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ISSN 1560-3547, Regular and Chaotic Dynamics, 2013, Vol. 18, No. 4, pp. 356­371. c Pleiades Publishing, Ltd., 2013.

Top ological Analysis of an Integrable System Related to the Rolling of a Ball on a Sphere
Alexey V. Borisov* and Ivan S. Mamaev**

Institute of Computer Science; Laboratory of Nonlinear Analysis and the Design of New Types of Vehicles, Udmurt State University, ul. Universitetskaya 1, Izhevsk, 426034 Russia A. A. Blagonravov Mechanical Engineering Research Institute of RAS, ul. Bardina 4, Moscow, 117334 Russia Institute of Mathematics and Mechanics of the Ural Branch of RAS, ul. S. Kovalevskoi 16, Yekaterinburg, 620990 Russia
Received Novemb er 16, 2012; accepted Decemb er 24, 2012

Abstract--A new integrable system describing the rolling of a rigid bo dy with a spherical cavity on a spherical base is considered. Previously the authors found the separation of variables for this system on the zero level set of a linear (in angular velo city) first integral, whereas in the general case it is not possible to separate the variables. In this paper we show that the foliation into invariant tori in this problem is equivalent to the corresponding foliation in the Clebsch integrable system in rigid bo dy dynamics (for which no real separation of variables has been found either). In particular, a fixed point of fo cus type is possible for this system, which can serve as a topological obstacle to the real separation of variables. MSC2010 numbers: 37J60, 37J35, 70H45 DOI: 10.1134/S1560354713040035 Keywords: integrable system, bifurcation diagram, conformally Hamiltonian system, bifurcation, Liouville foliation, critical perio dic solution

Contents
INTRODUCTION 1 2 3 4 EQUATIONS OF MOTION AND FIRST INTEGRALS THE SET S0 (FIXED POINTS) THE CRITICAL SET S1 (PERIODIC ORBITS) DISCUSSIONS AND OPEN PROBLEMS REFERENCES 357 357 360 363 369 370

* **

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

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INTRODUCTION In [16], S. A. Chaplygin integrated by quadratures the problem of rolling of a dynamically asymmetric balanced ball on an absolutely rough plane and p ointed out the main features of motion. This is a classical problem in nonholonomic mechanics, which still attracts the attention of researchers in various fields (see, e.g., [5, 6]). The dynamics of an analogous ball (usually called the Chaplygin ball) moving without slipping on the surface of a fixed sphere was studied in [10], where the integrability case was p ointed out which takes place only for some value of the ratio of the ball's radius to the sphere's radius (see b elow). In [15], the problem considered here is called the BMF-system. Other mechanical means of its realization are considered in [3]. It turned out that explicit integration of the new system presents considerable difficulties. On the zero level set of one of the known integrals, which is linear in momenta (and is an analog of the area integral), an explicit solution is presented in [2, 9]. The case of the non-zero constant of the ab ovementioned integral is still far from b eing solved. We only p oint out that a (quite complicated) conformally Hamiltonian representation of this system has b een found recently in [15]. However, this has not made it p ossible to make progress in its explicit integration. We note that for the integration of the classical Chaplygin ball problem with non-zero value of the area constant a nontrivial transformation of coordinates and time is used which reduces the system to a zero constant (prop osed by Chaplygin [16], discussed in [5], the generalization and geometry of this transformation is considered in [4]). A generalization of this transformation for the problem considered in this pap er is hardly p ossible, since it is substantially related to the vector integral of the angular momentum (this vector is fixed in absolute space), which exists for the Chaplygin problem but disapp ears in our problem. The problem of explicit integration more closely resembles here the Clebsch case in the Kirchhoff problem (which is well known in mechanics) describing the inertial motion of a simply connected rigid b ody in an infinite volume of an ideal fluid. Whereas on the zero level set of the area constant the separation of variables for the Clebsch case is sufficiently elementary (it is p erformed using sphero-conical coordinates, and the system itself is isomorphic to the classical Neumann system for the motion of a p oint over a sphere in a quadratic p otential), no real separation has b een p erformed so far for a non-zero constant, despite the fact that many famous mathematicians and mechanical engineers were concerned with this problem (see the b ooks [12, 13] published in the Russian language, where the classical works on this issue have b een gathered). One of the alternatives to the separation of variables and explicit solution in theta-functions are the methods of top ological analysis of integrable systems, which provide insight into the structure of foliation into tori and make it p ossible to find critical manifolds constituting the so-called particularly remarkable motions (in the terminology of G. App el'rot) and to investigate the stability of p eriodic tra jectories and equilibrium p ositions. For integrable Hamiltonian systems these methods are describ ed in [8, 14]. In [7, 17] (see also [18]), these results are set forth using the framework of a bifurcation complex (which is more illustrative for mechanics) and applications to a complete stability analysis are presented. It turns out that for integrable systems this analysis can b e p erformed without analyzing explicit quadratures. Since the ab ove-mentioned integrable system [15] is conformally Hamiltonian, all results [7] can b e applied in this case. This pap er presents a bifurcation analysis. In particular, it is shown that fixed p oints of focus typ e arise for some values of the parameters. The existence of fixed p oints of focus typ e (which are present, for example, in the Clebsch case) in phase space is apparently an obstruction to a real separation of variables (the Kotter variables for the Clebsch ¨ case are essentially complex). As further analysis will show, the system under consideration is in its top ological structure very similar to the Clebsch case, although it has a numb er of sp ecific features. 1. EQUATIONS OF MOTION AND FIRST INTEGRALS Consider a system describing the rolling of a rigid b ody whose spherical cavity is in contact with the spherical base (see Fig. 1). We shall assume that ­ the center of mass of the b ody coincides with the geometric center of the cavity O, ­ at the p oint of contact P the no-slip condition (absolutely rough surface) is satisfied, ­ the outer field is absent, ­ the radii of the base r and the cavity R are related as 1 : 2.
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Let us introduce a moving coordinate system Oe1 e2 e3 whose axes coincide with the principal axes of inertia of the rigid b ody; let be the normal at the p oint of contact(see Fig. 1) and let and M be the angular velocity of the b ody and the angular momentum relative to the p oint of contact. Then the equations of motion governing their evolution can b e represented in the form [2, 10] M = M â , = - â , = A(M + Z ), (AM , ) , A = diag(A1 ,A2 ,A3 ), Z = -1 D - ( , A ) (1.1)

