DYNAMIC BIFURCATION OF THE GINZBURGâLANDAU EQUATION ...
DYNAMIC BIFURCATION OF THE GINZBURG–LANDAU EQUATION TIAN MA, JUNGHO PARK, AND SHOUHONG WANG Abstract. We study in this article the bifurcation and stability of the solutions of the Ginzburg–Landau equation, using a notion of bifurcation called attractor bifurcation. We obtain in particular a full classification of the bifurcated attractor and the global attractor as λ crosses the first critical value of the linear problem. Bifurcations from the rest of the eigenvalues of the linear problem are obtained as well.
1. Introduction In this article, we consider the bifurcation of attractors and invariant sets of the complex Ginzburg–Landau (GL) equation, which reads ∂u − (α + iβ)4u + (σ + iρ)|u|2 u − λu = 0, ∂t where the unknown function u : Ω×[0, ∞) → C is a complex-valued function and Ω ⊂ Rn is an open, bounded, and smooth domain in Rn (1 ≤ n ≤ 3). The parameters α, β, σ, ρ, and λ are real numbers and (1.1)
(1.2)
α > 0,
σ > 0.
The initial condition for (1.1) is given by (1.3)
u(x, 0) = φ + iψ.
Also, (1.1) is supplemented with either the Dirichlet boundary condition, (1.4)
u|∂Ω = 0,
or the periodic boundary condition, (1.5)
Ω = (0, 2π)n and u is Ω-periodic.
The GL equation is an important equation in a number of scientific fields. It is directly related to the GL theory of superconductivity. In this context, the unknown function is the order parameter, the constants β and ρ are usually zero, and the bifurcation parameter λ is the GL parameter; see [11] and the references therein. 1991 Mathematics Subject Classification. 35, 37. Key words and phrases. Ginzburg–Landau equation, bifurcation, stability. The authors thank the two anonymous referees for their insightful comments. The work was supported in part by the Office of Naval Research, by the National Science Foundation, and by the National Science Foundation of China. 1
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T. MA, J. PARK, AND S. WANG
In fluid dynamics the GL equation is found, for example, in the study of Poiseuille flow, the nonlinear growth of convection rolls in the Rayleigh– Be´ nard problem and Taylor–Couette flow. In this case, the bifurcation parameter λ plays the role of a Reynolds number. The equation also arises in the study of chemical systems governed by reaction-diffusion equations. There are extensive studies from the mathematical point of view for the GL equation, and we refer in particular to [2, 3, 1, 5, 6, 7, 11] and the references therein for studies related to the global attractors, inertial manifolds, and soft and hard turbulences described by the GL equations. We study in this article the bifurcation and stability of the solutions of the complex GL equation. A nonlinear theory for this problem is established in this article using a notion of bifurcation called attractor bifurcation and its corresponding theorem developed recently by the authors in [9, 8]; see [10], a new book by two of the authors. The main objectives of this theory include (1) existence of bifurcation when the system parameter crosses some critical numbers, (2) dynamic stability of bifurcated solutions, and (3) the structure/patterns and their stability and transitions in the physical space. More precisely, the main theorem associated with the attractor bifurcation states that as the control parameter crosses a certain critical value when there are m + 1 (m ≥ 0) eigenvalues across the imaginary axis, the system bifurcates from a trivial steady state solution to an attractor with dimension between m and m + 1, provided the critical state is asymptotically stable. There are a few important features of the attractor bifurcation. First, the bifurcation attractor does not include the trivial steady state and is stable; hence it is physically important. Second, the attractor contains a collection of solutions of the evolution equation, including possibly steady states and periodic orbits as well as homoclinic and heteroclinic orbits. Third, it provides a unified point of view on dynamic bifurcation and can be applied to many problems in physics and mechanics. Fourth, from the application point of view, the Krasnoselskii–Rabinowitz theorem requires the number of eigenvalues m + 1 crossing the imaginary axis being an odd integer, and the Hopf bifurcation is for the case where m + 1 = 2. However, the new attractor bifurcation theorem obtained can be applied to cases for all m ≥ 0. In addition, the bifurcated attractor, as mentioned earlier, is stable, which is another subtle issue for other known bifurcation theorems. For the GL equation, bifurcation is obtained with respect to the parameter λ, and the main results obtained can be summarized as follows. First, for the GL equation with the Dirichlet boundary condition, let λ1 be the first eigenvalue of the elliptic operator −4. Our main results in this case include the following.
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1. If λ ≤ αλ1 , the trivial solution u = 0 is globally asymptotically stable. The global attractor of the GL equation consists exactly of the trivial steady state solution u = 0. 2. As λ crosses αλ1 , i.e., there exists an ² > 0 such that for any αλ1 < λ < αλ1 +², the GL problem bifurcates from the trivial solution an attractor Σλ . The bifurcated attractor Σλ attracts the open set L2 (Ω, C)/Γ, where Γ is the stable manifold of u = 0 having codimension two in L2 (Ω, C). More detailed structure of this bifurcated attractor can be classified as follows (see Figures 1.1 and 1.2.) (a) If |β| + |ρ| 6= 0, then the bifurcated attractor consists of exactly one stable limiting cycle, i.e., Σλ = S 1 , which is asymptotically stable. The global attractor Aλ is a two-dimensional (2D) disk consisting of the stable limiting cycle Σλ = S 1 , the (unstable) trivial steady state solution u = 0, and orbits connecting Σλ = S 1 and u = 0. In particular, if β 6= 0, then the bifurcation is a Hopf bifurcation to a stable limiting cycle. (b) If β = ρ = 0, then the bifurcated attractor Σλ has dimension between 1 and 2 and is a limit of a sequence of 2D annulus Mk with Mk+1 ⊂ Mk , i.e., Σλ = ∩∞ k=1 Mk . Again in this case, the global attractor Aλ is 2D, consisting of Σλ , u = 0 and the connecting orbits between them. 1
αλ 1
Σλ
λ
Figure 1.1. Bifurcation diagram for the GL equation with the Dirichlet boundary condition: (1) bifurcation appears at λ = αλ1 , (2) bifurcated attractor Σλ = S 1 is the boundary of the shaded region, and (3) the global attractor Aλ is the 2D disk, shown as the shaded region. Here the dotted line stands for the unstable trivial solution u = 0. 1Using a different method, we can in fact prove that Σ is also homeomorphic to S 1 , λ
which shall be reported elsewhere.
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T. MA, J. PARK, AND S. WANG
Γ
Γ
Figure 1.2. Phase space structure in L2 (Ω, C) × {λ} in the case where |β| + |ρ| = 6 0. Here the bifurcated attractor Σλ = S 1 is a stable limiting cycle.
Second, for the GL equations equipped with the periodic boundary condition, similar results can be obtained as well. In particular, in the case where |β| + |ρ| = 6 0, we prove that the bifurcated attractor Σλ is a sphere S 1 , containing no steady state solutions, and the global attractor Aλ is a 2 dimensional ball consisting of the trivial steady state u = 0, Σλ , and the orbits connecting them. Finally, bifurcation from any eigenvalue of the Laplacian can also be obtained as for the first eigenvalue. It is worth mentioning that the complete structure of the global attractor for the bifurcations from the first eigenvalue is obtained, while no such information is available for bifurcations from the rest eigenvalues. Important work on lower and upper bounds of the global attractor of the GL equation, together with their physical mechanisms, was done in the 1980’s in [2, 3, 1]. As mentioned earlier, the main objective of this article is to study bifurcation and transitions from the trivial solution. Hence we focus only on the local attractor near the trivial solution, which is part of the global attractor. Of course, near the first eigenvalue, complete information for both the global attractor and the bifurcated attractor is obtained in this article. For λ near other eigenvalues, the results here demonstrate only the transitions of the trivial solution and provide some partial information on the low bounds of the global attractor. As far as the dimension of the global attractor is concerned, our results are consistent with the work in [2, 3, 1]. The paper is organized as follows. In Section 2, we recall the attractor bifurcation theory. Sections 3 and 4 study the bifurcation of the GL equations for the Dirichlet boundary condition and for the periocic boundary, respectively. Section 5 deals with bifurcation from the rest of the eigenvalues.
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2. Abstract Bifurcation Theory 2.1. Preliminary. We recall in this section a general theory on attractor bifurcation for nonlinear evolution equations; see [9, 8]. Let H and H1 be two Hilbert spaces and H1 ,→ H be a dense and compact inclusion. We consider the nonlinear evolution equations du (2.1) = Lλ u + G(u, λ), dt (2.2) u(0) = u0 , where u : [0, ∞) → H is the unknown function, λ ∈ R is the system parameter, and Lλ : H1 → H are parameterized linear completely continuous fields depending continuously on λ ∈ R1 , which satisfy Lλ = −A + Bλ is a sectorial operator, A : H1 → H a linear homeomorphism, (2.3) B : H → H the parameterized linear compact operators. 1 λ It is easy to see that Lλ generates an analytic semigroup {e−tLλ }t≥0 . Then we can define fractional power operators Lαλ for any 0 ≤ α ≤ 1 with domain Hα = D(Lαλ ) such that Hα1 ⊂ Hα2 if α1 > α2 , and H0 = H. Furthermore, we assume that the nonlinear terms G(·, λ) : Hα → H for some 1 > α ≥ 0 are a family of parameterized C r bounded operators (r ≥ 1) continuously depending on the parameter λ ∈ R1 , such that G(u, λ) = o(kukHα ) ∀ λ ∈ R1 .
