2.9
6
Mathematical Problems in Engineering
where f1 : 0, R → X, f1 ηr : μGr, η, and f2 : W → C, f2 u, η, w : Cuη, η, w.
R
We define ξη : μ η Gη, ydy and then γη, w ξη 1 − μ/1 μ1 − η/Rn − w.
Let χu, η : uη ξη − 1 − μ/1 μη/Rn 1 − μ/μη/Rn , then the velocity of
η is written by
C u η , η, w
C χ u, η , w
1−2a0 − 2 χ u, η − 1−μ /μ 1μ 1−w
1
− .
σ
a0 χ u, η − 1−μ /μ 1μ 1−w 1−a0 − χ u, η − 1−μ /μ 1μ 1−w
2.10
Lemma 2.1. The functions f1 : 0, R → X, f2 : W → C and F : W → X ×C×R are continuously
differentiable with derivatives given by
∂G f1 η μ
·, η ,
∂z
1 − μ nηn−1
η
, η,
w
Cχ χ u, η , w u η η u
η ξ η η Df2 u, η, w u
μ 1 μ Rn
Cw χ u, η , w w,
DF u, η, w u
, η,
w
f2 u, η, w ·
f1
1 − μ nn − 1 n−2
η , 0, 0 η
η 1 μ Rn
1 − μ n n−1
Df2 u, η, w u
η , 1, 0
, η,
w
· f1 η 1 μ Rn
1 − μ n n−1
n n−1
−
η , 0, − n η
η,
1 μ Rn
R
2.11
where Cχ ∂C/∂χ and Cw ∂C/∂w.
The well posedness of solutions was shown in 23 applying the semigroup theory
24. Moreover, they obtained
using domains of fractional powers α ∈ 3/4, 1 of A and A
α
α that F : W ∩ DA → X × C × R is a continuously differentiable function, where DA
α
DA × 0, R × C.
Mathematical Problems in Engineering
7
3. Radially Symmetric Equilibrium Solutions and Linearization
The steady states are solutions of the following problem:
Au∗ ∗ n ∗ ∗ ∗ ∗ 1 − μ
1 − μ n ∗ n−1
η
∗
μ G ·, η∗ ,
w
1
−
C
u
,
η
η
,
η
−2w
1 μ Rn
1μ
R
0 C u∗ η ∗ γ η ∗ , w ∗ ,
∗
0 −2w 1 −
η∗
R
n
,
u∗ 0 0 u∗ R
3.1
∩ W.
for u∗ , η∗ , w∗ ∈ DA
Lemma 3.1. Define ξη : μ
0 < η < R.
R
η
Gη, ydy < 0. Then ξ η < 0, and ξ η μGη, η > 0 for
Proof. For n 2, the derivative of ξ is given by
ξ η μη −I0 η 1 μ K0 η 1 μ I1 η 1 μ K1 η 1 μ
− μη
K1 R 1 μ
2
2
.
I0 η 1 μ I1 η 1 μ
I1 R 1 μ
3.2
Let hη I0 η 1 μK0 η 1 μ − I1 η 1 μK1 η 1 μ. Then limη → 0 hη ∞, and
hR > 0. The derivative of h is
h η − 1 μ
I0 η 1 μ I1 η 1 μ
K1 η 1 μ
2I1 η 1 μ K0 η 1 μ
3.3
≤ −2 1 μ I1 η 1 μ
K1 η 1 μ − K0 η 1 μ
≤ 0.
Thus hη > 0, and thus ξ η < 0 for 0 < η < R. Moreover,
ξ η μG η, η
I1 η 1 μ
η .
