Chunhsien Lu

TAYLAN S ¸ENG ÜL Yiqiu Mao
email: ymao@indiana.edu

Taylan Sengul
email: taylan.sengul@marmara.edu.tr

Quan Wang

ON THE SPECTRAL INSTABILITY AND BIFURCATION OF 2D-QUASI-GEOSTROPHIC POTENTIAL VORTICITY EQUATION

Keywords: Spectral instability, Generalized Kolmogorov forcing, Hopf bifurcation

whether they are published or not. The documents may come

1. Introduction 2. Spectral problem 2.1. Existence of eigenvalues 2.2. Uniqueness of eigenvalues 2.3. Continuous dependence of eigenvalues on the control parameter 3. The existence of unstable point spectrum 4. Numerical computations of eigenvalue problem and critical control parameter 5. Dynamic bifurcation-nonlinear instability analysis 5.1. Reduced equations governing the Hopf bifurcation 5.2. Reduced equation governing double Hopf bifurcation 5.3. Transition led by the varying of aspect ratio 6. Numerical investigation of transition number and periodic solution 6.1. Numerical experiments on transition number 6.2. Numerical results on periodic solutions 6. [START_REF] Tollmien | General instability criterion of laminar velocity distributions[END_REF]

. Conclusion References 1. Introduction

The field of hydrodynamic stability has a long history, which can look back to Reynolds and Lord Rayleigh who studied stability and transition in laminar flows in the late 19th century. Because of its central role in many research efforts involving fluid flow, stability theory has grown into a relatively mature discipline, firmly based on a large body of knowledge and a vast body of literature. Nowadays, this is still a very active research field.

In the field of hydrodynamic stability, a fundamental problem is to establish certain criteria under which one can judge the stability of a steady-state flow. As the earliest attempt, for an inviscid incompressible parallel flow, it was Rayleigh who derived the famous inflection point criterion [START_REF] Rayleigh | On the stability, or instability, of certain fluid motions[END_REF]. The criterion says that if there are no inflection points in a steady parallel flow given by U (y), the flow must be linearly stable. More precisely, the criterion is expressed by

If the flow U (y) defined on [a, b] is unstable, then there exists a point y 0 ∈ (a, b) at which U (y 0 ) = 0. In 1950, Fjørtoft extended the Rayleigh's criterion and provided a relatively stronger necessary condition for the instability of a parallel flow, called Fjørtoft's criterion [START_REF] Fjø | Application of integral theorems in deriving criteria of stability for laminar flows and for the baroclinic circular vortex[END_REF], which is stated as follows:

If the flow U (y) defined on [a, b] is unstable, then there exists a point y 0 ∈ (a, b) at which U (y 0 ) = 0, and U (y)(U (y) -U (y 0 )) < 0 for some y ∈ [a, b]. Strictly speaking, above criteria provide only necessary conditions for a shear flow depending on one variable, which are not sufficient conditions and can not be applied to all unstable shear flows, see [START_REF] Tollmien | General instability criterion of laminar velocity distributions[END_REF][START_REF] Ellingsen | Stability of linear flow[END_REF]. Nevertheless, the Rayleigh's criterion and Fjørtoft's criterion offer an intuitive basis for the stability of a parallel flow. There are many extensions of the above criteria to judge the stability for compressible or incompressible flows, see [START_REF] Lin | On the stability of two-dimensional parallel flows. III. Stability in a viscous fluid[END_REF][START_REF] Lessen | On the stability of plane parallel laminar flows to two-and three-dimensional disturbances[END_REF][START_REF] Sun | General stability criterion for inviscid parallel flow[END_REF][START_REF] Ma | Stability and bifurcation of the taylor problem[END_REF][START_REF] Landahl | A note on an algebraic instability of inviscid parallel shear flows[END_REF][START_REF] Batchelor | Analysis of the stability of axisymmetric jets[END_REF]. For more details on various criteria for the stability of shear flow, we refer readers to [START_REF] Schmid | Stability and transition in shear flows[END_REF].

From numerous experiments, Reynolds found a nondimensional number controlling the transitions of a viscous flow which is now known as the Reynolds number. For many parallel flows, if the corresponding Reynolds number is increased, the flows lose their stability and break into different motions. To study the stability of a viscous incompressible parallel flow, Orr and Sommerfeld [START_REF] Orr | The stability or instability of the steady motions of a perfect liquid and of a viscous liquid, i[END_REF][START_REF] Sommerfeld | Ein beitrag zur hydrodynamischen erklärung der turbulenten flüssigkeitsbewegung[END_REF] initiated the study of the spectral problem via the Fourier normal mode theory. By searching for unstable solutions of the form e iα(x-ct) u(y), they derived the well-known Orr-Sommerfeld equation for linearized viscous fluids at a steady-state (shear flow), which is obtained from the Rayleigh equation by adding the viscous term. In general, it is impossible to find exact solutions of Orr-Sommerfeld equation. Hence, the general approach to find an unstable mode is using the asymptotic expansion with respect to the inverse of the Reynolds number where the first term solves the Rayleigh equation, see [START_REF] Drazin | Hydrodynamic Stability[END_REF] for a complete account of the physical literature on the subject. Whenever the corresponding asymptotic expansion is valid, for large Reynolds number, one can find an unstable mode and the corresponding parallel flow is guaranteed to be unstable. This method has been used to analyze the stability of general shear flows [START_REF] Grenier | Spectral instability of characteristic boundary layer flows[END_REF][START_REF] Grenier | Spectral instability of general symmetric shear flows in a two-dimensional channel[END_REF]. Yet, there are other methods such as the operator method and variational method for the stability analysis of a viscous flow. The basic requirements for these methods are that either the first eigenvalue must be real or the corresponding eigenvalue problem needs a variational structure, such as the Taylor-Couette flows and Rayleigh-Bénard convection [START_REF] Herron | The principle of exchange of stabilities for Couette flow[END_REF][START_REF] Guo | Critical rayleigh number in rayleigh-bénard convection[END_REF].

In this article, we are interested in the stability of the basic flow of form ψ 0 = -τ Ek 4 sin(ky), (1.1) which is a steady-state solution to the two dimensional (2D) forced quasi-geostrophic (QG) equation ∂ ∂t ∆ψ + J(ψ, ∆ψ) + ∂ψ ∂x = τ sin(ky) + E∆ 2 ψ (1.2) defined on the domain: 0, 2π a × [0, π], where ψ is the stream function, a, E, are all positive parameters, k ≥ 2 is an integer and τ is the intensity of the curl of forcing which is determined by the ratio of intensity of forcing to the horizontal scale. Besides, ∆ is the standard 2D Laplace operator, and the nonlinear operator (the Jacobian) J is defined by J(ψ, φ) := ∂ψ ∂x ∂φ ∂x -∂ψ ∂y ∂φ ∂x .

The equation (1.2) is basically the nondimensional vorticity equation of the 2D Navier-Stokes equation on a beta-plane. The basic velocity profile corresponding to (1.1) is (U (y), 0) = -τ Ek 3 cos(ky), 0 , whose structure is shown in Figure 1. Figure 1. Illustration of the steady-state flow ψ 0 as derived from the system (1.3). The blue solid curve represents the horizontal profile of the mean flow, while the black arrows represent the direction of the mean flow on the horizontal domain. On the dashed curve, the velocity is zero. (a) and (b) respectively correspond to the profile (U (y), 0) = -E -1 k -3 τ cos(ky), 0 with k = 3 and 4.

Next, we consider the deviation ψ = ψ -ψ 0 from the steady state (1.1). Substituting ψ = ψ + ψ 0 into (1.2) and omitting the primes, we obtain the governing equation for the deviation:

∂ ∂t ∆ψ + R k 3 cos(ky)∂ x ∆ψ + R k cos(ky)∂ x ψ + ∂ψ ∂x = E∆ 2 ψ -J(ψ, ∆ψ), (1.3) subjected to the boundary conditions ψ(x + 2π a , y) = ψ(x, y), ψ(x, 0) = ψ(x, π) = ∂ 2 y ψ(x, 0) = ∂ 2 y ψ(x, π) = 0, (1.4) where R is the control parameter related to the Reynolds number Re and the intensity of the curl of forcing, defined as

R = E τ, Re = E . (1.5)
An inviscid incompressible flow with structure as in Figure 1 is possibly unstable due to the Rayleigh and Fjortoft's conditions aforementioned. However, in this article we consider a viscous incompressible basic flow which is sustained by the corresponding forcing. A natural question is to know whether the basic flow is stable. The question has been answered for the special case k = 2, in other words when the system is driven by a Kolmogorov forcing, in [START_REF] Chen | Hopf bifurcation in quasi-geostrophic channel flow[END_REF]. In that paper, assuming L x and L y are the length scales of the domain, they show that the flow is stable for

Ly Lx > √ 3
2 , and becomes unstable at some R = R c for all

Ly

Lx < α 0 where

√ 3 4 < α 0 < √ 3
2 . Also, their numerical simulations as well as those in [START_REF] Dijkstra | Dynamic Transitions of Quasi-geostrophic Channel Flow[END_REF] suggest that a transition occurs not only for Ly Lx < α 0 but also for α 0 < Ly Lx < √ 3 2 . Although observed in numerical experiments, they are unable to prove this claim due to technical reasons. For the case of k > 2, i.e., the case of generalized Kolmogorov forcing, the problem is still open. In this paper, we extend the previous study to any positive integer k ≥ 2.

Although the methods of asymptotic expansions, the method of perturbation operator and the variational method work well for many classical hydrodynamical stability problems aforementioned, they are not suitable for this forced driven problem. In this article, we use a different method called method of continuous fractions. This method is probably first introduced in the seminal paper by Meshalkin and Sinai [START_REF] Mevsalkin | Investigation of the stability of a stationary solution of a system of equations for the plane movement of an incompressible viscous liquid[END_REF]. More precisely, we will show that the eigenvalue problem

E∆ 2 ψ - R cos(ky) k ∂ x ψ -∂ x ψ - R cos(ky) k 3 ∂ x ∆ψ = µ∆ψ (1.6)
associated with the equation (1.3), can be transferred into a family of algebraic equations with infinitely many variables whose solutions can be expressed in the form of continuous fractions. Making use of the basic property of continuous fractions [START_REF] Van Vleck | On the convergence of continued fractions with complex elements[END_REF][START_REF] Van Vleck | Errata: "On the convergence of continued fractions with complex elements[END_REF], one can exactly determine that there are finite number of eigenvectors possibly unstable. Then, by analyzing the asymptotic estimate of the ratio of the imaginary parts of eigenvalues to the control parameter R, we show that these eigenvalues continuously depend on R, and there is at least one of which whose real part indeed goes to infinity as the control parameter R approaches infinity. That implies the basic solution (1.1) will become unstable at a threshold R c . In other words, (1.1) is unstable for large Reynolds number Re.

Most of time, to write down the exact expression for each unstable eigenvalue is nearly impossible. In order to shed light on the bifurcation involved in the losing stability of the basic flows given by (1.1), we analyze the first eigenvalue solving equation (1.6) numerically. We find that there are at most two pairs of complex conjugate eigenvalues which become critical (real part changes sign). That is, both the Hopf and Hopf-Hopf (double Hopf) bifurcations are possible as results of the losing stability of the basic flows given by (1.1). To know the type of the bifurcation, we use a more natural method to reduce the QG equation to a system of ODEs. Furthermore, relying on the numerical estimates on the coefficients of the ODEs, we find that in the case of k = 3, 4, there exist both supercritical and subcritical Hopf bifurcations while only supercritical bifurcation occurs in the case of k = 5. In addition, for the case of double Hopf bifurcation, only supercritical bifurcation is allowed.

For the convenience of stating our main results on the instability of the basic solution (1.1) and the bifurcation associated with it, in the case a < √ k 2 -1, let us define the nonnegative function M (k, a) as follows

M (k, a) = max{F (k, a), G(k, a)} (1.7)
where

F (k, a) = min (m,s)∈K F (k, a, m, s), G(k, a) = min (m,s)∈K G(k, a, m, s), K = (m, s) ∈ Z + × Z + : k 2 -m 2 a 2 -s 2 > 0 ,
and functions F (k, a, m, s) and G(k, a, m, s) are defined as follows

F (k, a, m, s) =J m,s N k (k 2 -m 2 a 2 -(k -s) 2 ) 2 (m 2 a 2 + (k -s) 2 ) 2 + J m,s N k (k 2 -m 2 a 2 -(k -s) 2 ) 2 (m 2 a 2 + (k -s) 2 ) 2 m 2 a 2 + (2k -s) 2 -k 2 m 2 a 2 + (2k -s) 2 × m 2 a 2 + s 2 k 2 -m 2 a 2 -s 2 - m 2 a 2 + (k -s) 2 m 2 a 2 + (k -s) 2 -k 2 , (1.8) G(k, a, m, s) = (k 2 -m 2 a 2 -s 2 ) 2 (m 2 a 2 + s 2 ) 2 + (k 2 -m 2 a 2 -s 2 ) 2 (m 2 a 2 + s 2 ) 2 m 2 a 2 + s 2 + k 2 + 2ks m 2 a 2 + s 2 + 2ks × m 2 a 2 + s 2 k 2 -m 2 a 2 -s 2 + T k,s m 2 a 2 + (k -s) 2 k 2 -m 2 a 2 -(k -s) 2 , (1.9) in which T k,s = 0, if k 2 -m 2 a 2 -(k -s) 2 < 0 1, if k 2 -m 2 a 2 -(k -s) 2 ≥ 0 , N k = 0, if k = 2 1, if k ≥ 3 ,
and

J m,s = 1, if m 2 a 2 +s 2 k 2 -m 2 a 2 -s 2 ≥ m 2 a 2 +(k-s) 2 m 2 a 2 +(k-s) 2 -k 2 , 0, if m 2 a 2 +s 2 k 2 -m 2 a 2 -s 2 < m 2 a 2 +(k-s) 2 m 2 a 2 +(k-s) 2 -k 2 .
.

