Bifurcation Diagram of One Generalized Integrable Model of Vortex Dynamics
BBIFURCATION DIAGRAM OF ONE GENERALIZEDINTEGRABLE MODEL OF VORTEX DYNAMICSPavel E. Ryabov , , and Artemiy A. Shadrin Financial University under the Government of the Russian FederationLeningradsky prosp. 49, Moscow, 125993 Russia Institute of Machines Science, Russian Academy of SciencesMaly Kharitonyevsky per. 4, Moscow, 101990 Russia Udmurt State Universityul. Universitetskaya 1, Izhevsk, 426034 RussiaE-mail: [email protected], [email protected]
Abstract
The article is devoted to the results of a phase topology research on a generalized mathe-matical model, which covers such two problems as dynamics of two point vortices enclosed ina harmonic trap in a Bose-Einstein condensate and dynamics of two point vortices boundedby a circular region in an ideal fluid. New bifurcation diagrams are obtained and three-into-one and four-into-one tori bifurcations are observed for some values of the model’s physicalparameters. The presence of such bifurcations in the integrable model of vortex dynamicswith positive intensities indicates a complex transition and a connection between bifurca-tion diagrams in both limiting cases. In this paper, we analytically derive the equations,that define the parametric family of the generalized model’s bifurcation diagrams, includingbifurcation diagrams of the specified limiting cases. The dynamics of the general case’sbifurcation diagram is shown, using its implicit parametrization. The stable bifurcationdiagram, related to the problem of dynamics of two vortices bounded by a circular regionin an ideal fluid, is observed for particular values of parameters.
Keywords : completely integrable Hamiltonian system, bifurcation diagram, bifurcation ofLiouville tori, dynamics of point vortices, Bose-Einstein condensate
MSC2010 numbers : 76M23, 37J35, 37J05, 34A05
Received April 19, 2019 a r X i v : . [ n li n . S I] A p r INTRODUCTION
Integrable models of point vortices on a plane hold a central position in the analyticaldynamics of vortex structures. Studies of vortex dynamics in quantum physics have shownthat quantum vortices behave similarly to the thin vortex filaments studied in classical fluiddynamics. A special attention is paid to vortex structures in a Bose-Einstein condensate obtainedfor ultracold atomic gases [1]. In this paper we consider a generalized mathematical model whichcovers such two problems as dynamics of two point vortices enclosed in a harmonic trap in a Bose-Einstein condensate [2], [3], [4] and dynamics of two point vortices bounded by a circular regionin an ideal fluid [5], [6], [7]. This model leads to a completely Liouville integrable Hamiltoniansystem with two degrees of freedom, therefore topological methods used for such systems can beapplied. Topological methods have been successfully used to analyze the stability of absoluteand relative choreographies [6], [8], [9], [10], [11]. In integrable models, these motions are usuallyassociated with first integrals’ constant values for which these integrals, that are considered asfunctions of phase variables, are dependent in the sense of a linear dependence of differentials.The main role in the study of such dependence is played by a bifurcation diagram of momentummapping.The generalized mathematical model is described by a Hamiltonian system of differentialequations: Γ k ˙ x k = ∂H∂y k ( z , z ); Γ k ˙ y k = − ∂H∂x k ( z , z ) , k = 1 , , (1)where the Hamiltonian H has the form H = 12 (cid:104) Γ ln(1 − | z | ) + Γ ln(1 − | z | ) + Γ Γ ln (cid:18) [ | z − z | + (1 − | z | )(1 − | z | )] ε | z − z | c + ε ) (cid:19)(cid:105) . (2)Here, the Cartesian coordinates of k -th vortex ( k = 1 , ) with intensities Γ k are denoted by z k = x k + i y k . Physical parameter “ c ” expresses the extent of the vortices’ interaction, ε is aparameter of deformation. These parameters determine two limiting cases, namely, the modelof two point vortices enclosed in a harmonic trap in a Bose-Einstein condensate ( ε = 0 ) [2], [3],[4] and