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ISSN 1560-3547, Regular and Chaotic Dynamics, 2013, Vol. 18, Nos. 1­2, pp. 33­62. c Pleiades Publishing, Ltd., 2013.

The Dynamics of Vortex Rings: Leapfrogging, Choreographies and the Stability Problem
Alexey V. Borisov1,
2, 3 *

, Alexander A. Kilin
1

1, 2, 3 **

, and Ivan S. Mamaev1,

2, 3 ***

Institute of Computer Science, Udmurt State University, Universitetskaya 1, Izhevsk, 426034 Russia 2 A.A. Blagonravov Mechanical Engineering Research Institute of RAS Bardina str. 4, Moscow, 117334 Russia 3 Institute of Mathematics and Mechanics of the Ural Branch of RAS S. Kovalevskaja str. 16, Ekaterinburg, 620990 Russia
Received September 19, 2012; accepted December 21, 2012

Abstract--We consider the problem of motion of axisymmetric vortex rings in an ideal incompressible fluid. Using the topological approach, we present a metho d for complete qualitative analysis of the dynamics of a system of two vortex rings. In particular, we completely solve the problem of describing the conditions for the onset of leapfrogging motion of vortex rings. In addition, for the system of two vortex rings we find new families of motions where the relative distances remain finite (we call them pseudo-leapfrogging). We also find solutions for the problem of three vortex rings, which describe both the regular and chaotic leapfrogging motion of vortex rings. MSC2010 numbers: 76B47 DOI: 10.1134/S1560354713010036 Keywords: ideal fluid, vortex ring, leapfrogging motion of vortex rings, bifurcation complex, periodic solution, integrability, chaotic dynamics

In memory of H. Aref and V. V. Meleshko

Contents
INTRODUCTION 1 EQUATIONS OF MOTION FOR VORTEX RINGS 1.1 Axisymmetric Vortex Flows without Swirl 1.2 Toroidal Vortex Rings 1.3 Integrals of Motion and Reduction in Symmetry 2 TWO VORTEX RINGS 2.1 Leapfrogging and the Problem of Classifying Motions 2.2 The Case > 0 2.3 The Case < 0, P > 0 2.4 The Case < 0, P 0 3 THREE VORTEX RINGS CONCLUSION ACKNOWLEDGMENTS REFERENCES
* ** ***

34 35 35 36 38 40 40 43 49 54 56 59 60 60

E-mail: borisov@rcd.ru E-mail: aka@rcd.ru E-mail: mamaev@rcd.ru

33


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INTRODUCTION In this paper we address a number of issues involving the classical problem of motion of symmetric vortex rings in an ideal fluid. The problem of motion of two vortex rings was first considered by H. Helmholtz in his famous 1858 paper [29], which laid the foundations for the whole vortex dynamics. In particular, Helmholtz qualitatively described a particularly interesting effect, namely leapfrogging motion of vortex rings , i.e. the motion where the rings move along a common symmetry axis, alternately passing through each other. However, he did not present a theoretical analysis of this problem. W. M. Hicks [30­33] and F. W. Dyson [21, 22] explored, in more or less detail, the dynamics of this system and pointed out a few particular conditions for the onset of leapfrogging. Later S. A. Chaplygin also tried to explain the phenomenon of leapfrogging of vortex rings in his remarks to the Russian translation of H. Helmholtz's paper and presented his own analysis of this problem [18]. The general problem of description of the motion of two axisymmetric vortices and, in particular, their leapfrogging motion, was considered in several publications (see [16, 17, 28] etc.). It should be noted that the results of the above-mentioned papers concerning the analysis of this system are incomplete in spite of its integrability. Here we will try to make a complete qualitative analysis of the dynamics of a system of two rings, based on the topological approach and the concept of a bifurcation complex , and to obtain a complete description of the motion for different initial conditions and intensities of vortex rings. In addition to the classical leapfrogging, we point out some more families of periodic solutions to the reduced system, one of which corresponds to quite exotic motions called by us pseudoleapfrogging . Our approach uses the canonical Hamiltonian form of the equations of motion. In this case this form is not as fundamental as in the case of motion of point vortices on a plane and a sphere, because additional terms that account for the self-influence of the vortex must be added. The particular form of these terms depends on the choice of the model of vorticity distribution in the ring. Nevertheless, these terms do not affect the Hamiltonian nature of the system and the fundamental features of motion. We note that the Hamiltonian form of the equations of motion for vortex rings was first obtained by N. S. Vasilyev [62] (although a similar form of equations was used already by F. W. Dyson [22]). Afterwards the Hamiltonian nature of the system of vortex rings was rediscovered several times (see, e.g., [47, 49]). In addition to investigating the integrable problem of two vortex rings, this paper addresses the question of motion of three rings in an ideal fluid, which is much more complicated. This problem is no longer integrable. Chaotic motions are discussed in [8, 35], where a number of phase portraits are presented and a numerical investigation of the development of chaotic dynamics is performed. A rigorous proof of nonintegrability of the problem of three rings, which uses the method of splitting of separatrices, was given in [3, 4]. For a restricted problem where a vortex ring of infinitely small intensity moves in the field of two other vortex rings, a more detailed investigation of quasiperiodic solutions by methods of KAM theory was carried out in [7, 9]. However, in view of the complicated form of the Hamiltonian there is still no complete description of the dynamics of this system (even the construction of the Poincar´ e cross-section at the fixed levels of the Hamiltonian is nontrivial). In the present paper we try to fill up this gap where possible and explore a class of stable periodic orbits (of a reduced system). It turns out that these motions are closely related to the problem raised by J.C. Maxwell, who, after publication of H. Helmoltz's paper describing the leapfrogging motion of two rings, created a number of animations to illustrate the motion of rings having almost the same intensity. When observing the motion of three rings, J.C. Maxwell pointed out their joint leapfrogging (see his letter to W. Thomson [40]). We show that for the motion under investigation in the case of equal intensities there exists a uniformly moving coordinate system related to the center of vorticity of rings, in which the vortices execute a periodic motion along the same curve. Recall that such motions in celestial mechanics were pointed out in [19, 57], where they were called choreographies . In vortex dynamics, choreographies were discovered in [12], here we apply the methods of this paper to the study of analogous vortex rings. Numerical analysis shows that the choreographies in the three-vortex problem are stable in a rather large region of values of the energy and momentum integrals, and hence they can be observed in an experiment. Thus, we provide a theoretical justification for Maxwell's observation. The numerical analysis also shows that a stable leapfrogging motion of three vortex rings can also
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be observed at unequal intensities of vortices, moreover, there exist quite exotic motions of three rings, which we have called chaotic leapfrogging . In future it would be interesting to observe the existence of such motions experimentally. In conclusion, we note that in addition to the classical model of vortex rings (which we consider in this paper) there exists a more general solution to hydrodynamical equations due to W. M. Hicks [33], which describes axisymmetric spiral "vortex aggregates". This solution is distinctive in that it admits a swirl around the axis of symmetry. Some of the results of Hicks were rediscovered several times afterwards. For the history of this issue and a detailed list of literature see [39]; in this note the model of Hick's spiral vortex is extended to the case of interaction with a toroidal magnetic field. In [6] the restricted problem of the dynamics of a vortex ring in a flow with swirl in connection with the problem of vortex breakdown was examined. We note that this model is more complicated, so that for this model no complete description has been obtained even for the problem of two rings and no conditions for leapfrogging have been found. Apart from this, there is another model of a three-dimensional vortex singularity, namely the vorton model [46]; it can also be applied to describe vortex rings and more complicated closed vortex structures in three-dimensional space. The equations for vortons do not admit an invariant measure and do not preserve the energy integral. Numerical simulation, using the vorton model, of the interaction of axisymmetric vortex structures is performed in [26]; it is shown that if the number of vortons is sufficiently large, this model can be used to study the dynamics of vortex rings. We also note that both the standard model of vortex structures (which we explore in this note) and the vorton model, and other models, as opposed to point vortices on a plane, are not a weak solution to the hydrodynamical equations and are based on additional physical assumptions when these equations are derived. Although these models are not so fundamental, they provide an acceptable description of fundamental features observed in experiments. Below is an incomplete list of papers discussing other issues associated with the dynamics of vortex rings: [54­56] -- reduction, N vortex rings + rigid body interaction; [37, 38, 61] -- behaviour of an infinite system; [25, 45, 48, 59] -- existence and stability of solutions; [42, 52] -- experimental studies. 1. EQUATIONS OF MOTION FOR VORTEX RINGS 1.1. Axisymmetric Vortex Flows without Swirl First of all, we recall the basic hydrodynamical equations for axisymmetric flows without swirl [2, 36, 51]. Let us direct the axis Oz along the symmetry axis of the flow and choose the cylindrical coordinates z , r, . The velocity and vorticity of the flow can be represented as v = vz (z, r)ez + vr (z, r)er , = (z, r)e , vz vr - . = z r For brevity, here and in the sequel the dependence of the functions on time is not indicated. (Since there is no swirl, v = 0 and z = 0.) As is well known, in the case of an incompressible fluid (i.e. under the condition Âv = 0) the velocity can be expressed in terms of the streamfunction as vz = 1 (z, r) , r r vr = - 1 (z, r) . r z