The values of M (k, a) are shown in Figure 2 below. One can see that except for these values of a very close to √ k 2 -1, we have M (k, a) < 2, which is the key condition for our results, see the following theorem.

         µ m,1 (R c )      > 0, R > R c = 0, R = R c < 0, R < R c , m ∈ Z 0 ⊂ Z + , µ m,n (R c ) < 0, (m, n) ∈ Z + \ Z 0 × Z + , (1.10)
holds true (see Figure 6), where Z 0 is finite subset of Z and µ m,n is the eigenvalue solving (1.6) for each (m, n) ∈ Z + × Z + , then we have the following assertions:

(1) If Z 0 contains only one postive integer, then there is a parameter P (see (5.18)) whose real part determines a supercritical (subcritical) Hopf bifurcation on R > R c (R < R c ) if P < 0 ( P > 0). (2) If Z 0 contains two positive integers next to each other (see Table 6), then there are four parameters P i , i = 1, 2, 3, 4 (see (5.31)) describing the type of double Hopf bifurcation on R > R c . More precisely, if P i < 0 for i = 1, 2, 3, 4 and P 1 P 2 < P 3 P 4 , then the double Hopf bifurcation is supercritical, and as a result, two periodic solutions bifurcate from the basic flow (1.1). One of them is stable, the other one unstable, and their expressions are given by (5.34)- (5.35).

Proof. The first part of the theorem follows from Lemma 3.1 and Lemma 3.2. The second part is obtained from Theorem 5.1 and Theorem 5.2.

The rest of this article is organized as follows. In Section 2 we introduce and analyze the spectral problem. The instability of these eigenvalues are proved in Section 3. In Section 4, we give the numerical computations of eigenvalues and eigenvectors. The derivation of reduced equations for the Hopf and double Hopf bifurcations involved in the spectral instability and Bifurcation theorem are stated in Section 5. Section 6 contains the numerical results on the specific type of Hopf bifurcation and bifurcated periodic solutions, and conclusion.

Spectral problem

For the convenience, we introduce some notations and then rewrite (1.3) into an operator equation. For this purpose, let us use H 4 (Ω), H 2 (Ω), and L 2 (Ω) to denote the usual Sobolev and Lebesgue spaces and define H 1 , H 0 , and H -1 to be the Hilbert spaces

H 1 = {ψ ∈ H 4 (Ω) × H 4 (Ω)| ψ satisfies (1.4)}, H 0 = {ψ ∈ H 2 (Ω) × H 2 (Ω)| ψ satisfies (1.4)}, H -1 = L 2 (Ω) 2 ,
endowed with their natural inner products. We then introduce the differential operators L R :

H 1 → H -1 and A : H 0 → H -1 (G : H 0 × H 0 → H -1
) acting on ψ as follows:

Aψ = ∆ψ, G(ψ, φ) = -J(ψ, ∆φ), L R ψ = E∆ 2 ψ - R cos(ky) k ∂ x ψ -∂ x ψ - R cos(ky) k 3 ∂ x ∆ψ.
(2.1)

Given those notations, the equation (1.3) subjected to the boundary conditions (1.4) can be put into the following abstract operator form

dAψ dt = L R ψ + G(ψ, ψ). (2.2)
Note that because of the boundary conditions (1.4), A is an isomorphism between H 0 and H -1 . Hence, A -1 • L R is a bounded operator from H 1 into H 0 . Furthermore, due to the classical Sobolev embeddings, the inclusion H 1 → H 0 is dense and compact, and thus

A -1 • L R : D(A -1 • L R ) = H 1 ⊂ H 0 → H 0 is a completely continuous operator.
Then, the stability of the basic flows given by (1.1) is transferred into the analysis of the stability of zero solution to the preceding abstract system. Naturally, the eigenvalue problem

L R ψ = µ∆ψ (2.3) arises.
In what follows, we aim to search for the unstable point spectrum of L R for large control parameter R, which is the standard procedure in hydrodynamical stability analysis. The boundary conditions in (1.3) dictate that the eigenmodes have the following form

ψ m (x, y) = n∈N η m,n e iamx sin(ny), m ∈ Z, (2.4)
where η m,n = η -m,n ∈ C are the unknowns to be determined. By substituting the form (2.4) into the eigenvalue equation (2.3), for each m ∈ Z, we obtain an algebraic equation,

- n≥k R 2k (iam)η m,n sin((n + k)y) - n≥k R 2k (iam)η m,n sin((n -k)y) - n<k R 2k (iam)η m,n sin((n + k)y) + n<k R 2k (iam)η m,n sin((k -n)y) + n≥k R 2k 3 η m,n (iam)ξ m,n sin((n + k)y) + n≥k R 2k 3 η m,n (iam)ξ m,n sin((n -k)y) + n<k R 2k 3 η m,n (iam)ξ m,n sin((n + k)y) - n<k R 2k 3 η m,n (iam)ξ m,n sin((k -n)y) - n η m,n (iam) sin(ny) + n Eη m,n ξ 2 m,n sin(ny) = -µ n η m,n ξ m,n sin(ny), (2.5)
where

ξ m,n = a 2 m 2 + n 2 . (2.6)
By changing the index and matching the coefficients of terms corresponding to sin(ny) in (2.5), we arrive at (2.7)

- n≥2k R 2k (iam)η m,n-k sin(ny) - n≥1 R 2k (iam)η m,n+k sin(ny) - k+1≤n<2k-1 R 2k (iam)η m,n-k sin(ny) + 1≤n≤k-1 R 2k (iam)η m,k-n sin(ny) + n≥2k R 2k 3 η m,n-k (iam)ξ m,n-k sin(ny) + n≥1 R 2k 3 η m,n+k (iam)ξ m,n+k sin(ny) + k+1≤n<2k-1 R 2k 3 η m,n-k (iam)ξ m,n-k sin(ny) - 1≤n≤k-1 R 2k 3 η m,k-n (iam)ξ m,k-n sin(ny) - n≥1 η m,n (
From (2.7), we can see that for m = 0, the eigenvalues corresponding to ψ m are always negative, i.e. for each n ∈ N, µ = µ 0,n = -Eξ 0,n < 0. Thus, these eigenvalues corresponding to m = 0 do not affect the stability of the basic state. Now, we restrict our attention to positive integer values of m. Since the eigenvalues corresponding to -m and m, for positive m, can be paired by complex conjugate. From the identity (2.7), we find that

R 2k (iam) - R 2k 3 (iam)ξ m,k-n η m,k-n - R 2k (iam) - R 2k 3 (iam)ξ m,k+n η m,k+n + -(iam) + Eξ 2 m,n + µξ m,n η m,n = 0, if n ≤ k -1, - R 2k (iam) - R 2k 3 (iam)ξ m,2k η m,2k + -(iam) + Eξ 2 m,k + µξ m,k η m,k = 0, if n = k, (2.8) - R 2k (iam) - R 2k 3 (iam)ξ m,n-k η m,n-k - R 2k (iam) - R 2k 3 (iam)ξ m,n+k η m,n+k + -(iam) + Eξ 2 m,n + µξ m,n η m,n = 0, if n ≥ k + 1.
Let us denote,

A m,n = Ram 2k 3 (k 2 -ξ m,n ), B m,n = -(iam) + Eξ 2 m,n + µξ m,n .
(2.9)

We note that A m,n is real for any m, n and

A m,n > 0 if and only if a 2 m 2 + n 2 < k 2 , A m,n < 0 if and only if a 2 m 2 + n 2 > k 2 .
(2.10) Using (2.9), we can rewrite the identity (2.8) as a system of algebraic equations for the unknowns η m,n , which is

iA m,k-n η m,k-n -iA m,n+k η m,k+n + B m,n η m,n = 0, 1 ≤ n ≤ k -1, iA m,2k η m,2k -B m,k η m,k = 0, n = k, iA m,n-k η m,n-k + iA m,n+k η m,n+k -B m,n η m,n = 0, n ≥ k + 1. (2.11) Let η m,n = i 1+[ n k ] φ m,n /A m,n , (2.12)
and

d m,n = B m,n A m,n . (2.13)
In addition, letting n = kj + s, where 0 ≤ s ≤ k -1, then (2.11) can be rewritten as

φ m,2k + d m,k φ m,k = 0, n = k, φ m,(j+1)k + d m,jk φ m,jk -φ m,(j-1)k = 0, j ≥ 2, (2.14)
and

iφ m,k-s + φ m,k+s + d m,s φ m,s = 0, 1 ≤ s ≤ k -1, φ m,(j+1)k+s + d m,jk+s φ m,jk+s -φ m,(j-1)k+s = 0, 1 ≤ s ≤ k -1, j ≥ 1.
(2.15)

One can directly check above system has exactly k 2 + 1 independent systems of algebraic equations for each 0 ≤ s ≤ k 2 . Let ψ m,0 be the eigenfunction corresponding to (2.14)

(d m,(j+1)k-s ) φ m,(j+1)k-s 2 = 0, 1 ≤ s ≤ [k/2], (2.20) k-1 s=1 +∞ j=0 (d m,jk+s ) |φ m,jk+s | 2 = 0, (2.21)
where

d m,n = 2k 3 ξ m,n (Eξ m,n + µ) Ram (k 2 -ξ m,n ) , (2.22)
and (z) denotes the real part of z.

Proof. We multiply the first equation in (2.15) by φ m,k and the second equation by φ m,jk and then take summation over j ≥ 2 to get

+∞ j=1 d m,jk |φ m,jk | 2 - +∞ j=1 φ m,jk φ m,(j+1)k -φ m,(j+1)k φ m,jk = 0.
Taking the real part of the above identity yields (2.19).

Next, we multiply the second equation in (2.15) by φ m,jk+s by each j ≥ j 0 ≥ 1 and then take the summation over j for any 1 ≤ s ≤ k -1. If we take s → k -s in (2.24), add the resulting equation to (2.24) and finally take the real part, we get (2.20).

Finally summing (2.20) over s for 1 ≤ s ≤ k -1 we get (2.21).

Our first result is on the possible unstable modes and on the nonlinear stability of the system.

Lemma 2.2. For m = 0, if k 2 -a 2 m 2 -1 ≤ 0 then for any eigenvalue µ of (2.3) corresponding to an eigenvector ψ m given by (2.4), we have µ ≤ -E(m 2 a 2 + 1). In particular, if k 2 -a 2 -1 ≤ 0 then for any eigenvalue µ of (2.3), we have µ ≤ -E(a 2 + 1) and as a result, the system is nonlinearly stable.

Proof. First, suppose that k 2 -m 2 a 2 -1 < 0, we have

k 2 -ξ m,n ≤ k 2 -ξ m,1 = k 2 -m 2 a 2 -1 < 0, Now if we suppose that µ > -E(m 2 a 2 + 1) = -Eξ m,1 , then Eξ m,n + µ > E(ξ m,n -ξ m,1 ) ≥ 0
and by (2.22) we have d m,n < 0 which in turn yields by (2.19), (2.21) that φ m,n = 0 and by (2.12), η m,n = 0 for all m ∈ Z-{0}, n ∈ Z + which is a contradiction. By continuous dependence of the eigenvalues on system parameters, the result is still valid when

k 2 -a 2 m 2 -1 = 0. The second statement is just a consequence of the first statement since if k 2 -a 2 -1 ≤ 0 then k 2 -a 2 m 2 -1 ≤ 0 for all m = 0.
The below lemma shows that eigenmodes with only nonzero vertical Fourier coefficients concentrate on the integer multiples of the frequency of the external forcing are always stable.

Lemma 2.3. For an eigenvalue µ corresponding to the eigenvector ψ m,0 , m = 0, given in (2.16), we have µ ≤ -E(a 2 m 2 + 1).

Proof. Suppose on the contrary that µ + E(a 2 m 2 + 1) > 0. By definition (2.6), k 2 -ξ m,jk < 0 for all j ≥ 1 and as a result d m,jk < 0, for all j ≥ 1. Now, for any j ≥ 1, (2.19) implies that φ m,jk = 0 and by (2.12), we have η m,jk = 0 which leads to a contradiction.

From now on, we only need to focus on the case

a 2 < k 2 -1, (2.25)
which by Lemma 2.2 is a necessary (but not sufficient) condition for the existence of an unstable eigenmode. Under the assumption (2.25), we define r ∈ Z + as

0 < √ k 2 -1 a -1 ≤ r < √ k 2 -1 a , (2.26) so that k 2 -m 2 a 2 -1 > 0, ∀m, 1 ≤ m ≤ r, and k 2 -(r + 1) 2 a 2 -1 ≤ 0. Thus for each m, 1 ≤ m ≤ r, there exists q m ∈ Z + , 1 ≤ q m < k, (in fact 1 ≤ q m < √ k 2 (r+1) 2 -(k 2 -1)m 2 r+1
) for which

k 2 -m 2 a 2 -s 2 > 0, if 1 ≤ s ≤ q m , ≤ 0, if s > q m .
Let us define the index set

K = (m, s) ∈ Z + × Z + : 1 ≤ m ≤ r, 1 ≤ s ≤ q m .
Notice that under the assumption (2.25), (1, 1) ∈ K = ∅. Now, if we also assume

µ > -E(m 2 a 2 + 1), (2.27) then (2.22) implies that (d m,n ) > 0, if (m, n) ∈ K, (d m,n ) < 0, otherwise. (2.28)
As a consequence, we have the following lemma.