where D = mR2 , R is the radius of the cavity and Ai = (Ii + D)-1 , Ii are the principal moments of inertia of the b ody. Let Q b e the orthogonal matrix which defines the orientation of the b ody and whose rows corresp ond to the coordinates of the vectors ei in the fixed basis; its evolution is given by the Poisson equations Q = Q, where is the skew-symmetric matrix corresp on 0 = - 3 2 ding to the angular velocity: 3 - 2 0 1 . - 1 0 (1.2)

Fig. 1. A rigid b ody with a spherical cavity, rolling on a fixed ball.

The configuration space of the system under consideration is S 2 â SO(3), and the pair ( , Q) completely characterizes the p osition of the b ody. In what follows we will concentrate on the study of the six-dimensional reduced system (1.1), leaving aside the analysis of evolution of the orientation Q(t). We only note that for the complete system of equations (1.1) and (1.2) there exist three obvious first integrals expressing the constancy of the vector M in the fixed axes. These integrals can b e represented in the vector form QT M = const. For definiteness, we will assume throughout this pap er that the principal axes of inertia are chosen such that the inequalities 0 < D < I1 < I2 < I3 (1.3) are satisfied. Furthermore, we recall that the moments of inertia also satisfy the inequality of the triangles Ii + Ij Ik , and hence, in view of the inequalities (1.3), we need to require that I1 + I2 I3 . (1.4) Remark 1. The differences b etween the system (1.1) and the equations of the Chaplygin problem of the rolling motion of a ball on a plane are in the sign in the equations for and in the region of physical values of the moments of inertia: in the Chaplygin ball Ii < D, i = 1, 2, 3. The first integrals of the system (1.1) are given as follows: F0 = ( , ), F1 = (M , M ), F2 = ( , BM ), E= 1 (M , ), 2 (1.5)

where B = diag(B1 ,B2 ,B3 ), Bk = 1 (A-1 + A-1 - A-1 ), here i, j, k is a cyclic p ermutation of the i j k 2 indices 1, 2, 3. In addition, the system admits the invariant measure (D -1 - , A ) 2 dM d and, hence, according to the generalization of the Euler ­ Jacobi theorem [1], is integrable and conformal ly Hamiltonian , which allows one to use the top ological approach [7, 8] for its investigation. The explicit conformally Hamiltonian representation with the Poisson bracket linear in M was obtained in [15].
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The main goal of this paper is to indicate the critical trajectories of the system (1.1) using the integrals (1.5) and to construct its possible bifurcation diagrams and the corresponding bifurcation complexes. In this connection we recall some necessary definitions, following mainly [7]. First, we restrict (formally) the system (1.1) to the level surface of the integrals F0 and F2 : Mc = {(M , ) | 2 = 1, ( , BM ) = c}. (1.6)