The system of hydrodynamical equations governing the axisymmetric vortex flows of an incompressible fluid can be represented using the functions and as 1 1 1 + - =- 2 , t r z r r r z r r 1 2 1 + = -. r r r r z 2
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The solution to the second equation of this system with the help of the Green function can be represented as (z, r) = G(z, r, z, r)( z, r) dzdr. (1.2)

G(z, r, z, r) ­ the streamfunction of an infinitely thin circular vortex filament of unit intensity (whose vorticity is = (z - z ) (r - r)) ­ is given by 1/2 2 2 4r r rr - k K (k ) - E (k ) , k = , (1.3) G(z, r, z, r) = 2 k k (z - z )2 +(r + r)2 where K (k ) and E (k ) are the complete elliptic integrals of the first and second kind:
1 1

K (k ) =
0

dx , 1 - x2 1 - k 2 x2

E (k ) =
0

1 - k 2 x2 dx. 1 - x2

The expression for the streamfunction of an infinitely thin vortex ring in a somewhat different form was first obtained by J.C.Maxwell in 1893 [41]. Using Eqs. (1.1), it can be shown that the axisymmetric vortex flow with the streamfunction (1.2) possesses three conserved quantities =
S

(z, r) dz dr, E=
S

P=
S

(z, r)r2 dz dr, (1.4)

(z, r) (z, r) dz dr,

where is the full circulation of the velocity of the vortex flow, P is the absolute value of the momentum of the vortex flow [51], and E is its kinetic energy (here S is the cross-section of the region filled with fluid). 1.2. Toroidal Vortex Rings As is well known, in the case of plane-parallel flows the equations of vortex motion admit a "singular solution" governing the motion of point vortices on a plane. It is natural to conjecture that the system (1.1) also admits a solution of the form
N

=
i=1

i (z - Zi (t)) (r - Ri (t)),

(1.5)

where (x) is the Dirac delta function, which describes the superposition of coaxial "point" vortex rings with cylindrical coordinates Zi ,Ri . By the Helmholtz theorem on the transfer of vortex filaments, the functions Ri (t), Zi (t) must satisfy the equations of the form Ri (t) = vr (z, r)
r =Ri z =Z i

=-

1 Ri Zi

j G(Ri ,Zi ,Rj ,Zj ),
j =i

j G(Ri ,Zi ,Rj ,Zj ) + V
j =i (0) z

(1.6) (Zi ,Ri ).

Zi (t) = vz (z, r)
(0)

r =Ri z =Zi

1 = Ri Ri

Here, Vz induced (0) Vz = 0 filament

(Zi ,Ri ) describes the self-propulsion of a vortex, i.e. the velocity of a vortex filament by the filament itself in the absence of other vortices. As is well known, in the planar case, (0) and in the axisymmetric case, Vz = (which is due to the non-zero curvature of the [5]).
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Thus, there exist no singular solutions of the form (1.5) for axisymmetric vortex flows (1.1). Therefore, to describe the interaction of vortex rings in an ideal fluid, it is common to use the model of thin toroidal vortex rings. In this model each ring is a regular torus with a circular crosssection of radius ai whose center is at distance Ri from the symmetry axis (see Fig. 1), where Ri . The vorticity is taken to be zero everywhere outside the torus, while inside the torus the ai vorticity distribution depends only on the distance s to the center of the cross-section of the torus = (s) (Fig. 1).

Fig. 1. A vortex ring of radius R with cross-section radius a.

Remark. More complicated vortex ring models satisfying additional restrictions on the type of flow are presented in [22­24, 33]. The well-known Hill vortex [34, 36] is also a particular case of a vortex ring satisfying the stationarity requirement. There exist several models of thin vortex rings (e.g., a homogeneous ring, a hollow vortex ring etc.), which differ from each other by the distribution of vorticity (s). All of them are characterized by an approximately constant velocity of self-propulsion along the axis Oz without noticeable changes in form [36]. Without dwelling on each of them, we write down the general formula for the velocity of self-propulsion of thin toroidal vortex rings, which was obtained by P.G.Saffman [50, 51]:
a

V

(0) z

= 4R

8R 1 ln - +(a) , a 2

1 (a) = 2
0

2 (s) ds, s

(1.7)

where a is the cross-section radius of the ring (a R), R is the radius of the ring, (s) = s = 2 0 (s )s ds is the velocity circulation around the central part of the vortex ring of radius s, and (s) is the vorticity distribution of the cross-section. In the case of a hollow vortex ring, the vorticity inside the ring is absent, therefore, = 0 and (1.7) coincides with the Hicks formula [30, 31]. In the case of uniform vorticity distribution inside the ring, = 1 and (1.7) 4 coincides with the formula for the velocity obtained by Lord Kelvin [60]. Remark. A more general expression for the velocity of self-propulsion of the ring in a compressible fluid with an arbitrary vorticity distribution over the cross-section was derived in [20]. When several rings interact in the process of motion, their radii and thicknesses become variable, however (by the Helmholtz theorem of circulation and incompressibility of a fluid), the following relation is satisfied for each ring
2 Ri a2 = Bi = const, i

