Bifurcation Analysis of the Problem of Two Vortices on a Finite Flat Cylinder Текст научной статьи по специальности «Физика»

Russian Journal of Nonlinear Dynamics, 2024, vol. 20, no. 1, pp. 95-111. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd231209

NONLINEAR PHYSICS AND MECHANICS

MSC 2010: 76B47, 70H05, 37Jxx, 34Cxx

Bifurcation Analysis of the Problem of Two Vortices

on a Finite Flat Cylinder

A. A.Kilin, E. M. Artemova

This paper addresses the problem of the motion of two point vortices of arbitrary strengths in an ideal incompressible fluid on a finite flat cylinder. A procedure of reduction to the level set of an additional first integral is presented. It is shown that, depending on the parameter values, three types of bifurcation diagrams are possible in the system. A complete bifurcation analysis of the system is carried out for each of them. Conditions for the orbital stability of generalizations of von Karman streets for the problem under study are obtained.

Keywords: point vortices, ideal fluid, flat cylinder, bifurcation diagram, phase portrait, von Karman vortex street, stability, boundary, flow in a strip

1. Introduction

One of the approaches to investigating the dynamics of vortex structures is to describe it using a finite-dimensional model of point vortices in an ideal fluid. This model goes back to the work of Helmholtz [1] and has been well studied to date. The problems that have received the most attention in the literature are those of the motion of point vortices on a plane and a sphere (in a thin spherical fluid layer). Without claiming to provide a complete overview, we give here only a few references to relevant review papers [2-4].

Much less attention has been devoted to the dynamics of vortices on other curvilinear surfaces. For example, in [5] the stability of vortex rows on the surface of revolution was analyzed. The dynamics of point vortices on the Lobachevsky surface was studied in [6]. The dynamics of two vortices and some partial solutions of the problem of point vortices on a toroidal surface

Received November 22, 2023 Accepted December 20, 2023

The work of A.A.Kilin (Sections 1-3) was performed at the Ural Mathematical Center (Agreement No. 075-02-2023-933). The work of E. M. Artemova (Sections 4, 5) was supported by the framework of the state assignment or the Ministry of Science and Higher Education (No. FEWS-2020-0009).

Alexander A. Kilin kilin@rcd.ru Elizaveta M. Artemova artemova@rcd.ru

Ural Mathematical Center, Udmurt State University ul. Universitetskaya 1, Izhevsk, 426034 Russia

were considered in [7]. The cases of more complex surfaces (of nonconstant curvature, multiply connected) remain largely unexplored. We mention only a few papers in which the well-known results on vortex dynamics are generalized to the case of arbitrary surfaces. In [8] the equations of motion of point vortices are generalized to the case of an arbitrary surface, and Kimura's conjecture on the motion of a vortex dipole along a geodesic is proved. In the recent paper [9], an attempt is made to construct a model of point vortices interacting with a harmonic flow on an arbitrary compact surface.

Of special note is the class of problems of the motion of point vortices on the so-called "flat" cylinder and torus. These problems arise in describing the dynamics of periodic vortex structures on a plane after reduction by the corresponding discrete symmetry group. In this case, the infinite sequence of vortices is replaced by a single vortex and periodic boundary conditions are added. Research in this direction goes back to the work of Karman [10], which is devoted to the stability analysis of a vortex street arising behind the body moving in the fluid. Later, the dynamics of vortices on a "flat" cylinder was explored, for example, in [11-14]. In particular, a complete analysis of the motion of three vortices on a cylinder with zero total strength was made in [11]. And in [12, 13], the motion of singular (mirror-symmetric) configurations of four vortices was investigated.

Equations of vortex motion on a "flat" torus which arise in analyzing the dynamics of vortex lattices were obtained in [15]. In [16], the motion of three point vortices with zero total circulation in a periodic parallelogram was studied. The motion of two, three and four square lattices (or 2, 3, 4 vortices on a flat square torus) was examined in [17].