Lemma 2.4. If both (m, s) / ∈ K and (m, k -s) / ∈ K then the eigenmode ψ m,s given by (2.17), (2.18) is stable.

Proof. Suppose (2.27) holds. Note that (m, n) / ∈ K? for all n ≥ k. Thus in this case we find that (d m,jk+s ) < 0 and (d m,(j+1)k-s ) < 0 for all j ≥ 0. As a result of (2.20), we find that ψ m,s = 0 which is a contradiction. Sınce, if q m < [k/2] and q m < s ≤ [k/2] then both (m, s) and (m, k -s) are not in K, we have the following result.

Lemma 2.5. ψ m,s is stable if q m < [k/2] and q m < s ≤ [k/2].
By Lemma 2.4, the possible unstable eigenmodes are of the form ψ m,s for either (m, s) ∈ K or (m, k -s) ∈ K. The following lemma shows that for an unstable eigenmode ψ m,s , all of its Fourier coefficients must be non-zero.

Lemma 2.6. Suppose that µ > -Eξ m,1 , for each (m, s) ∈ K, If there exists j = j 0 ≥ 0 and such that φ m,j 0 k+s = 0, then φ m,jk+s = φ m,(j+1)k-s = 0, for all j ≥ 0.

Proof. If j 0 = 0 and 1 ≤ s ≤ k -1, using φ m,s = 0 and taking the real part of (2.24), we obtain

+∞ j=1 (d m,jk+s ) |φ m,jk+s | 2 = 0. (2.29) We have k 2 -ξ m,jk+s = k 2 (1 -j 2 ) -a 2 m 2 -s 2 < 0 for all j ≥ 1,
φ m,jk+s = 0, ∀j ≥ 1.
If j 0 ≥ 1 and φ m,j 0 k+s = 0 for some s such that 1 ≤ s ≤ k -1, by taking the real part of (2.23) and using φ m,j 0 k+s = 0, a similar argument as above shows that φ m,jk+s = 0, ∀j ≥ j 0 .

Recalling the last equation of (2.14), -φ m,(j-1)k+s + φ m,(j+1)k+s + d m,jk+s φ m,jk+s = 0, j ≥ 1 we can show that φ m,jk+s = 0, ∀j ≥ 0. by which and (2.14) we can also derive

φ m,jk+k-s = 0, for j 0 -1 ≥ j ≥ 0.
To illustrate Lemma 2.6, we consider an example.

Example 2.1. Suppose µ > -Eξ m,1 and k = 10. By Lemma 2.6 if η m,24 = 0 then we have

0 = η m,4 = η m,6 = η m,14 = η m,16 = η m,24 = η m,26 = • • • For convenience, through this article, we use [x 1 , x 2 , • • • ] to express the complex-valued con- tinuous fraction [x 1 , x 2 , x 3 • • • ] = 1 x 1 + 1 x 2 + 1 x 3 +••• , where x i ∈ C for all i ≥ 1.
The form of (2.15) enables us to consider the following groups of equations

d t η t + η k+t + iη k-t = 0, d jk+t η jk+t + η (j+1)k+t -η (j-1)k+t = 0, for j ≥ 1, (2.30) where 1 ≤ s ≤ k -1 and t ∈ {s, k -s}.
Lemma 2.7. Suppose that for 1 ≤ s ≤ k-1 we are given sequences , {d jk+s } ∞ j=0 and {d jk-s } ∞ j=1 which satisfy the conditions

(d jk+s ) < 0 and (d (j+1)k-s ) < 0, ∀j ≥ 1, k-1 s=1 +∞ j=0 |d jk+s | = +∞.
(2.31) Then, the system (2.30) has a nontrivial solution set {η jk+s = 0, η (j+1)k-s = 0, ∀j ≥ 0} if and only if the compatibility conditions

d s + i = -[d 3s , d 5s , • • • ], if k = 2s, d t = -[d k+t , d 2k+t , • • • ] -[d k-t , d 2k-t , • • • ], t ∈ {s, k -s} , if k = 2s, (2.32)
are satisfied. Particularly, one can derive that each element of any nontrivial solution set can be expressed in form of

η jk+t = η t α 1,t • • • α jk,t , t ∈ {s, k -s} , j ≥ 1, (2.33) where α jk,t = 1 d jk+t + 1 d (j+1)k+t +••• = 0, t ∈ {s, k -s} , j ≥ 1, (2.34)
Proof. We only need to prove the necessary and sufficient condition. For the proof of the second part, which is contained in the proof process of the first part. We first note that the assumptions (2.31) guarantee that the limit

[d jk+t , d (j+1)k+t , . . . ] = 1 d jk+t + 1 d (j+1)k+t +••• = α jk,t , t ∈ {s, k -s} (2.35)
exists for each j ≥ 1 and is nonzero by Theorem 2 in [START_REF] Van Vleck | On the convergence of continued fractions with complex elements[END_REF] so that the continued fractions given in the statement of the theorem indeed converges to a nonzero complex number. First, we prove the necessary part of the theorem. Now let us consider the case k = 2s. Then the equations (2.30) with t = s become

d s η s + η 3s + iη s = 0, d (2j+1)s η (2j+1)s + η (2j+3)s -η (2j-1)s = 0, j ≥ 1 (2.36)
from the first equation in (2.36) we get that

η 3s η s = -d s -i = 0. (2.37)
Moreover, we also have by the second equation in (2.36) that

η (2j+1)s η (2j-1)s = 1 d (2j+1)s + η (2j+3)s η (2j+1)s = 0, j ≥ 1 (2.
-i η s η k-s =d k-s + η 2k-s η k-s = 0, (2.40)
d s = - η k+s η s -i η k-s η s , (2.41)
η jk+t η (j-1)k+t = 1 d jk+t + η (j+1)k+t η jk+t = 0, t ∈ {s, k -s} , j ≥ 1. ( 2
d s = - 1 d k+s + η 2k+s η k+s - 1 d k-s + η 2k-s η k-s .
By making use of the iteration formula (2.42), we can see that

η 2k+t η k+t = 1 d 2k+t + 1 d 3k+t +••• , t ∈ {s, k -s},
inserting which into the preceding equation gives the second equation in (2.32).

In what follows, we prove the sufficient condition. In the case of k = 2s, the first equation in (2.32) and (2.35) give

d s + i = -[d 3s , d 5s , • • • ] = α 2s,s = 0. (2.43) Define ζ s = C 0 , ζ 2js+s = C 0 α s,s α 2s,s • • • α 2js,s
for any nonzero constant C 0 ∈ C, then by which and (2.35) we have

ζ 2js+s ζ 2(j-1)s+s = α 2js,s = 1 d (2j+1)s + η (2j+3)s η (2j+1)s = 0.
(2.44)

The (2.43) and (2.46)

d s ζ s + ζ 3s + iζ s = 0, d (2j+1)s ζ (2j+1)s + ζ (2j+3)s -ζ (2j-1)s = 0, j ≥ 1,
which means that (2.30) has nontrivial solution set in the case of k = 2s.

In the case of k = 2s, the second equation in (2.32) and (2.35) give

d t = -[d k+t , d 2k+t , • • • ] -[d k-t , d 2k-t , • • • ] = -α k,t + -1 d k-t + α k,k-t t ∈ {s, k -s} . (2.45)
Now, let's define

ζ t = C t , ζ jk+t = C t α k,t α 2k,t • • • α jk,t , t ∈ {s, k -s} , j ≥ 1.
where C t ∈ C is any nonzero complex-valued constant. then by which and (2.35) we have

ζ jk+t ζ (j-1)k+t = α jk,t = 1 d jk + α (j+1)k,t = 1 d jk + ζ (j+1)k ζ jk = 0, j ≥ 1. (2.46)
On the one hand, the equation (2.46) gives

d jk+t ζ jk+t + ζ (j+1)k+t -ζ (j-1)k+t = 0, j ≥ 1. (2.47)
On the other hand, it deduces from (2.45) and (2.46) that

d s = - ζ k+s ζ s + -1 d k-s + ζ k+k-s ζ k-s , (2.48)
which gives

d k-s + ζ k+k-s ζ k-s d s + ζ k+s ζ s = -1.
(2.49)

For any given nonzero

ζ k-s = C k-s ∈ C, if we chose ζ s = C s by ζ s = -iζ k-s -ζ k+s d s then we have from (2.49) d t ζ t + ζ k+t + iζ k-t = 0. t ∈ {s, k -s}.
(2.50)

Finally, (2.47) and (2.50) means {ζ jk+s = 0, ζ (j+1)k-s = 0, ∀j ≥ 0} is the nontrivial solution to the system (2.30). The proof is complete.

Now, denote

β m,jk+s = φ m,jk+s φ m,(j-1)k+s , β m,jk+k-s = φ m,jk+k-s φ m,(j-1)k+k-s , for j ≥ 1, (m, s) ∈ K. (2.51) (2.14) implies -i φ m,k-s φ m,s = d m,s + β m,k+s , -i φ m,s φ m,k-s = d m,k-s + β m,k+k-s , β m,jk+s = 1 d m,jk+s + β m,(j+1)+s , j ∈ N + , (m, s) ∈ K, β m,jk+k-s = 1 d m,jk+k-s + β m,(j+1)+k-s , j ∈ N + , (m, s) ∈ K. (2.52) Note that d m,n = -2k 3 (iam) Ram(k 2 -a 2 m 2 -n 2 ) + 2k 3 (m 2 a 2 + n 2 ) E(a 2 m 2 + n 2 ) + µ Rma(k 2 -m 2 a 2 -n 2 ) , k = 2, 3, • • • ; n = 1, 2, 3 • • • .
Apparently, for each k ≥ 2, we can choose a and m such that {d m,n | n = 1, 2, 3, • • • } satisfies (2.31). Therefore, from Lemma 2.7, we obtain the following proposition.

Proposition 2.1. For (m, s) ∈ K, the equation (2.15) has nontrivial solutions which are

{φ m,t , φ m,k+t , φ m,2k+t , • • • , } , if (m, s) ∈ K, t ∈ {s, k -s}, (2.53)
where

φ m,jk+t = φ m,t β m,t β m,k+t • • • β m,jk+t , j ≥ 1,
β m,jk+t = [d m,jk+t , d m,(j+1)k+t , . . . ],
if and only if

d m,s + i = -[d m,3s , d m,5s , . . . ], if k = 2s, d m,t = -[d m,k+t , d m,2k+t , . . . ] -[d m,k-t , d m,2k-t , . . . ], if k = 2t. ( 2
m,n = Ra(k 2 -m 2 a 2 -n 2 ), d m,n (µ) = 2k 3 (ξ m,n (Eξ m,n + µ) -iam) C m,n , µ ∈ C, n ∈ Z + . ( 2
D(µ) = -[d m,3s (µ), d m,5s (µ), . . . ] -i, k = 2s, H(µ) = -[d m,k+s (µ), d m,2k+s (µ), . . . ] -[d m,k-s (µ), d m,k+k-s (µ), . . . ],
and

W (µ) : C → C as W (µ) = Cm,sD(µ) 2k 3 ξm,s -Eξ m,s + iam ξm,s , if k = 2s, Cm,sH(µ) 2k 3 ξm,sd m,k-s (µ) -Eξ m,s + iam ξm,s , if d m,s > 0 and d m,k-s < 0, The inequalities |D(µ)| ≤ 1 | d m,k+s (µ)| + 1 < 1 2k 3 ξ m,k+s C m,k+s ((s + k) 2 -s 2 ) + 1 < +∞, |H(µ)| ≤ 1 | d m,k+s (µ)| + 1 | d m,k-s (µ)| < +∞,
where we use |Eξ m,k+s + µ| = Eξ s + µ + (s + k) 2 -s 2 and the condition Eξ s + µ > 0, d m,jk+k-s < 0 (j ≥ 0), d m,jk+s < 0 (j ≥ 1), and | d m,k±s (µ)| > C for some C and all µ with µ > -Eξ m,s , guarantee the boundedness (in µ) of W (µ), i.e.,

|W (µ)| ≤ max |C m,s | |D(µ)| 2k 3 ξ m,s + Eξ m,s + am ξ m,s , |C m,s | |H(µ)| 2k 3 ξ m,s |d m,k-s (µ)| + Eξ m,s + am ξ m,s = M.
If we define the set D 1 as follows

D 1 = {µ ∈ C | µ > -Eξ m,s , |µ| ≤ M },
then, the Brouwer Fixed Point Theorem implies that W m,s has a fixed point µ = µ ms in D.

That is,

µ ms = W (µ ms ),
which imply that (2.54) hold true at µ = µ ms .

Case ii: d m,s > 0 and d m,k-s > 0. Without loss of generality, we can assume that s < k -s, otherwise we can switch s and k -s below. In this case, we can not use

1 | d m,k-s| to control the complex-valued function [d m,k-s (µ), d m,2k-s (µ), . . . ]
due to d m,jk+k-s < 0 and d m,k-s > 0 for j ≥ 1. Note that the second identity of (2.54) is equivalent to

d m,s (µ)d m,k-s (µ) = -1 -d m,s (µ)[d m,2k-s (µ), d m,3k-s (µ), . . . ] - [d m,k-s (µ), d m,2k-s (µ), . . . ] [d m,k+s (µ), d m,2k+s (µ), . . . ] , (2.56)
Using right-hand side of the preceding identity, we can a define complex-valued function

U (µ) = -1 -d m,s (µ)[d m,2k-s (µ), d m,3k-s (µ), . . . ] - [d m,k-s (µ), d m,2k-s (µ), . . . ] [d m,k+s (µ), d m,2k+s (µ), . . . ] which is bounded. That is, |U (µ)| ≤1 + |d m,s | | d m,2k-s | + |d m,k-s | + 1 | d m,2k-s| | d m,k+s | < +∞,
F (µ) = C m,s U (µ) 2k 3 ξ m,s d m,k-s (µ)
.