the model of two point vortices bounded by a circular region in an ideal fluid ( c = 0 , ε = 1 ) [5], [6], [7]. The phase space P is defined as a direct product of two open disks of radius2 with the exception of vortices’ collision points P = { ( z , z ) : | z | < , | z | < , z (cid:54) = z } . The Poisson structure on the phase space P is given in the standard form { z k , ¯ z j } = − k δ kj , (3)where δ kj is the Kronecker delta.The system (1) admits an additional first integral of motion, the angular momentum ofvorticity , F = Γ | z | + Γ | z | . (4)The function F together with the Hamiltonian H forms on P a complete involutive set ofintegrals of system (1). According to the Liouville-Arnold theorem, a regular level surface of thefirst integrals is a nonconnected union of two-dimensional tori filled with conditionally periodictrajectories. The momentum mapping F : P → R is defined by setting F ( x ) = ( F ( x ) , H ( x )) .Let C denote the set of all critical points of the momentum mapping, i.e., points at which rank d F ( x ) < . The set of critical values Σ = F ( C ∩ P ) is called the bifurcation diagram .In works [12] and [13] the bifurcation diagram was analytically investigated at c = 1 and ε = 0 . In [14] and [15] a reduction to a system with one degree of freedom was performed and abifurcation of three tori into one was found at c > and ε = 0 . This bifurcation was observedearlier by Kharlamov [16] while studying a phase topology of the Goryachev-Chaplygin-Sretenskyintegrable case in rigid body dynamics. In Fomenko, Bolsinov, and Matveev’s work [17] it wasfound as a singularity in a 2-atom form of a Liouville foliation’s singular layer. In Oshemkov andTuzhilin’s work [18], devoted to the splitting of saddle singularities, such a bifurcation was foundto be unstable and its perturbed foliations were presented. In the situation where the physicalparameter of vortices’ intensity ratio is experiencing integrable perturbation, said bifurcationcomes down to the bifurcation of two tori into one and vice versa [14]. In another limitingcase ( c = 0 , ε = 1 ), the bifurcation analysis of dynamics of two point vortices bounded by acircular domain in an ideal fluid is performed [6], [7]. In these limiting cases completely differentbifurcation diagrams were obtained. In the case of a positive vortex pair a new bifurcationdiagram is obtained for which the bifurcation of four tori into one is observed [19]. The presence3f three-into-one and four-into-one tori bifurcations in the integrable model of vortex dynamicswith positive intensities indicates a complex transition and connection between two bifurcationdiagrams in both limiting cases. A. V. Borisov suggested to study both these integrable modelsand to find out how the bifurcation diagrams of both limiting cases are related. In this paperwe analytically derive the equations that define a parametric family of bifurcation diagrams ofthe generalized model (1) containing bifurcation diagrams of the specified limiting cases. In thegeneral case reduction to a system with one degree of freedom allows us to apply level curves ofcorresponding Hamiltonian in order to observe different kinds of Liouville tori bifurcations. We define the polynomial expressions F and F from phase variables: F = x y − y x , (5) F = cx (Γ x + Γ x )( x + y − x ( x x −
1) + x y ][( x − x + x y ]++Γ [( x − x + x y ] (cid:110) εx ( x + y − + x ( x − x )( x + y )[ x ( x x −
1) + x y ] (cid:111) +Γ x ( x + y − (cid:110) x ( x − x )( x + y )( x ( x x −
1) + x y ) + ε [( x − x + x y ] (cid:111) , (6)and denote by N the closure of system’s set of solutions: F = 0 , F = 0 . (7)Then the theorem below is true. Theorem 1.