Other related problems include those of point vortices moving in a strip or in a rectangular region. Problems of this type are usually investigated by the image method. In this method, the influence of boundaries is described by adding image vortices symmetric relative to the boundaries of the region. The strength of the images reverses sign at each reflection from the boundary. As a result, for example, in the case of a strip, to each vortex one associates two vortex rows with opposite vortex strengths. The coordinates of these vortex rows are related to each other. Thus, the analysis of the dynamics of N vortices in a strip is equivalent to the study of the dynamics of N pairs of vortex rows on some invariant manifold. Or, which is the same, it is equivalent to the study of the dynamics of N vortex pairs on a "flat" cylinder on some invariant manifold. In a similar way, the dynamics of N vortices in a rectangular region is equivalent to the dynamics of N quadruples of vortex lattices (more precisely, to their dynamics on some invariant manifold). There has been little research in this direction. An analysis of the dynamics of a vortex pair in a strip is carried out in [18]. Nonintegrability of the problem of the motion of a vortex pair in a rectangular region is shown in [19].

This paper is concerned with the dynamics of two vortices on a "flat" cylinder of finite length. This problem is equivalent to the problem of the motion of two vortex rows in a strip of fluid. In [20], a model describing the interaction of point vortices on a flat finite cylinder is proposed. We use this model to perform a complete bifurcation analysis of the problem of two vortices of an arbitrary strength.

This paper is organized as follows. In Section 2 we present equations of motion and perform their reduction to the level set of a first integral. In Section 3 we carry out a parametric analysis of the problem and find regions in the parameter space which correspond to different types of bifurcation diagrams. In Section 4 we present all possible types of bifurcation diagrams and phase portraits of the reduced system. In Section 5 we present an interpretation of the results obtained from the viewpoint of the stability of vortex streets for different parameters and initial conditions.

2. Mathematical model

2.1. Equations of motion

Consider the motion of two vortices of strengths r1 and r2 in an ideal incompressible fluid on a flat cylinder. Denote by R the length of the circle of the cylinder's base and by L its height. Also, by virtue of the arbitrariness of the choice of units of measurement, we will assume that L = n (see Fig. 1). The continuous line in Fig. 1 denotes the solid walls of the cylinder, and the dotted line denotes the boundaries that are identified (periodic boundaries).

Fig. 1. A schematic representation of vortices on a cylinder

To describe the motion of the vortices, we introduce a fixed coordinate system Oxy as shown in Fig. 1. Let (x1, y1) and (x2, y2) denote the coordinates of the first and the second vortex, respectively. The equations of vortex motion on a flat cylinder were derived in [20] and, for two vortices, have the form

¿1 = 7ZX(x 1 - x2, Vi, V2) + TZ E

sin(2yi)

4n

r_i

47T

r

4vr Z-, cosh(nE) - cos(2y~y ^ = 4^Y{Xl ~ ^ Vl> ^

n=

+ ^0

¿2 = TzMx2 ~ V2, Vl) + JZ E

£2 sin(2y2) • _ £1 _

4vr 2L, cosh(nR) - cos(2y2)' y'2 An {X'2 Xl' y'2> Vlh

(2.1)

where X and Y are functions that describe the interaction of the vortices and have the form

+^0

X(x, y^ y2) = E

Y(x, y^ y2) =

+^0 E

(sm{y1 + y2) - sin(y1 - y2)) cosh(.T - nR) - sin(2y2) (cosh(x — nR) — cos(y1 — y2))(cosh(x — nR) — cos(y1 + y2)):

_sinh(.r - nR){cos{y1 - y2) - cos{y1 + y2))_

(cosh(x — nR) — cos(y1 — y2))(cosh(x — nR) — cos(y1 + y2))

Remark 1. The convergence of the series X(x, y1, y2) and Y(x, y1, y2) was proved in [17]. The convergence of the series