We have

|F (µ)| ≤ |C m,s | |U (µ)| 2k 3 |ξ m,s | |d m,k-s (µ)| < +∞.
In a similar way to case i, one can show that there exists µ = µ ms such that

µ ms = F (µ ms ),
i.e., (2.54) hold true in this case.

Remark 2.1. The Lemma 2.8 and Proposition Proposition 2.1 guarantee the existence of nontrivial solution to equation (2.15), and µ ms is the corresponding eigenvalue of the operator L R defined in (2.1), the corresponding eigenfunctions are given by (2.17)-(2.18).

Uniqueness of eigenvalues.

Lemma 2.9. Under the condition of Lemma 2.8, if (m, k -s) / ∈ K, then there corresponds a unique eigenvalue µ ms of the operator L R satisfying µ ms > -E(m 2 a 2 + s 2 ) and solving (2.54).

Proof. Here, only prove the case of k = 2s, for the case k = 2s, the proof is same. Suppose that µ ms,1 and µ ms,2 are two different values which satisfies (2.54), and for j ≥ 1, let's denote

β + m,jk+s (µ ms,1 ) = 1 d m,jk+s (µ ms,1 ) + 1 d m,(j+1)k+s (µ ms,1 ) + • • • = 1 d m,jk+s (µ ms,1 ) + β + m,(j+1)k+s (µ ms,1 )
, and

β - m,jk+s (µ ms,2 ) = 1 d m,jk+s (µ ms,2 ) + 1 d m,(j+1)k+s (µ ms,2 ) + • • • = 1 d m,jk+s (µ ms,2 ) + β - m,(j+1)k+s (µ ms,2 )
. By (2.51), we have

β + m,k+s (µ ms,1 ) -β - m,k+s (µ ms,2 ) = β + m,k+s (µ ms,1 )β - m,k+s (µ ms,2 )[d m,k+s (µ ms,2 ) + β - m,2k+s (µ ms,2 ) -d m,k+s (µ ms,1 ) -β + m,2k+s (µ ms,1 )] = - 2k 3 ξ m,k+s C m,k+s φ m,k+s (µ ms,1 )φ m,k+s (µ ms,2 ) φ m,s (µ ms,1 )φ ms (µ ms,2 ) (µ ms,1 -µ ms,2 ) -β + m,k+s (µ 1 )β - m,k+s (µ 2 )[β + m,2k+s (µ ms,1 ) -β - m,2k+s (µ ms,2 )]
By induction, it deduces that φ m,s (µ ms,1 )φ m,s (µ ms,2 ) β + m,k+s (µ ms,1 ) -β - m,k+s (µ ms,2 )

= ∞ j=1
(-1) j 2k 3 ξ m,jk+s C m,jk+s φ m,jk+s (µ ms,1 )φ m,jk+s (µ ms,2 )(µ ms,1 -µ ms,2 ).

(2.57)

In a similar way, we have

φ m,k-s (µ ms,1 )φ m,k-s (µ ms,2 ) β + m,k+k-s (µ ms,1 ) -β - m,k+k-s (µ ms,2 ) = ∞ j=1 (-1) j 2k 3 ξ m,jk+k-s C m,jk+k-s φ m,jk+k-s (µ ms,1 )φ m,jk+k-s (µ ms,2 )(µ ms,1 -µ ms,2 ).
(2.58)

On the other hand, from (2.52) we have

-i φ m,k-s (µ ms,1 ) φ m,s (µ ms,1 ) = d m,s (µ ms,1 ) + β + m,k+s (µ ms,1 ), -i φ m,s (µ ms,1 ) φ m,k-s (µ ms,1 ) = d m,k-s (µ ms,1 ) + β + m,k+k-s (µ ms,1 ), -i φ m,k-s (µ ms,2 ) φ m,s (µ ms,2 ) = d m,s (µ ms,2 ) + β - 1,s (µ ms,2 ), -i φ m,s (µ ms,2 ) φ m,k-s (µ ms,2 ) = d m,k-s (µ ms,2 ) + β - m,
β + m,k+k-s (µ ms,1 ) -β - m,k+k-s (µ m,s,2 ) + 2k 2 ξ m,k-s C m,k-s (µ ms,1 -µ ms,2 ) = i φ m,k-s (µ ms,1 )φ m,s (µ ms,2 ) -φ m,s (µ ms,1 )φ m,k-s (µ ms,2 ) φ m,k-s (µ ms,1 )φ m,k-s (µ ms,2
) .

(2.61)

Therefore, it derives from (2.60) and (2.61) that

φ m,s (µ ms,1 )φ m,s (µ ms,2 ) β + m,k+s (µ ms,1 ) -β - m,k+s (µ ms,2 ) + φ m,k-s (µ ms,1 )φ m,k-s (µ ms,2 ) β + m,k+k-s (µ ms,1 ) -β - m,k+k-s (µ m,s,2 ) = - 2k 2 ξ m,s C m,s φ m,s (µ ms,1 )φ m,s (µ ms,2 ) + 2k 2 ξ m,k-s C m,k-s φ m,k-s (µ ms,1 )φ m,k-s (µ ms,2 ) (µ ms,1 -µ ms,2 )
by which and making use of (2.57)-(2.58) one can obtain

-2k 2 ξ m,s C m,s φ m,s (µ ms,1 )φ m,s (µ ms,2 )- 2k 2 ξ m,k-s C m,k-s φ m,k-s (µ ms,1 )φ m,k-s (µ ms,2 ) (µ ms,1 -µ ms,2 ) = ∞ j=1
(-1) j 2k 3 ξ m,jk+s C m,jk+s φ m,jk+s (µ ms,1 )φ m,jk+s (µ ms,2 )(µ ms,1 -µ ms,2 )

+ ∞ j=1 (-1) j 2k 3 ξ m,jk+k-s C m,jk+k-s φ m,jk+k-s (µ ms,1 )φ m,jk+k-s (µ ms,2 )(µ ms,1 -µ ms,2 ).
(2.62)

Now, based on the preceding inequality, it deduces

|µ ms,1 -µ ms,2 | < -C m,s ξ m,k-s 2ξ m,s C m,k-s |φ m,k-s (µ m,s,1 )| 2 |φ m,s (µ ms,1 )| 2 + |φ m,k-s (µ ms,2 )| 2 |φ m,s (µ ms,2 )| 2 + 2 l=1 +∞ j=1 -C k m,s ξ m,jk+s 2ξ m,s C m,jk+s |φ m,jk+s (µ ms,l )| 2 |φ m,s (µ ms,l )| 2 + 2 l=1 +∞ j=1 -C k m,s ξ m,jk+k-s 2ξ m,s C m,jk+k-s φ k m,jk+k-s (µ ms,l ) 2 |φ m,s (µ ms,l )| 2 |µ ms,1 -µ ms,2 | , (2.63)
where we have used (2.65)

(d m,s (µ ms,l )) > 0, (d m,k-s (µ ms,l )) < 0, l = 1, 2, ( 2
we have the identity

(d m,s (µ ms,l )) = -(d m,k-s (µ ms,l )) |φ m,k-s (µ ms,l )| 2 |φ m,s (µ ms,l )| 2 - +∞ j=1 (d m,jk+s (µ ms,l )) |φ m,jk+s (µ ms,l )| 2 |φ m,s (µ ms,l )| 2 - +∞ j=1 (d m,jk+k-s (µ ms,l )) |φ m,jk+k-s (µ ms,l )| 2 |φ m,s (µ ms,l )| 2 , l = 1, 2.
(2.66) by which we arrive at where we use the ineqality

-C m,s ξ m,k-s ξ m,s C m,k-s |φ m,k-s (µ ms,1 )| 2 |φ m,s (µ ms,1 )| 2 + |φ m,k-s (µ ms,2 )| 2 |φ m,s (µ m,s,2 )| 2 + 2 l=1 +∞ j=1 -C m,s ξ m,jk+s ξ m,s C m,jk+s
-C m,s ξ m,jk+p ξ m,s C m,jk+p < (d m,jk+p (µ ms,l )) (d m,s (µ ms,l )) , 1 ≤< p < k -1.
Therefore, (2.63) and (2.67) imply that

|µ ms,1 -µ ms,2 | < |µ ms,1 -µ ms,2 | ,
which leads a contradiction. Hence, µ ms,1 = µ ms,2 = µ ms .

2.3. Continuous dependence of eigenvalues on the control parameter. We have known that for R > 0 and

0 < m 2 a 2 < k 2 -1, k = 2, 3, • • • ,
There exists a unique µ ms (R) solving the corresponding implicit function F m,s (R, µ) = 0, and µ ms > -Eξ m,s , where

F m,s (R, µ ms ) =d m,s + [d m,k+s (µ ms ), d m,2k+s (µ ms ), • • • ] + [d m,k-s (µ ms ), d m,k+k-s (µ ms ), • • • ] = 0, s ∈ K, k = 2s or F m,s (R, µ ms ) =d m,s + i + [d m,3s (µ ms ), d m,5s (µ ms ), • • • ] = 0, k = 2s.
(2.68) And for each (m, s) ∈ K, µ ms (R) is one eigenvalue of the operator L R . In what follows, our goal is to show that µ ms (R) is continuously dependent on R.

Lemma 2.10. For each (m, s) ∈ K), the eigenvalue µ ms (R) given in Lemma 2.8 is continuously dependent on R.

Proof. Here, we only prove the case of k = 2s; for the case of k = 2s, the proof is same. Based on the Implicit Function Theorem, we only need to show ∂F m,s ∂µ > 0.

(2.69)

From (2.57) and (2.58), we obtain that

(φ m,s ) 2 ∂β m,k+s ∂µ = ∞ j=1 (-1) j 2k 3 ξ m,jk+s C m,jk+s (φ m,jk+s ) 2 , (φ m,k-s ) 2 ∂β m,k+k-s ∂µ = ∞ j=1 (-1) j 2k 3 ξ m,jk+k-s C m,jk+k-s (φ m,jk+k-s ) 2 .
(2.70)

Upon performing straight forward calculation, we can obtain

(φ m,k-s ) 2 ∂D m,s ∂µ (d m,k-s + β m,,k+k-s ) 2 = ∂d m,s ∂µ + ∂β m,k+s ∂µ (d m,k-s + β m,k+k-s ) 2 (φ m,k+k-s ) 2 - ∂d m,k-s ∂µ + ∂β m,k+k-s ∂µ (φ m,k+k-s ) 2 = - ∂d m,s ∂µ + ∂β m,k+s ∂µ (φ m,s ) 2 - ∂d m,k-s ∂µ + ∂β m,k+k-s ∂µ (φ m,k-s ) 2 = - ∂d m,s ∂µ (φ m,s ) 2 - ∂d m,k-s ∂µ (φ m,k-s ) 2 - ∞ j=1 (-1) j 2k 3 ξ m,jk+s C m,jk+s (φ m,jk+s ) 2 - ∞ j=1 (-1) j 2k 3 ξ m,jk+k-s C m,jk+k-s (φ m,jk+k-s ) 2 = - 2k 3 ξ m,s C m,s (φ m,s ) 2 - 2k 3 ξ m,k-s C m,k-s (φ m,k-s ) 2 - ∞ j=1 (-1) j 2k 3 ξ m,jk+s C m,jk+s (φ m,jk+s ) 2 - ∞ j=1 (-1) j 2k 3 ξ m,jk+k-s C m,jk+k-s (φ m,jk+k-s ) 2 ,
(2.71) by which and making using of (2.65), one can see that

(φ m,k-s ) 2 ∂D m,s ∂µ (d m,k-s + β m,k+k-s ) 2 > 2k 3 ξ m,s C m,s (φ m,s ) 2 + 2k 3 ξ m,k-s C m,k-s |φ m,k-s | 2 + ∞ j=1 2k 3 ξ m,jk+s C m,jk+s |φ m,jk+s | 2 + ∞ j=1 2k 3 ξ m,jk+k-s C m,jk+k-s |φ m,jk+k-s | 2 > 0,
which means (2.69) holds true, where we have used the identity (2.66).

The existence of unstable point spectrum

In the previous section, we have shown that for each (m, s) ∈ K, there exists at least one eigenvalue µ ms of L R satisfying µ ms > -Eξ m,s . In this section, our objective is to show that there exists a certain element of K such that the corresponding point spectrum of L R can be positive if some additional condition can be satisfied. Proof. We use the proof by contradiction to prove the theorem. One can assume that lim R→+∞ (µ ms (R)) < +∞, for all (m, s) ∈ K.

From previous section, we know that the eigenvector of L R corresponding to µ ms (R) is

ψ m,s (x, y) = +∞ j=0
η m,kj+s e imax sin((kj + s)y)

+ +∞ j=0 η m,kj+k-s e imax sin((kj + k -s)y), (3.3)
where η m,jk+s = i 1+j φ m,jk+s /A m,jk+s and in which {φ m,sk+s } solving the equations

iφ m,k-s + φ m,k+s + d m,s φ m,s = 0, 1 ≤ s ≤ k -1, iφ m,s + φ m,k+k-s + d m,k-s φ m,k-s = 0, 1 ≤ s ≤ k -1, φ m,(j+1)k+s + d m,jk+s φ m,jk+s -φ m,(j-1)k+s = 0, 0 ≤ s ≤ k -1, j ≥ 1, φ m,(j+1)k+k-s + d m,jk+k-s φ m,jk+k-s -φ m,(j-1)k+k-s = 0, 0 ≤ s ≤ k -1, j ≥ 1.
(3.4)

We know that there must be

|φ m,s | |φ m,k-s | < 1 or |φ m,s | |φ m,k-s | ≥ 1.