The set of critical points C of the moment mapping F coincides with the (7) system’s set of solutions. The set N is a two-dimensional invariant submanifold of the system (1) with the Hamiltonian (2) .Proof. To prove the first statement of the theorem it is necessary to find the phase space pointswhere the rank of the moment map is not maximal. With the help of direct computations onecan verify that the Jacobi matrix of the moment map has zero minors of the second order at thepoints z ∈ P , the coordinates of which satisfy the equations of system (7). Therefore C = N .4he fact that the relations (7) are invariate might be prooved by the following chain of correctequalities: ˙ F = { F , H } F =0 = σ F , ˙ F = { F , H } F =0 = − x y x + y σ σ F , where polynomial functions σ k from phase variables have the following explicit form: σ = 1( x − x ) x ( x + y − x ( x x −
1) + x y ][( x − x + x y ] ,σ = c ( x + y − (cid:110) Γ (cid:104) x (cid:16) x (cid:0) ( x − x − x (cid:1)(cid:17) + x x (cid:0) (2 x − x − x (cid:1) y + x y (cid:105) −− Γ x ( x + y )( − x x − x + 3 x ( x + y )) (cid:111) − Γ ( x + y ) (cid:110) x + x ( x + y )(1 + x + y )++ x x (2 ε ( x + y − − x + y )) (cid:111) − Γ x ( x + y − (cid:110) ε [( x − x + x y ]++( x + y )[2 x x (1 + x + y ) − x − x ( x + y )] (cid:111) . Let Γ = 1 , Γ = a, ε = 0 , c (cid:54) = 0 . (8)Substitution of (8) parameters’ values in (6) leads to the expression F = a ( x + y ) x −− ( x + y )[( x + y )( a − c ) + c ] x x + ( x + y )[( ac − x + y − − a ] x x +[( x + y )( x + y + a − c −
1) + b ] x x − ac ( x + y − x . (9)In the case of a positive vortex pair Γ = 1 , Γ = 1 , ε (cid:54) = 0 , c (cid:54) = 0 , (10)the equation (6) takes the form F = ( x + x ) F , where F = [ x ( x + y ) − x ] (cid:110) ( x + x )( x + y )[ x ( x + y ) − cx ] + x [( c − x + y ) x + cx ] (cid:111) + ε ( x + y − x ( x + y ) − x ][( x + y )( x − x x + x ) − x ] . F = 0 breaks down into two subsystems, taking into account (5): x + x = 0 , y + y = 0 and F = 0 , F = 0 , each of which is a two-dimensional invariant submanifold of the system (1) with the Hamiltonian(2) and specified as in (10) parameters’ values.It should be noted that the expression (6) for another limiting case c = 0 , ε = 1 , also disintegrates in the case of a positive vortex pair ( Γ = Γ = 1 ): F = ( x + x ) F , where F = − x ( x + y − ++ x ( x + y ) (cid:110) − x − x ( x + y ) − x ( x + y ) (3 x − x ) + x x [2 − y − x + 2( x + y ) ] (cid:111) . To determine the bifurcation diagram Σ as the image of the critical points’ set C of themomentum mapping F , it is convenient to change to polar coordinates x = r cos θ , y = r sin θ , x = r cos θ , y = r sin θ . (11)Substitution of (11) into the first equation of the system (7) results in an equation sin( θ − θ ) =0 , i.e. θ − θ = 0 or θ − θ = π . Next, we restrict vortex intensities to positive values, i.e. weassume that Γ > and Γ > . In contrast to the case of the intensities with opposite signs[12], in this particular situation the equation θ − θ = 0 is impossible no matter which valuesparameters c and ε take. In case of θ = θ + π , the second equation of the system (7) is reducedto W ( r , r ) = 0 , (12)6here W ( r , r ) = (1 − r )(1 − r ) (cid:110) [ c (1 + r r ) + ε ](Γ r − Γ r ) − ε (Γ r − Γ r ) (cid:111) −− r r ( r + r )(1 + r r )[Γ (1 − r ) − Γ (1 − r )] . Substituting (11) into the Hamiltonian (2) and the vorticity moment (4) in the case where θ = θ + π , leads to the following values of the first integrals: h = 12 { Γ ln(1 − r ) + Γ ln(1 − r ) } + Γ Γ ln (cid:104) (1 + r r ) ε ( r + r ) c + (cid:15) (cid:105) ,f = Γ r + Γ r . (13)This system (13) together with the equation (12) defines an implicit bifurcation diagram on theplane R ( f, h ) . In some special cases, it was possible to find an explicit parametrization of the bifurcationdiagram.Let ε = 0 . After reduction by a non-zero factor of r r , the equation (12) takes the form c (1 − r )(1 − r )(Γ r − Γ r ) − r r ( r + r )[Γ (1 − r ) − Γ (1 − r )] = 0 . (14)The algebraic curve (14) might be parametrized in the form of r = 1 √ (cid:115) c (Γ − Γ t )(1 + t ) + ( t + 1)[(Γ − Γ ) t ± √D ] t [Γ t ( c + t + t ) − Γ (1 + t + ct )] ,r = t · r , where D = [ c (1 − t )(Γ − Γ t ) + (Γ + Γ ) t ] − Γ t . The corresponding bifurcation diagram Σ is given as a curve on the plane R ( f, h ) : Σ : f = (Γ + Γ t ) r ,h = 12 (cid:110) Γ ln(1 − r ) + Γ ln(1 − t r ) − c Γ Γ ln[(1 + t ) r ] (cid:111) ,r = c (Γ − Γ t )(1 + t ) + ( t + 1)[(Γ − Γ ) t ± √D ]2 t [Γ t ( c + t + t ) − Γ (1 + t + ct )] . (15)7n the case of a vortex pair of positive intensities, i.e. Γ = Γ = 1 , after substitution (11)and θ = θ + π in (9) the critical set C also takes the simple form θ = θ + π ; r = r ; c (1 − r )(1 − r ) − r r ( r + r ) = 0 . (16)The last equation of the system (16) coincides with the equation in the paper [3] on P. 225301-2derived entirely from other considerations. Thus, our conclusion explains that the equation [3]on p. 225301-2 defines the radii of critical vortex motions.In this case the corresponding bifurcation diagram Σ consists of two curves γ and γ , where γ : h = ln (cid:16) − f (cid:17) − c f ) , < f < γ : h = 12 ln (cid:20) s ( s − c + s − (cid:21) − c ln (cid:20) cs c + s − (cid:21) ,f = cs − s − c + s − c + s − , s ∈ (cid:16) √ c )2+ √ c (cid:105) . (17)For the values of the physical parameter c > , the curve γ has a cusp at s = (cid:2) − c + √ c ( c − (cid:3) ( c − c − ,which coincides with the point of tangency, when c = 3 and s = √ c )2+ √ c .The parameterized curve (15) also have the cusps points that satisfy the equation a c + a c + a c + a = 0 , (18)where a = 4Γ Γ (1 − t ) t (Γ − Γ t ) ,a = 2(1 − t ) t (Γ − Γ t ) (cid:2) + Γ ) t (Γ − Γ t ) − (Γ + Γ )(1 + t )(Γ − Γ t ) (cid:3) ; a = − t ( t + t −
2) + 4Γ t (2 t − t −
1) + Γ Γ (1 + 2 t − t + 24 t − t ) −− Γ t ( − − t + 10 t − t + t ) + Γ Γ t ( −
16 + 24 t − t + 2 t + t )++4Γ Γ t ( − t − t + 3 t + t ) − Γ t (2 − t + 10 t − t + 2 t ); a = − − Γ ) (Γ + Γ ) t [ − Γ t + Γ t (2 + t ) + Γ (1 + 2 t )] . Moreover, the discriminant of the left side polynomial in the equation (18) describes a situ-ation where the cusps points “merge” into one and one of the branches becomes smooth, which8eads to a bifurcation diagram describing the interaction of two point vortices in an ideal fluidinside a circular cylinder [6].As an addition, we investigate the stability features of critical circles whose radii satisfy (16)and which lie in the preimage of the bifurcation curves (15) and (17). In this case, it is sufficientto determine the type (elliptic/hyperbolic) in any one of the points ( f, h ) on a smooth branchof the curve Σ [20].The type of a critical point x with rank in an integrable system with two degrees of freedomcan be calculated the following way. One should specify the first integral F , such that dF ( x ) = 0 and dF (cid:54) = 0 in a neighborhood of this point. The point x is a fixed point for the Hamiltonianvector field sgrad F and it is possible to calculate the linearization of this field at a given point –the operator A F at the point x . This operator will have two zero eigenvalues and the remainingfactor of the characteristic polynomial is µ − C F , where C F = trace ( A F ) . When C F < we getthe point of a type “center” (the corresponding periodic solution is elliptic, it is a stable periodicsolution in phase space, the limit of the concentric family of two-dimensional regular tori), andfor C F > we get the point of a type “saddle” (the corresponding periodic solution is hyperbolicand there are motions, asymptotic to this solution, lying on two-dimensional separatrix surfaces).Here, explicit expressions for C F are presented only for bifurcation curves γ and γ : γ : C F = (4 − c ) f + 4 cf − c, < f < γ : C F = ( c − s + 2( c − c − s − c − , s ∈ (cid:16) √ c )2+ √ c (cid:105) . Fig. 1 a), b) show an enlarged fragment of the bifurcation diagram in the case of the identicalintensities and a = 1 , while the parameter c > and the deformation parameter ε = 0 . Thesigns “ + “ and “ − “ correspond to elliptic (stable) and hyperbolic (unstable) periodic solutionsin the phase space. As expected, the type change occurs at the cusp A and the point of tangency B , both depicted on the bifurcation diagram Σ .Let Γ = Γ = 1 , ε (cid:54) = 0 . (19)9 ) =1;1 1.1 1.2 1.3 - - - - A BC + + + + + + + + + ++ + + + + + + + + ++++++++++++++++ --- - - - - - - - - - - - - - - - - - - - - - - (cid:2) T (cid:3) (cid:4) (cid:3) T (cid:3)(cid:5) T (cid:3) a - - - - h =-5.1 - - - - a c d eb (cid:4) (cid:3) T (cid:3) (cid:2) T (cid:3) (cid:2) T (cid:3) (cid:3) T (cid:3) (cid:2) T (cid:3) (cid:3) T (cid:3) (cid:5) T (cid:3) b) a= c Figure 1: a) Enlarged fragment of the bifurcation diagram Σ where Γ = Γ = 1 and c > , ε = 0 ;b) Σ -perturbation where Γ = 1 , Γ = 1 . and c > , ε = 0 .After substituting (11), (19) and θ = θ + π into (6), the critical set C also takes a simple form θ = θ + π ; r = r ;(1 + r r )[( r + r )( r r + c ) − ( c − r r − c ]++ ε (1 − r )(1 − r )( r + r r + r −