+w

E

n=— 00

sm^yj

cosh(nR) — cos(2y4)'

i = 1, 2

is proved in a similar way. The series in Eqs. (2.1) were summed up to terms of order no less than £ = 10~16.

n=

n=

Equations (2.1) can be represented in the Hamiltonian form

- -L^K - -L^IL - -L^IL - 1 dH

with the Hamiltonian

H = r^a y^ ^ cosh(.T1 - x2 - nR) - cos(y1 + y2) 4vr cosh(.T1 — x2 — nR) — cos (y1 — y2)

+ £i v lncosh(nR) ~ cos(2Vi) | r2 y^ ln cosh (nR) - cos(2 y2) 8vr ^ cosh (nR) 8vr ^ cosh (nR) '

and the Poisson bracket

! r n ! |

Vj} = "F-'

i

where 5ij is the Kronecker symbol. In addition to the energy integral, which coincides with the Hamiltonian (2.2), the equations of motion (2.1) admit an additional first integral

P= (23)

Since the coordinates y are bounded, the values of the integral P are defined in the region P £

e (z - |r||+|r^, | + |Fil2|ral))-

In the system of equations (2.1) we will assume that the vortex with the strength larger in absolute value is denoted by index 1, and that ri = 1. This can always be achieved by renumbering the vortices and by rescaling time as t = r1i. As a result, in the system (2.1) and the integrals (2.2), (2.3) there remain two parameters: the length of the cylinder R and the

r

relative strength 7 = p2- e [—1, 1] \ {0}. 2.2. Reduction

Let us perform a reduction to the fixed level set of the integral P. To do so, we transform from the variables (x1, y1, x2, y2) to the variables (£, n, Z, P) as follows:

£ = C = x1 +x2, P=y±±p>l. (2.4)

The reverse change of variables has the form

C + £ „ C-£ P-v

2 2 7

It can be seen from the expressions (2.4) that this change of variables is reversible for all values of the parameter 7. The new variables (£, n, Z, P) have been chosen so that their Poisson bracket is canonical.

In the new variables, on the fixed level set of the integral P = p, the equations of motion for the variables £ and n decouple, form a reduced system, and have the following form:

^ ~ &rj ~ 4TT ^

sin(2 (p + rj))

cosh(??ií) - cos(2(p + r¡)) cosh(nR) - cos

((7 - 1) sin fr+y-" - (7 + 1) sin cosh(e - nR)

+ 'cosh(£ - nR) - cos Mltoliz) (COsh(e - nR) - cos

7 sin — sin(2(p + rf))

cosh(£ - nR) - cos Mlz£±!Z j (COsh(£ - nR) - cos

V =

Y

{p+v)i+p-y

T

m _

di

sinh(£ - nR) (cos - cos

(2.5)

4vr (cosh(£ - nR) - cos (cosh(e - nR) - cos

where H is the Hamiltonian (2.2) written in the new variables (2.4)

7 cosh(£ - nR) - cos ^l+r-1

4vr J^ Cosh(C - nR) - cos MM

J_ ^ lncosh(nfl)-cos(2(p + ?7)) 7^ ^ ln cosh(nfl) - cos 87T ^ cosh(??,E) 87T ^ cosh(??,E)

T»--rv~) v ' T»--rv-i V '

The evolution of the variable ( is defined by an additional quadrature of the form

^ ~ dp ~ 4tt ^

sin(2 (p + rj)) 7 sin ^r1

cosh (nR) - cos(2 (p + tj)) Cosh(nR) - cos

((7 + 1) sin (P+V)l+P-V _ (7 _ 1) sin cosh(^ _

cosh(¿¡ — nR) — cos Í£±!Zl^_£±!í j (cosh(¿¡ — nR) — cos

(p+vYi+p-y

7 sin + sin(2(p + r?))

cosh(£ - nR) - cos (COsh(£ - nR) - cos b+vh-tv-v^j

This quadrature allows one to restore the trajectories of the vortices in absolute space using the solutions to the system of equations (2.5).