Case i:

|φm,s|

< 2(k 2 -m 2 a 2 -s 2 ) 2 (m 2 a 2 + s 2 ) 2 + 2(k 2 -m 2 a 2 -s 2 ) 2 (m 2 a 2 + s 2 ) 2 m 2 a 2 + s 2 + k 2 + 2ks m 2 a 2 + s 2 + 2ks × m 2 a 2 + s 2 k 2 -m 2 a 2 -s 2 + T k,s m 2 a 2 + (k -s) 2 k 2 -m 2 a 2 -(k -s) 2
=2G(k, m, a, s), see [START_REF] Rayleigh | On the stability, or instability, of certain fluid motions[END_REF].

Case ii:

|φm,s| |φm,k-s| < 1 and φ m,k-s = 1. For very large R we have

(d m,s (µ ms )) + (d m,k-s (µ ms )) -(d m,k+k-s (µ ms )) > |φ m,k+k-s (µ ms )| 2 = |iφ m,s + d m,k-s | 2 = iφ m,s + 2k 3 ξ m,k-s Iµ ms Ra (k 2 -ξ m,k-s ) 2 (3.7)
which gives the estimate

2k 3 Iµ ms Rma 2 < 2(k 2 -m 2 a 2 -(k -s) 2 ) 2 (m 2 a 2 + (k -s) 2 ) 2 + 2(k 2 -m 2 a 2 -(k -s) 2 ) 2 (m 2 a 2 + (k -s) 2 ) 2 m 2 a 2 + (2k -s) 2 -k 2 m 2 a 2 + (2k -s) 2 × m 2 a 2 + s 2 k 2 -m 2 a 2 -s 2 - m 2 a 2 + (k -s) 2 m 2 a 2 + (k -s) 2 -k 2 = 2F (k, m, a, s), see (1.8), (3.8)
where the condition

m 2 a 2 + s 2 k 2 -m 2 a 2 -s 2 ≥ m 2 a 2 + (k -s) 2 m 2 a 2 + (k -s) 2 -k 2 .
should be satisfied, due to (d m,s (µ ms )) + (d m,k-s (µ ms )) ≥ 0.

From above, we have seen that Note that M (k, a) < 2 guarantees |γ ms | < 2 for (m, s) equal to some (q, p) ∈ K.

Let us denote lim R→+∞ φ q,jk+k-p = φ j , lim R→+∞ φ q,jk+p = φ j , j ≥ 0, and lim

R→+∞ d q,jk+p = d j i = ξ q,jk+p iγ qp C q,jk+p , lim R→+∞ d q,jk+k-p = d j i = ξ q,jk+k-p iγ qp C q,jk+k-p , j ≥ 0. lim j→+∞ |d q,jk+p | = lim j→+∞ = |d q,jk+k-p | = γ qp < 2.
Otherwise, one can choose a subsequence {R k |R k → +∞}. Then, we have

i φ 0 + φ 1 + i d 0 φ 0 = 0, i φ 0 + φ 1 + i d 0 φ 0 = 0, φ j+1 + i d j φ j -φ j-1 = 0, , j ≥ 1, φ j+1 + i d j φ j -φ j-1 = 0, , j ≥ 1.
from which we have

φ 1 φ 1 = -i d 0 -i -i -i d 0 φ 0 φ 0 , φ j+1 φ j = -i d j 1 1 0 φ j φ j-1 , j ≥ 1, φ j+1 φ j = -i d j 1 1 0 φ j φ j-1 , j ≥ 1.
(3.9) Denote

M j = -i d j 1 1 0 , X j+1 = φ j+1 φ j , N j = -i d 1 1 0 , Y j+1 = φ j+1 φ j , (3.10)
then

X j+1 = M j X j = M j M j-1 X j-1 = M j M j-1 • • • M 1 X 1 , Y j+1 = N j X j = N j N j-1 Y j-1 = N j Y j-1 • • • N 1 Y 1
The characteristic polynomial of M j is

λ 2 + i d j λ -1 = 0.
The eigenvalues of M j are

λ j = -d j + 4 -( d j ) 2 2 , λj = -d j -4 -( d j ) 2 2 = -d j -λ j = - 1 λ j .
Similarly, the characteristic polynomial of N j is

β 2 + i d j β -1 = 0.
The eigenvalues of N j are

β j = -d j + 4 -( d j ) 2 2 , βj = -d j -4 -( d j ) 2 2 = -d j -β j = - 1 β j .
Note that the corresponding diagonalization is

M j P j = P j Λ j ,
where

P j = 1 1 λ j + i d j -λ j , Λ j = λ j 0 0 -λ j -i d j , (P j ) -1 = 1 4 -( d j ) 2 λ j 1 λ j + id j -1 . Define Q j = (P j+1 ) -1 P j Λ j ,
then we have

(P j+1 ) -1 X j+1 = Λ j+1 Q j • • • Q 1 (P 1 ) -1 X 0 . (3.11)
Direct calculation of this matrix product gives

Q j = W j     1 + λ j λ j+1 1 - λ j+1 λ j λ j λ j+1 -1 -1 - 1 λ j+1 λ j     = U j    (1 + λ j λ j+1 )λ j+1 λ j+1 λ j (λ j -λ j+1 ) λ j -λ j+1 -(1 + λ j λ j+1 ) 1 λ j    , where W j = 1 4 -( d j+1 ) 2 , U j = 1 1 + (λ j+1 ) 2 . Next, denote λ = lim j→∞ λ j , then Q = lim j→∞ Q j = λ 0 0 -1/λ .
Thus, we have |λ j | = 1 for large enough j. In this case the eigenvalues of (Q j ) * Q j are given by

σ ± j = |1 + λ j λ j+1 | 2 + |λ j -λ j+1 | 2 |1 + (λ j+1 ) 2 | 2 ± |1 + λ j λ j+1 ||λ j -λ j+1 | |1 + (λ j+1 ) 2 | 2
Note also that we have

(Q j • • • Q 1 ) -1 2 ≤ j l=1 1 σ - l
A sufficient condition for the convergence of the infinite product is that

∞ l=1 1 -σ - l σ - l < ∞. (3.12)
On the other hand, we have

2|1 + λ j λ j+1 | 2 = 4 -d j d j+1 + (4 -( d j ) 2 )(4 -( d j+1 ) 2 ), |1 + λ j λ j+1 | 2 ≤ 1 + (λ j+1 ) 2 2 , |λ j -λ j+1 | ≤ C( d j -d j+1 ),
and thus for large j, we have

1 -σ - j σ - j ≤ C( d j d j+1 ) ≤ C(jk + p) -2 , which implies lim j→+∞ (Q j • • • Q 1 ) 2 > C > 0.
Therefore, using (3.11) Proof. Now, we use proof by contradiction to the lemma. Assuming otherwise, i.e.,Iµ ms = 0, we derive a contradiction. To this end, notice that for any positive integer b

Id m,n d m,n = -am E(a 2 m 2 + n 2 )(a 2 m 2 + n 2 ) > -am E (a 2 m 2 + (n + b) 2 ) (a 2 m 2 + (n + b) 2 ) = Id m,n+b d m,n+b .
Combining upper inequality and (2.54), using induction and notice all d's have negative real parts except d m,s , we see

Id m,k-s d m,k-s < Id m,s d m,s ≤ max I[d m,k+s , d m,2k+s , • • • ] [d m,k+s , d m,2k+s , • • • ] , I[d m,k-s , d m,2k-s , • • • ] [d m,k-s , d m,2k-s , • • • ] < max Id m,k+s d m,k+s , Id m,k-s d m,k-s ≤ Id m,k-s d m,k-s ,
which leads to a contradiction and hence Iµ ms = 0. which is equivalent to

R < 2Ek 4 G 1 (a, k)G 2 (a, k)G 3 (a, k)G 4 (a, k)
where

G 1 (a, k) = 2s + k a 2 m 2 + E 2 (a 2 m 2 + s 2 ) 4 , G 2 (a, k) = 3k -2s a 2 m 2 + E 2 [a 2 m 2 + (k -s) 2 ] 4 , G 3 (a, k) = 2a 2 m 2 + k 2 + 2ks + 2s 2 k 2 -m 2 a 2 -s 2 , G 4 (a, k) = 2a 2 m 2 + 4k 2 -4ks + 2s 2 k 2 -m 2 a 2 -(k -s)
(d m,s + [d m,k+s , d m,2k+s , • • • ]) × (d m,k-s + [d m,2k-s , d m,3k-s , • • • ]) = -1,
and this is true for both k = 2s and k = 2s, hence

|d m,s + [d m,k+s , d m,2k+s , • • • ]| × |d m,k-s + [d m,2k-s , d m,3k-s , • • • ]| = 1. (3.15)
We assume the contrary where Iµ = 0, then both

[d m,k+s , d m,2k+s , • • • ] and [d m,2k-s , d m,3k-s , • • • ]
have negative real parts while

I[d m,k+s , d m,2k+s , • • • ] [d m,k+s , d m,2k+s , • • • ] < tan(α k+s )
and

I[d m,2k-s , d m,3k-s , • • • ] [d m,2k-s , d m,3k-s , • • • ] < tan(α 2k-s )
hold true for the same reason as in previous lemma. Then taking into account d m,s and d m,k-s have positive real parts, α s > α k+s and according to a geometric identity for triangles we have the following:

|d m,s + [d m,k+s , d m,2k+s , • • • ]| sin α s -arctan I[d m,k+s ,d m,2k+s ,••• ] [d m,k+s ,d m,2k+s ,••• ] = |d m,s | sin θ
θ is an unknown angle, then

|d m,s + [d m,k+s , d m,2k+s , • • • ]| ≥ |d m,s | sin α s -arctan I[d m,k+s , d m,2k+s , • • • ] [d m,k+s , d m,2k+s , • • • ] > |d m,s | sin(α s -α k+s ),
and same inequality holds when s is replace by k -s. Then we reach a contradiction between (3.14) and (3.15) hence Iµ ms = 0. Remark 3.2. The criterion in the Lemma 3.2 for critical R c is satisfied for all the numerical calculation in later chapters.

Remark 3.3. The Lemma 3.1 and Lemma 3.2 together mean the imaginary part of the first eigenvalue is different from zero when it becomes critical.

Numerical computations of eigenvalue problem and critical control parameter

In the previous section, we have proven that if the two conditions a < k 2 -1 and M (k, a) < 2 hold, then the eigenvalue problem (2.3) must have a spectrum point µ qp ((q, p) ∈ K) first becoming critical (real part becomes zero) when the control parameter R crosses some critical value, and Iµ qp = 0 at the critical value. The eigenvector ψ q,p corresponding to µ qp is then given by one of (2.17) and (2.18) with (m, s) = (q, p), whose explicit expressions can be obtained by relying on solving the algebraic equations (2.52) and the specific value of µ qp . Thus, µ qp need to first be solved from the implicit function F q,p (µ qp ) = 0 defined by (2.68). But this is very difficult due to the complicated expression of F q,p . Hence, we have to count on numerical computations.

In this section we give numerical computations of the eigenvalue problem and critical values of R with different a, k and E. Following the numerical method introduced in [START_REF] Shen | Spectral methods: algorithms, analysis and applications[END_REF], each eigenfunction ψ m,s can be approximated by the function in form

ψ N m (x, y) = e imax Σ N -4 m=0 p m ω m (2y/π -1) , (4.1)
where ω m (m ∈ N) are of the form

ω m = L m + 4 k=1 a m,k L m+k ,
in which L m are the Legendre polynomials. Note that the boundary conditions Thus R c is the critical value at which the basic flow becomes unstable. The variation of R c with the length scale α is typically as shown in Figure 4-Figure 5. From these two figures we see that there exists α = α s depending only on the Ekman number E at which the marginal stability curves for k = 3 and k = 4 intersect. Moreover, for α < α s , the marginal stability curves are monotonically located, i.e. R c increases as k is increased. Hence, for aspect ratio α smaller than α s , a larger R value is needed for the basic flow corresponding to k + 1 to become unstable than that corresponding to k. Since R is proportional to the intensity of wind, a physical conclusion is that the basic flow driven by the wind forcing term-τ sin((k + 1)y) is easier to maintain its stability than that driven by the wind-τ sin(ky) if they have same intensity. For

ψ(x, 0) = ψ(x, π) = ∂ 2 y ψ(x, 0) = ∂ 2 y ψ(x, π) = 0,
α s ≤ α < √ 2, R c is larger for k = 3 than for k = 4, due to fact that all eigenvalues have negative real part if α ≥ √ 2, i.e., R c → +∞ as α approaches √ 2 . Similarly, when α < √ 15/2 and α is close √ 15/2, R c is larger for k = 4 than for k = 5. When α approaches √ k 2 -1/2 (k = 3, 4, 5), R c → +∞,
this leads to a unusual feature, which is shown in Figure 11 and Figure 12, where the corresponding transition number P could have positive real part near this point. The physical conclusion involved in this special case are summarized in conclusion in the final section and we omit it here.