1) = 0 . The corresponding bifurcation diagram Σ is defined on the plane R ( f, h ) and consists oftwo curves γ and γ , where γ : h = ln (cid:16) − f (cid:17) −
12 ( c + ε ) ln(2 f ) + ε ln (cid:18) f (cid:19) , < f < γ : h = ln (cid:16) x ε (cid:112) x − z − c (cid:17) ,f = z − xz + 2 ,z = x [( ε + c )( x −
1) + 1]( ε + 1) x − ε , x ∈ (1; x ] . Here x denotes the root of the equation ( z − x ) = 4( x − , x > . (20)Fig. 2 and 3 show the bifurcation diagram and its enlarged fragment in the case of (19) forthe parameter values ε = 28 , c = 12 . Note that the curve γ has the cusps points A, B , and C
10s the point of tangency between γ and γ for the specified parameter values, where x = √ (cid:114) (cid:113) − √ (cid:113) √ ≈ , f C = 257 (cid:104)
45 + √ (cid:112)
45 + 19 √ − (cid:113) √ (cid:105) ≈ , h C ≈ − , . Shown on Fig. 3 a), the tangent point C satisfies (20). - - (cid:2) (cid:3) T (cid:4) (cid:4) T (cid:4) h f (cid:2) ++ + + + + +++ - - - - - - - ----- Figure 2: Bifurcation diagram Σ with Γ = Γ = 1 and c = 12 , ε = 28 . C (cid:2) T (cid:2) (cid:3) T (cid:2) (cid:4) (cid:4) (cid:2) T (cid:2) (cid:2) T (cid:2) (cid:5) T (cid:2) (cid:3) T (cid:2) A BS h f + + + + + + + + +++ + + ++ ++++++ +++++ + + + + + + + + + +- - - - - - - - - - - - - - - - - - - - - - - - - - - - - -- - - - - - - - - - - - - - - a) b) Figure 3: Enlarged fragment of the bifurcation diagram with Γ = Γ = 1 c = 12; ε = 28 .Again, the signs “ + “ and “ − “ correspond to elliptic (stable) and hyperbolic periodicsolutions in phase space. The type change occurs at the cusp points A and B , as well as at the11oint of tangency C , all shown on the bifurcation diagram Σ . For clarity, an explicit expressionof the coefficient C F is shown. It is responsible for the type (elliptic/hyperbolic) of the smoothbranch of the curve γ : γ : C F = (4 − c + 3 ε ) f + 2( c + 4 − ε ) f + 4( c + 5 ε ) f − c + ε ) , < f < . At C F < we get the point of a type “center”, and at C F > we get the point of type “saddle”. Here we give a fragment of the bifurcation diagram Σ (Fig. 4) and its dynamics (Fig. 5)for the general case, using implicit parametrization in the form of the equation (12) and depen-dencies (13). The program of an interactive bifurcation diagram visualization was written inthe Python programming language, using interactive environment of
Jupyter Notebook . Havingspecified values of the parameters Γ = 1 , Γ = 1 , c = 12 , ε = 28 , numerical methodswere implemented in order to solve the polynomial equations (12) for r , given r ∈ (0; 1) .Thus on a plane of coordinates ( f, h ) the bifurcation diagram Σ was plotted in the form ofdependencies (13). When the physical parameter of the intensity ratio is perturbed (for clarity, a = Γ Γ = 1 , ), the “separation” of the point C is observed (on Fig. 3(a) it corresponds to thepoint of tangency C ). This leads to the perturbation of a part of the bifurcation diagram. Thissituation is typical for the perturbation of bifurcation diagrams of integrable systems that havepoints of tangency between bifurcation sheets. For example, such pattern holds for bifurcationdiagrams of Kovalevskaya’s top integrable cases and its generalization to Kovalevskaya-Yehiagyrostat in a rigid body dynamics [21].Fig. 5 shows fragments of the bifurcation diagram’s change in dynamics using interactivevisualization program described above. Along with the change of parameters, such as the ratioof intensities a = Γ Γ , the interaction parameter of vortices c and the