(2.4)
Actually, in this paper we need only the following conditions for the operator Lλ = −A + Bλ , which ensure that Lλ is a sectorial operator. Let there be a eigenvalue sequence {ρk } ⊂ C and an eigenvector sequence {ek , hk } ⊂ H1 of A : Azk = ρk zk , zk = ek + ihk , Reρk → +∞ as k → ∞, (2.5) |Imρ /(Reρ + a)| ≤ C for some constants a, C > 0, k
k
such that {ek , hk } is a basis of H. Condition (2.5) implies that A is a sectorial operator. Hence we can define fractional power operator Aα with domain Hα = D(Aα ). Then for the operator Bλ : H1 −→ H, we assume that there is a constant 0 ≤ θ < 1 such that (2.6)
Bλ : Hθ −→ H bounded ∀ λ ∈ R.
Let {Sλ (t)}t≥0 be an operator semigroup generated by the equation (2.1) which enjoys the following properties: (i) For any t ≥ 0, Sλ (t) : H → H is a linear continuous operator. (ii) Sλ (0) = I : H → H is the identity on H and (iii) For any t, s ≥ 0, Sλ (t + s) = Sλ (t) · Sλ (s).
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T. MA, J. PARK, AND S. WANG
Then the solution of (2.1) and (2.2) can be expressed as u(t) = Sλ (t)u0 ,
t ≥ 0.
Definition 2.1. A set Σ ⊂ H is called an invariant set of (2.1) if S(t)Σ = Σ for any t ≥ 0. An invariant set Σ ⊂ H of (2.1) is called an attractor if Σ is compact, and there exists a neighborhood U ⊂ H of Σ such that for any ϕ ∈ U we have (2.7)
lim distH (u(t, ϕ), Σ) = 0.
t→∞
The largest open set U satisfying (2.7) is called the basin of attraction of Σ. Definition 2.2. (1) We say that (2.1) bifurcates from (u, λ) = (0, λ0 ) an invariant set Ωλ if there exists a sequence of invariant sets {Ωλn } of (2.1), 0 ∈ / Ωλn , such that lim λn = λ0 ,
n→∞
lim max |x| = 0.
n→∞ x∈Ωλn
(2) If the invariant sets Ωλ are attractors of (2.1), then the bifurcation is called attractor bifurcation. (3) If Ωλ are attractors and are homotopy equivalent to an m-dimensional sphere S m , then the bifurcation is called an S m -attractor bifurcation. 2.2. Center Manifold Theorems. We assume that the spaces H1 and H can be decomposed into ( H1 = E1λ ⊕ E2λ , dim E1λ < ∞, near λ0 ∈ R1 , (2.8) eλ ⊕ E eλ, E eλ = E λ, E e λ = closure of E λ in H, H=E 1 2 1 1 2 2 where E1λ and E2λ are two invariant subspaces of Lλ ; i.e., Lλ can be decomposed into Lλ = Lλ1 ⊕ Lλ2 such that for any λ near λ0 , e1λ , Lλ1 = Lλ |E λ : E1λ → E 1 (2.9) Lλ = Lλ | λ : E λ → E eλ, 2 2 2 E 2
Lλ2
where all eigenvalues of possess negative real parts, and all eigenvalues of Lλ1 possess nonnegative real parts at λ = λ0 . Thus, for λ near λ0 , (2.1) can be rewritten as dx = Lλ1 x + G1 (x, y, λ), dt (2.10) dy = Lλ y + G (x, y, λ), 2 2 dt where u = x + y ∈ H1 , x ∈ E1λ , y ∈ E2λ , Gi (x, y, λ) = Pi G(u, λ), and ei are canonical projections. Furthermore, we let Pi : H → E E2λ (α) = E2λ ∩ Hα , with α given by (2.4).
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Theorem 2.3 (Center Manifold Theorem, [4]). Assume (2.4)–(2.6),(2.8), and (2.9). Then there exist a neighborhood of λ0 given by |λ − λ0 | < δ for some δ > 0, a neighborhood Bλ ⊂ E1λ of x = 0, and a C 1 function h(·, λ) : Bλ → E2λ (α), depending continuously on λ, such that (1) h(0, λ) = 0, Dx h(0, λ) = 0; (2) the set n o Mλ = (x, y) ∈ H1 | x ∈ Bλ , y = h(x, λ) ∈ E2λ (α) , called center manifold, is locally invariant for (2.1), i.e. for any u0 ∈ Mλ , uλ (t, u0 ) ∈ Mλ , ∀ 0 ≤ t < tu0 , for some tu0 > 0, where uλ (t, u0 ) is the solution of (2.1); and (3) if (xλ (t), yλ (t)) is a solution of (2.10), then there are a βλ > 0 and kλ > 0 with kλ depending on (xλ (0), yλ (0)) such that kyλ (t) − h(xλ (t), λ)kH ≤ kλ e−βλ t . If we consider only the existence of the local center manifold, then conditions in (2.9) can be modified in the following fashion. Let the operator Lλ = Lλ1 ⊕ Lλ2 and Lλ2 be decomposed into λ L = Lλ21 ⊕ Lλ22 , 2 λ Eλ = Eλ ⊕ Eλ , E e2λ = E e21 ⊕ E e22 , 2 21 22 (2.11) λ eλ dim E21 = dim E21 < ∞, λ λ λ e2i L2i : E2i →E are invariant (i = 1, 2), such that at λ = λ0 e λ have zero real parts, eigenvalues of Lλ1 : E1λ → E 1 λ λ λ e21 (2.12) eigenvalues of L21 : E21 → E have positive real parts, λ λ e22 eigenvalues of Lλ22 : E22 →E have negative real parts. Then we have the following center manifold theorem. Theorem 2.4. Assume (2.4)–(2.6), (2.8), (2.11), and (2.12). Then the conclusions (1) and (2) in Theorem 2.3 hold true. 2.3. Attractor Bifurcation. A complex number β = α1 +iα2 ∈ C is called an eigenvalue of Lλ : H1 → H if there are x, y ∈ H1 such that Lλ z = βz, z = x + iy, or, equivalently, Lλ x = α1 x − α2 y, Lλ y = α2 x + α1 y.
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Now let the eigenvalues (counting the multiplicity) of Lλ be given by β1 (λ), β2 (λ), · · · , βk (λ) ∈ C. Suppose that (2.13)
< 0 Reβi (λ) = = 0 >0
(2.14)
if λ < λ0 , if λ = λ0 if λ > λ0 ,
Reβj (λ0 ) < 0
(1 ≤ i ≤ m + 1),
∀ m + 2 ≤ j.
Let the eigenspace of Lλ at λ0 be m+1 o [ n E0 = u ∈ H1 | (Lλ0 − βi (λ0 ))k u = 0, k = 1, 2, · · · . . i=1
It is known that dim E0 = m + 1. Let H1 = H = Rn . The following attractor bifurcation theorem was proved in [9]. Theorem 2.5 (Attractor Bifurcation Theorem). Let H1 = H = Rn , the conditions (2.13) and (2.14) hold true, and u = 0 be a locally asymptotically stable equilibrium point of (2.1) at λ = λ0 . Then the following assertions hold true. (1) Equation (2.1) bifurcates from (u, λ) = (0, λ0 ) an attractor Aλ for λ > λ0 , with m ≤ dim Aλ ≤ m + 1, which is connected as m > 0. (2) The attractor Aλ is a limit of a sequence of (m + 1)-dimensional annulus Mk with Mk+1 ⊂ Mk . In particular, if Aλ is a finite simplicial complex, then Aλ has the homotopy type of S m . (3) For any uλ ∈ Aλ , uλ can be expressed as uλ = vλ + o(kvλ kH1 ),
vλ ∈ E0 .
(4) If G : H1 → H is compact and the equilibrium points of (2.1) in Aλ are finite, then we have the index formula ( X 2 if m = even, ind [−(Lλ + G), ui ] = 0 if m = odd. u ∈A i
λ
(5) If u = 0 is globally stable for (2.1) at λ = λ0 , then for any bounded open set U ⊂ H with 0 ∈ U there is an ε > 0 such that as λ0 < λ < λ0 + ε, the attractor Aλ bifurcated from (0, λ0 ) attracts U/Γ in H, where Γ is the stable manifold of u = 0 with codimension m + 1. In particular, if (2.1) has a global attractor in H then U = H. Remark 2.6. As H1 and H are infinite dimensional Hilbert spaces, if (2.1) satisfies the conditions (2.4)–(2.6),(2.13), and (2.14) and u = 0 is a locally (global) asymptotically stable equilibrium point of (2.1) at λ = λ0 , then the assertions (1)–(5) of Theorem 2.5 hold; see [9, 8].