At the critical value R c , based on our numerical results in Figure 6, we assume that

         µ m,1 (R c )      > 0, R > R c = 0, R = R c < 0, R < R c , m = m 0 , µ m,n ≤ µ m,1 (R c ) < 0, m ∈ Z + , ≥ 1, m = m 0 (4.2)
which is called the principle of exchange stability (PES) condition. Our numerical experiments (see Figure 3) show that for any specified E and k, there are some discrete values of α at which the critical curve R c,m 0 and R c,m 0 +1 intersect each other for some m 0 , which indicates that at these discrete points the eigenvalues µ m 0 ,1 and µ m 0 +1,1 become critical at same time with the increasing of control parameter R. In Table 1, we list some of these discrete points, the corresponding R c , the eigenvalues mu m 0 ,1 and mu m 0 +1,1 . We call these points as double Hopf points at which there are two pairs of complex conjugate eigenvalues becoming critical. Except these discrete double Hopf α s points, the rest of α points are (single) Hopf points. Hence Hopf points are generic while double Hopf points are non-generic. Table 6. The parameters at which there exist double Hopf bifurcations.

Parameters

R c µ m 0 ,1 µ m 0 +1,1 E = 0.001, α = 0.391885, m 0 = 1, k = 3 3.0255 0.130952i 0.165939i E = 0.001, α = 0.239395, m 0 = 2, k = 3 2.9947 0.152166 0.194089i E = 0.001, α = 0.478736, m 0 = 1, k = 4 4.0310 0.115708i 0.202386i E = 0.001, α = 0.286763, m 0 = 2, k = 4 3.9943 0.130893i 0.160132i E = 0.001, α = 0.204844, m 0 = 3, k = 4 3.9852 0.136496i 0.157190i E = 0.001, α = 0.623391, m 0 = 1, k = 5 6.3168 0.076243i 0.117755i E = 0.001, α = 0.370582, m 0 = 2, k = 5 6.1400 0.087923i 0.112281i E = 0.001, α = 0.264076, m 0 = 3, k = 5 6.0965 0.092319i 0.109612i

E = 0.005, α = 0.380220, m 0 = 1, k = 3 4.4315 0.106538i 0.179346i E = 0.005, α = 0.226793, m 0 = 2, k = 3 4.2596 0.125852i 0.168606i E = 0.005, α = 0.500889, m 0 = 1, k = 4 7.9683 0.058971i 0.1142060i E = 0.005, α = 0.294956, m 0 = 2, k = 4

Dynamic bifurcation-nonlinear instability analysis

We have showed that if a 2 < k 2 -1 and M (k, a) < 2, then there exists a critical R c above which the basic state (1.1) becomes linearly unstable. From Figure 2, we see that except for these values of a very close to √ k 2 -1, the condition M (k, a) < 2 holds true. Besides, each eigenvector of the operator L R corresponding to the eigenvalue first becoming unstable for large R, must be expressed in the form

ψ m (x, y) = e iamx Ψ m (y) (5.1)
where Ψ m (y) is in form of

Ψ m (y) = j∈N +
η m,jk+s sin((jk + s)y)

+ j∈N +
η m,jk+k-s sin((jk + k -s)y), for some (m, s) ∈ K.

The numerical experiments shown in the previous section display that the PES condition holds true at the critical value R c , which means that there both Hopf and double Hopf bifurcations are possible. To obtain more detailed information, including supercritical or subcritical Hopf bifurcation, the number of periodic solutions bifurcated from the basic solution and the transition type, we need to reduce the PDE governing the QG equation to a system of ODEs, called the reduced equations. In [START_REF] Kieu | On the Hopf (double Hopf) bifurcations and transitions of two-layer western boundary currents[END_REF], to obtain a corresponding reduced system, the authors use an iteration method to derive the second order approximation of the center manifold function when the control parameter is close to criticality. In the present work, we shall apply the iteration method to obtain the reduced equations for the system (1.3) in the vicinity of R c . The method is easier and can be used for other general evolution equations whose linear part is a completely continuous operator.

As discussed in the previous section, both Hopf and double Hopf bifurcations are possible depending on the choice of the aspect ratio. In this section, we will focus on the associated dynamical transition analysis in the case of Hopf and double bifurcations at the threshold R c . More precisely, we want to know whether the corresponding Hopf bifurcation is supercritical or subcritical, which correspond to two types of dynamical phase transition-continuous transition and jump transition from the basic state (1.1) to a new state when R > R c . The new state is a stable periodic solution or another type of attractor, see [START_REF] Ma | Phase transition dynamics[END_REF].

ϕ 1 = e iam 0 x Ψ m 0 (y) (5.2)
whose dual eigenvector is

ϕ * 1 = e iam 0 x Ψ * m 0 (y), (5.3)
and they satisfy the normalization condition:

(∆ϕ 1 , ϕ * 1 ) = 1. We denote the center-unstable space as H c = {ηϕ 1 + ηϕ 1 |η ∈ C}. Then the center manifold function Φ is a function from H c to H ⊥ c , i.e. Φ = Φ(ξ) where

ξ = 2 l=1 η l φ l , φ l = e iam l x Ψ m l (y), η 1 = η 2 , φ 1 = φ 2 = ϕ 1 , m 1 = -m 2 = m 0 . (5.4)
Let us define the projections P c and P s from H 1 to H c and H ⊥ c , respectively. then the equation (2.2) can be decomposed into

dξ dt = A -1 L R ξ + P c A -1 G(ξ + φ, ξ + φ), dφ dt = A -1 L R φ + P s A -1 G(ξ + φ, ξ + φ), (5.5)
where we have used ψ = ξ + φ, ξ = P c ψ, φ = P s ψ.

It is well known that for the center manifold function Φ, its derivative vanishes at ξ = 0, i.e., ∇Φ(ξ)| ξ=0 = 0. This allows us to take the Taylor expansion of Φ as follows:

Φ = Φ 2 + o(|η| 2 ) = 2 l,l =1 η l η l Φ 2,ll + o(|η| 2 ).
(5.6)

Following the approach in [START_REF] Kieu | On the Hopf (double Hopf) bifurcations and transitions of two-layer western boundary currents[END_REF], one can obtain that Φ 2,ll solves

2 l,l =1 η l η l L R Φ 2,ll -2 2 l,l =1
η l η l β l AΦ 2,ll = -P s G(ξ, ξ).

(5.7)

Note that

G(ξ, ξ), ξ = 2 l,l ,l =1 η l η l η l 2π a 0 e ia(m l +m l -m l )x dx π 0 G m l ,m l (Ψ m l , Ψ m l )Ψ m l dy = 0 for ξ, ξ ∈ H c with ξ = 2 l=1 η l e iam l x Ψ m l , ξ = 2 l=1
η l e iam l x Ψ m l , see (5.4), due to 2π a 0 e ia(m l +m l -m l )x dx = 0, l, l , l ∈ {1, 2}.

Hence,

P s G(ξ, ξ) = G(ξ, ξ) = 2 l,l =1 η l η l G(φ l , φ l ),
and (5.7) becomes

2 l,l =1 η l η l L R Φ 2,ll -2 2 l,l =1 η l η l β l AΦ 2,ll = - 2 l,l =1
η l η l G(φ l , φ l ).

(5.8)

Comparing the coefficients on both sides of (5.8), one can find that Φ 2,ll solves

L R Φ 2,ll -2β l AΦ 2,ll = -G(φ l , φ l ) = i e ia(m l +m l )x G m l ,m l Ψ m l , Ψ m l , l = l , 2L R Φ 2,ll -2(β l + β l )AΦ 2,ll = -G(φ l , φ l ) -G(φ l , φ l ) = i e ia(m l +m l )x G m l ,m l Ψ m l , Ψ m l + G m l ,m l Ψ m l , Ψ m l , l = l , (5.9)
in which

-G(φ l , φ l ) = J(φ l , ∆φ l ) = ∂φ l ∂x ∂∆φ l ∂y - ∂φ l ∂y ∂∆φ l ∂x = i e ia(m l +m l )x G m l ,m l Ψ m l , Ψ m l , (5.10)
G m l ,m l Ψ m l , Ψ m l = am l Ψ m l d 2 dy 2 -a 2 m 2 l Ψ m l -am l Ψ m l d 2 dy 2 -a 2 m 2 l Ψ m l . (5.11)
The expression (5.10) means that solutions to the equations (5.9) take the form Φ 2,ll (x, y) = e ia(m l +m l )x Ψ 2,ll (y) (5.12)

Theorem 5.1. If M (k, a) < 2, a < √ k 2 -1,
and the first eigenvalue of L R is simple and the condition (4.2) holds true, then there exists a supercritical Hopf bifurcation at R c in the system (1.3) when P < 0, and a stable periodic solution ψ p bifurcated from the zero solution on R > R c , whose expression is

ψ p =2 A m 0 e iBm 0 t+imax Ψ m 0 + 2A 2 m 0 Φ 2,11 e i2Bm 0 t + Φ 2,12 + o(A 2 m 0 ), (5.21)
where

Ψ m 0 = +∞ j=0
(η m 0 ,jk+s sin((jk + s)y) + η m 0 ,jk+k-s sin((jk + k -s)y)) , Φ 2,11 and Φ 2,12 = Φ 2,21 solve (5.9). When P > 0, the Hopf bifurcation is subcritical, and a unstable periodic orbit ψ p given by (5.21) bifurcates on R < R c such that ψ p → 0 as R → R c , and there is no periodic solution bifurcating from 0 on R > R c .

5.2.

Reduced equation governing double Hopf bifurcation. In this subsection, we modify the method introduced in the preceding subsection to derive the reduced equation for the case of double Hopf bifurcation. Let us denote the first two eigenvalues of L R as µ 1 = µ 11 + iµ 12 and µ 2 = µ 21 + iµ 22 , whose real parts change sign at the same threshold R c . Based on the numerical results in the previous section, see Table 6, the corresponding eigenvectors be given by ϕ 1 = e iam 0 x Ψ m 0 (y), ϕ 2 = e ia(m 0 +1)x Ψ m 0 +1 (y) (5.22) whose dual eigenvectors are

ϕ * 1 = e iam 0 x Ψ * m 0 (y), ϕ * 2 = e ia(m 0 +1)x Ψ * m 0 +1 (y) (5.23)
where Ψ m (y) = λ l φ l , φ l = e ian l x Ψ n l (y),

λ 1 = η 1 , λ 2 = η 2 , λ 3 = η 1 , λ 4 = η 2 , φ 1 = φ 3 = ϕ 1 , φ 2 = φ 4 = ϕ 2 . n 1 = m 0 , n 2 = -m 0 , n 3 = m 0 + 1, n 4 = -m 0 -1.
Let P c and P s be the projectors from H to H c and H ⊥ c , respectively. The equation (2.2) can be decomposed into

dξ dt = A -1 L R ξ + P c A -1 G(ξ + φ, ξ + φ), dφ dt = A -1 L R φ + P s A -1 G(ξ + φ, ξ + φ), (5.24)
where we have used ψ = ξ + φ, ξ = P c ψ, φ = P s ψ.

Denote Φ = Φ 2 (ξ) + o(ξ 2 ) = 4 l,l =1 Φ 2,ll λ l λ l + o(ξ 2 ), (5.25)
The reduced equation can be rewritten as

∂λ 1 ∂t = µ 1 λ 1 + λ 1 (P 1 |λ 1 | 2 + P 2 |λ 3 | 2 ) + o(|λ| 3 ), ∂λ 3 ∂t = µ 3 λ 3 + λ 3 (P 3 |λ 1 | 2 + P 4 |λ 3 | 2 ) + o(|λ| 3 ).
(5.32)

For the coefficients in above reduced system, based on our numerical experiments, see Table 7, we assume P i < 0, i = 1, 2, 3, 4; P 1 P 2 < P 3 P 4 .

(5.33)

Making use of the reduced system and assumptions on coefficients, we can obtain the following theorem:

Theorem 5.2. Under the conditions of Theorem 5.1 and if there are two pairs of conjugate eigenvalues becoming critical at R c , then the system (1.3) has a supercritical Hopf bifurcation, and there are two periodic solutions ψ p,1 and ψ p,2 bifurcated from zero solution on R > R c , which can be respectively given by

ψ p,1 =2 A m 0 e iBm 0 t+im 0 ax Ψ m 0 + 2A 2 m 0 Φ 2,11 e i2Bt + Φ 2,12 + o(A 2 m 0 ), (5.34)
ψ p,2 =2 A m 0 +1 e iB m 0 +1 t+i(m 0 +1)ax Ψ m 0 +1 + 2A 2 m 0 +1 Φ 2,33 e i2B m 0 +1 t + Φ 2,34 + o(A 2 m 0 +1 ), (5.35)
where Proof. From the center manifold reduction introduced previously, for the double Hopf case, we have already known that the bifurcation in the infinite dimension system (1.3) has been reduced to that in the four dimension system (5.32). It is from [START_REF] Kuznetsov | Elements of applied bifurcation theory[END_REF] (Page.356-366) well known that the ODEs (5.32) bifurcates to two periodic solutions if the condition (5.33) holds true. The two periodic solutions are respectively given by

A m = µ m,1 | P | 1 2 , B m = Iµ m,1 + IP µ m,1 | P | , Ψ m = +∞ j=0 (η m,
(λ p,1 , 0) = A n 1 e iBn 1 t + o(A n 1 ), 0 , n 1 = m 0 , (0, λ p,2 ) = A n 3 e iBn 3 t + o(A n 3 ), 0 , n 3 = m 0 + 1.
Using the same method in [START_REF] Kieu | On the Hopf (double Hopf) bifurcations and transitions of two-layer western boundary currents[END_REF][START_REF] Kuznetsov | Elements of applied bifurcation theory[END_REF], one can show that (λ p,1 , 0) is stable, but (0, λ p,2 ) is unstable.