On the fixed level set of the integral p e (f (7 ~~ IyI)> § + f (7 ~~ |7|)) the reduced system (2.5) is defined on the cylinder

Mp = {(£ n) I £ mod ^ n e (nmin, nmax)} - ^

where nmin and nmax depend on the values of the first integral p and the parameters R and 7 as follows:

n=—oo

1. when y > 0

p G (o, |(1 + 7)) , Vmin = P~ min(2p, ytt), r?max = -p + min(2j9, vr);

2. when y < 0

(Yn 1 1

P G (Y' 2 J ' = T,miai = 2 ~ 77r' ^ ~ 2 max(2p ~ 7r' 77r^

It follows from (2.4) that the boundaries of the cylinder Mp (n = nmin and n = nmax) correspond to the approach of the vortices to the boundaries yi = 0, n. Thus, on the boundaries of the cylinder Mp the flow (2.5) is not defined.

3. Types of bifurcation diagrams

Consider the question of classification of possible types of motion depending on the system parameters and the values of first integrals. In the general case, the pattern of motion depends on the number and types of fixed points of the reduced system (2.5) and on its singularities.

The fixed points of the system (2.5) lie on the vertical straight lines £ = 0 and £ = This follows from the oddness of the hyperbolic sine function sinh in Eq. (2.5) for n and from the periodicity of the system under study in the variable £. Numerical experiments show that no fixed points arise in the system when £ has other values. The coordinates n of the fixed points on the straight lines £ = 0 and £ = § are defined from the solution of the equations

£

dH „ , . _ dHR/2

ç=o dn

= 0 and £

£=R/2 dn

(3.1)

respectively, where WQ = TL(£ = 0), T-LR^2 = H (£ = Equations (3.1) are represented using series. In the general case, it is not possible to construct their explicit solution. So, in what follows we will solve Eqs. (3.1) numerically. In the particular case y = ±1 the explicit expressions for n were obtained and analyzed in [20].

For a fixed value of the parameters y and R the system of equations

h = Ho(p, n, R, y),

&H0 „ (3.2)

dn

0

defines, on the plane of first integrals (p, h), bifurcation curves corresponding to fixed points of the reduced system which lie on the straight line £ = 0. A similar system,

h = 'HR/2(P, n, R ^^

dHR/2 (3.3)

dn

defines curves corresponding to the fixed points lying on the straight line C = "f • All curves (3.2), (3.3) on the plane (p, h) form a bifurcation diagram.

Remark 2. The bifurcation diagram can also be constructed as an image of a critical set of the integral map z = (£, n, Z, P) ^ ^(z) = (P(z), H(z)) on the plane of first integrals. However, in this case the explicit reduction to the fixed level set of first integrals allows one to reduce the problem to investigating the functions H0 and Wp,^.

The type of bifurcation diagram depends on the asymptotics and singular points of the bifurcation curves at which the bifurcation curves are tangent to each other. These points are defined by the systems of equations

^^ = 0, ^^ = o. (3.4)

dn dn2

Qualitative changes in the diagram can occur for two reasons:

1. Changes in the asymptotics of the functions H0 and HR/2.

2. Birth (disappearance) of singular points on the bifurcation curves.

To analyze the first reason, we consider the dependence of the singularities of the function H on parameter values. The singularities of the Hamiltonian correspond to the approach of one or both vortices to the boundaries of the cylinder and to the approach of the vortices to each other. As shown in [20], the asymptotics corresponding to the approach of the vortices to the boundary do not change as the parameters vary, lim H = —to. And the asymptotics corresponding

to the approach of the vortices to each other depends on the sign of y, i.e., lim H =

V1^V2, X1

= sign(Y) to. Thus, on the parameter plane (y, R) the straight line y = 0 separates different types of bifurcation diagrams.