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3. Bifurcation of the GL Equation with Dirichlet Boundary Condition As mentioned in the introduction, we study in this article attractor bifurcation of the GL equation under either the Dirichlet or the periodic boundary conditions. We start with the GL equation with the Dirichlet boundary condition. Let H k (Ω, C) = {u1 + iu2 | uj ∈ H k (Ω), j = 1, 2}, H01 (Ω, C) = {u ∈ H 1 (Ω, C) | u|∂Ω = 0}, where H k (Ω) is the usual real-valued Sobolev space. Let λ1 be the first eigenvalue of −4 with the Dirichlet boundary condition (1.4). Then we have the following main bifurcation theorem for the GL equations with the Dirichlet boundary condition. Theorem 3.1. (1) If λ ≤ αλ1 , u = 0 is a globally asymptotically stable equilibrium point of (1.1) with (1.4). (2) As λ crosses αλ1 , i.e., for any αλ1 < λ < αλ1 + ² for some ² > 0, the problem (1.1) with (1.4) bifurcates from (u, λ) = (0, αλ1 ) an attractor Σλ . (3) The bifurcated attractor Σλ has dimension between 1 and 2 and is a limit of a sequence of 2D annulus Mk with Mk+1 ⊂ Mk ; i.e. Σλ = ∩∞ k=1 Mk . (4) If β 6= 0, then the bifurcation is a Hopf bifurcation, i.e., Σλ = S 1 , which is asymptotically stable (limiting cycle). (5) If β = 0 and ρ 6= 0, then the bifurcated attractor Σλ is a periodic orbit, which is a limiting cycle. (6) Moreover, for each αλ1 < λ < αλ1 + ², the bifurcated attractor Σλ attracts the open set L2 (Ω, C)/Γ, where Γ is the stable manifold of u = 0 having codimension two in L2 (Ω, C). Proof. We proceed in several steps as follows. Step 1. Let u = u1 + iu2 . The GL problem (1.1) with (1.3) can be equivalently written as follows: ∂u1 = α4u1 − β4u2 + λu1 − σ|u|2 u1 + ρ|u|2 u2 , ∂t ∂u2 (3.1) = β4u1 + α4u2 + λu2 − σ|u|2 u2 − ρ|u|2 u1 , ∂t u1 (x, 0) = φ(x), u2 (x, 0) = ψ(x). We shall apply Theorems 2.3 and 2.5 to prove this theorem. Let H1 = H 2 (Ω, C) ∩ H01 (Ω, C),
H = L2 (Ω, C).
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The mappings Lλ = −A + Bλ and G : H1 → H are defined as µ ¶ α4u1 − β4u2 −Au = , β4u1 + α4u2 ¶ µ u1 Bλ u = λ , u2 µ ¶ −σ|u|2 u1 + ρ|u|2 u2 Gu = . −σ|u|2 u2 − ρ|u|2 u1 It is known that H1/2 = H01 (Ω, C). By the Sobolev embedding theorems and 1 ≤ n ≤ 3, the mapping G : H1/2 → H is C ∞ . The condition (2.4) is fulfilled. Let {λk } ⊂ R and {ek } ⊂ H 2 (Ω) ∩ H01 (Ω) be the eigenvalues and eigenvectors of −4 with the Dirichlet boundary condition (1.4) ½ − 4ek = λk ek , ek |∂Ω = 0. We know that 0 < λ1 < λ2 ≤ · · · ≤ λk ≤ · · · ,
λk → ∞ as k → ∞,
L2 (Ω).
and {ek } is an orthogonal basis of It is easy to see that the eigenvalues of A are given by αλk ± iβλk , k = 1, 2, · · · with the corresponding eigenvectors zk = ek + iek , and {ek , iej |1 ≤ k, j < ∞} is an orthogonal basis of H. Thus the conditions (2.5) and (2.6) are valid for A and Bλ . The eigenvalues of Lλ = −A + Bλ are as follows: (3.2)
(λ − αλk ) ± iβλk ,
k = 1, 2, · · · .
In addition, the spaces H and H1 can be decomposed into the form e2 , H1 = E1 ⊕ E2 and H = E1 ⊕ E E1 = {x1 e1 + iy1 e1 | x1 , y1 ∈ R}, ∞ ∞ X X E2 = { (xk + iyk )ek | λ2k (x2k + yk2 ) < ∞}, e2 = { E
k=2 ∞ X
k=2 ∞ X
k=2
k=2
(xk + iyk )ek |
(x2k + yk2 ) < ∞},
and the operator Lλ is decomposed into ( Lλ = Lλ1 ⊕ Lλ2 , e2 . Lλ1 = Lλ |E1 : E1 → E1 , Lλ2 = Lλ |E2 : E2 → E Thus the conditions (2.8) and (2.9) are satisfied.
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By the center manifold theorem, the attractor bifurcation of (1.1) with (1.4) is equivalent to that of the bifurcation equations
(3.3)
dx1 = (λ − αλ1 )x1 + βλ1 y1 + P1 G1 (x1 + iy1 + h), dt dy1 = −βλ x + (λ − αλ )y + P G (x + iy + h), 1 1 1 1 1 2 1 1 dt
where h = h1 + ih2 is the center manifold function satisfying h(x1 , y1 ) = o(|x1 | + |y1 |), and P1 Gi (u) (i = 1, 2) are given by Z P1 G1 (u) = [−σ|u|2 u1 + ρ|u|2 u2 ]e1 dx, ZΩ P1 G2 (u) = [−σ|u|2 u2 − ρ|u|2 u1 ]e1 dx, (3.4) Ω
∞ X (xk + iyk )ek . u = u1 + iu2 = k=1
Step 2. Now we show that u = 0 is a globally asymptotically stable equilibrium of (2.1) for λ ≤ αλ1 . In fact, from (3.1) we can derive that Z Z 1 d 2 |u| dx = (−α|∇u|2 + λ|u|2 − σ|u|4 )dx 2 dt Ω Ω Z ≤ − [(αλ1 − λ)|u|2 + σ|u|4 ]dx Ω
which implies that u = 0 is globally stable. Step 3. We know that (1.1) has a global attractor; see [12]. Obviously, for the eigenvalues (3.2) of Lλ , the conditions (2.13) and (2.14) for λ0 = αλ1 are satisfied. Therefore, by Remark 2.6, (2.1) bifurcates from (u, λ) = (0, αλ1 ) an attractor Σλ which attracts H/Γ. Step 4. We now prove that Σλ = S 1 . Obviously, as β 6= 0 the bifurcation is the typical Hopf bifurcation. Therefore, we have to consider only the case where β = 0. In this case, the bifurcation equations (3.3) read
(3.5)
dx = εx + P1 G1 (xe1 + h1 + iye1 + ih2 ), dt dy = εy + P G (xe + h + iye + ih ), 1 2 1 1 1 2 dt
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T. MA, J. PARK, AND S. WANG
where ε = λ − αλ1 > 0 sufficiently small. By (3.4) we have Z P1 G1 (x, y) = [−σu31 + ρu32 − σu22 u1 + ρu21 u2 ]e1 dx Ω
= (by h(x, y) = o(|x| + |y|)) = a(−σx3 + ρy 3 − σy 2 x + ρx2 y) + o(|x|3 + |y|3 ), P1 G2 (x, y) = a(−σy 3 − ρx3 − σx2 y − ρy 2 x) + o(|x|3 + |y|3 ), where u1 = xe1 + h1 (x, y), u2 = ye1 + h2 (x, y), and Z a= e41 (x)dx > 0. Ω
Thus, the bifurcation equations (3.5) lead to dx = εx − a(σx3 − ρy 3 + σy 2 x − ρx2 y) + o(|x|3 + |y|3 ), dt (3.6) dy = εy − a(σy 3 + ρx3 + σx2 y + ρy 2 x) + o(|x|3 + |y|3 ). dt We can see that the attractor Σλ has no nonzero singular point; i.e., the singular point u = 0 of (1.1) with (1.4) is unique provided |ρ| + |β| 6= 0, because from (3.1) we have ¸ Z · Z ∂u2 ∂u1 − u1 dx = [β|∇u|2 + ρ|u|4 ]dx. u2 ∂t ∂t Ω Ω By Theorem 2.5, Σλ has the homotopy type of S 1 ; hence Σλ contains at least one periodic orbit provided ρ 6= 0. Take the polar coordinate system x = r cos θ, Then (3.6) becomes
2 2 dr = ε − aσr + o(r ) , dθ aρr r(0) = r0 .
(3.7)
From (3.7) it follows that aρ 2 (r (2π) − r2 (0)) = 2 Because
r2
= 2
y = r sin θ.
r2 (θ, r0 )
r (θ, r0 ) =
Z
2π
[ε − aσr2 + o(r2 )]dθ.
0
is
C∞
r02
+ R(θ) · o(|r0 |2 ),
on r0 ≥ 0, we have the Taylor expansion R(0) = 0.
Hence we get aρ 2 (r (2π) − r2 (0)) = 2πε − 2πaσr02 + o(|r0 |2 ). 2 Obviously the initial values r0 > 0 in (3.7) satisfying (3.8)
2πε − 2πaσr02 + o(|r0 |2 ) = 0
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are corresponding to the periodic orbits of (3.6). It is easy to see that the solution r02 > 0 of (3.8) near r0 = 0 is unique. Thus we deduce that Σλ is a periodic orbit provided ρ 6= 0. The proof is complete. ¤ 4. Bifurcation of the GL Equation with Periodic Boundary Condition For the GL equation with periodic boundary condition, the first eigenspace is larger than that in the Dirichlet boundary condition case, and to proceed, we need the following function spaces: k Hper (Ω, C) = {u ∈ H k (Ω, C) | u satisfy (1.5)}.