Based on the center manifold reduction introduced before, in the vicinity of R = R c , each solution ψ of (1.3) can be decomposed into

ψ = ξ + Φ(ξ) = ξ + 4 l,l =1 Φ 2,ll λ l λ l , ξ = 4 l=1 λ l φ l , λ 1 = λ 2 , λ 3 = λ 4 + o(ξ 2 ), φ 1 = φ 2 = ϕ 1 , φ 3 = φ 4 = ϕ 2
where λ 1 and λ 3 solve the system of ODEs (5.32). Now, let (λ 1 , λ 2 ) = (λ p,1 , 0) , (0, λ p,2 ), respectively. After substituting them into

ξ + 4 l,l =1 Φ 2,ll λ l λ l ,
we arrive at (5.34) and (5.35).

Remark 5.1. In fact, the periodic solution ψ p,1 is the ψ p given in Theorem 5.2.

5.3.

Transition led by the varying of aspect ratio. Although Theorem 5.2 focuses on the bifurcation at the value of the aspect ratio α where there are two pairs of complex conjugate eigenvalues becoming critical at the same critical control parameter R c . Based on it and Theorem 5.1, we can still analyze the Hopf bifurcation near the double Hopf point. Numerically, we have known that there are several discrete values of α in [0.2, √ k 2 -1/2) at which there exist double Hopf bifurcations, the number of these discrete values (double Hopf point) depends on k and E, see Table 6. Let α 0 be one of these discrete value, and let m 0 be the integer such that µ m 0 ,1 = max m∈Z { µ m,1 } for α > α 0 . Then, at α = α 0 , it has µ m 0 +1,1 (R c ) = µ m 0 ,1 (R c ) = 0, and µ m 0 +1,1 = max m∈Z { µ m,1 } for α < α 0 . For α in the vicinity of α 0 , there exists a change in the number of periodic solutions led by the increasing of the control parameter R describing the relation between stable periodic solution and the varying of aspect ratio, see Figure 10, which is introduced in the forthcoming two paragraphs.

For α close to α 0 and α < α 0 , we have

R c = R c,m 0 +1 < R c,m 0 , where R c,m 0 is very close to R c = R c,m 0 +1 . If R c < R < R c,m 0 , we have µ m 0 ,1 < 0, µ m 0 +1,1 > 0
which means the reduced equations will be (5.19), and there is only one periodic solution ψ = ψ p,2 bifurcated from zero point of the system (1.3), which is stable and goes to zero as α approaches α 0 due to R c,m 0 +1 = R c,m 0 at α = α 0 , see Figure 10-(

a). If R = R c,m 0 , µ m 0 ,1 = 0, µ m 0 +1,1 > 0,
In this case, using the reduced system (5.32), one can show that there is one periodic solution ψ = ψ p,2 which is stable. If R c,m 0 < R and R very close to R c,m 0 , we have

µ m 0 ,1 > 0, µ m 0 +1,1 > 0
which means the reduced equations will be (5.32), and there are two periodic solutions ψ p,1 and ψ p,2 bifurcated from zero point of the system (1.3), where ψ p,1 is stable but ψ p,2 unstable, see Figure 10-(b).

For α close to α 0 and α > α 0 , we have

R c = R c,m 0 < R c,m 0 +1 , where R c,m 0 +1 is very close to R c = R c,m 0 . If R c < R < R c,m 0 +1 , we have µ m 0 ,1 > 0, µ m 0 +1,1 < 0
which means the reduced equations should be (5.19), and there is only one periodic solution ψ = ψ p,1 bifurcated from zero point of the system (1.3), which is stable and goes to zero as α approaches α 0 due to R c,m 0 +1 = R c,m 0 at α = α 0 , see Figure 10-(a). If R = R c,m 0 , then

µ m 0 ,1 > 0, µ m 0 +1,1 = 0,
In this case, using the reduce system (5.32), one can show that there is one periodic solution

ψ = ψ p,1 which is stable. If R c,m 0 +1 < R and R very close to R c,m 0 +1 , then µ m 0 ,1 > 0, µ m 0 +1,1 > 0
which means the reduced equations should be (5.32), and there are two periodic solutions ψ p,1 and ψ p,2 bifurcated from zero point of the system (1.3), where ψ p,1 is stable but ψ p,2 unstable, see Figure 10-(b).

From the Figure 10 we can see that there exists a symmetry-breaking led by the increasing of the control parameter R and varying of α near the double Hopf point α 0 . The symmetrybreaking is in the sense that for α < α 0 and 0 < (α 0 -α)/α 0 1, when R increases from the value below R c,m 0 to that above it, the stable periodic solution changes from ψ p,2 to ψ p,1 while for α > α 0 and 0 < (α -α 0 )/α 0 1, when R increases from the value below R c,m 0 +1 to that above it, the stable periodic solution is always ψ p,1 . The symmetry-breaking reveals a type of jump phase transition in the sense that for α < α 0 and 0 < (α 0 -α)/α 0 1, when R increases from the value below R c,m 0 to that above it, the topological structure of flows changes from the structure given by ψ = ψ 0 + ψ p,2 to that given by ψ = ψ 0 + ψ p,1 when R ≥ R c,m 0 , and R = R c,m 0 is a jump point of the observed periodic solution In this section, we numerically estimate the real part of the complex number P defined by (5.18), which entirely determines the type of Hopf bifurcation, called transition number. We have shown that P is entirely determined by the nonlinear coupling of the first eigenvector and its dual eigenvector. Since the nonlinear term is proportional to the Rossby number , so is P . It is known that the QG equation (1.3) is an approximation of the shallow water equation for small around 10 -2 . For the purpose of illustration, we only give the numerical estimates for the real parts of P when = 0.01, which is a standard choice in atmospheric and oceanic dynamics. E is the Ekman number which is also a small number around 10 -3 . Here, we only take E ∈ {0.001, 0.005}. For k, we only focus on k = {3, 4, 5}. For the aspect ratio α = a/2, a natural choice is that α is neither too small nor too large so that the rectangular domain is not skewed in one direction. As a necessary condition for instability, we require that α < √ k 2 -1/2. Then, we can take α ∈ [0.2, √ k 2 -1/2). The numerical results with mesh size = 0.02 are shown in Figure 11 and Figure 12.

ψ = ψ 0 + ψ p,2 , R c < R ≤ R c,m 0 , α < α 0 , 0 < (α 0 -α)/α 0 1, ψ 0 + ψ p,1 , R > R c,m 0 , α < α 0 , 0 < (α 0 -α)/α 0 1. due to ψ p,2 -ψ p,1 = ψ p,2 > 0 at R = R c,m 0 .
From Figure 11 and Figure 12 we can see that in the case of the Hopf bifurcation, the complex number P has negative real part for k = 5 with all α ∈ [0.2, √ 6) while its real part is positive for some aspect ratios α near √ k 2 -1/2 when k = 3, 4. This indicates that for k = 5 the Hopf bifurcation in the system (1.3) is supercritical whereas it allows subcritical Hopf bifurcation in case of k = 3, 4. From the perspective of phase transition dynamics associated with the subcritical Hopf bifurcation, see [START_REF] Ma | Phase transition dynamics[END_REF], there exists a jump transition in (1.3) at R = R c and there exists a subcritical bifurcation on R s < R c at which there is a separation of periodic orbits. In other words, at R = R s , another physical state of the system (1.2) different from the basic state (1.1) emerges although (1.1) is still linearly stable.

As we mentioned before, the system also allows double Hopf bifurcations for special aspect ratios as given in Table 6. The numerical estimates on the coefficients P i (i = 1, • • • , 4) of the reduced equation is (5.32) are shown in Table 7, all have negative real parts and P 1 P 2 < P 3 P 4 . Hence, the double Hopf bifurcation occurring in the system (1.3) is also supercritical, and the stable periodic solution is same as in the Hopf bifurcation case. which is the reduced equation in the Hopf bifurcation case. Hence, the Hopf case and double Hopf are same in the sense that the stable periodic solution is the physically observed one.

Table 7. The coefficients in reduced equations for double Hopf case corresponding to parameters in Table 6, where the Rossy number = 0.01.

P 1 P 2 P 3 P 4 -0.0014 + 0.0049i -0.0085 + 0.0122i -0.0079 + 0.0031i -0.0107 + 0.0179i -0.0019 + 0.0062i -0.0096 + 0.0135i -0.0099 + 0.0047i -0.0035 + 0.0130i -0.0017 + 0.0083i -0.0156 + 0.0294i -0.0125 + 0.0033i -0.0060 + 0.0291i -0.0023 + 0.0108i -0.0163 + 0.0289i -0.0166 + 0.0072i -0.0048 + 0.0216i -0.0027 + 0.0120i -0.0169 + 0.0289i -0.0194 + 0.0104i -0.0045 + 0.0194i -0.0164 + 0.0257i -0.0875 + 0.0442i -0.0721 -0.0218i -0.0796 + 0.0287i -0.0211 + 0.0312i -0.0929 + 0.0493i -0.0905 -0.0033i -0.0536 + 0.0431i -0.0235 + 0.0333i -0.0951 + 0.0524i -0.0988 + 0.0107i -0.0446 + 0.0428i -0.0019 + 0.0023i -0.0064 + 0.0011i -0.0053 -0.0026i -0.0061 -0.0010i -0.0025 + 0.0025i -0.0071 + 0.0015i -0.0068 -0.0015i -0.0056 + 0.0012i -0.0026 + 0.0027i -0.0101 + 0.0073i -0.0087 -0.0002i -0.0081 + 0.0048i -0.0034 + 0.0037i -0.0102 + 0.0073i -0.0098 + 0.0023i -0.0069 + 0.0053i -0.0037 + 0.0042i -0.0099 + 0.0073i -0.0099 + 0.0037i -0.0063 + 0.0054i -0.0117 -0.0074i -0.0216 + 0.0172i -0.0426 + 0.0022i -0.0235 + 0.0003i -0.0124 + 0.0172i -0.0504 + 0.0768i -0.0226 + 0.0247i -0.0331 + 0.0060i -0.0195 + 0.0212i -0.0645 + 0.0720i -0.0381 + 0.0299i -0.0363 + 0.0095i -0.0237 + 0.0223i -0.0696 + 0.0679i -0.0464 + 0.0335i -0.0374 + 0.0117i -0.0264 + 0.0226i -0.0717 + 0.0649i -0.0514 + 0.0363i -0.0378 + 0.0132i

In what follows, for k = 3, 5, α = 0.5, E = 0.001 and = 0.01, we illustrate the observed periodic state which is the addition of the basic solution (1.1) and bifurcated periodic solution given by (5.21) obtained from the perturbation equation (1.3), i.e., ψ = ψ p + ψ 0 , shown in Figure 13 and Figure 14. We see that the observed periodic state contains regular periodic pattern moving from the positive direction of x-axis to negative one. More physically speaking, for control parameter R > R c , the structure of flows given by the forced shear shown in Figure 1 will break into periodic circulation patterns moving from eat to west. Besides, it deduces from Figure 13 and Figure 14 that the number of circulation patterns contained in the new state is given by k × m 0 , where m 0 corresponds to the first eigenvalue µ m 0 ,1 and eigenvector ϕ 1 , see Figure 7 when the control parameter R is greater a threshold R c . The control parameter R is proportional to Reynolds number and the intensity τ of the curl of the forcing. This implies that the stronger forcing the shear flows become more unstable, and the small viscosity they are more easier to turn to unstable. Secondly, using numerical computations we find that there exist the Hopf and double Hopf bifurcations involved in the losing stability of the shear flows at the threshold R c .

To put insight into more details in the bifurcations, we use a more natural method to reduce the QG equation (1.3) to a ODEs (5.19) or (5.32) determined by the choice of aspect ratio. Upon performing numerical estimates on coefficients in the ODEs for k = 3, 4, 5, it is found that in case of k = 3, both supercritical and subcritical Hopf bifurcations occur. But Only supercritical Hopf bifurcation is allowed in the case of k = 4, 5. We also examine the type of double Hopf bifurcation, finding that there exist two periodic solutions bifurcated from the shear flows on R > R c , one is stable, the other one is unstable, and the stable is the limit of stable periodic solution in Hopf case when aspect ratio approaches the double Hopf point.

From numerical results, we also know that there are many large scale circulation patterns contained in the solution consisting of the adding of the shear flows and bifurcated periodic solution. Intuitively speaking, when R > R c the shear flows disappears, which breaks into periodic flow structure with many circulation patterns. The number of circulation patterns are given by k × m 0 , where m 0 is related to the bifurcated periodic solution. This can be used to understand a lot of periodic patterns formed in atmosphere and ocean. That is, the intensity of wind and viscosity play an important role in the periodic circulation patterns formed in the atmospheric and oceanic system.

Figure 2 .Theorem 1 . 1 . 2 - 1 , 5 . 1 . 2 .

 21121512 Figure 2. The graphs of function M (k, a) with k = 3, 4, 5 and [0.4, √ k 2 -1). Theorem 1.1. If a ≥ √ k 2 -1, then the basic solution (1.1) is always stable. If a < √ k 2 -1 and M (k, a) < 2, then there exists a critical control parameter R c > 0 such that the basic solution (1.1) is linearly stable R < R c and linearly unstable if R > R c . Proof. The theorem can be obtained from the Lemma 2.2 and Theorem 3.1. Remark 1.1. The condition M (k, a) < 2 is only a necessary technical requirement and in fact the basic solution becomes linearly unstable at a critical R c for any a < √ k 2 -1, see Figure 4 and Figure 5. Theorem 1.2. Each eigenvalue solving (1.6) at R c has nonzero imaginary part. Furthermore, suppose the principle of exchange of stability condition     

  iam) sin(ny) + n≥1 Eη m,n ξ 2 m,n sin(ny) = -µ n≥1 η m,n ξ m,n sin(ny).