To define the parameters at which singular points of bifurcation curves are born (disappear), it is necessary to add to Eqs. (3.4) the conditions of their degeneracy

d3"^o,_R/2 _

dn3

Thus, it is necessary to solve two systems of equations

u . = 0 d2n0 = d3H0 =

drj ' drj2 ' dr/3 '

and

ddH D/o d H R/O d

~ljir = 0' -ftf = 0' -af = °- (3-6)

Equations (3.5) and (3.6) define on the plane (y, R) the curves cr0 and &R/2 which separate different types of bifurcation diagrams.

The numerical analysis shows that the system (3.5) has no solutions for the fixed points lying on the straight line £ = 0. Thus, the regions on the plane (p, h) which correspond to different types of bifurcation diagrams are separated only by the curves (3.6).

The numerical solution of the system (3.6) is shown in Fig. 2. In the same figure, regions with different numbers of singular points or different asymptotics are hatched:

1. The cross-hatched region (labeled I in Fig. 2): on the bifurcation curves there exist two singular points (cusp points) (i.e., there exist one to three fixed points for £ = §), and there exist one or two fixed points for £ = 0.

Fig. 2. Existence regions of fixed points on the parameter plane (j, R)

2. The left-hatched region (labeled II in Fig. 2): there are no singular points on the bifurcation curves, the syste points for £ = 0.

curves, the system always has one fixed point for £ = ^, and there exist one or two fixed

3. The right-hatched region (labeled III in Fig. 2): there are no singular points on the bifurcation curves, the system always has one fixed point for £ = ^, and there exist no fixed points for £ = 0. Also, a change of the asymptotics of the Hamiltonian occurs in region III (as opposed to regions I and II).

When Eqs. (3.6) are solved numerically, the curve &R/2 is defined with some accuracy. In addition, near the straight line y = 0 one can observe a narrow region with a different type of bifurcation diagram (shown in gray in Fig. 2). However, a successive increase in the accuracy of the solution of the equations shows that the straight line 7 = 0 does bound region I and no additional regions with other types of diagrams arise near it. The appearance of the narrow region is due to calculation errors. An illustration of such a numerical convergence of the boundary y = 0 is presented in Fig. 3. Figure 3 shows regions on the plane (y, p) with different numbers of fixed points of the reduced system with R = 1. In the figure one can see that the region caused by errors becomes smaller as the accuracy of finding the coordinate n of the fixed points is increased.

Next, we carry out a bifurcation analysis of the system for the parameters (y, R) lying in each of the three regions in Fig. 2.

4. Bifurcation analysis 4.1. Region I

Bifurcation diagrams for R = 1 and y = — 0.5 are presented in Fig. 4. Figure 4a shows bifurcation curves s1 and s2 corresponding to fixed points for £ = 0, and Fig. 4b shows bifurcation curves e1; e2, s3 for fixed points with £ = F°r chosen parameter values the branches s1 and s2 are very close to the branches e1, e2. Therefore, for convenience they are depicted in different figures. A general schematic representation of the bifurcation diagram, with the scale varied accordingly, is given in Fig. 5.

one root

two roots

three roots

not physical area

-0.4

-0.2 -0.15 -0.1 -0.05 0 -0.2 -0.15 -0.1 -0.05 0

7

7

(a) max(<fy) « 0.003

(b) max(<S?7) «0.001

-0.2 -0.15 -0.1 -0.05 0

7

(c) max(<fy)« 0.0003

Fig. 3. Regions with different numbers of fixed points of the reduced system with £ = with different accuracy Sn of finding the coordinate of the fixed points

(a) (b)

Fig. 4. Bifurcation curves for R = 1, 7 = —0.5 for the fixed points (a) £ = 0, (b) £ = -77, (c) an enlarged fragment of the bifurcation diagram with cusp points

The red vertical straight lines in Figs. 5 and 4 denote critical values of p at which the system acquires a singularity corresponding to coincidence of vortices. The continuous lines denote the bifurcation curves corresponding to stable fixed points, and the dotted lines denote those corresponding to unstable fixed points. Gray denotes the region where motions are impossible.