(1.8)

where Bi are the constants characterizing the volumes of the rings. We shall assume that for the Ri ) and are far enough away from each other whole time of motion the vortices remain thin (ai not to affect the toroidicity of the form and the vorticity distribution inside the rings. Then, in the
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a linear approximation in R , the vortex rings will interact as infinitely thin vortex filaments (1.5), and by (1.6) and (1.7) their velocities can be represented as

1 Ri = - Ri Zi 1 Zi = Ri Ri

j G(Ri ,Zi ,Rj ,Zj ),
j =i

j G(Ri ,Zi ,Rj ,Zj ) +
j =i

i 4Ri

ln

8Ri 1 - - (ai ) , ai 2

(1.9)

where G is given by (1.3). A more detailed derivation of the equations of motion for the rings can be found, for example, in [2]. The most complete results have been obtained for the dynamics of homogeneous vortex rings. Therefore, throughout this paper we shall confine ourselves to this case by setting 1 = . 4 Remark 1. It is straightforward to obtain analogous results for other models of vortex rings with the corresponding (a). The equations of motion (1.9) can be represented in Hamiltonian form Zi = {Zi ,H } = 1 H , i Ri Ri ij , i Ri
3/2

1 H Ri = {Ri ,H } = - , i Ri Zi

where the Poisson bracket of coordinates Zi ,Ri satisfies the relations {Zi ,Rj } = {Zi ,Zj } = {Ri ,Rj } = 0,

and the Hamiltonian can be written as 1 H= 2
N

Ri
i=1

2 i

8 Ri ln Bi

-

7 4

+
i=j

i j G(Ri ,Zi ; Rj ,Zj ).

(1.10)

The equations of motion for vortex rings in Hamiltonian form were first presented in [62]. 1.3. Integrals of Motion and Reduction in Symmetry In the general case, Eqs. (1.9) possess, in addition to the Hamiltonian, another first integral of motion
N

P=
i=1

2 i Ri ,

(1.11)

which is, up to a constant factor, identical with the second integral in (1.4). The existence of this integral is related to the invariance of the system relative to parallel translations along the axis Oz , while the Hamiltonian (1.10) depends only on the differences (Zi - Zj ). This symmetry can also be used to reduce the system by one degree of freedom. The reduction can be performed by various means. Here we describe the method similar to the Jacobi method (see, e.g., [53]) in the N -body problem of celestial mechanics. We confine ourselves to the case N i = 0. To do this, we determine the center of vorticity of the entire system of i=1 rings on the axis Oz
N N

0 =
i=1

i Zi
i=1

i ,
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(1.12)
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and instead of the coordinates Zi we introduce the coordinates of the vorticity centers of the groups of vortices
i i

i = Zi+1 -
j =1

j Zj
j =1

j ,

i = 1 ...N - 1,

(1.13)

which determine (although not immediately) the positions of vortex rings on the axis Oz relative to the center of vorticity. We supplement the variables i with canonically conjugate variables i (i. e. {i ,j } = ij ), which are given by
i i i i+1

i =

R

2 i+1

-
j =1

j R

2 j j =1



j



i+1 j =1



j j =1

j .

(1.14)

For brevity we denote Z = (Z1 ,... ,ZN ), R = (R1 ,... ,RN ), = (1 ,... ,N -1 ), = (1 ,... ,N -1 ). It can be shown by direct calculation that the transformation (Z , R) (,0 , ,P ) is reversible and the following proposition holds Prop osition 1. The equations of motion of the reduced system have the canonical form H , i = i i = - H , i i = 1 ...N - 1, (1.15)

where the Hamiltonian H = H (, ,P ) depends parametrical ly on the value of the first integral (1.11). For a solution of the reduced system (t), (t), the dynamics of a ful l system is restored by means of an additional quadrature H . 0 = P (1.16)

We note that, unlike the celestial mechanics problem, in this case the center of vorticity moves nonuniformly: its velocity depends on the specific tra jectory and is given by (1.16). A similar transformation is used in the problem of N point vortices on a plane and a sphere for symmetry reduction [13]. It follows from (1.13) and (1.14) that the reduced system (1.15) actually describes the motion in a coordinate system related to the center of vorticity. This suggests an interesting geometric interpretation of absolute motion for periodic solutions to the reduced system (1.15). This interpretation is based on the concept of choreography , which was introduced by C. Moore, A. Chenciner and C. Sim´ [19, 44, 57] for the N -body problem of celestial mechanics (for the case o of point vortices on a plane, choreographies are pointed out in [12]). Prop osition 2. Let ((t), (t)) be a periodic solution to the reduced system (1.15) with period T . Then, in a moving coordinate system related to the vorticity center (1.12), the vortex rings move along closed (general ly different) trajectories with period T . Thus, periodic solutions to the reduced system correspond to relative choreographies of vortex rings [12, 57]. Proof. To prove the proposition, it suffices to show that the coordinates of vortex rings in the moving coordinate system Zi = Zi - 0 and Ri are periodic functions of time with period T . To do this, we represent Zi as
N N

Zi =
j =1

j (Zi - Zj )
j =1

j ,

(1.17)

where all relative distances Zi - Zj are expressed in terms of Jacobi's variables . Thus, the coordinates Zi (i = 1 ...N ) depend only on the variables of the reduced system and are periodic
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functions of time with period T . By (1.11) and (1.14), at a fixed value of the integral P = P0 = const, all coordinates Ri are expressed in terms of the coordinates of the reduced system and the quantity P0 and, consequently, also are periodic functions of time with period T . Thus, on the plane of variables (Z ,R) all vortex rings move along closed (generally different) tra jectories, with one and the same period T . 2. TWO VORTEX RINGS 2.1. Leapfrogging and the Problem of Classifying Motions We consider the problem of motion of two vortex rings. This problem was first investigated by Helmholtz in [29], where he pointed to the existence of the phenomenon of leapfrogging motion of vortex rings. Later, this problem was studied in more detail by Hicks [32], Gr¨ [27], Dyson [21] obli and others ( further references can be found in [43]. The criterion for the onset of leapfrogging in the case of two vortex rings of the same sign and volume was pointed out by Hicks; this criterion can be formulated as follows: Suppose that at an initial instant the vortices lie in the same plane and the ratio of their radii is equal to = R1 . Then there is a critical value 0 , such that for > 0 leapfrogging R2 arises, otherwise the vortices fly apart infinitely; this critical value 0 corresponds to the motion where the vortices at infinity have equal velocities. For vorticities of different signs the technique, due to Hicks, for identifying the leapfrogging criterion becomes inapplicable; below we give a detailed analysis of conditions for the onset of leapfrogging depending on the values of the first integrals and point out the corresponding criteria. In addition, for vorticities of the same sign the Hicks criterion is also violated at insufficiently large values of P . We provide a sufficiently detailed classification for motions of two vortex rings depending on the values of the first integrals of the system. The assertion formulated by Hicks was later used to determine the condition for the existence of leapfrogging in the case of rings of unequal volume [28, 43], where the motions of a system of two vortex rings were partially classified depending on the initial conditions of the reduced system. Let us make a change of variables
2 p1 = R1 ,

p2 = R

2 2

and choose the units of measurement and the numeration of the rings in such a manner that 1 = 1, B1 = 1, 2 = (-1, 1], B2 = (0, ). Remark 2. The case = -1 corresponds to is not applicable. i = 0, therefore, the reduction described above