I1 R 1 μ K1 η 1 μ − I1 η 1 μ K1 R 1 μ
I1 R 1 μ
3.4
8
Mathematical Problems in Engineering
Since Ii are increasing functions and Ki are decreasing functions i 0, 1,
I1 R 1 μ K1 η 1 μ − I1 η 1 μ K1 R 1 μ
I1 R 1 μ − I 1 η 1 μ
K1 η 1 μ
3.5
K1 η 1 μ − K1 R 1 μ
I1 η 1 μ > 0,
and thus ξ η μGη, η > 0 for 0 < η < R. For n 3, the derivative of ξ is
√
μ
ξ η R 1 μ 1 1 − 1 − 2η 1 μ 2η2 1 μ e2η 1μ
M2
√
√ μ
−
R 1 μ − 1 e2R 1μ − 1 2η 1 μ 2η2 1 μ e2R−η 1μ ,
M2
3.6
√
√
2R 1μ
2η 1μ
2
2η
1
μ
1
μR
1
μ
1
R
1
μ
−
1e
.
Since
e
≥ 1 where
M
2
2η 1 μ 2η2 1 μ, we have ξ η ≤ −μ/M2 1 R 1 μ4η4 1 μ2 < 0 for 0√< η < R.
Moreover, ξ η μGη, η > 2μ/M2 R − η 1 μη 1 μ 1 η 1 μ − 1e2η 1μ > 0
for 0 < η < R.
Theorem 3.2. Suppose that (i) 0 < 1/2 − a0 < 2μ − 1/2μ and ξ η 1 − μ/2μ1 μn/Rn ηn−1 < 0 or (ii) 0 < 2μ−1/2μ < 1/2−a0 and ξ η1−μ/2μ1μn/Rn ηn−1 > 0.
0 with
Then the stationary problem of 2.7 has the only stationary solution u∗ , η∗ , w∗ for all σ /
u∗ 0 and 2w∗ 1 − η∗ /Rn ,η∗ ∈ 0, R. The linearization of F at 0, η∗ , w∗ is
DF 0, η∗ , w∗ u
, η,
w
⎛ ⎞
∗
∗ ∗
∗ ∗
1 − μ n ∗ n−1
4
u
η
,
w
,
w
η
w
γ
η
η
γ
η
w
η
η
w
⎜ σ
⎟
μ
Rn
⎜
⎟
⎜
⎟
⎜
⎟
∗ 1 − μ n ∗ n−1
1−μ
⎜
⎟
n ∗ n−1
⎜
⎟
· μG ·, η 2w
n η
η
−
η
n
⎜
⎟
1
μ
R
1
μ
R
⎜
⎟.
⎜
⎟
⎜ 4
⎟
∗
∗ ∗
∗ ∗
1
−
μ
n
⎜
⎟
η γη η , w η γw η , w w
η∗ n−1 η w
⎜ σ u
⎟
n
μ
R
⎜
⎟
⎝
⎠
n ∗ n−1
−2w
− nη
η
R
3.7
The pair 0, η∗ , w∗ corresponds to a unique steady state v∗ , η∗ , w∗ of 2.1 for σ /
0 with v∗ r ∗
∗
gr, η , w .
Proof. From the system 3.1, η∗ and w∗ are solutions of the following equations:
∗
u 0,
∗
C 0, η , w
∗
∗
0 2w 1 −
η∗
R
n
.
3.8
Mathematical Problems in Engineering
9
We only check the existence of η∗ of 2.10 and 3.8, and thus we let
1−μ
1
Γ η : − a0 − χ 0, η 2
μ 1μ
n 1 − η/R
1−
.
2
3.9
Then
ΓR 1
− a0 ,
2
Γ0 1−μ
1
− a0 − ξ0 ,
2
2μ 1 μ
Γ η −ξ η −
1−μ
n n−1
η .
2μ 1 μ Rn
3.10
Since ξ η < 0 and ξ0 μ/1 μ for n 2, 3, there is a unique η∗ ∈ 0, R when Γ η < 0,
Γ0 > 0, ΓR < 0, or Γ η > 0, Γ0 < 0, ΓR > 0.