To classify and analyze the solutions of the system under study and possible types of vortex motion on a flat cylinder, we can use an approach based on the construction of a bifurcation complex. We recall that, in a bifurcation complex, to each region in the bifurcation diagram one associates a number of leaves that coincides with the number of connectedness components of the integral manifold at given values of the integrals from this region. These leaves are glued together along the bifurcation curves corresponding to unstable fixed points of the reduced system. And the bifurcation curves corresponding to stable fixed points are the boundaries of the leaves. In the problem we consider here, the bifurcation complex is a fairly complex multisheeted surface, and so we do not present its explicit form here. We restrict ourselves to pointing out the number

i Pi Pi P3 p

/ T2 T2 \

/ / / / / / / / / / / / / / / / / / / / / / / ! 2T2 / / / / / / / / \ N r / ! / / ■ \ / ^ X2Î2A / w \ 3t2/N3IA / v \ / \ \ ! 4TT2 \ \ • 3T2 \ \ \ N \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2T2 \\ \ \ \ \ \ \ \\

Fig. 5. A schematic representation of the bifurcation diagram

of connectedness components (tori T2) in the bifurcation diagrams (see Figs. 4, 5, 7, 9). Also, we present schematic profiles of the sections formed by the intersection of the bifurcation complex with the plane p = const, along with phase portraits.

Figure 6 shows possible types of phase portraits for different values of the integral p. The red lines in the phase portraits denote unstable invariant manifolds (separatrices) of saddle fixed points. The green points denote singularities corresponding to coincidence of the vortices. We recall that the upper and lower boundaries of the phase portrait also correspond to singularities where one or both vortices approach the walls of the cylinder. Besides the variable £ is defined on the interval [0, R). However, for ease of representing the critical points lying on the boundary £ = 0, we will draw a doubled phase portrait on the interval £ G [—R, R).

We now consider the evolution of the phase portrait of the reduced system (2.5) as the value of the integral p is varied.

1. When p ^ 0, the system (2.5) has an unstable fixed point s1 with £ = 0 and a stable one, e1, with £ = If. A typical view of the phase portrait is shown in Fig. 6a. Since there are many fine details in Fig. 6 (as well as in Fig. 8), we present, in addition to the phase portraits themselves, their enlarged fragments near the fixed points.

2. When 0 < p < p1, the second unstable fixed point s2, with £ = 0, and a singularity at the point ^0, Pjïpj) arise. A typical view of the phase portrait is shown in Fig. 6b.

3. When p = p1, where p1 is defined from Eqs. (3.4) for HR/2, a tangent bifurcation occurs,

giving rise to a stable (e2) and an unstable (s3) fixed point with { = -f • The corresponding point on the bifurcation diagram (see Figs. 4, 5) is shown by a circle marker.

4. When p1 < p < p2, the system has three (two stable e12 and an unstable s3) fixed points with £ = ^ and two unstable ones, s12, with £ = 0. A typical view of the phase portrait is shown in Fig. 6c.

5. When p = p2, where p2 is defined from the equation

-1 -0.5 0 0.5

-rV f1

1) X

-0-fr

0.4

~T2

-00 —oo

-1 -0.5 0 0.5 (a) p = —0.3

1 -0.5 0 0.5 1 (c) p = 0.135

AV

— OO —00 —00

-1 -0.5 0 0.5 1 (b) p = 0.05

-1 -0.5 0 0.5 1 (e) p = p3 «0.3927

Fig. 6. Phase portraits of the system for R =1, 7 = —0.5 (left) and schematic profiles of the sections of the bifurcation complex (right)

a nonlocal bifurcation occurs, resulting in a merging of the séparatrices corresponding to the fixed points s2 and s3. This leads to a rearrangement of the phase portrait, but the number and the types of fixed points in the system remain unchanged. The corresponding point on the bifurcation diagram (see Figs. 4, 5) is shown by a circle marker.