Then the main result in this section is the following theorem. Theorem 4.1. For the GL equation (1.1) with the periodic boundary condition (1.5), we have the following assertions. (1) (a) As λ > α, the problem (1.1) with (1.5) bifurcates from (u, λ) = (0, α) an invariant set Σλ . Σλ has dimension between 4n − 1 and 4n and is a limit of a sequence of 4n annulus Mk with Mk+1 ⊂ Mk ; i.e., Σλ = ∩∞ k=1 Mk . (b) If |ρ| + |β| = 6 0, then Σλ contains no steady state solutions of (1.1) with (1.5). (2) (a) As λ ≤ 0, u = 0 is globally asymptotically stable. (b) As λ > 0 the problem (1.1) with (1.5) bifurcates from (u, λ) = (0, 0) an attractor Σλ ⊂ L2 (Ω, C). The bifurcated attractor Σλ has dimension between 1 and 2, and is a limit of a sequence of 2m-dimensional annulus Mk with Mk+1 ⊂ Mk , i.e. Σλ = ∩∞ k=1 Mk . (c) If ρ 6= 0 then Σλ is a periodic orbit. (d) Σλ attracts L2 (Ω, C)/Γ, where Γ is the stable manifold of u = 0 with codimension two in L2 (Ω, C). 2 (Ω, C), H = L2 (Ω, C) and the mappings L and Proof. Let H1 = Hper λ per G : H1 → H be as defined in the previous section. Similar to the proof of Theorem 3.1, Lλ and G satisfy the conditions in Theorem 2.3. We know that the eigenvalue problem ½ − 4ek = λk ek , (4.1) ek (x + 2kπ) = ek (x)
has an eigenvalue sequence λ0 = 0 < λ1 ≤ λ2 ≤ · · · ≤ λk ≤ · · · ,
λk → ∞ as k → ∞,
and an eigenvector sequence {ek } which constitutes a common orthogonal basis of H1 and H. The second eigenvalue λ1 = 1 has multiplicity 2n, i.e., λ1 = · · · = λ2n , with the first eigenvectors sin xj ,
cos xj
(x = (x1 , · · · , xn ) ∈ Ω = (0, 2π)n ).
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T. MA, J. PARK, AND S. WANG
Eigenvalues of Lλ are as in (3.2), and the second eigenvalue Λ1 = (λ − α) ± iβ has multiplicity 4n. For simplicity, let e2j−1 = sin xj ,
e2j = cos xj ,
(j = 1, · · · , n).
Then the spaces H and H1 can be decomposed into the following form e2 , H = E1 ⊕ E 2n X E1 = { (z1j + iz2j )ej | z1j , z2j ∈ R}, j=1
e2 = E1⊥ . E Then the bifurcation equations of (1.1) with (1.5) are given by dZ1 = (λ − α)Z1 + βZ2 + P G1 (u), dt (4.2) dZ2 = −βZ + (λ − α)Z + P G (u), 1 2 2 dt where u = u1 + iu2 and (u1 , u2 ) = (Z1 + h1 (Z1 , Z2 ), Z2 + h2 (Z1 , Z2 )), (Z1 , Z2 ) =
2n X (z1j , z2j )ej . j=1
e2 is the center manifold function satisfying Here h = h1 + ih2 : E1 → E (4.3)
h(Z1 , Z2 ) = o(|Z1 | + |Z2 |)
and P G1 (u) =
2n X
Z ej
j=1
P G2 (u) =
2n X j=1
Ω
[−σ|u|2 u1 + ρ|u|2 u2 ]ej dx,
Z ej
Ω
[−σ|u|2 u1 − ρ|u|2 u2 ]ej dx.
By Theorem 2.5, we infer from (4.3) that the problem (1.1) and (1.5) bifurcates from (u, λ) = (0, α) an invariant set Σλ . The proof is complete. ¤ Remark 4.2. In fact, the invariant set Σλ of (1.1) with (1.5) is a sphere S 4n−1 ; namely, Σλ is homeomorphic to a sphere S 4n−1 . The topological structure of an attractor of vector fields should be stable provided some nondegenerate conditions hold. we shall discuss the topic elsewhere.
DYNAMIC BIFURCATION OF THE GINZBURG–LANDAU EQUATION
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5. Bifurcation of invariant sphere S m More generally, for the GL equations we have the bifurcation theorem of invariant sphere S m (m ≥ 1) at any eigenvalue. Theorem 5.1. Let λk be an eigenvalue of −4 with the boundary condition (1.4), or (1.5), which has multiplicity m ≥ 1. Then, as λ > αλk , the problem (1.1) with (1.4), or (1.1) with (1.5), bifurcates from (u, λ) = (0, αλk ) an invariant set Σλ . This invariant set Σλ has dimension between 2m − 1 and 2m and is a limit of a sequence of 2m dimensional annulus Mk with Mk+1 ⊂ Mk ; i.e., Σλ = ∩∞ 6 0, then there is no singular k=1 Mk . If |β| + |ρ| = point in Σλ . Proof. We denote the eigenvectors of −4 corresponding to λk by {e∗1 , · · · , e∗m }. Thus, the space H1 and H defined in Theorem 3.1 or Theorem 4.1 can be decomposed into ⊥ H1 = Em ⊕ Em ,
⊥ em ⊕ E em H=E ,
Em = span{e∗i + ie∗j | 1 ≤ i, j ≤ m}, ⊥ Em = {u ∈ H1 | hu, viH1 = 0 ∀v ∈ Em },
em = Em , E
e ⊥ = {u ∈ H | hu, viH = 0 ∀v ∈ E em }. E m
By the center manifold theorem, the bifurcation equations of (1.1) with (1.4), or (1.1) with (1.5), at λ = λk are equivalent to
(5.1)
∂v1 = α4v1 − β4v2 + λv1 + P G1 (v + h(v)), ∂t ∂v2 = β4v + α4v + λv + P G (v + h(v)), 1 2 2 2 ∂t
⊥ is the center where λ is near λk , v = v1 + iv2 ∈ Em , and h : Em → Em manifold function, G = (G1 , G2 ) : H1 → H defined as in Theorem 3.1 or em is the projection. Theorem 4.1, and P : H → E The equations (5.1) are a system of ordinary differential equations with order 2m: dZ1 = (λ − α)Z1 + βZ2 + [−σ|Z|2 Z1 + ρ|Z|2 Z2 ] + o(|Z|3 ), dt (5.2) dZ2 = −βZ + (λ − α)Z + [−σ|Z|2 Z − ρ|Z|2 Z ] + o(|Z|3 ), 1 2 2 1 dt
where Z = Z1 + iZ2 . The eigenvalues of the linear part are still (λ − αλk ) ± iβλk , with multiplicity 2m.
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T. MA, J. PARK, AND S. WANG
By Theorem 2.5 it suffices to prove that v = 0 is asymptotically stable for (5.2) at λ = αλk . For λ = αλk , we infer from (5.2) that Z Z 1d 2 |v| dx = G(v + h(v))vdx 2 dt Ω Ω = (by h(v) = o(|v|) Z G(v)vdx + o(|v|4 ) = Ω Z |v|4 dx + o(|v|4 ), = −σ Ω
which implies that v = 0 is asymptotically stable for (5.2) at λ = αλk . The proof is complete. ¤ References [1] M. Bartuccelli, P. Constantin, C. R. Doering, J. D. Gibbon, and M. Gis¨ lt, Hard turbulence in a finite-dimensional dynamical system?, Phys. Lett. A, selfa 142 (1989), pp. 349–356. [2] , On the possibility of soft and hard turbulence in the complex Ginzburg-Landau equation, Phys. D, 44 (1990), pp. 421–444. [3] C. R. Doering, J. D. Gibbon, D. D. Holm, and B. Nicolaenko, Low-dimensional behaviour in the complex Ginzburg-Landau equation, Nonlinearity, 1 (1988), pp. 279– 309. [4] D. Henry, Geometric theory of semilinear parabolic equations, vol. 840 of Lecture Notes in Math., Springer-Verlag, Berlin, 1981. ´c ˇ, Ginzburg-Landau dynamics with a time-dependent [5] H. G. Kaper and P. Taka magnetic field, Nonlinearity, 11 (1998), pp. 291–305. [6] I. Kukavica, An upper bound for the winding number for solutions of the GinzburgLandau equation, Indiana Univ. Math. J., 41 (1992), pp. 825–836. [7] , Hausdorff length of level sets for solutions of the Ginzburg-Landau equation, Nonlinearity, 8 (1995), pp. 113–129. [8] T. Ma and S. Wang, Attractor bifurcation theory and its applications to RayleighB´enard convection, Commun. Pure Appl. Anal., 2 (2003), pp. 591–599. [9] , Dynamic bifurcation of nonlinear evolution equations, Chinese Ann. Math.Ser. B, 2004. [10] , Bifurcation Theory and Applications, World Scientific, to appear in 2005. [11] Q. Tang and S. Wang, Time dependent Ginzburg-Landau equations of superconductivity, Phys. D, 88 (1995), pp. 139–166. [12] R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, vol. 68 of Appl. Math. Sci. 68, Springer-Verlag, New York, second ed., 1997. (TM) Department of Mathematics, Sichuan University, Chengdu, People’s Republic of China and Department of Mathematics, Indiana University, Bloomington, IN 47405 (JP) Department of Mathematics, Indiana University, Bloomington, IN 47405 E-mail address: [email protected] (SW) Department of Mathematics, Indiana University, Bloomington, IN 47405 E-mail address: [email protected]
1. Introduction In this article, we consider the bifurcation of attractors and invariant sets of the complex Ginzburg–Landau (GL) equation, which reads ∂u − (α + iβ)4u + (σ + iρ)|u|2 u − λu = 0, ∂t where the unknown function u : Ω×[0, ∞) → C is a complex-valued function and Ω ⊂ Rn is an open, bounded, and smooth domain in Rn (1 ≤ n ≤ 3). The parameters α, β, σ, ρ, and λ are real numbers and (1.1)
(1.2)
α > 0,
σ > 0.