The formula for DF0, η∗ , w∗ follows from Lemma 2.1, the relation Cχ 0, η∗ , w∗ 4/σ, and Cw 0, η∗ , w∗ 41 − μ/σμ1 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 that a0 and μ satisfy 0 < 1/2 − a0 < 2μ − 1/2μ and ξ η 1 −
μ/2μ1 μn/Rn ηn−1 < 0. One defines for 1 ≥ α > 3/4 the operator B that is a linear
as
α to DA
operator from DA
B :
σ
DF 0, η∗ , w∗ .
4
3.11
One then defines 0, η∗ , w∗ to be a Hopf point for 2.7 if there exists an 0 > 0 and a C1 -curve
−0 τ ∗ , τ ∗ 0 −→ λτ, φτ ∈ C × D A
C
3.12
τB such that
YC denotes the complexification of the real space Y of eigendata for −A
τBφτ λτφτ, −A
τBφτ λτ φτ,
i −A
ii λτ ∗ iβ with β > 0,
τ ∗ B \ {±iβ},
iii Reλ /
0 for all λ in the spectrum of −A
0 transversality,
iv Re λ τ ∗ /
where τ 4/σ.
4. Hopf Bifurcation Analysis
We will show that there is a Hopf bifurcation from the curve σ → 0, η∗ , w∗ of radially
symmetric stationary solution. The linearized eigenvalue problem of 2.7 is
u, η, w τB u, η, w λI3 u, η, w ,
−A
4.1
10
Mathematical Problems in Engineering
where I3 is a 3 by 3 identity matrix. This is equivalent to
1 − μ
n n−1 w n η∗
η
A λu τ u η∗ γη η∗ , w∗ η γw η∗ , w∗ w μ
R
1 − μ n ∗ n−1
1 − μ
n n−1 −2w − n η∗
· μG ·, η∗ η ,
η
n
1μR
1μ
R
1 − μ
n n−1 w n η∗
λη τ u η∗ γη η∗ , w∗ η γw η∗ , w∗ w η ,
μ
R
λw −2w −
4.2
n ∗ n−1
η.
η
Rn
Our main theorem is stated as follows.
Theorem 4.1. Suppose that a0 and μ satisfy 0 < 1/2 − a0 < 2μ − 1/2μ and ξ η 1 −
μ/2μ1 μn/Rn ηn−1 < 0, the problem 2.7, and 2.1, has a unique stationary solution
u∗ , η∗ , w∗ , where u∗ 0 and w∗ 1/21 − n/Rn η∗ n−1 , and v∗ , η∗ , w∗ , respectively, for all
τ ∗ B has a purely imaginary pair
τ > 0. Then there exists a unique τ ∗ such that the linearization −A
∗
∗ ∗
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.
τ ∗ B has a unique pair of purely imaginary
Theorem 4.2. For τ ∗ ∈ R \ {0}, the operator −A
∗
∗ ∗
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 denote bn n/Rn η∗ n−1 , n 2, 3. We assume without loss of
τ ∗ B with eigenvalue
generality that β > 0 and Φ∗ is the normalized eigenfunction of −A
∗
1
iβ. We have to show that Φ , iβ can be extended to a C -curve τ → φτ, λτ of eigendata
First, we note that if
τB with Reλ τ ∗ /
0. For this, let Φ∗ : ψ0 , η0 , w0 ∈ DA.
for −A
0 and w0 /
0, for
w0 0, then η0 0 vice versa in the last equation of 4.2. We see that η0 /
otherwise, by 4.2, Aiβψ0 iβ μG·, η∗ η0 1−μ/1μbn 1−μ/1μ w0 0,
which is not possible because A is symmetric. So without loss of generality, let η0 1. Define
E : DAC × C × C × R −→ XC × C × C,
Eu, w, λ, τ
⎛
∗
∗ 1 − μ
1−μ
λu
−
τ
u
η
b
γ
w
w
μG
·,
η
γ
A
b
η
w
n
n
⎜
μ
1μ
⎜
⎜
1−μ
⎜
⎜
2w bn : ⎜
1
μ
⎜
⎜
∗
1−μ
⎜
λ
−
τ
u
η
γ
w
w
γ
b
η
w
n
⎝
μ
λw 2w bn
⎞
⎟
⎟
⎟
⎟
⎟
⎟.