6. When p2 < p < p3, the number and the types of fixed points remain unchanged compared to the interval p G (p1, p2). A typical view of the phase portrait is shown in Fig. 6d.

7. When p = p3 = f (1 + 7), the unstable fixed points s1 and s2 lie on the same level of energy (albeit on different connectedness components). A typical view of the phase portrait is shown in Fig. 6e.

8. As the value of the integral p is increased further, similar bifurcations occur, but in the reverse sequence.

Remark 3. Note that the system under consideration has the symmetry yi ^ n — yi. In the variables P, n this symmetry has the form

n n

2(1-7)-'?, P^-(1 + y)-P

The presence of this symmetry leads to the coincidence of the phase portraits at the level sets P = p and P = f (1 + 7) — p up to vertical reflection. From this it also follows that the bifurcation diagram is symmetric with respect to the straight line p = p3. In addition, the fixed points ex and e2 correspond to identical configurations of vortices located near the opposite walls of the cylinder. Similarly, the fixed points sx and s2 correspond to the identical vortex configurations.

4.2. Region II

Figure 7 shows a bifurcation diagram on the plane of first integrals (p, h) for R = 3, 7 = = —0.8. The curves denoted by s1 and s2 correspond to fixed points with £ = 0, and the curve e corresponds to those with C = §• In the bifurcation diagram and the phase portraits, use is made of the notation introduced earlier.

Fig. 7. Bifurcation diagram for R =3, 7 = —0.8

Consider the evolution of the phase portraits of the reduced system (2.5) as the value of the integral p is varied.

00 —oo

-3-2-10 1 2 3 (a) p= —0.1

_ AV_

;

r^rtlTf

— 00 —oo —oo

\l

•22-

-3-2-10 1 2 3

\ WT

-3-2-10 1 2 ?"

\ — oo — oo — oo (b) p = 0.1

Jk

-3-2-10 1 2 3 (c) p = p3~ 0.1571

Fig. 8. Phase portraits of the system for R =3, y = —0.8 (left) and a schematic section of the bifurcation complex (right)

1. When p ^ 0 the system (2.5) has an unstable fixed point s1 with £ = 0 and a stable one,

e, with £ = §. A typical view of the phase portrait is shown in Fig. 8a.

2. When 0 < p < p1, the second unstable fixed point s2 with £ = 0 and a singularity ^0, Pyq^j arise. A typical view of the phase portrait is shown in Fig. 8b.

3. When p3 = f (1+7), a nonlocal bifurcation occurs, resulting in a merging of the séparatrices of the fixed points s1 and s2. The corresponding phase portrait is shown in Fig. 8c. The corresponding point on the bifurcation diagram (see Fig. 7) is shown by a circle marker.

4. As the value of the integral p is increased further, similar bifurcations occur, but in the reverse sequence.

4.3. Region III

Figure 9 shows a bifurcation diagram of the system and a typical view of the phase portrait for R = 1, y = 0.5. In the bifurcation diagram and the phase portrait, use is made of the notation introduced earlier. In this case, the phase portrait does not change qualitatively as the value of the integral p is varied.

kh

0.40.3 0.2 0.10

-0.1 -0.2-0.3-0.4-

T2 /

/ \ / \ / \ / \ / N

/ \ / \

/ \ p

-/-1-1-1-r\-►

/ 0.5 1 1.5 2\

/ \ / \ / \ / \

! 2T2 \

1 iV

0.8

0.6

-1 -0.5—( 1-0^5-1

-O.'i

-0 4

-oo —oo

(b)

(a)

Fig. 9. Bifurcation diagram (a) and a typical phase portrait with a schematic section of the bifurcation complex (b) for R =1, 7 = 0.5

5. Stability of von Karman streets in a strip of fluid

We note that the fixed points of the reduced system correspond to steady-state solutions of the complete system in which the vortices form some constant configuration rotating uniformly with constant velocity about the cylinder's axis. Also, the fixed points with £ = 0 correspond to configurations where the vortices lie above each other. And the fixed points with C = "f correspond to configurations where the vortices lie on opposite sides of the cylinder. In terms of vortex rows these configurations can be interpreted as symmetric vortex streets (where vortices of different rows lie strictly above each other) and staggered vortex streets (where vortices of different rows are displaced relative to each other by half the distance between the vortices). In some studies, the latter configuration is also called the Karman configuration.