The four-dimensional (conformally Hamiltonian) system thus obtained p ossesses a pair of quadratic integrals F1 and E , which determine the integral mapping of the phase space Mc onto the plane of values of the first integrals, x (x) = F1 (x),E (x) = (f, h) R2 , x Mc .

The region of possible motions (Mc ) is a complete image of the phase space on the plane of first integrals (f, h). Each p oint (f, h) (Mc ) corresp onds to the integral manifold of the system Mc h = {F1 (x) = f, E (x) = h}, which, generally sp eaking, can contain several connected f, comp onents. The set of critical points of the integral map is defined as S = {x Mc | rank d(x) < 2}, it is an invariant set entirely filled with critical tra jectories. The set S breaks up into two disjoint subsets: S = S0 S1 , S0 = {x Mc | rank d(x) = 0}, S1 = {x Mc | rank d(x) = 1}. In a typical situation, S0 consists of isolated p oints, and S1 is a one-parameter family (or several families) of closed curves which are p eriodic solutions of the original system (1.1), or consist entirely of the fixed p oints (1.1), or include asymptotic tra jectories going into fixed p oints. The images of the corresp onding sets on the plane of first integrals are denoted similarly: = (S ), 0 = (S0 ), 1 = (S1 ).

The bifurcation diagram of an integrable system is defined to b e the region of p ossible motions (Mc ) depicted on the plane of first integrals (f, h) together with the image of the critical set and the indication of the images 0 and 1 . For this system (1.1) in the case ( , BM ) = 0 (i.e. c = 0) a separation of variables has b een found in [2]. It was shown that the only p ossible critical p eriodic solutions (i.e. the set S1 ) are determined by the conditions i = j = 0, k = 0, (i, j, k) = (1, 2, 3). Hence, , that is, the b ody rolls along a large circle on a supp orting sphere whose plane coincides with one of the principal planes of inertia of the b ody. On the plane of first integrals, the bifurcation curves (the set 1 ) are given by k : h = Ak f, k = 1, 2, 3. (1.7)

In this case, the set S0 coincides with the fixed p oint = 0, which on the plane of first integrals (f, h) corresp onds to the origin of coordinates (i.e. 0 = {(0, 0)}). The region of p ossible motions (M0 ) lies b etween the upp er straight line 1 and the lower straight line 3 (the corresp onding bifurcation diagram is shown in Fig. 2a). It can b e shown that each regular p oint (f, h) (M0 ) corresp onds to a pair of invariant tori of the system (1.1) (the bifurcation complex is depicted in Fig. 2b).
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Fig. 2. The bifurcation diagram and the bifurcation complex in the case (BM , )=0.

2. THE SET S0 (FIXED POINTS) We b egin the description of the critical set of the system in the case c = 0 with fixed p oints and the analysis of their typ e and stability. The condition rank d = 0 implies that the differentials (gradients) of b oth integrals F2 and E , restricted to Mc , vanish. In terms of the initial variables M , it can b e rewritten using the undetermined multipliers as F0 F2 E = 1 + 1 , M M M F0 F2 F1 = 2 + 2 , M M M F0 = 2 = 1, F2 E F0 F2 = 1 + 1 , F1 F0 F2 = 2 + 2 , = ( , BM ) = c,

where i , i are the undetermined multipliers. Using these equations, it can b e shown that for p oints of the set S0 the vectors M , , are collinear M and, in addition, each of these vectors is an eigenvector of the matrix A. Hence, we conclude that the set S0 coincides with the fixed p oints of the system (1.1). We finally find that there are three pairs of fixed p oints defined by the relations
± Pk : = ±ek ,

M = Ik = ±ek ,

k = 1, 2, 3,

(2.1)

where ek are the unit vectors of the principal axes of inertia.
± For each of the solutions Pk according to (1.1) and (1.2) we conclude that one of the axes of inertia of the b ody (directed along ek ) remains constant relative to the fixed passes through the p oint of contact. The b ody uniformly rotates ab out this axis in one or h direction with angular velocity k = 2c Bk , where h, c are the values of the first integrals an entry of the matrix B.