-

  φ m,(j 0 -1)k+s φ m,j 0 k+s + +∞ j=j 0 d m,jk+s |φ m,jk+s | 2 + +∞ j=j 0 -φ m,jk+s φ m,(j+1)k+s + φ m,(j+1)k+s φ m,jk+s = 0. (2.23) Now we multiply the first equation in (2.15) by φ m,s , choose j 0 = 1 in (2.23) and add the resulting equations to find that iφ m,k-s φ m,s + +∞ j=0 d m,jk+s |φ m,jk+s | 2 + +∞ j=0 -φ m,jk+s φ m,(j+1)k+s + φ m,(j+1)k+s φ m,jk+s = 0 (2.24)

  .55) Case i: For d m,s > 0 with k = 2s, or d m,s > 0 and d m,k-s < 0. Let us further define,

  due to d m,jk+k-s < 0 and d m,jk+s < 0 for j ≥ 1. Note that s < k -s, d m,s > 0 and d m,k-s > 0 means that |d m,k-s | > C > 0 for some constant C depending on a and m, and Eξ m,s + µ > 0. Define the complex-valued function

2 + 2 +

 22 .64) In view of (d m,s (µ ms,l )) |φ m,s (µ ms,l )| 2 + (d m,k-s (µ ms,l )) |φ m,k-s (µ ms,l )| +∞ j=1 (d m,jk+s (µ ms,l )) |φ m,jk+s (µ ms,l )| +∞ j=1 (d m,jk+k-s (µ ms,l )) |φ m,jk+k-s (µ ms,l )| 2 = 0, l = 1, 2.

2 + 2 l=1 +∞ j=1 - 2 < - 2 l=1( 2 - 2 l=1 2 - 2 l=1 2 ( 2

 22j=122222222 |φ m,jk+s (µ ms,l )| 2 |φ m,s (µ ms,l )| C m,s ξ m,jk+k-s ξ m,s C m,jk+k-s |φ m,jk+k-s (µ ms,l )| 2 |φ m,s (µ s,l )| d m,k-s (µ ms,l )) (d m,s (µ ms,l )) |φ m,k-s (µ ms,l )| 2 |φ m,s (µ s,l )| +∞ j=1 (d m,jk+s (µ s,l )) (d m,s (µ ms,l )) |φ m,jk+s (µ ms,l )| 2 |φ m,s (µ ms,l )| +∞ j=1 (d m,jk+k-s (µ ms,l )) (d m,s (µ ms,l )) |φ m,jk+k-s (µ ms,l )| 2 |φ m,s (µ ms,l )| 2 =

Theorem 3 . 1 .

 31 If a < √ k 2 -1 and M (k, a) < 2, then there exists (q, p) ∈ K such that for the coressponding eigenvalue µ qp (R) given in Lemma 2.8, we have lim R→+∞ (µ qp (R)) = +∞. (3.1)

2k 3 2k 3

 33 Iµm s Rma is bounded for all R > 0. Let lim R→+∞ Iµ ms Rma = γ ms .

Remark 3 . 1 .Lemma 3 . 1 .

 3131 leads to contradiction. That is, lim R→+∞ (µ qp (R)) = +∞. (3.13)holds true. If k=2, the case ii in the proof does not occur. Let µ ms (R) with R > 0, k ≥ 2, (m, s) ∈ K be the eigenvalue such that µ ms (R) = 0, if d m,s > 0 and d m,k-s < 0, then we have Iµ m,s = 0.

Lemma 3 . 2 .

 32 Let µ ms (R) with R > 0, k ≥ 2, (m, s) ∈ K be the eigenvalue such that µ ms = 0, d m,s > 0 and d m,k-s > 0, if |d m,s ||d m,k-s | sin(α s -α k+s ) sin(α k-s -α 2k-s ) > 1 (3.14)

Figure 3 .Figure 4 .Figure 5 .

 345 Figure 3. Illustration of critical curve R c,m at which µ ms = 0, where E = 0.005, k = 4.

Figure 6 .

 6 Figure 6. Numerical verification of PES condition, i.e. µ m 0 ,1 (R c , a) > 0, where E = 0.005..

  7.5369 0.075962i 0.107989i E = 0.005, α = 0.209615, m 0 = 3, k = 4 7.4317 0.082089i 0.104762i E = 0.005, α = 1.043049, m 0 = 1, k = 5 18.6877 0.196584i 0.001885i E = 0.005, α = 0.650002, m 0 = 2, k = 5 15.7648 0.303154i 0.205841i E = 0.005, α = 0.468238, m 0 = 3, k = 5 14.8312 0.278923i 0.212416i E = 0.005, α = 0.365319, m 0 = 4, k = 5 14.4379 0.267586i 0.216835i E = 0.005, α = 0.299336, m 0 = 5, k = 5 14.2373 0.261069i 0.219949i Finally, we show the spatial structure of the critical eigenvectors in Figure 7-Figure 9.

Figure 7 .Figure 8 .Figure 9 .

 789 Figure 7. Illustration of critical eigenvector, where E = 0.001, α = 0.5, k = and R c = 2.989563, µ 1,1 = max m∈Z { µ m,1 }.

5. 1 .

 1 Reduced equations governing the Hopf bifurcation. For the convenience of reference, let us denote the first eigenvalue of L R as µ 1 = µ 11 + iµ 12 , and the corresponding eigenvector as

  j∈N + η m,jk+s sin(ny), (m, s) ∈ K, m = m 0 , m 0 + 1, and they satisfy the normalization conditions: (∆ϕ 1 , ϕ * 1 ) = 1, (∆ϕ 2 , ϕ * 2 ) = 1. Denote the center-unstable space as H c = {η 1 ϕ 1 + η 1 ϕ 1 + η 2 ϕ 2 + η 2 ϕ 2 |η 1 , η 2 ∈ C}. Then the center manifold function Φ is a function from H c to H ⊥ c , i.e., Φ = Φ(ξ) where ξ = 4 l=1

Figure 10 . 6 . 1 .

 1061 Figure 10. The change in the number of periodic solutions led by the increasing of R and the varying of α. The blue solid line while the dashed line represents unstable periodic solution.

Figure 11 .Figure 12 . 6 . 2 .

 111262 Figure 11. The values of transition number for E = 0.001, = 0.01..

-Figure 9 . 6 . 3 .

 963 Conclusion. In this work, firstly, we prove the instability arising in driven shear flows governed by quasi-geostrophic (QG) equation with a generalized Kolmogorov forcing f = -τ k cos(ky)

  hence by(2.22), we have d m,jk+s < 0 for all j ≥ 1. Hence, by (2.29), we must have

  (µ ms,1 )φ m,k-s (µ ms,2 ) -φ m,k-s (µ ms,1 )φ m,s (µ ms,2 ) φ m,s (µ ms,1 )φ m,s (µ ms,2 ) ,

	from which we obtain that
	β + m,k+s (µ ms,1 ) -β -m,k+s (µ ms,2 ) + = i φ m,s (2.60) 2k 2 ξ m,s (µ ms,1 -µ ms,2 ) C m,s
	(2.59)

k+k-s (µ ms,2 )

  Under the assumption (3.2), it deduce from that for very large R, we have the inequality(d m,s (µ ms )) + (d m,k-s (µ ms )) -(d m,k+s (µ ms )) > |φ m,k+s (µ ms )| 2 = iφ m,k-s + 2k 3 ξ m,s Iµ ms Ra (k 2 -ξ m,s )

		2
		(3.6)
	from which we derive that
	2k 3 Iµ ms	2
	Rma

|φm,k-s| ≥ 1. In this case, we can always choose eigenvector ψ m (x, y) such that φ m,s = 1. From (2.65) we have (d m,s (µ ms )) + (d m,k-s (µ ms )) -(d m,k+s (µ ms )) > |φ m,k+s (µ ms )| 2 = |iφ m,k-s + d m,s | 2 . (3.5)

  2 , then we have Iµ ms = 0. Here α n = arctan

	am E(m 2 a 2 +n 2 ) 2 .
	Proof. Notice from equation (2.54) we can derive that

Table 4 .

 4 Eigenvalues with m = 2, α = 1.3, E = 0.001, k = 4 and R = 50.

	Eigenvalue	µ
	µ 2,1	-0.568751022907150 + 3.431513355301365i
	µ 2,2	-0.568768379271038 + 3.431487850188656i
	µ 2,3	-0.580902355593918 + 3.442050974812954i
	µ 2,4	-0.581038750307876 + 3.442149661154682i
	µ 2,5	-0.603649874854427 -3.406751918968515i

Table 5 .

 5 Eigenvalues with m = 2, α = 1.3, E = 0.001, k = 5 and R = 50.

	Eigenvalue			µ
	µ 2,1	-0.338433272641911 + 0.521561532470981i
	µ 2,2	-0.365592537325345 -0.159493634815083i
	µ 2,3	-0.381233485344943 + 0.326380342862074i
	µ 2,4	-0.486219835936696 + 0.038666904426508i
	µ 2,5	-0.502968082353312 + 1.521444333443289i
	100
	95
	90
	85
	80
	75
	70
	65
	60
	55
	50
	0.2	0.4	0.6	0.8	1	1.2	1.4

  jk+s sin((jk + s)y) + η m,jk+k-s sin((jk + k -s)y)) , Φ 2,11 , Φ 2,12 = Φ 2,21 and Φ 2,33 , Φ 2,34 = Φ 2,43 are solved from(5.26). Particularly, ψ p,1 is stable while ψ p,2 is unstable.

Acknowledgments The work of Quan Wang was supported by the National Science Foundation of China (NSFC) (No. 11771306).

Figure 13. Illustration of periodic solution ψ = ψ 0 + ψ p , where E = 0.001, α = 0.5, k = 3 and R 2,c = 3.4163 > 3.089562 > R c = 2.989563.

Figure 14. Illustration of periodic solution ψ = ψ 0 + ψ p , where E = 0.001, α = 0.5, k = 5 and R = 6.167881 > R c = 6.067881.

mean that a n,k can be determined through ω m (±1) = ω m (±1) = 0, which are given by a n,1 = a n,3 = 0, a n,4 = -1 -a n,2 . a n,2 = 2(2n + 5)(n 2 + 5n + 9) (n + 3)(n + 4)(2n + 7) .

For the sake of convenience, in the rest of this article, we use the new symbol {µ m,n |(m, n) ∈ Z × Z} to express the eigenvalue set of the eigenvalue problem (2.3). Suppose all eigenvalues have been ordered as µ m,1 ≥ µ m,2 ≥ µ m,3 → -∞, m ∈ Z, then let us denote m 0 = arg max{ µ m,1 : m ∈ Z}.

so that µ m 0 ,1 ≥ µ m,1 , for all m ∈ Z. That is µ m 0 ,1 is the first eigenvalue, and we denote the corresponding first eigenvector as ϕ 1 .

The discussion in the previous section means µ m,1 = µ -m,1 , then we focus our attention to positive integers m. The domain used in present work is [0, 2π/a] × [0, π], for the sake of convenience, we use α to represent the ratio of the length in y-direction of the domain to its length in x-direction, i.e., α := π/(2π/a) = a/2. In what follows, we choose the values of aspect ratio α in the interval [0.2, √ k 2 -1/2), and examine the relation of α and the number of eigenvalues whose real parts change sign (moving from negative half plane to the positive one).

For the purpose of illustration, we list some eigenvalues with specified control parameters at first, which are shown in Table -0.516445191916328 -4.285863662499639i

Table 2. where for each (l, l

(5.13) in which

(5.14)

Upon obtaining the second order approximation Φ 2 of the center manifold function, the reduced equation takes the form of

which is equal to

By denoting a j,l,l = G(φ j , Φ 2,ll ) + G(Φ 2,ll , φ j ), ϕ * 1 , we rewrite the preceding reduced equation as

a j,l,l η j η l η l + o(|η 1 | 3 ).

(5.17)

Straightforward calculation one can note that the function

has a factor e ia(m j +m l +m l -m 0 )x which due to integration on [0, 2π/a] implies that

Let us define

and rewrite (5.17) as

It is well known that the system of ODEs (5.19) bifurcates from zero point to a stable periodic solution on R > R c if P < 0, and the periodic solution takes the form of (5.20) in which

Based on the center manifold reduction aforementioned and (5.20), we get following theorem:

Using the method as in the previous subsection, we have

(5.26)

Right hand side of above equations imply that the solutions can be expressed as

where for each (l, l ) ∈ {1, 2, 3, 4} × {1, 2, 3, 4}, Ψ 2,ll solves

(5.28)

Therefore, the reduced equation can be derived as

(5.29)

Let us denote a j,l,l ,p = G(φ j , Φ 2,ll ), ϕ * p + G(Φ 2,ll , φ j ), ϕ * p . Making use of

The system of ODEs (5.29) becomes

a j,l,l ,p λ j λ l λ l + o(|λ| 3 ), p = 1, 3.

(5.30) Upon performing direct calculation, one can note that the function G(φ j , Φ 2,ll ) + G(Φ 2,ll , φ j ) ϕ * p has a factor e ia(n j +n l +n l -np)x which means if n j + n l + n l -n p = 0 then a j,l,l ,p = 0.

Let P i (i = 1, 2, 3, 4) be defined as (5.31)