As in the case of an infinite cylinder (with no walls) [21], in the problem we consider here, both symmetric and staggered configurations exist for any parameter values of the problem. And the bifurcation analysis of the system carried out by us allows several statements to be formulated about the stability of the above-mentioned configurations, more precisely, about their orbital stability, which follows from the stability of the corresponding fixed points of the reduced system.

Proposition 1. Staggered stationary configurations of vortex rows with arbitrary strengths of opposite signs in a strip of fluid exist at any distance between the rows and are orbitally stable if

a. the parameters (y, R) lie in region I in Fig. 2 and the distance between vortex rows is larger than some critical

\Vl -y21 > Ay*(7, R) =Pl(l - i) + Vl (l + i where p1 and are solutions of the system (3.4) for HR/2.

b. the parameters (7, R) lie in region II in Fig. 2.

A Ay*

A Ay*

0.2-

0.5

-1 -0.8 -0.6 -0.4 -0.2 0 (a) R= 1

-1 -0.8 -0.6 -0.4 -0.2 0 (b) R = 3

Fig. 10. Dependence Ay*(j) for (a) R = 1, (b) R = 3

Examples of the dependence Ay*(7) for R = 1 and R = 3 are given in Fig. 10.

Proposition 2. Symmetric stationary configurations of vortex rows with arbitrary strengths of opposite signs in a strip of fluid exist at any distance between the rows and are always unstable.

Proposition 3. Symmetric stationary configurations of vortex rows with arbitrary strengths of the same sign in a strip of fluid do not exist, and staggered configurations exist at any distance between the rows and are always unstable.

In fact, these statements generalize the results on the orbital stability of von Kármán vortex streets to the case of arbitrary vortex strengths and addition of the boundaries of the fluid layer under consideration.

The main results on the stability of von Kármán streets and similar configurations can be found in [10, 22-24]. A study of the complete (not only orbital) linear stability of von Kármán streets in a strip of fluid for the case of vortex strengths equal in absolute value, but opposite in sign is presented in [25]. In this paper authors show that there exist staggered streets which are linearly stable even for a complete system. The instability of staggered streets in a strip of fluid for equal vortex strengths is established in [26].

We also note that, in this paper, we have considered only the stability of vortex streets with respect to row-preserving perturbations. To analyze the stability with respect to more general perturbations, one can, for example, following Kármán, break down each row into two (or n) subrows (of a larger period) and consider the stability of steady-state solutions for the problem of four (or 2n) vortices on a cylinder. A similar analysis for vortex streets in a strip of fluid in the case of vortex strengths equal in absolute value, but opposite in sign is made in [27].

6. Conclusion

In this paper we have examined in detail the integrable case of two vortices on a finite flat cylinder. We have presented an explicit form of the equations and proposed a procedure

of reduction to the level set of an additional first integral. We have carried out a complete bifurcation analysis of the system and found that, depending on the parameter values, three types of bifurcation diagrams are possible. We have considered each type of diagrams separately and constructed possible phase portraits. Also, we have presented a generalization of the results on the stability of von Kärmän streets to the case of motion of vortex streets of arbitrary strengths in a strip of fluid.

Possible avenues for further research may include an analytical proof of some results obtained here numerically. It would also be interesting to investigate problems of the existence and stability of stationary configurations of a larger number of vortices on a cylinder, and, in particular, to analyze the stability of a von Kärmän street with vortices of different strengths in a strip of fluid with respect to more general perturbations.

Conflict of interest

The authors declare that they have no conflict of interest.

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