principal axes and the other and Bk is

On the level surface Mc (1.6), for each of the fixed p oints (2.1) we choose the local variables x = (x1 ,x2 ,x3 ,x4 ), where i = x1 , Mi = x3 , i = x2 , k = ± Mk = ± 1 - x2 - x2 , 1 2 bk 1 - x2 - x2 1 2 ,
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Mj = x4 ,

c - bi x1 x3 - bj x2 x4

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here and in the sequel i, j, k is a cyclic p ermutation of 1, 2, 3. The characteristic p olynomial of the ± linearized system on Mc for each of the p oints Pk can b e represented in the form Ik (D - Ik + Ii + Ij ) 4 Ik (D - Ik + Ii + Ij + ak 2c 2c (D - Ik )(D - Ik + Ii + Ij )+2Ii Ij - 2DIk (Ik - , bk = ak = (D + Ii )(D + Ij ) (D + k () = From the typ e of the characteristic p olynomial we conclude that
± the types of the fixed points Pk depend not on the value of the integral c but only on the parameters D, Ii , i = 1, 2, 3.

)

2



+ bk , (2.2)

Ii )(Ik - Ij ) . Ii )(D + Ij )

We recall that on the plane of coefficients of the characteristic p olynomial the typ es of fixed p oints are separated by the curves b = 0 and a2 - 4b = 0 (see Fig. 3). We find from (2.2) and (1.3) that b2 < 0. Hence,
± P2 are points of sadd le­center type (i.e. the rotations of the midd le axis of inertia are always unstable).

For the analysis of rotations ab out the other axes we sp ecify the values k = (D + Ii )2 (D + Ij )2 (a2 - 4bk ), k k = (D + Ii )(D + Ij )ak , which determine the signs of the discriminant and the coefficient ak of the characteristic p olynomial (2.2). ± For the fixed p oints P1 defining the rotation ab out the smallest axis, we represent these values in the form 1 = D(I2 + I3 - 2I1 )+ I2 (I3 - I1 )+ I3 (I2 - 1) + (I1 - D)2 , 1 = 4DI1 (I3 - I2 )2 +(I1 - D)2 (I1 - D )2 +(2D + I2 + I3 )(I2 + I3 - 2I1 ) . Using the inequalities (1.3), we notice that 1 > 0, 1 > 0, that is
± P1 are points of center­center type. ± To investigate the typ e of p oints P3 , we rewrite the quantities 3 , 3 in the form

Fig. 3. The typ es of fixed p oints on the plane of values of the coefficients of the characteristic p olynomial.

3 = 4DI3 (I3 - D)2

2 2 - -1 , 4DI3 (I3 - D)2 = I1 + I2 - (I3 - D),

3 =

12 - 2 +(I3 - D)2 - 4DI3 , 2

= I2 - I1 .

Hence, for a fixed value of I3 on the plane of parameters I1 , I2 the equations 3 = 0 and 3 = 0 determine the ellipse and the hyp erb ola with center at the p oint I1 = I2 = I3 - D and the principal axes turned relative to the coordinate axes by (see Fig. 4). 4 In what follows we assume, without loss of generality, that D = 1. For a fixed value of I3 , according to (1.3) and (1.4), the region of p ossible values I1 and I2 is ­ for I3 2 -- a rectangular triangle b ounded by segments of the straight lines I2 = I3 , I1 = 1 and I1 = I2 (see Fig. 5a),
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Fig. 4. A typical view of the curves 3 = 0 and 3 = 0 on the plane (I1 ,I2 ) for various values of I3 (D = 1).