As a result, we represent the Hamiltonian of the system of two vortex rings as H= 11 p 2 1
/2

ln 8p1

3/4

-

7 4

+

1 21 p2 2

/2

ln

8p2
1/2

3/4

-

7 4

+

1 1/4 1/4 p p C (k ), 12 (2.1)
2

C (k ) =

2 2 - k K (k ) - E (k ), k k

k=

4p1 p2 (p1
1/2

1/2 1/2

+ p2 )2 +(Z1 - Z2 )

.

In this case the integral P becomes P = p1 + p2 . (2.2)

Using (1.13) and (1.14), we define the new variables and , such that p1 ,p2 ,Z1 ,Z2 are expressed in terms of them as follows: P P - , p2 = + , Z2 - Z1 = . (2.3) p1 = 1+ 1+
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Here, is determined on the entire real axis and has the meaning of the distance between the rings that is measured along the common axis. At a fixed value of the integral P the quantity varies within the following limits: P P , at > 0, 1+ 1+ -P , at < 0 and P > 0, 1+ P , at < 0 and P < 0. 1+ - Note that is proportional to the difference of the squares of the rings' radii. Substituting these variables into (2.1), we obtain a Hamiltonian system with one degree of freedom, whose equations of motion can be written in canonical form H (, , P ) , = =- H (, , P ) , (2.4)

and the Hamiltonian depends parametrically on the quantities , and the value of the integral P (i.e. (2.4) is a three-parameter family of one-degree-of-freedom systems). Fig. 2 shows typical phase portraits of a reduced system for different and . The points denote the equilibrium positions of the reduced system. The empty circles (at the edges of the phase portraits) denote infinitely far equilibrium positions, which we shall describe below, and the bold lines are unstable invariant manifolds (separatrices). Owing to the integrability of the system of two vortex rings, we use the approach developed in [10] for classification of the types of motion according to the values of the first integrals H, P . This approach is based on an analysis of a bifurcation diagram of the system used for the construction of a bifurcation complex, which is one of the simplest topological invariants of the system [11]. In this case, different leaves of the bifurcation complex correspond to different types of vortex motion. We recall that in a bifurcation complex, one assigns a number to each region on the diagram. The number corresponds to the number of isolated invariant manifolds in The leaves are glued together along the corresponding bifurcation curves. Roughly sp bifurcation complex is analogous to the Riemann surface of the analytic function of variable. The use of this complex allows a natural classification of possible motions of rings. bifurcation the region. eaking, the a complex two vortex

Since the motion is not compact in this case, it is more convenient to investigate a reduced system with one degree of freedom that depends on the value of the integral P as a parameter. To illustrate the method, in what follows we shall confine our attention to the case where the vortex rings have equal volumes = 1. For the more general case, where is not equal to 1, the picture does not change qualitatively. To construct a bifurcation complex, it is necessary to consider: · the fixed points of the reduced system , which are the critical points of the Hamiltonian H (, ) and are determined from the system of equations H = 0, H = 0, (2.5)

· singularities of the system , which correspond to the points where either the radius of one of the vortices tends to zero or the vortices merge and H (, ) , · behaviour of the system at infinity .
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Fig. 2. A typical view of phase portraits of the reduced system of two vortex rings at different values of parameters.

We note that in [10] only the compact case was considered; owing to non-compactness of the integral manifolds of the system under investigation, it is necessary to consider the singularities and the behaviour of the tra jectories at infinity and to generalize correspondingly the determination of the bifurcation complex. A bifurcation analysis in the noncompact case is presented, for example, in W. M. Alexeev [1], where bifurcation diagrams are constructed for the problem of two fixed centers. We shall use an approach based on the concept of an infinitely far equilibrium position as a point with coordinates (±, ), for which the standard conditions for the first derivatives of the Hamiltonian to be equal to zero are satisfied: H =
= , ±

H

= 0.
= , ±

Physically these "equilibrium positions" correspond to such motions in which the vortex rings move with constant velocities and are so far away from each other that their interaction may be ignored. For a bifurcation analysis using this approach, it is necessary to expand the concept of a connected component of the level of the first integrals (leaf of a bifurcation complex) as follows: two trajectories belong to the same connected component of the level of the first integrals (lie on the same leaf of a bifurcation complex) if their arbitrarily smal l neighborhoods intersect during the motion along the phase flow.
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Such a generalization is necessary to ensure that the tra jectories asymptotic to unstable, infinitely far equilibrium positions (separatrices) belong to the same leaf of the bifurcation complex as the equilibrium position itself (although formally they do not intersect). We construct bifurcation diagrams on the plane of the first integrals (P, H ) and the corresponding bifurcation complexes. It can be shown that there are three cases which differ qualitatively from each other: 1) > 0, 2) < 0,P > 0, 3) < 0,P 0. 2.2. The Case > 0
The Bifurcation Complex

According to (2.2), for > 0 the value of the momentum integral varies within the interval P (0, +). As will be shown below, possible values of the Hamiltonian belong to the interval H (Hmin (P ), +), where Hmin (P ) is the minimal value of the Hamiltonian for a fixed values of P . Fixed p oints of the reduced system. Fixed points of the Hamiltonian vector field correspond to motions of the vortex rings where the dimensions and relative locations of the rings remain constant. Note that the rings move along a common axis as a unit with constant velocity given by (1.16). As was pointed out above, to find the fixed points, it is necessary to explore in detail the critical points of the Hamiltonian H (, ) at different values of the quantities and P . A typical example of the dependence H (, ) for given and P , which corresponds to the phase portrait in Fig. 2a, is given in Fig. 3.

Fig. 3. The surface H (, ) at = 0.8, P = 2.