deformation parameter ε ,the formation of triangular regions, bounded by pieces of bifurcation curves, is observed. Theseregions experience various deformations (e.g., some triangular regions disappear) and finally wecan observe a stable bifurcation diagram which corresponds to the problem of dynamics of twovortices bounded by a circular region in an ideal fluid [6].12 =-3.225 (cid:2) h f a b c d e (cid:3) T (cid:4) (cid:3) T (cid:4) (cid:4) T (cid:4) (cid:4) T (cid:4)(cid:5) T (cid:4) (cid:5) T (cid:4) (cid:2) h =-3.225 (cid:3) T (cid:4) h f Figure 4: Enlarged fragment of the bifurcation diagram perturbation for Γ = 1; Γ = 1 , c = 12; ε = 28 . Here we restrict intensities’ values by positive ones, i.e. throughout this section we assumethat the intensity parameters Γ and Γ have a positive sign. Let’s perform an explicit reductionto a system with one degree of freedom. In order to perform this for the system (1) withHamiltonian (2), one should substitute phase variables ( x k , y k ) to new variables ( u, v, α ) usingthe formulas below x = 1 √ Γ [ u cos( α ) − v sin( α )] , y = 1 √ Γ [ u sin( α ) + v cos( α )] ,x = 1 √ Γ (cid:112) f − u − v cos( α ) , y = 1 √ Γ (cid:112) f − u − v sin( α ) . The physical variables ( u, v ) are Cartesian coordinates of one of the vortices in a coordinatesystem that is associated with another vortex rotating around the center of vorticity. Thechoice of such variables is suggested by the presence of the integral of the angular momentumof vorticity (4), which is invariant under the rotation group SO (2) . The existence of a one-parameter symmetry group allows to perform a reduction to a system with one degree of freedomin a similar fashion as in mechanical systems with symmetry [16]. Backward substitution U = (cid:112) Γ x x + y y (cid:112) x + y , V = (cid:112) Γ y x − x y (cid:112) x + y , f h f h f h f h f h f h f Figure 5: Dynamics of the bifurcation diagram Σ in the general case.leads to canonical variables with respect to the bracket (3): { U, V } = −{ V, U } = 1 , { U, U } = { V, V } = 0 . The system with respect to the variables ( u, v ) is Hamiltonian: ˙ u = ∂H ∂v , ˙ v = − ∂H ∂u , (21)with Hamiltonian H = 12 (cid:110) Γ ln (cid:16) − u + v Γ (cid:17) + Γ ln (cid:16) − f Γ + u + v Γ (cid:17) −− Γ Γ ( c + ε ) ln (cid:104)(cid:16) u √ Γ − (cid:112) f − u − v √ Γ (cid:17) + v Γ (cid:105) − + ε Γ Γ ln (cid:104)(cid:16) − u (cid:112) f − u − v √ Γ Γ (cid:17) + v ( f − u − v )Γ Γ (cid:105) . (cid:111) (22)14he rotation angle α ( t ) of the rotating coordinate system satisfies the differential equation ˙ α = Γ Γ − f + u + v + c Γ Γ (cid:112) Γ R ( u, v ) Q ( u, v ) + ε Γ Γ (cid:112) Γ (Γ − u − v ) (cid:112) f − u − v R ( u, v ) Q ( u, v ) , where R ( u, v ) = Γ ( f − u − v ) − u Γ ,Q ( u, v ) = √ Γ u (cid:112) f − u − v (cid:2) Γ ( u + v ) − Γ ( f − u − v ) (cid:3) ++ √ Γ ( f − u − v ) (cid:2) Γ ( f − u − v ) − Γ ( u − v ) (cid:3) R ( u, v ) = √ Γ (cid:112) f − u − v (Γ + u + v ) − √ Γ u (Γ + f − u − v ) ,Q ( u, v ) = Γ (cid:2) Γ (Γ + 4 u ) + ( u + v ) (cid:3) ( f − u − v ) + Γ ( u + v ) (cid:2) Γ + ( f − u − v ) (cid:3) −− u √ Γ Γ (cid:112) f − u − v (Γ + u + v )(Γ + f − u − v ) . The fixed points of the reduced system (21) are determined by the critical points of thereduced Hamiltonian (22) and correspond to the relative equilibria of vortices in the system (1).For a fixed value of an integral of the moment of vorticity f , the regular levels of the reducedHamiltonian are compact and motions occur along closed curves. It can be shown that thecritical values of the reduced Hamiltonian determine the bifurcation diagram (12), (13).Note some interesting special cases. For a segment of the bifurcation curve ( AB ) (Fig. 1), themotion on the plane ( u, v ) occurs along a curve that is topologically structured as S ˙ ∪ S ˙ ∪ S (Fig. 6b)), and the integral critical surface is a trivial bundle over S with the layer S ˙ ∪ S ˙ ∪ S .When passing through a section of the curve ( AB ) in the case of c > (Fig. 1), the bifurcationof three tori into one occurs as follows