­ for I3 > 2 -- a quadrangle b ounded by the straight lines I2 = I3 , I1 = 1, I1 = I2 and I1 + I2 = I3 (see Figs. 5b­5d). It is straightforward to show that for each fixed I
3

­ the ellipse 3 = 0 is tangent to the straight line I2 = I3 at the p oint of intersection with the - c hyp erb ola 3 = 0 (see Fig. 5); the abscissa of this p oint of intersection is equal to I1 = 3II33+11 c and under the conditions (1.3) satisfies the inequality 1 < I1 < I3 , ­ for I3 > 2 the segment of the straight line I1 + I2 = I3 b ounding the region of p ossible I1 , I2 lies inside the ellipse 3 = 0 (see Figs. 5b­5d), ­ the branches of the hyp erb ola 3 = 0 change their p osition relative to the principal axes for I3 = 3 + 2 2 (see Fig. 5c). Analyzing the p ossible location of the curves 3 = 0, 3 = 0 and the b oundaries of the region of p ossible values of the moments of inertia, we conclude that
± for each fixed value of I3 the fixed points P3 can be of one of the fol lowing three types: sadd le­ sadd le, focus­focus and center­center (the corresponding regions in Fig. 5 are denoted by the letters S, F and C ).

We finally summarize the results obtained:
± Theorem 1. The set S0 of the system (1.1) coincides with its set of fixed points Pk , k = 1, 2, 3, which are determined by Eqs. (1.7). The types of fixed points do not depend on the values of the

first integrals but are determined by the values of the parameters I , k = 1, 2, 3. The types of fixed k points in the case where the moments of inertia satisfy the relations (1.3) and (1.4) are:
± P1 -- center­center, ± P2 -- sadd le­center, ± P3 -- of one of three types: sadd le­sadd le, focus­focus, center­center (depending on the abovementioned conditions).

D

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Fig. 5. A typical view of the regions on the plane I1 ,I2 , which corresp ond to different typ es of the fixed ± point P3 , for various fixed values of I3 (D = 1).

3. THE CRITICAL SET S1 (PERIODIC ORBITS) By analogy with the analysis of the Clebsch case in [7], in this problem we shall construct a set of critical p eriodic orbits using the method of undetermined multipliers. The condition rank d = 1 on the manifold Mc (1.6) can b e represented as E F0 F1 F2 F0 F1 F2 E = 0 +1 +2 , = 0 +1 +2 , M M M M (n, n) = 1, ( , BM ) = c. Taking into account the relations
E M

= ,

a change of undetermined multipliers 0 = obtain a system of linear homogeneous equations

E = Z 2 (det

Z , we make the substitution Z = N and A)0 , 1 = (det A)1 , 2 = 2Z (det A)2 ; we

(A - 1 (det A)E)M +(A - 22 (det A)B)N = 0, (A - 22 (det A)B)M +(A - 0 (det A)E)N = 0.
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Since the matrices A and B are diagonal, this system breaks up into three indep endent systems of two equations of the following form: (Ai - 1 det A)Mi +(Ai - 22 Bi det A)Ni = 0, i = 1, 2, 3. (Ai - 22 Bi det A)Mi +(Ai - 0 det A)Ni = 0, (3.2)

For each of these systems, the vanishing of the determinant of the corresp onding coefficient matrix is the condition for the existence of a non-zero solution: gi = (Ai - 0 det A)(Ai - 1 det A) - (Ai - 22 Bi det A)2 = 0, i = 1, 2, 3. It can b e shown that only two of these three equations are indep endent. Thus, two typ es of solutions are p ossible. 1. Three indep endent two-parameter families of solutions for the undetermined multipliers 0 , 1 , 2 , for which gk = 0, gi = 0, gj = 0, where i, j, k is a cyclic p ermutation of the indices 1, 2, 3. Such a choice of undetermined ± multipliers corresp onds to the fixed p oints Pk described above (see (2.1)). 2. Two two-parameter families are satisfied simultaneously with ± = 0
i ±

of solutions for 0 , 1 , 2 for which the following relations g1 = g2 = g3 = 0, (3.3) Ji 2 - 22 2 2
i

Ji 2 - 22 ± 2 2

R0 (2 ),

± = 1

R0 (2 ), (3.4)

R0 (2 ) = 2 (2 - J1 )(2 - J2 )(2 - J3 ), where Ji = Ii + D, and + corresp onds to the upp er sign in the formula and - to the lower one. We show that the solutions of the second typ e corresp ond to several one-parameter families of p eriodic solutions. According to (3.3), each system (3.2) has a nontrivial solution, so that the general solution can b e represented as M = -(A - 22 (det A)B)-1 (A - 0 (det A)E)N . (3.5) Substituting this relation into Eq. (1.1), which defines Z , and noting that N = Z and 0 is defined according to (3.4), we obtain one-parameter families of quadrics ( , C± (2 ) ) = D C (2 ) = A A - 22 (det A)B
± -1 -1