⎟
⎟
⎟
⎠
4.3
Mathematical Problems in Engineering
11
τB with
The equation Eu, w, λ, τ 0 is equivalent to λ being an eigenvalue of −A
∗
eigenfunction u, 1, w. By 4.2, we have Eψ0 , w0 , iβ, τ 0 which is equivalent to
∗ 1 − μ
A iβ ψ0 iβ μG ·, η bn w0 ,
1μ
∗
1−μ
∗
iβ τ ψ0 η γη γw w0 bn w0 ,
μ
4.4
iβw0 −2w0 − bn .
To apply the implicit function theorem to E, we have to check that E is in C1 and that
Du,w,λ E ψ0 , w0 , iβ, τ ∗ ∈ LDAC × C × C, XC × C × C is an isomorphism.
4.5
In addition, the mapping
, w,
λ
Du,w,λ E ψ0 , w0 , iβ, τ ∗ u
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
0
A iβ u
λψ
1−μ
1−μ
1−μ
w
bn 2
w
μG ·, η∗ −τ ∗ u
η∗ γw w
μ
1μ
1μ
1−μ
w
η∗ γw w
λ − τ ∗ u
μ
0 iβw
λw
2w
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
4.6
is a compact perturbation of the mapping
u
, w,
λ −
→ A iβ u
, w,
λ
4.7
which is invertible. In order to verify 4.5, it suffices to show that the system
Du,w,λ E ψ0 , w0 , iβ, τ ∗ u
, w,
λ 0,
4.8
which are
∗ 1 − μ
∗
1−μ
1−μ
∗
A iβ u
λψ0 τ u
w
μG ·, η bn − 2
w,
η γw w
μ
1μ
1μ
1−μ
λ τ ∗ u
w
,
η∗ γw w
μ
0 −iβw
− 2w,
λw
4.9
12
Mathematical Problems in Engineering
necessarily implies that u
0, w
0, and λ 0. We define φ : ψ0 − μG·, η∗ 1 − μ/1 μbn 1 − μ/1 μw0 , then the first equation of 4.9 is given by
iβ 1 − μ w.
A iβ u
λφ
1μ
4.10
Since v, η, w, λ ψ0 , 1, w, iβ solves 4.2, φ is a solution to the equation
A iβ φ −μδη∗ − 1 − μ bn w0 ,
4.11
1−μ
iβ
φ η∗ bn w0 ,
∗
τ
μ
4.12
2 iβ w0 −bn .
4.13
Multiply 4.11 by r n−1 n 2, 3 and integrate, then
R
−φrr −
0
R
n−1
φr μ 1 iβ φ r n−1 dr −μδη∗ − 1 − μ bn w0 r n−1 dr
r
0
4.14
which implies that
R
0
∂ n−1 −r φr μ 1 iβ
∂r
n−1 Rn
r n−1 φ dr −μ η∗
− 1 − μ bn w0 .
n
4.15
Therefore we obtain
μ 1 iβ
n−1 Rn
− 1 − μ bn w0 .
r n−1 φ −μ η∗
n
4.16
Multiply 4.10 by r n−1 n 2, 3 and 4.11 by r n−1 φ. Now we integrate the resulting equation
to obtain
n−1
1 − μ Rn
·
w.
−λ r n−1 φ iβ
μ 1 iβ
r u
4.17
1μ n
iβrφ2 .
Multiply 4.10 and 4.11 by r n−1 φ, and then eliminate the term λA
Integrating the resulting equation, we have
η∗ n−1 φ η∗ λ 1 − μ bn w0 0 λμ
n−1 ∗ u
η μ 1 iβ
r n−1 φ μ η∗
n−1
1−μ w
μ 1 iβ
r φ
1μ
∗ n−1
λ
∗
1−μ
1−μ
μ η
w
μ 1 iβ
− λφ η − iβ
w
1μ
τ∗ μ 1 μ
1 − μ bn w0 μ 1 iβ
iβ
r n−1 u
4.18
Mathematical Problems in Engineering
13
by 4.16 and 4.17. Using 4.9, 4.12, and 4.13 in the above equation, then we obtain
0 λ
μ 1 i2β ∗ ∗ ∗ 1 − μ
1−μ
bn
bn − μG η , η ξ η
−
τ∗
μ
μ 2 iβ2
.