T → S × (cid:0) S ˙ ∪ S ˙ ∪ S (cid:1) → T . With the help of thereduced Hamiltonian level curves in Fig. 6, this bifurcation is clearly visible ( h = − . for a) f = 1 . ; b) f = 1 . ; c) f = 1 . ).For another special case of the bifurcation diagram (Fig. 3 b), where the bifurcation curves γ and γ intersect at the point S ( x S = 1 , f = 0 , h = − , ), themovement occures on the plane ( u, v ) and goes by the curve which is topologically arranged as S ˙ ∪ S ˙ ∪ S ˙ ∪ S (Fig. 7 a). In this case the integral critical surface is a trivial bundle over S witha layer of S ˙ ∪ S ˙ ∪ S ˙ ∪ S . During cross-over of the point S , moving along the line h = h on thebifurcation diagram Σ , the bifurcation of four tori into one occures T → S × (cid:0) S ˙ ∪ S ˙ ∪ S ˙ ∪ S (cid:1) T . With the help of the reduced Hamiltonian (22) level curves, this four-into-one bifurcationis clearly seen on Fig. 7. The following parameters were used here: Γ = Γ = 1 , c = 12 , ε = 28 . - - - - uv - - - - uv - - - - v u a ) b ) c ) Figure 6: Level curves of reduced Hamiltonian H for Γ = Γ = 1 and c > , ε = 0 . - - - - v u - - - - - v u - - - v ua ) Figure 7: Level curves of reduced Hamiltonian H along the line h = h for Γ = Γ = 1 and c = 12 , ε = 28 .Let’s consider an integrable perturbation of the physical parameter of the intesities’ ratio a = Γ Γ when the deformation parameter ε is absent, i.e. equal to zero. In this case, a perturbationof a special layer of Liouville foliation is observed, i.e. in the terminology of singularities, theatom D desintegrates into two atoms of type B [18]. Fig. 8 corresponds to the section of thebifurcation diagram on Fig. 1 b) along the line h = h = − . . Here, for specified values ofsecond integral’s parameter f = a – e , the perturbation is clearly presented (Fig. 8 (b) (d) of thesaid feature (see Fig. 6 (b))). This result is also confirmed by [18].Finally, for an interactive fragment of the bifurcation diagram perturbation (Fig. 4) for Γ = - - - v ua - - - - v ub - - - - c v u - - - - c v u - - - - dv u - - - - ev u Figure 8: Level curves of reduced Hamiltonian H for Γ = 1; Γ = 1 , and c > .
1; Γ = 1 , c = 12; ε = 28 (here the parameters are arbitrary), the corresponding contourlines of the reduced Hamiltonian H for the selected values of the second integral f = a – e areshown on Fig. 9.In conclusion it should be mentioned, that implemented for arbitrary intensities Γ , Γ , thephysical parameter c and the deformation parameter ε , computer model of absolute dynamicsof the vortices in the fixed coordinate system described by (1) and (2) is based on the analyticalresults of this publication (an implicit parametrization of the bifurcation diagrams (12), (13), thereduction to the system with one degree of freedom (21), the stability analysis). A classificationof bifurcation diagrams (study of points of tangency bifurcations, cusps points, depending on thevalues of physical parameters a, c, ε ) might be a separate research topic with the aim of creatinga topological atlas for the generalized integrable model of vortex dynamics considered here. ACKNOWLEDGMENTS
The authors thank to Prof. A. V. Borisov for useful discussions.17 - - - av u - - - - v ub - - - - cv u - - - - dv u - - - - ev u - - - - v u Figure 9: Level curves of reduced Hamiltonian H for Γ = 1; Γ = 1 . and c = 12 , ε = 28 along the line h = h − . at f = a – e . FUNDING
The work of P. E. Ryabov was supported by RFBR grant 17-01-00846 and was carried outwithin the framework of the state assignment of the Ministry of Education and Science of Russia(project no. 1.2404.2017/4.6).
CONFLICT OF INTEREST
The authors declare that they have no conflicts of interest.
AUTHORS’ CONTRIBUTIONS
Interactive visualization of the bifurcation diagram is made by A. A. Shadrin based on theequations of a bifurcation diagram (12), (13) and reduction to the system with one degree offreedom in general case (21). Authors P. E. Ryabov and A. A. Shadrin were involved in writing18he text of the paper. All authors participated in discussing the results.