Using (2.1) and (2.3), it can be shown that the Hamiltonian (2.1) is an even function in (i.e. H (, ) = H (, - )), which monotonically decreases on the interval (0, ) for each fixed . Thus, all fixed points of the reduced system lie on the axis = 0, and their coordinates on this axis are determined from the equation H = 0.
=0

(2.6)

Analyzing the asymptotics of the function f ( ) = H (, 0) and the sign of its second derivatives f ( ), it is straightforward to show that for > 0 Eq. (2.6) always has two solutions a > 0
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and b < 0, which correspond to the fixed points a and b in Fig. 2, both points being unstable. When the integral P changes its value on the plane of the first integrals (P, H ), two families of the fixed points a and b correspond to the bifurcation curves a : Ha (P ) and b : Hb (P ) (see Fig. 5). In the problem at hand singularities arise when the radius of one of the vortices tends to P P zero. These cases correspond to the straight lines = min = - 1+ (p2 = 0) and = max = 1+ (p1 = 0), which bound the phase portrait above and below (see Fig. 2). On the bifurcation diagram, these singularities correspond to the curves given by c : Hc (P ) = H (, max ) = d : Hd (P ) = H (, min 1 2
3/2

P

1/2

ln 8 ln 8P

P
3/4

3/4

- 7 . - 4

7 , 4

1 P )= 2

(2.7)

1/2

The case where two rings merge to form a single ring is another singularity of the system. On the phase portrait, the single ring corresponds to the point = 0, = 0 (see Fig. 2, 6), at which the energy H (, ) +. Thus, the region of possible motions on the plane of the first integrals is not bounded above by the value of the energy H . Sp ecial features of a reduced system at infinity ( ±). For a correct description of noncompact bifurcations, which correspond to reconstructions of the phase portrait at infinity, we shall use the approach described above. Remark 3. In some cases the behaviour of tra jectories at infinity can be described using the compactification of the phase space [15], but in this case this leads to a nonphysical identification of the tra jectories corresponding to entirely different motions. We first find the coordinates of infinitely far fixed points (equilibrium positions) as extrema of the functions 1 2 1/2 1 1/2 7 7 3/4 3/4 p1 + p2 . (2.8) ln 8p1 - ln 8p2 - H ( ) = lim H (, ) = 2 4 2 4 Using the equations of motion (1.6), it is straightforward to show that H = V2 - V1 = 0, (2.9)

where V2 and V1 are the velocities of vortex rings at infinity which are directed along a common axis of symmetry. Remark 4. The above-mentioned leapfrogging criterion due to Hicks, which requires the velocities of the rings to be equal at infinity, is in effect a condition for the limit tra jectories of the system to be critical. Certain infinitely far critical points also define the boundary of the leapfrogging region at arbitrary intensities (i.e. = 1) and volumes of the rings ( = 1). It can be shown that for P which does not attain some critical value P crit the function H ( ) has only one extremum (minimum), while for P > P crit it has three extrema. The value P crit is determined from the system of equations H = 0, After some transformations for P P
crit crit

2 H = 0. 2 2 (1 + 3/2 ) 1 + , 3 (1 + )
3/2

(2.10)

we find 1+ exp 16 (2.11)

=

and the value of is found from the equation 2( - )(1 + ln = (1 + )

)

.
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Fig. 4. The graph of the function H ( ) for = 0.8.

A typical view of the function H ( ) for P < P crit and P > P crit is shown in Fig. 4. The point e in Fig. 4 corresponds to the stable infinitely far fixed point e in Fig. 2, and the energy He corresponding to it is a global minimum of the function H (, ) and determines the boundary of the region of possible motions on the plane (P, H ). The boundary is given by the curve e . The points f and g in Fig. 4 appear at P > P crit and correspond to the unstable and stable infinitely far fixed points, respectively. These points correspond to the bifurcation curves f , g . A general view of the bifurcation diagram for = 0.8 is shown in Fig. 5. The region of possible values of the integrals is shown in grey. The solid lines show bifurcation curves, which correspond to the stable fixed (including infinitely far) points of the reduced system, unstable equilibrium positions correspond to the dashed lines. In order to construct a bifurcation complex from the bifurcation diagram in Fig. 5, we assign to all regions (which are separated from each other by bifurcation curves) such a number of various leaves as is the number of connected components (taking into account the definition given on p. 42) which the lines of the level of the Hamiltonian H (, , P ) have at corresponding values of the first integrals P , H . The bifurcation curves shown with solid lines are free edges of these leaves. The leaves are glued together along the dashed bifurcation curves. The points of intersection of these curves P and P correspond to the merging of asymptotic tra jectories to various unstable fixed points. It is more convenient to imagine this complex in three-dimensional space R3 = {P, H, Q}, where the coordinate Q is a quantity that allows one to distinguish different leaves (for example, some initial coordinate for the tra jectories of the reduced system); schematic profiles of sections of the resulting ob ject formed by the intersection of the ob ject with the plane P = const are shown in Fig. 6. In the case of interest, the bifurcation complex consists of eight leaves. We denote them by sequences of the letters corresponding to the bifucation curves (or to their segments) by which this leaf is bounded. The order of letters corresponds to the enumeration of bifurcation curves during the by-pass of the boundary of the leaf counterclockwise (see Fig. 7). We write down a full list of leaves of the bifurcation complex: Oba, abe1 , abe2 , ca, db, fb, bf g1 , bf g2 . In the name of the first leaf of the bifurcation complex, O means the vertical straight line P = 0. Fig. 7 schematically shows the bifurcation leaves, which are glued together as follows. The leaves abe1 , abe2 , ca, db are glued to the leaf Oba along the curve a and a part of the curve b (P < P ); the leaves fb and bf g1 , bf g2 are glued to the leaf Oba along the curve f and, in addition, they are glued to each other and to the leaf db along a part of the curves b (P > P ). Only one type of motion of vortex rings is possible on each of the above-listed leaves, except the first one, and the leaf Oba can be divided into three regions with different types of possible motions: ab, fba, Oabf .
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Fig. 5. A general view of the bifurcation diagram (on the left) and its enlarged fragments (on the right) for a system of two vortex rings at = 0.8. The solid lines show the boundaries of the leaves of a bifurcation complex, and the dashed lines show the bifurcation curves along which the leaves are glued together. The arrows below in the left figure indicate the values of the integral P , for which the next figure (Fig. 6) shows the schematic cross-sections of the bifurcation complex and the corresponding phase portraits of the reduced system.

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Fig. 6. Various types of phase portraits and the corresponding profiles of a bifurcation complex for = 0.8 and various values of P . The small Latin letters denote the fixed points of a reduced system (the position of "infinitely far fixed points" is marked with empty circles), the bold lines show the tra jectories asymptotic to the unstable fixed points, and the diagonally shaded areas are the leapfrogging regions.

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Fig. 7. The diagram (not to scale) of the leaves of a bifurcation complex for the case > 0.