4.19
Suppose that λ / 0. Then the real and imaginary parts of 4.19 are given by
1 − μ bn β4 9β2 12
μ 1 ∗ ∗ ∗ · − μG η , η ξ η
0
−
2 ,
τ∗
μ
4 β2
1−μ
1
2bn
0 ∗ −
·
2 .
τ
μ
4 β2
4.20
From these equations, we have
1 − μ bn β4 9β2 10 − 2μ
∗ ∗ ∗
·
0 μG η , η ξ η .
2
μ
4 β2
4.21
This leads to a contradiction that the right hand side is positive for β > 0 and 0 < μ < 1.
Therefore, we should have λ 0. Thus w
0 and u
0.
Theorem 4.3. Under the same condition as in Definition 3.3, 0, η∗ , w∗ , τ ∗ satisfies the transversality condition. Hence this is a Hopf point for 2.7.
Proof. By implicit differentiation of Eψ0 τ, wτ, λτ, τ 0,
Du,w,λ E ψ0 , w0 , iβ, τ ∗ ψ0 τ ∗ , w τ ∗ , λ τ ∗ ⎞
⎛
∗ ∗ 1 − μ
∗
1−μ
1−μ ∗
μ
G
η
b
b
w
,
η
γ
w
ψ
η
γ
τ
n
0
η
w 0
n
⎟
⎜
1μ
μ
μ
⎟
⎜
⎟
⎜
⎟
⎜
∗
1−μ
1−μ ∗
⎜
⎟.
⎟
⎜
bn w τ ψ0 η γη γw w0 ⎟
⎜
μ
μ
⎠
⎝
0
4.22
14
Mathematical Problems in Engineering
: w τ ∗ , and λ : λ τ ∗ satisfy the equations
This means that the functions u
: ψ0 τ ∗ , w
1−μ
1−μ
1−μ
0 − τ∗ u
w
μ G η∗ , η∗ bn 2
w
η∗ γw w
A iβ u
λψ
μ
1μ
1μ
1−μ
1−μ
1−μ
ψ0 η∗ γη γw w0 bn w0
bn ,
μG η∗ , η∗ μ
μ
1μ
1−μ
1−μ
w
ψ0 η∗ γη γw w0 λ − τ ∗ u
η∗ γw w
bn w0 ,
μ
μ
0 2 iβ w
0.
λw
4.23
By letting φ : ψ0 − μG·, η 1 − μ/1 μbn 1 − μ/1 μw0 , then
iβ 1 − μ w.
A iβ u
λφ
1μ
4.24
∗
iβ
1−μ
λ
w
∗2 .
u
η
γ
w
∗
τ
μ
τ
4.25
From 4.9 and 4.23, we obtain
We now multiply 4.24 by r n−1 φ and 4.11 by r n−1 u
and then subtract these two equations,
then we get
1 − μ n−1
n−1 φ2 μr n−1 u
wr
rδη∗ 1 − μ bn w0 r n−1 u
iβ
φ.
λr
1μ
4.26
Comparing to 4.11 and then integrating, we have
n−1 ∗ φ η λ − λ 1 − μ bn w0 0 −μ η∗
− 1 − μ bn w0 μ 1 iβ
r
n−1
n−1 ∗ u
η μ 1 iβ
r n−1 φ − μ η∗
1−μ w
μ 1 iβ
u
− iβ
1μ
4.27
r
n−1
φ.
Using 4.16 and 4.17 in the above equation, then
n−1
∗
1−μ
η∗ μ 1 iβ u
λφ
0 μ η∗
w
.