= 1,

A - ± (det A)E + A, 0
+

(3.6)

where ± is defined by (3.4). 0 Since in the chosen coordinate system the matrices C ± ± ± diag(C1 ,C2 ,C3 ), we finally obtain:

and C

-

are diagonal, i.e. C± =

the critical periodic orbits of the system (1.1) on the fixed level set of the first integrals F0 = 1, F2 = c and D = 1 are given as the intersection of one-parameter families of quadrics with a sphere:
2 1 ± C1

+

2 2 ± C2 1

+

2 3 ± C3

= 1,

2 2 2 1 + 2 + 3 = 1,

C

± k

± = Ck )-

Jk (1 ± Rk ), = 2

R0 . Rk = 2 (2 - Jk )
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(3.7)

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2 2 We express any pair of quantities i , j from (3.7) and, using (3.5), we substitute them into (1.5) for the integrals F1 , F2 and E . Since the curves (3.7) are the tra jectories of the system, we obtain the following expressions which are indep endent of the coordinates :

E = Z 2 U ± (2 ),

F1 = Z 2 V ± (2 ),

F2 = ZW ± (2 ),

where the functions U ± , V ± and W ± dep end only on 2 and the parameters D, Ik (they are rather unwieldy, so we do not explicitly present them here). Hence, it follows that on the critical tra jectories (3.7) the function Z (M , ) remains constant. Denoting the constants of the integrals by E = h and F1 = f , we obtain on the fixed level F2 = c bifurcation curves given in parametric form as follows: h = c2 U ± (2 ) , (W ± (2 ))2 f = c2 V ± (2 ) . (W ± (2 ))2 (3.8)

Remark 2. It follows from this representation that on the plane of values of the integrals R2 = {(f, h)} the relative p ositions of the bifurcation curves are indep endent of the quantity c. Only their dimensions dep end on it. We examine the conditions for the existence of solutions in the system (3.7) more closely and construct a bifurcation diagram. First of all, we note that the coefficients in the quadric (3.6) are real if (3.9) 2 (-, 0) (J1 ,J2 ) (J3 , +). We recall the conditions for the existence of intersection of the quadrics (3.7):
± ± ± ­ if all coefficients Ci± > 0, then one must require that min C1 , C2 , C3 ± ± ± 1, and max C1 , C2 , C3 ± ± ­ if one of the coefficients Ci± < 0, then min Cj , Ck ± ± ­ if two coefficients Ci± , Cj < 0, then Ck

1

1, j, k = i,

1, k = i, j .

Fig. 6. A typical view of the functions Rk , k = 1, 2, 3.
± To analyze the b ehavior of the coefficients Ck (2 ) we will require a numb er of prop erties of the functions Rk (2 ) under the condition that the inequalities (1.3) are satisfied:

on the interval (-, 0): all functions Rk > 1, k = 1, 2, 3, and monotonically increase, on the interval (J1 ,J2 ): the function R1 monotonically decreases from + to 0, the function R2 monotonically decreases from 0 to -, the function R3 is negative, vanishes at the ends of (1) - the interval and has the only minimum, R3 = -R3 , inside at the p oint 2 = J3 ,
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on the interval (J3 , +): the functions R1 and R2 monotonically increase from 0 to 1, the func(2) + tion R3 monotonically decreases from + to R3 on the interval J3 ,J3 and monotonically
+ increases from R3 to 1 on the interval J3 , + , (2)

where the p oints of the minimum 2 = J3 (J1 ,J2 ) and 2 = J3 R3 (2 ) are defined as the roots of the equation
2 - 2J3 2 + J3 J3 = 0, 2 ± and the values of R3 are ± R3 = J3 =

(1)

(2)

(J3 , +) of the function

J1 J2 , J1 + J2 - J3

J2 (J3 - J1 ) ± J3 Ji Jj , Ji + Jj - Jk

J1 (J3 - J2 )

,

- + 0 < R3 < R3 < 1.

Furthermore, each equation |Rk | = 1 has the unique solution 2 = Jk on the set (3.9): Jk = J1 ,J2 (J1 ,J2 ), J3 (J3 , +).