Classification of Motions

Altogether, there exist three basic qualitatively different types of tra jectories. We shall briefly describe each of them. 1. Repulsion of rings. At t = - the rings are infinitely far away from each other and, as time increases, move towards each other (in the system of the center of vorticity). After approaching each other up to a minimum distance, the rings repulse each other and fly apart again. A monotone decrease of the radius of one ring and increase of the radius of the other ring occur during the whole motion. The larger P , the less the radii of the rings change. This type of motion corresponds to the leaves abe1 , abe2 , bf g1 and bf g2 . 2. Single passage of one ring through another. In this case, after approaching each other the rings do not repulse but pass through each other alternately and then fly apart to infinity. This type of motion corresponds to the leaves ca, db, fb, and the regions ab and fba on the leaf Oba. We note that till the interaction (t -) the dimensions of each of the rings are the same as after the interaction (t +); nevertheless, during the interaction they can change depending on the leaf and the initial conditions. Thus, for example, motions are possible for which the larger ring first becomes a smaller one, passes through the second ring and then becomes a larger one again (the region ab and a part of the region fba on the leaf Oba). 3. Leapfrogging. In the problem of vortex rings, nonuniform motion in which the rings pass through each other alternately is called leapfrogging. This type of motion is of particular interest from the viewpoint of dynamics. Therefore, we discuss it in more detail.
A Condition for the Existence of Leapfrogging

In terms of the tra jectories of the reduced system, leapfrogging corresponds to closed tra jectories surrounding the point = 0, = 0. The diagonally shaded areas in Fig. 6 denote the corresponding regions of phase portraits, and the bold lines denote segments of the profiles of a bifurcation complex. On the bifurcation diagram, the region of the existence of leapfrogging lies on the leaf Oba and is shaded as well. As can be seen from the diagram, the condition for the existence of leapfrogging depends on the value of P . Below we state a full condition for leapfrogging, which generalizes the results of Hicks.
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Theorem 1. At > 0 the motion of two vortex rings is leapfrogging if two conditions are satisfied simultaneously: 1. the initial conditions correspond to the leaf Oba of the bifurcation complex, 2. H > max(Ha (P ),Hb (P ),Hf (P )). We note that Hicks' requirement that the velocities at infinity are equal is applicable only in the region P > P , where P is calculated from the equation Hb (P ) = Hf (P ). Thus, Theorem 1 supplements Hicks' requirement in the region P < P . Remark 5. We note that the model of thin vortex rings implies that the radii of the rings are 1, p2 4/3 . In the case considerably larger than their cross-section, i.e. in our notation p1 > 0 this leads to a situation where the physically meaningful motions of vortex rings lie in the region of large momenta P 1+ 4/3 . When the intensities of the rings are equal or nearly equal (i.e. when is close to unity) the leapfrogging region is bounded by the curve Hf (P ) at large P and, consequently, the Hicks criterion is satisfied. However, as decreases, the critical value P crit at which the solution Hf appears, tends very rapidly to infinity, and the curve Hb (P ) gets into the region of large P (see Fig. 5). Thus, for sufficiently thin rings but of considerably different vorticity, Theorem 1 generalizes the Hicks criterion for the onset of leapfrogging.

2.3. The Case < 0, P > 0
The Bifurcation Complex

By analogy with the previous case the structure of the phase portrait and the classification of motions of the system is determined by fixed points, singularities and the behavior at infinity. We consider them successively. Fixed p oints. As in the case > 0, all fixed points lie on the axis = 0, and their coordinates are determined from (2.6). In the case at hand, Eq. (2.6) can have either two solutions or none depending on the value of the integral P . In case there exist two solutions, one of them corresponds to an unstable fixed point and the other to a stable one. Thus, for < 0, P > 0 there exist orbitally stable motions in which both rings move as a unit along a common axis with constant velocity. The critical values of P at which a pair of fixed points arises (or vanishes), are found from a solution to the system of equations H0 (, P ) = 0, where H0 (, P ) = H (, , P )|
=0

2 H0 (, P ) = 0, 2

(2.13)

.

crit crit A numerical analysis shows that the system (2.13) has at least two solutions P1 and P2 . crit < P < P crit Eq. (2.6) has no solutions, while for P < P crit or P > P crit it has two solutions. For P1 2 1 2 On the bifurcation diagram (Fig. 8) and on the phase portraits (Fig. 10), the corresponding crit families of fixed points are denoted by the letters a and b for P < P1 and by the letters e and f crit . for P > P2

Remark 6. For a complete proof of the absence of other solutions P crit (at = 1), it is necessary to numerically construct the intersection of two-dimensional surfaces (2.13) in three-dimensional space R3 = {(, P, )}.
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Fig. 8. A general view of the bifurcation diagram (on the left) and its enlarged fragments (on the right) for a system of two vortex rings at = -0.5. The solid lines are the boundaries of leaves of the bifurcation complex, while the dashed lines are the bifurcation curves along which the leaves are glued together. The arrows below in the left figure indicate the values of the integral P , for which Fig. 6 schematically shows the sections of the bifurcation complex and the corresponding phase portraits of the reduced system.

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Fig. 9. The diagram (not to scale) of the leaves of a bifurcation complex for < 0, P > 0.

Singularities of the system. In the case < 0, P > 0 there exist only two singularities. The first one is the case = 0, = 0 where two rings merge to form a single ring, with H (, ) - (whereas in the case > 0 considered above the energy goes to +). At < 0 and a fixed P > 0 the radii of the rings can be arbitrarily large, which corresponds to an infinitely large value of the energy (H (, ) +). Therefore, in the case at hand the entire half-plane P > 0 is the region of possible motions. The second singularity is the case where the radius of the second ring goes to zero. On the phase portraits, this case corresponds to the boundary of the region of possible motions in the form of a straight line = max = -P , and on the bifurcation diagram, this case corresponds to the curve 1+ d : Hd (P ) = H (, max ) = 1 P 2
1/2

ln 8P

3/4

-

7 , 4

(2.14)

which is the boundary of one of the leaves of the bifurcation complex. Sp ecial features at infinity. As in the case > 0, analysis of the behavior at infinity involves the investigation of the function H ( ) = lim H (, ). As the numerical analysis shows, in the


case considered here the function H ( ) possesses for all values of P at least one minimum that corresponds to an unstable infinitely far equilibrium position. On the phase portraits (Fig. 10), the corresponding equilibrium positions are denoted by the letter c and the bifurcation curve corresponding to them (Fig. 8) is denoted by c . Since the equilibrium position is unstable, the same bifurcation curve corresponds to two separatrices which are denoted on the phase portraits by bold lines. Fig. 8 shows a general view of the bifurcation diagram. To construct a bifurcation complex, we analyze the phase portraits for various fixed values of P = const and construct all possible crosssections of the complex (Fig. 10). On the basis of the information obtained we conclude that in the case at hand the complex already consists of five leaves Oace-, Oce+, Oba, dac and ef . The signs + and - in the name determine which part of the plane of the first integrals belongs to the leaf (above or below the given boundary). The leaves are schematically shown in Fig. 9 and are glued together as follows. The leaves Oba and ef are glued to the leaf Oace- along the curves
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Fig. 10. Various types of phase portraits and the corresponding profiles of the bifurcation complex for = -0.5 and various values of P . The small Latin letters denote the fixed points of the reduced system (the position of the "infinitely far fixed points" is marked by empty circles), the bold lines denote the tra jectories asymptotic to unstable fixed points, and the diagonally shaded areas are the leapfrogging regions.

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a and e ; (P < P < is glued to the curve

the leaf dac is glued to the leaf P ) and is glued to the leaf ef the leaf Oba along a part of the c (P < P < P ) and a part of

Oace- along parts of the curves a (P < P ) and c along a part of the curve c (P > P ); the leaf Oce+ curve c (P < P ), to the leaf Oace- along a part of the curve e (P > P ).