The initial condition for (1.1) is given by (1.3)
u(x, 0) = φ + iψ.
Also, (1.1) is supplemented with either the Dirichlet boundary condition, (1.4)
u|∂Ω = 0,
or the periodic boundary condition, (1.5)
Ω = (0, 2π)n and u is Ω-periodic.
The GL equation is an important equation in a number of scientific fields. It is directly related to the GL theory of superconductivity. In this context, the unknown function is the order parameter, the constants β and ρ are usually zero, and the bifurcation parameter λ is the GL parameter; see [11] and the references therein. 1991 Mathematics Subject Classification. 35, 37. Key words and phrases. Ginzburg–Landau equation, bifurcation, stability. The authors thank the two anonymous referees for their insightful comments. The work was supported in part by the Office of Naval Research, by the National Science Foundation, and by the National Science Foundation of China. 1
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T. MA, J. PARK, AND S. WANG
In fluid dynamics the GL equation is found, for example, in the study of Poiseuille flow, the nonlinear growth of convection rolls in the Rayleigh– Be´ nard problem and Taylor–Couette flow. In this case, the bifurcation parameter λ plays the role of a Reynolds number. The equation also arises in the study of chemical systems governed by reaction-diffusion equations. There are extensive studies from the mathematical point of view for the GL equation, and we refer in particular to [2, 3, 1, 5, 6, 7, 11] and the references therein for studies related to the global attractors, inertial manifolds, and soft and hard turbulences described by the GL equations. We study in this article the bifurcation and stability of the solutions of the complex GL equation. A nonlinear theory for this problem is established in this article using a notion of bifurcation called attractor bifurcation and its corresponding theorem developed recently by the authors in [9, 8]; see [10], a new book by two of the authors. The main objectives of this theory include (1) existence of bifurcation when the system parameter crosses some critical numbers, (2) dynamic stability of bifurcated solutions, and (3) the structure/patterns and their stability and transitions in the physical space. More precisely, the main theorem associated with the attractor bifurcation states that as the control parameter crosses a certain critical value when there are m + 1 (m ≥ 0) eigenvalues across the imaginary axis, the system bifurcates from a trivial steady state solution to an attractor with dimension between m and m + 1, provided the critical state is asymptotically stable. There are a few important features of the attractor bifurcation. First, the bifurcation attractor does not include the trivial steady state and is stable; hence it is physically important. Second, the attractor contains a collection of solutions of the evolution equation, including possibly steady states and periodic orbits as well as homoclinic and heteroclinic orbits. Third, it provides a unified point of view on dynamic bifurcation and can be applied to many problems in physics and mechanics. Fourth, from the application point of view, the Krasnoselskii–Rabinowitz theorem requires the number of eigenvalues m + 1 crossing the imaginary axis being an odd integer, and the Hopf bifurcation is for the case where m + 1 = 2. However, the new attractor bifurcation theorem obtained can be applied to cases for all m ≥ 0. In addition, the bifurcated attractor, as mentioned earlier, is stable, which is another subtle issue for other known bifurcation theorems. For the GL equation, bifurcation is obtained with respect to the parameter λ, and the main results obtained can be summarized as follows. First, for the GL equation with the Dirichlet boundary condition, let λ1 be the first eigenvalue of the elliptic operator −4. Our main results in this case include the following.
DYNAMIC BIFURCATION OF THE GINZBURG–LANDAU EQUATION
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1. If λ ≤ αλ1 , the trivial solution u = 0 is globally asymptotically stable. The global attractor of the GL equation consists exactly of the trivial steady state solution u = 0. 2. As λ crosses αλ1 , i.e., there exists an ² > 0 such that for any αλ1 < λ < αλ1 +², the GL problem bifurcates from the trivial solution an attractor Σλ . The bifurcated attractor Σλ attracts the open set L2 (Ω, C)/Γ, where Γ is the stable manifold of u = 0 having codimension two in L2 (Ω, C). More detailed structure of this bifurcated attractor can be classified as follows (see Figures 1.1 and 1.2.) (a) If |β| + |ρ| 6= 0, then the bifurcated attractor consists of exactly one stable limiting cycle, i.e., Σλ = S 1 , which is asymptotically stable. The global attractor Aλ is a two-dimensional (2D) disk consisting of the stable limiting cycle Σλ = S 1 , the (unstable) trivial steady state solution u = 0, and orbits connecting Σλ = S 1 and u = 0. In particular, if β 6= 0, then the bifurcation is a Hopf bifurcation to a stable limiting cycle. (b) If β = ρ = 0, then the bifurcated attractor Σλ has dimension between 1 and 2 and is a limit of a sequence of 2D annulus Mk with Mk+1 ⊂ Mk , i.e., Σλ = ∩∞ k=1 Mk . Again in this case, the global attractor Aλ is 2D, consisting of Σλ , u = 0 and the connecting orbits between them. 1
αλ 1
Σλ
λ
Figure 1.1. Bifurcation diagram for the GL equation with the Dirichlet boundary condition: (1) bifurcation appears at λ = αλ1 , (2) bifurcated attractor Σλ = S 1 is the boundary of the shaded region, and (3) the global attractor Aλ is the 2D disk, shown as the shaded region. Here the dotted line stands for the unstable trivial solution u = 0. 1Using a different method, we can in fact prove that Σ is also homeomorphic to S 1 , λ
which shall be reported elsewhere.
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T. MA, J. PARK, AND S. WANG
Γ
Γ
Figure 1.2. Phase space structure in L2 (Ω, C) × {λ} in the case where |β| + |ρ| = 6 0. Here the bifurcated attractor Σλ = S 1 is a stable limiting cycle.
Second, for the GL equations equipped with the periodic boundary condition, similar results can be obtained as well. In particular, in the case where |β| + |ρ| = 6 0, we prove that the bifurcated attractor Σλ is a sphere S 1 , containing no steady state solutions, and the global attractor Aλ is a 2 dimensional ball consisting of the trivial steady state u = 0, Σλ , and the orbits connecting them. Finally, bifurcation from any eigenvalue of the Laplacian can also be obtained as for the first eigenvalue. It is worth mentioning that the complete structure of the global attractor for the bifurcations from the first eigenvalue is obtained, while no such information is available for bifurcations from the rest eigenvalues. Important work on lower and upper bounds of the global attractor of the GL equation, together with their physical mechanisms, was done in the 1980’s in [2, 3, 1]. As mentioned earlier, the main objective of this article is to study bifurcation and transitions from the trivial solution. Hence we focus only on the local attractor near the trivial solution, which is part of the global attractor. Of course, near the first eigenvalue, complete information for both the global attractor and the bifurcated attractor is obtained in this article. For λ near other eigenvalues, the results here demonstrate only the transitions of the trivial solution and provide some partial information on the low bounds of the global attractor. As far as the dimension of the global attractor is concerned, our results are consistent with the work in [2, 3, 1]. The paper is organized as follows. In Section 2, we recall the attractor bifurcation theory. Sections 3 and 4 study the bifurcation of the GL equations for the Dirichlet boundary condition and for the periocic boundary, respectively. Section 5 deals with bifurcation from the rest of the eigenvalues.
DYNAMIC BIFURCATION OF THE GINZBURG–LANDAU EQUATION
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2. Abstract Bifurcation Theory 2.1. Preliminary. We recall in this section a general theory on attractor bifurcation for nonlinear evolution equations; see [9, 8]. Let H and H1 be two Hilbert spaces and H1 ,→ H be a dense and compact inclusion. We consider the nonlinear evolution equations du (2.1) = Lλ u + G(u, λ), dt (2.2) u(0) = u0 , where u : [0, ∞) → H is the unknown function, λ ∈ R is the system parameter, and Lλ : H1 → H are parameterized linear completely continuous fields depending continuously on λ ∈ R1 , which satisfy Lλ = −A + Bλ is a sectorial operator, A : H1 → H a linear homeomorphism, (2.3) B : H → H the parameterized linear compact operators. 1 λ It is easy to see that Lλ generates an analytic semigroup {e−tLλ }t≥0 . Then we can define fractional power operators Lαλ for any 0 ≤ α ≤ 1 with domain Hα = D(Lαλ ) such that Hα1 ⊂ Hα2 if α1 > α2 , and H0 = H. Furthermore, we assume that the nonlinear terms G(·, λ) : Hα → H for some 1 > α ≥ 0 are a family of parameterized C r bounded operators (r ≥ 1) continuously depending on the parameter λ ∈ R1 , such that G(u, λ) = o(kukHα ) ∀ λ ∈ R1 .