Using the ab ove-mentioned prop erties of the functions Rk , k = 1, 2, 3, we conclude that on the + - + interval (-, 0) all coefficients Ck > 1, Ck < 0, and on the interval (J3 , +) all Ck > 1. Thus, the system (3.7) admits the solutions for the family + only on the interval (J1 ,J2 ), for the family - on the set (J1 ,J2 ) (J3 , +). Let us consider each of these cases separately.
+ The family + . A typical view of the functions Ck on the interval (J1 ,J2 ) is shown in Fig. 7. In this case the solutions of the system (3.7) exist on the interval P 2 (J2 ,J2 ),

(3.10)

+ + P where J2 is the root of the equation C2 = 1, and J2 is the root of the equation C2 = 0. To show + this, we have to make sure that J2 lies on the left of the roots of the equation C3 = 1 (if such exist): indeed, in view of the inequalities (1.3) we obtain

C

+ 3 2 =J

2

1 = (J1 + J3 - J2 ) > 1. 2

Remark 3. It can b e shown that at the p oints 2 = Jk the equalities 1 - - - 2 = J1 : C1 = 0, C2 = C3 = (J2 + J3 - J1 ), 2 1 + + + 2 = J2 : C2 = 0, C1 = C3 = (J1 + J3 - J2 ), 2 1 - - - 2 = J3 : C3 = 0, C1 = C2 = (J1 + J2 - J3 ), 2 are satisfied, and the quadric is flattened into a circular disk. ± P The left end J2 of the interval (3.10) corresp onds to the fixed p oints P2 , and any value of 2 ± inside (3.10) corresp onds to a pair of p eriodic orbits. The only exception is the case where P3 are + p oints of saddle­saddle typ e. Then a pair of roots of the equation C3 = 1 (see Fig. 7b) arises inside P P the interval (3.10). We denote these roots by J3 and J3 ; they corresp ond to a pair of fixed p oints and to the separatrix curves connecting them.

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Fig. 7. A typical view of the functions C is shaded.

+ k

on the interval (J1 ,J2 ). The interval of the existence of solutions

P Substituting 2 (J2 ,J2 ) into (3.8) on the plane R2 = {(f, h)}, we obtain

the bifurcation curve

+

emanating from the point P2 (see Figs. 8­10).

± + If the p oint P3 is of saddle­saddle typ e, then the curve 1 passes through P3 , and the values P P 2 (J3 , J3 ) corresp ond to the "dovetail" ("smile", see Fig. 13). - The family - on the interval (J1 ,J2 ). A typical view of the functions Ck in this case is shown in Fig. 8. In this case the solutions of the system (3.7) exist only for the interval P 2 (J1 ,J1 ),

(3.11)

- - P where, as ab ove, J1 and J1 are the roots of the equations C1 = 0 and C1 = 1, resp ectively.

Fig. 8. A typical view of the functions C is shaded.

- k

on the interval (J1 ,J2 ). The interval of the existence of solutions

± The right end of the interval (3.11) corresp onds to the fixed p oints P1 , and any value of inside (3.11) also corresp onds to a pair of p eriodic orbits. P Substituting 2 (J1 ,J1 ) into (3.8), we obtain

2

the bifurcation curve

- 1

emanating from the point P1 (see Figs. 8­10).
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- ± Fig. 9. The view of the curves Ck (2 ) in cases where P3 is of saddle­saddle (a) and focus­focus typ e (b). The interval of the existence of solutions is shaded.

- ± Fig. 10. The view of the functions Ck (2 ) under the condition that P3 is of center­center typ e. The intervals of the existence of solutions are shaded.

As will b e evident, this curve b ounds ab ove the region of p ossible values of the integrals on the plane R2 = {(f, h)}.
± The family - on the interval (J3 , +). Dep ending on the typ e of p oints P3 , three kinds of - relative p ositions of the curves Ck (2 ), k = 1, 2, 3 (see Figs. 9 and 10) are p ossible. ± If P3 of saddle­saddle typ e (see Fig. 9a), then the interval of the existence of solutions of the system (3.8) is: P 2 (J3 ,J1 ), ± if P3 is of focus­focus typ e (see Fig. 9b), then the corresp onding interval is P 2 (J1 ,J3 ), P where, as ab ove, J1 and J3 are the roots of the equations P in the previous case, the value 2 = J1 corresp onds to the inside the intervals (3.12) and (3.13) corresp onds to a pair of ± corresp onds to a pair of p oints P2 connected by separatrix

(3.12)

(3.13)
- - C1 = 1 and C3 = 0, resp ectively. As ± fixed p oints P1 , and each value of 2 P curves except for 2 = J2 . This value curves.