Classification of Motions

Let us briefly dwell on the classification of possible types of motion. In the case we consider here there exist three basic types of motion. 1. Single passage of one ring through another. This type of motion is analogous to the observed one in the case > 0 and is observed on the leaves Oce+, dac, in the region Oca on the leaf Oba and in the region ec on the leaf ef . 2. Leapfrogging. The bifurcation diagram (Fig. 8) and the phase portraits (Fig. 10) show the regions corresponding to leapfrogging motions (diagonally shaded areas). In this case the criterion for the onset of leapfrogging is described by the following theorem. Theorem 2. The motion of two vortex rings whose intensities have different signs at P > 0 is leapfrogging motion in the case where the initial conditions correspond to the leaf Oace- of the bifurcation complex. 3. Oscil lations with propulsion. This is another type of periodic constrained relative motion of two vortex rings. On the phase portraits these motions lie around the stable equilibrium positions b and f . As was stated above, these equilibrium positions correspond to the ring configurations at which they are located in each other and move as a unit along the axis with constant velocity. Solutions lying around the equilibrium positions are motions in which the larger ring moves along the axis with some nonconstant velocity, and the smaller ring oscillates inside the larger one, alternately overtaking and lagging behind it. In this case the larger ring can be said to capture the smaller one. On the bifurcation diagram (Fig. 8), the regions corresponding to these solutions are represented by the vertically shaded areas. As can be seen from Fig. 8, on the plane of the first integrals there exist two regions in which the above-mentioned motion occurs. The first region lies on the leaf Oba in the region Obac, and the second one lies on the leaf ef in the region cef . The difference between these solutions is that in the region Obac the ring with negative intensity is the larger one, while in the region cef it is the ring with positive intensity that is larger. Moreover, in the latter case the dimensions of the rings remain almost constant all the time, while in the former case they change considerably during motion. We also note that in the former case the closest approach of the rings occurs with the maximal velocity. As a result, during fast motion the impression arises that the rings pass through each other, while in fact this is not the case. We call such motions pseudo-leapfrogging . The above solutions are illustrated in Fig. 11 showing a typical view of the tra jectories of vortex rings in a fixed coordinate system and in a coordinate system related to the center of vorticity. We note that the tra jectories of rings in absolute space depend not only on the form of the curves along which the rings move in a system of the center of vorticity but also on the velocity of the center of velocity, which is given by (1.16). In particular, if one selects the corresponding initial conditions and intensities of the rings, the displacement of the center of vorticity within one period of the reduced system may become equal to zero. Thus, the vortex rings will execute leapfrogging motion while staying in the same place. In the terminology of [19, 44, 57] such motions can be called absolute choreographies. Remark 7. We note that in the case < 0 the condition for the rings to be thin does not lead to explicit imposition of restrictions on the value of the integral P , since it can be close to zero even at large radii of the rings. Thus, the region of physically meaningful motions lies on the entire half-plane P > 0.
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Fig. 11. Tra jectories of vortex rings on the plane (Z, R) in a system of the center of vorticity (on the left) and in a moving space (on the right) for = -0.5 and for various values of P : a) P = 0.001 (the region Obac in Fig. 8), capture of a negative vortex ring by a positive one, b) P = 5 (the region cef in Fig. 8), capture of a positive vortex ring by a negative one, c) P = 0.02 and E < Ea (the region Oace- in Fig. 8), leapfrogging, d) P = 0.02 and Ea < E < Ec (the region Oace- in Fig. 8), leapfrogging.

2.4. The Case < 0, P
The Bifurcation Complex

0

Fixed p oints of the reduced system. As the analysis of solutions to (2.6) shows, two options are possible in the case considered depending on the value of . 1. For values of exceeding some critical value = -0.8775 (which generally depends on ), for all values of P there exists only one stable fixed point, which we shall denote by the letter a, and the corresponding bifurcation curve will be denoted by a (see Fig. 12, 15).

Fig. 12. The bifurcation diagram for a system of two vortex rings for = -0.5. The solid lines denote the boundaries of the leaves of the bifurcation complex, and the dashed lines denote the bifurcation curves along which the leaves are glued together. The region of possible motions is shown in grey.

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Fig. 13. The bifurcation diagram (on the left) and its enlarged fragment (on the right) for a system of two vortex rings at = -0.95. The solid lines denote the boundaries of the leaves of the bifurcation complex, and the dashed lines denote the bifurcation curves along which the leaves are glued together. The region of possible motions is shown in grey.

Fig. 14. The diagram (not to scale) of the leaves of the bifurcation complex for < , P

0.

crit 2. For < , depending on the value of P , there can exist either one stable fixed point (P > P1 crit crit crit and P < P2 ) or two stable points and one unstable fixed point (P2 < P < P1 ). On the phase portraits and the bifurcation diagram, we shall denote the stable points by the letters a and d and the unstable point by e (see Fig. 13, 16). Singularities of the system. In this case the conditions < 0 and P < 0 imply that the radius of the ring with negative intensity is always larger than that of the ring with positive

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Fig. 15. The phase portrait and the profile of the bifurcation complex for = -0.5, P = -1. The small Latin letters denote the fixed points of the reduced system, the position of infinitely far fixed points is marked with empty circles, the bold lines denote the tra jectories asymptotic to unstable fixed points, the vertically shaded areas are the regions in which one ring is captured by the other.

intensity; thus, there remains only one singularity, namely the case where the radius of the first ring goes to zero. On the phase portraits for this case the boundary of the region of possible motions P is = max = 1+ , and on the bifurcation diagram for this case corresponds to the curve c : Hc (P ) = 1 2
2

P

ln 8

P

3/4

-

7 , 4

which is the boundary of one of the leaves of the bifurcation complex. Behavior at infinity. As for < 0 and P > 0, in the case at hand there exists only one unstable infinitely far equilibrium position (which we shall denote by the letter b) for all values of P . A general view of the bifurcation diagram is shown in Fig. 12 for > and in Fig. 13 for < . The corresponding phase portraits and cross-sections of the bifurcation complex are given in Fig. 15 and 16. As can be seen from Fig. 12 and 15, in the case > there exists only one type of the phase portrait of the reduced system, and the bifurcation complex consists of three leaves: aOb, bO+ and bOc. In the case < there exist four possible types of phase portraits (see Fig. 16), and the complex consists already of four leaves aObe, bO+, deb and bOc (see Fig. 13).
Possible Types of Motions

As was stated above, in the case of interest one vortex ring is always larger than the other, consequently, for < 0 and P < 0 there do not exist any leapfrogging motions . In this case all possible motions are restricted to two types: 1. Single passage of one ring through the other (the leaves bO+ and bOc). 2. Capture of the first ring by the second one (the leaves aOb at > and the leaves aObe and deb at < ). 3. THREE VORTEX RINGS We now consider the problem of motion of three coaxial vortex rings. By Proposition 1 (p. 39), the equations of motion (1.9) for three vortex rings can be reduced to a two-degree-of-freedom system by changing 1 = Z2 - Z1 , 2 = Z3 - 1 Z1 +2 Z2 , 1 +2
2 2 1 = (R2 - R1 )

2 =

1 2 1 R1 2 R3 - 1

1 2

, +2 +2 R +2

2 2

3 (1 +2 ) . 1 +2 +3
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Fig. 16. Various types of phase portraits and the corresponding profiles of the bifurcation complex for = -0.95 and various values of P . The small Latin letters denote the fixed points of the reduced system, the position of infinitely far fixed points is marked with empty circles, the bold lines denote the tra jectories asymptotic to unstable fixed points, the vertically shaded areas are the regions in which oscillations with propulsion occur.