(2.4)
Actually, in this paper we need only the following conditions for the operator Lλ = −A + Bλ , which ensure that Lλ is a sectorial operator. Let there be a eigenvalue sequence {ρk } ⊂ C and an eigenvector sequence {ek , hk } ⊂ H1 of A : Azk = ρk zk , zk = ek + ihk , Reρk → +∞ as k → ∞, (2.5) |Imρ /(Reρ + a)| ≤ C for some constants a, C > 0, k
k
such that {ek , hk } is a basis of H. Condition (2.5) implies that A is a sectorial operator. Hence we can define fractional power operator Aα with domain Hα = D(Aα ). Then for the operator Bλ : H1 −→ H, we assume that there is a constant 0 ≤ θ < 1 such that (2.6)
Bλ : Hθ −→ H bounded ∀ λ ∈ R.
Let {Sλ (t)}t≥0 be an operator semigroup generated by the equation (2.1) which enjoys the following properties: (i) For any t ≥ 0, Sλ (t) : H → H is a linear continuous operator. (ii) Sλ (0) = I : H → H is the identity on H and (iii) For any t, s ≥ 0, Sλ (t + s) = Sλ (t) · Sλ (s).
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T. MA, J. PARK, AND S. WANG
Then the solution of (2.1) and (2.2) can be expressed as u(t) = Sλ (t)u0 ,
t ≥ 0.
Definition 2.1. A set Σ ⊂ H is called an invariant set of (2.1) if S(t)Σ = Σ for any t ≥ 0. An invariant set Σ ⊂ H of (2.1) is called an attractor if Σ is compact, and there exists a neighborhood U ⊂ H of Σ such that for any ϕ ∈ U we have (2.7)
lim distH (u(t, ϕ), Σ) = 0.
t→∞
The largest open set U satisfying (2.7) is called the basin of attraction of Σ. Definition 2.2. (1) We say that (2.1) bifurcates from (u, λ) = (0, λ0 ) an invariant set Ωλ if there exists a sequence of invariant sets {Ωλn } of (2.1), 0 ∈ / Ωλn , such that lim λn = λ0 ,
n→∞
lim max |x| = 0.
n→∞ x∈Ωλn
(2) If the invariant sets Ωλ are attractors of (2.1), then the bifurcation is called attractor bifurcation. (3) If Ωλ are attractors and are homotopy equivalent to an m-dimensional sphere S m , then the bifurcation is called an S m -attractor bifurcation. 2.2. Center Manifold Theorems. We assume that the spaces H1 and H can be decomposed into ( H1 = E1λ ⊕ E2λ , dim E1λ < ∞, near λ0 ∈ R1 , (2.8) eλ ⊕ E eλ, E eλ = E λ, E e λ = closure of E λ in H, H=E 1 2 1 1 2 2 where E1λ and E2λ are two invariant subspaces of Lλ ; i.e., Lλ can be decomposed into Lλ = Lλ1 ⊕ Lλ2 such that for any λ near λ0 , e1λ , Lλ1 = Lλ |E λ : E1λ → E 1 (2.9) Lλ = Lλ | λ : E λ → E eλ, 2 2 2 E 2
Lλ2
where all eigenvalues of possess negative real parts, and all eigenvalues of Lλ1 possess nonnegative real parts at λ = λ0 . Thus, for λ near λ0 , (2.1) can be rewritten as dx = Lλ1 x + G1 (x, y, λ), dt (2.10) dy = Lλ y + G (x, y, λ), 2 2 dt where u = x + y ∈ H1 , x ∈ E1λ , y ∈ E2λ , Gi (x, y, λ) = Pi G(u, λ), and ei are canonical projections. Furthermore, we let Pi : H → E E2λ (α) = E2λ ∩ Hα , with α given by (2.4).
DYNAMIC BIFURCATION OF THE GINZBURG–LANDAU EQUATION
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Theorem 2.3 (Center Manifold Theorem, [4]). Assume (2.4)–(2.6),(2.8), and (2.9). Then there exist a neighborhood of λ0 given by |λ − λ0 | < δ for some δ > 0, a neighborhood Bλ ⊂ E1λ of x = 0, and a C 1 function h(·, λ) : Bλ → E2λ (α), depending continuously on λ, such that (1) h(0, λ) = 0, Dx h(0, λ) = 0; (2) the set n o Mλ = (x, y) ∈ H1 | x ∈ Bλ , y = h(x, λ) ∈ E2λ (α) , called center manifold, is locally invariant for (2.1), i.e. for any u0 ∈ Mλ , uλ (t, u0 ) ∈ Mλ , ∀ 0 ≤ t < tu0 , for some tu0 > 0, where uλ (t, u0 ) is the solution of (2.1); and (3) if (xλ (t), yλ (t)) is a solution of (2.10), then there are a βλ > 0 and kλ > 0 with kλ depending on (xλ (0), yλ (0)) such that kyλ (t) − h(xλ (t), λ)kH ≤ kλ e−βλ t . If we consider only the existence of the local center manifold, then conditions in (2.9) can be modified in the following fashion. Let the operator Lλ = Lλ1 ⊕ Lλ2 and Lλ2 be decomposed into λ L = Lλ21 ⊕ Lλ22 , 2 λ Eλ = Eλ ⊕ Eλ , E e2λ = E e21 ⊕ E e22 , 2 21 22 (2.11) λ eλ dim E21 = dim E21 < ∞, λ λ λ e2i L2i : E2i →E are invariant (i = 1, 2), such that at λ = λ0 e λ have zero real parts, eigenvalues of Lλ1 : E1λ → E 1 λ λ λ e21 (2.12) eigenvalues of L21 : E21 → E have positive real parts, λ λ e22 eigenvalues of Lλ22 : E22 →E have negative real parts. Then we have the following center manifold theorem. Theorem 2.4. Assume (2.4)–(2.6), (2.8), (2.11), and (2.12). Then the conclusions (1) and (2) in Theorem 2.3 hold true. 2.3. Attractor Bifurcation. A complex number β = α1 +iα2 ∈ C is called an eigenvalue of Lλ : H1 → H if there are x, y ∈ H1 such that Lλ z = βz, z = x + iy, or, equivalently, Lλ x = α1 x − α2 y, Lλ y = α2 x + α1 y.
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T. MA, J. PARK, AND S. WANG
Now let the eigenvalues (counting the multiplicity) of Lλ be given by β1 (λ), β2 (λ), · · · , βk (λ) ∈ C. Suppose that (2.13)
< 0 Reβi (λ) = = 0 >0
(2.14)
if λ < λ0 , if λ = λ0 if λ > λ0 ,
Reβj (λ0 ) < 0
(1 ≤ i ≤ m + 1),
∀ m + 2 ≤ j.
Let the eigenspace of Lλ at λ0 be m+1 o [ n E0 = u ∈ H1 | (Lλ0 − βi (λ0 ))k u = 0, k = 1, 2, · · · . . i=1
It is known that dim E0 = m + 1. Let H1 = H = Rn . The following attractor bifurcation theorem was proved in [9]. Theorem 2.5 (Attractor Bifurcation Theorem). Let H1 = H = Rn , the conditions (2.13) and (2.14) hold true, and u = 0 be a locally asymptotically stable equilibrium point of (2.1) at λ = λ0 . Then the following assertions hold true. (1) Equation (2.1) bifurcates from (u, λ) = (0, λ0 ) an attractor Aλ for λ > λ0 , with m ≤ dim Aλ ≤ m + 1, which is connected as m > 0. (2) The attractor Aλ is a limit of a sequence of (m + 1)-dimensional annulus Mk with Mk+1 ⊂ Mk . In particular, if Aλ is a finite simplicial complex, then Aλ has the homotopy type of S m . (3) For any uλ ∈ Aλ , uλ can be expressed as uλ = vλ + o(kvλ kH1 ),
vλ ∈ E0 .
(4) If G : H1 → H is compact and the equilibrium points of (2.1) in Aλ are finite, then we have the index formula ( X 2 if m = even, ind [−(Lλ + G), ui ] = 0 if m = odd. u ∈A i
λ
(5) If u = 0 is globally stable for (2.1) at λ = λ0 , then for any bounded open set U ⊂ H with 0 ∈ U there is an ε > 0 such that as λ0 < λ < λ0 + ε, the attractor Aλ bifurcated from (0, λ0 ) attracts U/Γ in H, where Γ is the stable manifold of u = 0 with codimension m + 1. In particular, if (2.1) has a global attractor in H then U = H. Remark 2.6. As H1 and H are infinite dimensional Hilbert spaces, if (2.1) satisfies the conditions (2.4)–(2.6),(2.13), and (2.14) and u = 0 is a locally (global) asymptotically stable equilibrium point of (2.1) at λ = λ0 , then the assertions (1)–(5) of Theorem 2.5 hold; see [9, 8].