References [1] Fetter, A. L., Rotating trapped Bose-Einstein condensates,
Rev. Mod. Phys. , 2009, vol. 81,no. 2, pp. 647–691.[2] Torres, P. J., Kevrekidis, P. G., Frantzeskakis, D. J., Carretero-Gonzalez, R., Schmelcher,P. and Hall, D. S., Dynamics of vortex dipoles in confined Bose–Einstein condensates,
Phys. Lett. A. , 2011, vol. 375, pp. 3044–3050.[3] Navarro, R., Carretero-Gonz´alez, R., Torres, P. J., Kevrekidis, P. G., Frantzeskakis, D. J.,Ray, M. W., Altunta¸s, E. and Hall, D. S., Dynamics of Few Co-rotating Vortices in Bose-Einstein Condensates,
Phys. Rev. Lett. , 2013, vol. 110, no. 22, pp. 225301-1–6.[4] Koukouloyannis, V. and Voyatzis, G. and Kevrekidis, P. G., Dynamics of three noncoro-tating vortices in Bose–Einstein condensates,
Phys. Rev. E. , 2014, vol. 89, no. 4, pp. 042905-1–14.[5] Greenhill, A. G., Plane vortex motion //
Quart. J. Pure Appl. Math. , 1877/78, vol. 15,no. 58, pp. 10–27[6] Kilin, A. A., Borisov, A. V. and Mamaev, I. S., The Dynamics of Point Vortices Insideand Outside a Circular Domain, in
Basic and Applied Problems of the Theory of Vortices
Borisov, A. V. and Mamaev, I. S. and Sokolovskiy, M. A. (Eds.), Izhevsk: Regular andChaotic Dynamics, Institute of Computer Science, 2003, pp. 414-440 (Russian).[7] Kilin, A. A., Borisov, A. V., Mamaev I. S., The Dynamics of Point Vortices Inside andOutside a Circular Domain, in
Mathematical methods of vortex structure dynamics
Borisov,A. V. and Mamaev, I. S.(Eds.) M.-Izhevsk: Regular and Chaotic Dynamics„ Institute ofComputer Science, 2005, pp. 148–173 (Russian).[8] Borisov, А. V. and Kilin, A. A., Stability of Thomson’s Configurations of Vortices on aSphere,
Regular and Chaotic Dynamics , 2000, vol. 5, no. 2, pp. 189–200.199] Borisov, A. V., Mamaev, I. S. and Kilin, A. A., Absolute and relative choreographies in theproblem of point vortices moving on a plane,
Regular and Chaotic Dynamics , 2004, vol. 9,no. 2, pp. 101–111.[10] Borisov, A. V., Kilin, A. A. and Mamaev, I. S., The Dynamics of Vortex Rings: Leapf-rogging, Choreographies and the Stability Problem,
Regular and Chaotic Dynamics , 2013,vol. 18, nod. 1–2, pp. 33–62.[11] Borisov, A. V., Ryabov, P. E. and Sokolov, S. V., Bifurcation analysis of the motion ofa cylinder and a point vortex in an ideal fluid,
Mathematical Notes , 2016, vol. 99, no. 6,pp. 834–839.[12] Sokolov, S. V. and Ryabov, P. E., Bifurcation Analysis of the Dynamics of Two Vortices in aBose–Einstein Condensate. The Case of Intensities of Opposite Signs,
Regular and ChaoticDynamics , 2017, vol. 22, no. 8, pp. 979–998.[13] Sokolov, S. V. and Ryabov, P. E. Bifurcation Diagram of the Two Vortices in a Bose-EinsteinCondensate with Intensities of the Same Signs,
Doklady Mathematics , 2018, vol. 97, no. 3,pp. 1–5.[14] Ryabov, P. E. Bifurcations of Liouville Tori in a System of Two Vortices of Positive Intensityin a Bose-Einstein Condensate,
Doklady Mathematics , 2019, vol. 99, no. 2, pp. 1–5.[15] Ryabov, P. E., Sokolov, S. V. Phase Topology of Two Vortices of Identical Intensities in aBose-Einstein Condensate,
Rus. J. Nonlin. Dyn. , 2019, vol. 15, no. 1, pp. 59–66.[16] Kharlamov, M. P.,
Topological Analysis of Integrable Problems of Rigid Body Dynamics ,Leningrad: Leningr. Gos. Univ., 1988 (Russian).[17] Bolsinov, A. V., Matveev, S. V. and Fomenko, A. T., Topological classification of integrableHamiltonian systems with two degrees of freedom. List of systems of small complexity,
Russian Mathematical Surveys , 1990, vol. 45, no. 2, pp. 59–94.[18] Oshemkov, A. A. and Tuzhilin, M. A., Integrable perturbations of saddle singularities of rank0 of integrable Hamiltonian systems,
Sbornik: Mathematics , 2018, vol. 209, no. 9, pp. 1351–1375. 2019] Ryabov P. E. On bifurcation of the four Liouville tori in one generalized integrable modelof the vortex dynamics, https://arxiv.org/abs/1903.09945 (Russian).[20] Bolsinov, A. V., Borisov, A. V. and Mamaev, I. S., Topology and Stability of IntegrableSystems,
Russian Math. Surveys , 2010, vol. 65, no. 2, pp. 259–318; see also:
Uspekhi Mat.Nauk , 2010, vol. 65, no. 2, pp. 71–132.[21] Kharlamov, M. P., Ryabov, P. E., Kharlamova, I. I., Topological Atlas of the Kovalevskaya-Yehia Gyrostat,