Using the standard approach (see, e.g., [14]) for a reduced system, one can construct the Poincar´ e cross-section at the level of the Hamiltonian H (1 ,2 ,1 ,2 ,P ) = const. As a secant we choose the plane 1 = 0. The section of the level of the Hamiltonian formed by the intersection of this level with the plane defines in the three-dimensional space R3 = {(2 ,2 ,1 )} a rather complicated (noncompact) two-dimensional surface Mh,P0 = (2 ,2 ,1 ) H (1 ,2 ,1 ,2 ,P )
1 =0 P =P0

= h . It i s

obvious that this, rather arbitrary choice of the plane of section does not guarantee a global recurrence of tra jectories back onto the surface of section. In addition, a part of the tra jectories can go to infinity, which makes analysis of the mapping difficult. However, it is sufficient to search for periodic tra jectories. Fig. 17 shows a series of Poincar´ maps. For convenience, they are pro jected onto the plane e (2 ,2 ) in the case of three equal vortex rings. The bold points show the fixed points of the map, which correspond to periodic solutions of the reduced system. By Proposition 2 (p. 39), these periodic solutions correspond to motions of vortex rings where in the system of the center of
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vorticity they move along closed tra jectories, i.e., they are relative choreographies. Examples of such tra jectories corresponding to different periodic solutions of the reduced system are depicted in Fig. 18.

Fig. 17. The Poincar´ maps of the problem of three vortex rings for 1 = 2 = 3 = 1, B1 = B2 = B3 = 1, e P = 10 and different values of the Hamiltonian H . The regions in which motion is impossible are shown in grey. The small Latin letters denote the fixed points of the map.

At large values of energy the dynamics of the system of three vortex rings breaks down -- just as in the case of two rings -- into constrained and unconstrained motion. The boundary between them is the loop of separatrices asymptotic to the unstable fixed point b (Fig. 17). As can be seen from the figure, the constrained motion can be both regular and chaotic. As energy decreases, the invariant tori are destroyed and the chaotic layers grow both around the origin of coordinates and in the vicinity of the point b. After the destruction of the last invariant torus separating the chaotic layers, they merge (see Fig. 17b), and questions can be raised about the chaotic dissipation, the probability of one ring being captured by a pair of others etc. [58]. We note that practically on the entire interval of the change of energy in the vicinity of the origin of coordinates there exist stable fixed points denoted on the phase portraits by the letters a and a . Next, the corresponding tra jectories of vortex rings are shown in a system related to the center of vorticity (Fig. 18a) and in a fixed coordinate system (Fig. 18b). As can be seen from Fig. 18a and 18b, the solutions a represent a threefold leapfrogging motion of vortex rings. The assumptions of its existence were made already by J.C.Maxwell [40]. As the analysis of the Poincar´ map shows, e these periodic solutions corresponding to threefold leapfrogging motion exist and are stable in a wide range of values of the integrals P and H . This points, among other things, to the possibility of experimentally observing this phenomenon. Another property of the periodic solutions a and a is the connectedness of the corresponding choreography: in a system of the center of vorticity all three rings move along one and the same tra jectory lagging behind by 1 of a period (Fig. 18a). 3 We note that in addition to the solutions a, almost for periodic solutions c, which are obtained from each other by and their tra jectories in a system of the center of vorticity. the leapfrogging motion as well, but in this case the rings (see Fig. 18c). all values of energy there exist six stable combinatorial rearrangements of vortices These periodic tra jectories correspond to move already along different tra jectories

Thus, in the system of three equal vortex rings in a wide range of values of the integrals P and H there exist stable periodic motions in the system of the center of vorticity which correspond to a regular leapfrogging motion of the rings in a fixed system, i.e. motion in which each of the rings alternately passes through the two others.
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Fig. 18. Tra jectories of vortex rings in a system related to the center of vorticity (Fig. a, b, c) and in a fixed coordinate system (Fig. d, e). Fig. (a) and (d) correspond to the periodic solution a in Fig. 17b, Fig. (b) and (c) correspond to the solutions b and c, respectively, and Fig. (e) corresponds to the chaotic leapfrogging motion of vortex rings. Fig. (d) shows the positions of vortices at times ti .

As has already been stated above, for large values of energy there exist bounded chaotic tra jectories (see Fig. 17a). These tra jectories correspond to motions of vortex rings where they chaotically pass through each other without scattering to infinity. Such motions can be called chaotic leapfrogging . An example of the tra jectories of vortex rings, which correspond to chaotic leapfrogging, in a fixed coordinate system is shown in Fig. 18e. Thus, in the problem of three equal vortex rings, for large values of energy there exists a region of initial conditions that corresponds to chaotic leapfrogging. We note that at small values of energy the chaotic leapfrogging can be observed as well. However, in this case there are no invariant tori separating the constrained and unconstrained motion (see Fig. 17b), therefore, after a certain time period the vortex rings can dissipate and go to infinity. CONCLUSION In this paper, we have presented a complete analysis of the dynamics of two arbitrary vortex rings using the classical finite-dimensional model of vortex interaction of coaxial vortices and have partially studied the interaction of three vortex rings of equal intensity. In both cases, the most interesting motion to consider is the leapfrogging motion of vortices described already
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by Helmholtz. Although the equations of motion are approximate and are derived under some unobvious assumptions, the main results of this paper can undoubtedly be checked experimentally or refined in examining more exact models (e.g., the vorton model or modeling by directly counting the Navier ­ Stokes equations). It would be interesting to carry out this analysis, since the vortex ring structures are constantly observed when viscous fluid flows around axisymmetric bodies and are responsible for the onset of vortex resistance. Recently there have been a number of proposals to use vortex rings for practical purposes. ACKNOWLEDGMENTS The authors thank S. M. Ramodanov for reading the manuscript and for useful remarks. This research was supported by the RFBR grant 11-01-91056-NTsNI a, the Target Programme "Development of Scientific Potential of Higher Schools" (State contract 1.2953.2011, 2012­2014); the Federal Target Programme "Scientific and Scientific-Pedagogical Personnel of Innovative Russia" (State contract 14.B37.21.1935, 2009­2013); grant of leading scientific schools NSh-2519.2012.1. A. A. Kilin's research was supported by the grant of the President of the Russian Federation for the Support of Young Russian Scientists­Doctors of Science (MD-2324.2013.1). REFERENCES
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