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3. Bifurcation of the GL Equation with Dirichlet Boundary Condition As mentioned in the introduction, we study in this article attractor bifurcation of the GL equation under either the Dirichlet or the periodic boundary conditions. We start with the GL equation with the Dirichlet boundary condition. Let H k (Ω, C) = {u1 + iu2 | uj ∈ H k (Ω), j = 1, 2}, H01 (Ω, C) = {u ∈ H 1 (Ω, C) | u|∂Ω = 0}, where H k (Ω) is the usual real-valued Sobolev space. Let λ1 be the first eigenvalue of −4 with the Dirichlet boundary condition (1.4). Then we have the following main bifurcation theorem for the GL equations with the Dirichlet boundary condition. Theorem 3.1. (1) If λ ≤ αλ1 , u = 0 is a globally asymptotically stable equilibrium point of (1.1) with (1.4). (2) As λ crosses αλ1 , i.e., for any αλ1 < λ < αλ1 + ² for some ² > 0, the problem (1.1) with (1.4) bifurcates from (u, λ) = (0, αλ1 ) an attractor Σλ . (3) The bifurcated attractor Σλ has dimension between 1 and 2 and is a limit of a sequence of 2D annulus Mk with Mk+1 ⊂ Mk ; i.e. Σλ = ∩∞ k=1 Mk . (4) If β 6= 0, then the bifurcation is a Hopf bifurcation, i.e., Σλ = S 1 , which is asymptotically stable (limiting cycle). (5) If β = 0 and ρ 6= 0, then the bifurcated attractor Σλ is a periodic orbit, which is a limiting cycle. (6) Moreover, for each αλ1 < λ < αλ1 + ², the bifurcated attractor Σλ attracts the open set L2 (Ω, C)/Γ, where Γ is the stable manifold of u = 0 having codimension two in L2 (Ω, C). Proof. We proceed in several steps as follows. Step 1. Let u = u1 + iu2 . The GL problem (1.1) with (1.3) can be equivalently written as follows: ∂u1 = α4u1 − β4u2 + λu1 − σ|u|2 u1 + ρ|u|2 u2 , ∂t ∂u2 (3.1) = β4u1 + α4u2 + λu2 − σ|u|2 u2 − ρ|u|2 u1 , ∂t u1 (x, 0) = φ(x), u2 (x, 0) = ψ(x). We shall apply Theorems 2.3 and 2.5 to prove this theorem. Let H1 = H 2 (Ω, C) ∩ H01 (Ω, C),
H = L2 (Ω, C).
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T. MA, J. PARK, AND S. WANG
The mappings Lλ = −A + Bλ and G : H1 → H are defined as µ ¶ α4u1 − β4u2 −Au = , β4u1 + α4u2 ¶ µ u1 Bλ u = λ , u2 µ ¶ −σ|u|2 u1 + ρ|u|2 u2 Gu = . −σ|u|2 u2 − ρ|u|2 u1 It is known that H1/2 = H01 (Ω, C). By the Sobolev embedding theorems and 1 ≤ n ≤ 3, the mapping G : H1/2 → H is C ∞ . The condition (2.4) is fulfilled. Let {λk } ⊂ R and {ek } ⊂ H 2 (Ω) ∩ H01 (Ω) be the eigenvalues and eigenvectors of −4 with the Dirichlet boundary condition (1.4) ½ − 4ek = λk ek , ek |∂Ω = 0. We know that 0 < λ1 < λ2 ≤ · · · ≤ λk ≤ · · · ,
λk → ∞ as k → ∞,
L2 (Ω).
and {ek } is an orthogonal basis of It is easy to see that the eigenvalues of A are given by αλk ± iβλk , k = 1, 2, · · · with the corresponding eigenvectors zk = ek + iek , and {ek , iej |1 ≤ k, j < ∞} is an orthogonal basis of H. Thus the conditions (2.5) and (2.6) are valid for A and Bλ . The eigenvalues of Lλ = −A + Bλ are as follows: (3.2)
(λ − αλk ) ± iβλk ,
k = 1, 2, · · · .
In addition, the spaces H and H1 can be decomposed into the form e2 , H1 = E1 ⊕ E2 and H = E1 ⊕ E E1 = {x1 e1 + iy1 e1 | x1 , y1 ∈ R}, ∞ ∞ X X E2 = { (xk + iyk )ek | λ2k (x2k + yk2 ) < ∞}, e2 = { E
k=2 ∞ X
k=2 ∞ X
k=2
k=2
(xk + iyk )ek |
(x2k + yk2 ) < ∞},
and the operator Lλ is decomposed into ( Lλ = Lλ1 ⊕ Lλ2 , e2 . Lλ1 = Lλ |E1 : E1 → E1 , Lλ2 = Lλ |E2 : E2 → E Thus the conditions (2.8) and (2.9) are satisfied.
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By the center manifold theorem, the attractor bifurcation of (1.1) with (1.4) is equivalent to that of the bifurcation equations
(3.3)
dx1 = (λ − αλ1 )x1 + βλ1 y1 + P1 G1 (x1 + iy1 + h), dt dy1 = −βλ x + (λ − αλ )y + P G (x + iy + h), 1 1 1 1 1 2 1 1 dt
where h = h1 + ih2 is the center manifold function satisfying h(x1 , y1 ) = o(|x1 | + |y1 |), and P1 Gi (u) (i = 1, 2) are given by Z P1 G1 (u) = [−σ|u|2 u1 + ρ|u|2 u2 ]e1 dx, ZΩ P1 G2 (u) = [−σ|u|2 u2 − ρ|u|2 u1 ]e1 dx, (3.4) Ω
∞ X (xk + iyk )ek . u = u1 + iu2 = k=1
Step 2. Now we show that u = 0 is a globally asymptotically stable equilibrium of (2.1) for λ ≤ αλ1 . In fact, from (3.1) we can derive that Z Z 1 d 2 |u| dx = (−α|∇u|2 + λ|u|2 − σ|u|4 )dx 2 dt Ω Ω Z ≤ − [(αλ1 − λ)|u|2 + σ|u|4 ]dx Ω
which implies that u = 0 is globally stable. Step 3. We know that (1.1) has a global attractor; see [12]. Obviously, for the eigenvalues (3.2) of Lλ , the conditions (2.13) and (2.14) for λ0 = αλ1 are satisfied. Therefore, by Remark 2.6, (2.1) bifurcates from (u, λ) = (0, αλ1 ) an attractor Σλ which attracts H/Γ. Step 4. We now prove that Σλ = S 1 . Obviously, as β 6= 0 the bifurcation is the typical Hopf bifurcation. Therefore, we have to consider only the case where β = 0. In this case, the bifurcation equations (3.3) read
(3.5)
dx = εx + P1 G1 (xe1 + h1 + iye1 + ih2 ), dt dy = εy + P G (xe + h + iye + ih ), 1 2 1 1 1 2 dt
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T. MA, J. PARK, AND S. WANG
where ε = λ − αλ1 > 0 sufficiently small. By (3.4) we have Z P1 G1 (x, y) = [−σu31 + ρu32 − σu22 u1 + ρu21 u2 ]e1 dx Ω
= (by h(x, y) = o(|x| + |y|)) = a(−σx3 + ρy 3 − σy 2 x + ρx2 y) + o(|x|3 + |y|3 ), P1 G2 (x, y) = a(−σy 3 − ρx3 − σx2 y − ρy 2 x) + o(|x|3 + |y|3 ), where u1 = xe1 + h1 (x, y), u2 = ye1 + h2 (x, y), and Z a= e41 (x)dx > 0. Ω
Thus, the bifurcation equations (3.5) lead to dx = εx − a(σx3 − ρy 3 + σy 2 x − ρx2 y) + o(|x|3 + |y|3 ), dt (3.6) dy = εy − a(σy 3 + ρx3 + σx2 y + ρy 2 x) + o(|x|3 + |y|3 ). dt We can see that the attractor Σλ has no nonzero singular point; i.e., the singular point u = 0 of (1.1) with (1.4) is unique provided |ρ| + |β| 6= 0, because from (3.1) we have ¸ Z · Z ∂u2 ∂u1 − u1 dx = [β|∇u|2 + ρ|u|4 ]dx. u2 ∂t ∂t Ω Ω By Theorem 2.5, Σλ has the homotopy type of S 1 ; hence Σλ contains at least one periodic orbit provided ρ 6= 0. Take the polar coordinate system x = r cos θ, Then (3.6) becomes
2 2 dr = ε − aσr + o(r ) , dθ aρr r(0) = r0 .
(3.7)
From (3.7) it follows that aρ 2 (r (2π) − r2 (0)) = 2 Because
r2
= 2
y = r sin θ.
r2 (θ, r0 )
r (θ, r0 ) =
Z
2π
[ε − aσr2 + o(r2 )]dθ.
0
is
C∞
r02
+ R(θ) · o(|r0 |2 ),
on r0 ≥ 0, we have the Taylor expansion R(0) = 0.
Hence we get aρ 2 (r (2π) − r2 (0)) = 2πε − 2πaσr02 + o(|r0 |2 ). 2 Obviously the initial values r0 > 0 in (3.7) satisfying (3.8)
2πε − 2πaσr02 + o(|r0 |2 ) = 0
DYNAMIC BIFURCATION OF THE GINZBURG–LANDAU EQUATION
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are corresponding to the periodic orbits of (3.6). It is easy to see that the solution r02 > 0 of (3.8) near r0 = 0 is unique. Thus we deduce that Σλ is a periodic orbit provided ρ 6= 0. The proof is complete. ¤ 4. Bifurcation of the GL Equation with Periodic Boundary Condition For the GL equation with periodic boundary condition, the first eigenspace is larger than that in the Dirichlet boundary condition case, and to proceed, we need the following function spaces: k Hper (Ω, C) = {u ∈ H k (Ω, C) | u satisfy (1.5)}.