η − iβ
1μ
4.28
By substituting 4.12, 4.13, and 4.23, we have
iβ
μ 1 iβ ∗ 2 λ
τ
1−μ
μ 1 2iβ
1−μ
bn
bn − μG η∗ , η∗ − ξ η∗ −
∗
τ
μ
μ 2 iβ2
.
4.29
Mathematical Problems in Engineering
15
The real part of λ is given by
2 2
β 2
λ μ 1 Q − βP ,
Re
λ
β
μ
1
∗
τ
4.30
where
1 − μ bn
∗ ∗ ∗ 1 − μ
4 − β2
μ1
b
·
P
−
μG
η
,
η
−
ξ
η
−
n
2 ,
τ∗
μ
μ
4 β2
2β 1 − μ
4bn β
·
−
2 ,
∗
τ
μ
4 β2
1 − μ bn β 4
μ1 β
∗ ∗ ∗ 2
β
β
μG
η
,
η
9β
8
−
4μ
.
ξ
η
μ 1 Q − βP ∗
2
τ
μ 4 β2
4.31
Q
Since μ 1Q − βP > 0 for 0 < μ < 1, we have Re λ > 0. Therefore, Re λ 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 point τ ∗ exists uniquely.
Theorem 4.4. Under the same condition as in Definition 3.3, 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 → Eu, 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 μbn 1 − μ/1 μw0 ,
A iβ φ −μδη∗ − 1 − μ bn w0 ,
1−μ
iβ
φ η∗ μG η∗ , η∗ ξ η∗ bn w0 ,
∗
τ
μ
2 iβ w0 −bn .
4.32
The second equation becomes
iβ
1−μ
1
−μGβ η∗ , η∗ μG η∗ , η∗ ξ η∗ −
1 − μ bn w0 bn w0 ,
∗
τ
μ 1 iβ
μ
4.33
16
Mathematical Problems in Engineering
where Gβ is a Green’s function of the differential operator Aiβ. The real and imaginary parts
of this above equation are given by
1 − μ bn β 1 − μ β 2
1 − μ bn β
∗ ∗
β
−μ Im Gβ η , η 2
τ∗
μ 4 β2
4 β2
1 μ β2
1 − μ bn β
−μ Im Gβ η∗ , η∗ 1 μ β2 3μ 1 ,
2
μ 4 β2
1 μ β2
1 − μ bn 2 β 2
∗
0 −μ Re Gβ μ , μ μG η , η ξ η μ 4 β2
1 − μ bn μ 1 2 β 2 β 2
−
2
μ 1 β2 4 β2
∗
∗
−μ Re Gβ μ , μ
∗
∗
∗
∗
∗
μG η , η
∗
4.34
1 − μ bn β
∗
ξ η 2
μ 4 β2
1 μ β2
× β4 3β2 2 1 μ .
Since Im Gβ 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
1 − μ bn β
T β : −μ Re Gβ μ∗ , μ∗ μG η∗ , η∗ ξ η∗ 2
μ 4 β2
1 μ β2
× β4 3β2 2 1 μ
4.35
,
then T ∞ μ Gη∗ , η∗ ξ η∗ > 0 and T 0 ξ η∗ 1 − μbn /2μ1 μ < 0 by assumption.
If we show that T β < 0, then the existence of β is proved:
2 1 − μ bn β
∗ ∗ T β −μ Re Gβ η , η
2
2 2
μ 4 β2
1 μ β2
×
μ2 2μ 2 β4 4 1 μ 1 2μ β2 2 1 μ 1 4μ − μ2 .
4.36
Since 14μ−μ2 > 0 for 0 < μ < 1 and Re Gβ η∗ , η∗ < 0 in 17, Lemma 12, we have T β > 0
for all β > 0.
There is a unique pure imaginary eigenvalue β > 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.
Mathematical Problems in Engineering
17
Acknowledgment
This paper was supported by Kyonggi University Grant 2008.
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