Then the main result in this section is the following theorem. Theorem 4.1. For the GL equation (1.1) with the periodic boundary condition (1.5), we have the following assertions. (1) (a) As λ > α, the problem (1.1) with (1.5) bifurcates from (u, λ) = (0, α) an invariant set Σλ . Σλ has dimension between 4n − 1 and 4n and is a limit of a sequence of 4n annulus Mk with Mk+1 ⊂ Mk ; i.e., Σλ = ∩∞ k=1 Mk . (b) If |ρ| + |β| = 6 0, then Σλ contains no steady state solutions of (1.1) with (1.5). (2) (a) As λ ≤ 0, u = 0 is globally asymptotically stable. (b) As λ > 0 the problem (1.1) with (1.5) bifurcates from (u, λ) = (0, 0) an attractor Σλ ⊂ L2 (Ω, C). The bifurcated attractor Σλ has dimension between 1 and 2, and is a limit of a sequence of 2m-dimensional annulus Mk with Mk+1 ⊂ Mk , i.e. Σλ = ∩∞ k=1 Mk . (c) If ρ 6= 0 then Σλ is a periodic orbit. (d) Σλ attracts L2 (Ω, C)/Γ, where Γ is the stable manifold of u = 0 with codimension two in L2 (Ω, C). 2 (Ω, C), H = L2 (Ω, C) and the mappings L and Proof. Let H1 = Hper λ per G : H1 → H be as defined in the previous section. Similar to the proof of Theorem 3.1, Lλ and G satisfy the conditions in Theorem 2.3. We know that the eigenvalue problem ½ − 4ek = λk ek , (4.1) ek (x + 2kπ) = ek (x)
has an eigenvalue sequence λ0 = 0 < λ1 ≤ λ2 ≤ · · · ≤ λk ≤ · · · ,
λk → ∞ as k → ∞,
and an eigenvector sequence {ek } which constitutes a common orthogonal basis of H1 and H. The second eigenvalue λ1 = 1 has multiplicity 2n, i.e., λ1 = · · · = λ2n , with the first eigenvectors sin xj ,
cos xj
(x = (x1 , · · · , xn ) ∈ Ω = (0, 2π)n ).
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T. MA, J. PARK, AND S. WANG
Eigenvalues of Lλ are as in (3.2), and the second eigenvalue Λ1 = (λ − α) ± iβ has multiplicity 4n. For simplicity, let e2j−1 = sin xj ,
e2j = cos xj ,
(j = 1, · · · , n).
Then the spaces H and H1 can be decomposed into the following form e2 , H = E1 ⊕ E 2n X E1 = { (z1j + iz2j )ej | z1j , z2j ∈ R}, j=1
e2 = E1⊥ . E Then the bifurcation equations of (1.1) with (1.5) are given by dZ1 = (λ − α)Z1 + βZ2 + P G1 (u), dt (4.2) dZ2 = −βZ + (λ − α)Z + P G (u), 1 2 2 dt where u = u1 + iu2 and (u1 , u2 ) = (Z1 + h1 (Z1 , Z2 ), Z2 + h2 (Z1 , Z2 )), (Z1 , Z2 ) =
2n X (z1j , z2j )ej . j=1
e2 is the center manifold function satisfying Here h = h1 + ih2 : E1 → E (4.3)
h(Z1 , Z2 ) = o(|Z1 | + |Z2 |)
and P G1 (u) =
2n X
Z ej
j=1
P G2 (u) =
2n X j=1
Ω
[−σ|u|2 u1 + ρ|u|2 u2 ]ej dx,
Z ej
Ω
[−σ|u|2 u1 − ρ|u|2 u2 ]ej dx.
By Theorem 2.5, we infer from (4.3) that the problem (1.1) and (1.5) bifurcates from (u, λ) = (0, α) an invariant set Σλ . The proof is complete. ¤ Remark 4.2. In fact, the invariant set Σλ of (1.1) with (1.5) is a sphere S 4n−1 ; namely, Σλ is homeomorphic to a sphere S 4n−1 . The topological structure of an attractor of vector fields should be stable provided some nondegenerate conditions hold. we shall discuss the topic elsewhere.
DYNAMIC BIFURCATION OF THE GINZBURG–LANDAU EQUATION
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5. Bifurcation of invariant sphere S m More generally, for the GL equations we have the bifurcation theorem of invariant sphere S m (m ≥ 1) at any eigenvalue. Theorem 5.1. Let λk be an eigenvalue of −4 with the boundary condition (1.4), or (1.5), which has multiplicity m ≥ 1. Then, as λ > αλk , the problem (1.1) with (1.4), or (1.1) with (1.5), bifurcates from (u, λ) = (0, αλk ) an invariant set Σλ . This invariant set Σλ has dimension between 2m − 1 and 2m and is a limit of a sequence of 2m dimensional annulus Mk with Mk+1 ⊂ Mk ; i.e., Σλ = ∩∞ 6 0, then there is no singular k=1 Mk . If |β| + |ρ| = point in Σλ . Proof. We denote the eigenvectors of −4 corresponding to λk by {e∗1 , · · · , e∗m }. Thus, the space H1 and H defined in Theorem 3.1 or Theorem 4.1 can be decomposed into ⊥ H1 = Em ⊕ Em ,
⊥ em ⊕ E em H=E ,
Em = span{e∗i + ie∗j | 1 ≤ i, j ≤ m}, ⊥ Em = {u ∈ H1 | hu, viH1 = 0 ∀v ∈ Em },
em = Em , E
e ⊥ = {u ∈ H | hu, viH = 0 ∀v ∈ E em }. E m
By the center manifold theorem, the bifurcation equations of (1.1) with (1.4), or (1.1) with (1.5), at λ = λk are equivalent to
(5.1)
∂v1 = α4v1 − β4v2 + λv1 + P G1 (v + h(v)), ∂t ∂v2 = β4v + α4v + λv + P G (v + h(v)), 1 2 2 2 ∂t
⊥ is the center where λ is near λk , v = v1 + iv2 ∈ Em , and h : Em → Em manifold function, G = (G1 , G2 ) : H1 → H defined as in Theorem 3.1 or em is the projection. Theorem 4.1, and P : H → E The equations (5.1) are a system of ordinary differential equations with order 2m: dZ1 = (λ − α)Z1 + βZ2 + [−σ|Z|2 Z1 + ρ|Z|2 Z2 ] + o(|Z|3 ), dt (5.2) dZ2 = −βZ + (λ − α)Z + [−σ|Z|2 Z − ρ|Z|2 Z ] + o(|Z|3 ), 1 2 2 1 dt
where Z = Z1 + iZ2 . The eigenvalues of the linear part are still (λ − αλk ) ± iβλk , with multiplicity 2m.
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T. MA, J. PARK, AND S. WANG
By Theorem 2.5 it suffices to prove that v = 0 is asymptotically stable for (5.2) at λ = αλk . For λ = αλk , we infer from (5.2) that Z Z 1d 2 |v| dx = G(v + h(v))vdx 2 dt Ω Ω = (by h(v) = o(|v|) Z G(v)vdx + o(|v|4 ) = Ω Z |v|4 dx + o(|v|4 ), = −σ Ω
which implies that v = 0 is asymptotically stable for (5.2) at λ = αλk . The proof is complete. ¤ References [1] M. Bartuccelli, P. Constantin, C. R. Doering, J. D. Gibbon, and M. Gis¨ lt, Hard turbulence in a finite-dimensional dynamical system?, Phys. Lett. A, selfa 142 (1989), pp. 349–356. [2] , On the possibility of soft and hard turbulence in the complex Ginzburg-Landau equation, Phys. D, 44 (1990), pp. 421–444. [3] C. R. Doering, J. D. Gibbon, D. D. Holm, and B. Nicolaenko, Low-dimensional behaviour in the complex Ginzburg-Landau equation, Nonlinearity, 1 (1988), pp. 279– 309. [4] D. Henry, Geometric theory of semilinear parabolic equations, vol. 840 of Lecture Notes in Math., Springer-Verlag, Berlin, 1981. ´c ˇ, Ginzburg-Landau dynamics with a time-dependent [5] H. G. Kaper and P. Taka magnetic field, Nonlinearity, 11 (1998), pp. 291–305. [6] I. Kukavica, An upper bound for the winding number for solutions of the GinzburgLandau equation, Indiana Univ. Math. J., 41 (1992), pp. 825–836. [7] , Hausdorff length of level sets for solutions of the Ginzburg-Landau equation, Nonlinearity, 8 (1995), pp. 113–129. [8] T. Ma and S. Wang, Attractor bifurcation theory and its applications to RayleighB´enard convection, Commun. Pure Appl. Anal., 2 (2003), pp. 591–599. [9] , Dynamic bifurcation of nonlinear evolution equations, Chinese Ann. Math.Ser. B, 2004. [10] , Bifurcation Theory and Applications, World Scientific, to appear in 2005. [11] Q. Tang and S. Wang, Time dependent Ginzburg-Landau equations of superconductivity, Phys. D, 88 (1995), pp. 139–166. [12] R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, vol. 68 of Appl. Math. Sci. 68, Springer-Verlag, New York, second ed., 1997. (TM) Department of Mathematics, Sichuan University, Chengdu, People’s Republic of China and Department of Mathematics, Indiana University, Bloomington, IN 47405 (JP) Department of Mathematics, Indiana University, Bloomington, IN 47405 E-mail address: [email protected] (SW) Department of Mathematics, Indiana University, Bloomington, IN 47405 E-mail address: [email protected]