Thus, b oth cases (3.12) and (3.13) corresp ond to the bifurcation curve
- 2

emanating from the point P1 and passing through the point P3 .
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± If P3 is of center­center typ e, then the solutions of the system (3.8) exist on a pair of intervals P P P 2 (J3 ,J3 ) (J3 ,J1 ), P where J3 and J3 are the roots of the equation C - 3

= 1. In this case we obtain that

- P P the values 2 (J3 ,J1 ) correspond to the bifurcation curve 3 emanating from the point P1 and ending at the point P3 , - P the values 2 (J3 ,J3 ) correspond to the bifurcation curve 4 emanating from P3 and going into .

Thus, in this case p ossible typ es of bifurcation diagrams coincide with analogous typ es in the Clebsch case (the only difference is that the typ e of diagram dep ends not on the value of the linear integral F1 = c but only on the parameters Ik /D, k = 1, 2, 3). Moreover, it follows from the ab ove that for fixed c, each p oint in the region of p ossible values of the integrals corresp onds to a pair of integral manifolds E1 = h and F1 = f , hence, the typ es of bifurcation complexes also coincide with the typ es of bifurcation complexes in the Clebsch case (see Figs. 14, 15 and 16 in [7]). Because of this, we do not present them here.

Fig. 11. Bifurcation diagram in the case where ± P3 is of center­center typ e (D = 1, I1 = 7, I2 = 8, I3 = 9). The arrows indicate the direction of increase in the parameter 2 .

Fig. 12. Bifurcation diagram in the case where ± P3 is of focus­focus typ e (D = 1, I1 = 2.5, I2 = 3.7, I3 = 5). The arrows indicate the direction of increase in the parameter 2 .

We finally obtain the following result: Theorem 2. The type of the bifurcation diagram for the system (1.1) depends only on the values of
i the parameters D and does not depend on the value of the integral ( , BM ) = c. When the physical restrictions (1.3) and (1.4) are satisfied, three types of diagrams are possible, which are determined by the type of the fixed point P3 (permanent rotation about the largest axis), they are represented in Figs. 11, 12 and 13. The bifurcation complexes coincide with the corresponding bifurcation complexes of the Clebsch case.

I

4. DISCUSSIONS AND OPEN PROBLEMS As is evident from the ab ove analysis, the BMF-system has a structure that is top ologically very similar to that of the Clebsch case. It would b e worthwhile to elucidate the question of their actual top ological equivalence and p erhaps to find an explicit algebraic transformation relating b oth cases (such formulae are well known, for example, for the Clebsch case and the Schottky ­ Manakov top on SO(4)). Issues of the absolute dynamics of a ball are of the most interest in mechanics (the b ehavior of the p oint of contact and similar problems were partially solved by Chaplygin for the motion of a
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± Fig. 13. Bifurcation diagram in the case where P3 is of saddle­saddle typ e, and a magnified view of the region in the neighb orhood of this saddle p oint (D = 1, I1 = 1.1, I2 = 2.95, I3 = 3). The arrows indicate the direction of increase in the parameter 2 .

ball on a plane [15] (see also [15])). Of particular interest are the stable p eriodic motions, which suggest a practical application of the integrable system (we note that the integral F2 has a "latent" symmetry origin and is not related to any natural conservation laws). In [5], it is p ointed out that the integrability of the system (1.1) is preserved when the field of the Bruns problem V = 1 (, A-1 ) is added. In this sense some sup erp osition of the problem 2 considered in this pap er and of the Clebsch system is obtained. It would b e interesting to p erform bifurcation analysis of this general system -- p erhaps it p ossesses a considerably richer top ological structure. The authors are grateful to A. V. Bolsinov for helpful suggestions and remarks. REFERENCES
1. Bolsinov, A. V., Borisov, A. V., and Mamaev, I. S., Hamiltonization of Nonholonomic Systems in the Neighborho o d of Invariant Manifolds, Regul. Chaotic Dyn., 2011, vol. 16, no. 5, pp. 443­464. 2. Borisov, A. V., Fedorov, Yu. N., and Mamaev, I. S., Chaplygin Ball over a Fixed Sphere: An Explicit Integration, Regul. Chaotic Dyn., 2008, vol. 13, no. 6, pp. 557­571. 3. Borisov, A. V., Kilin, A. A., and Mamaev, I. S., Rolling of a Homogeneous Ball over a Dynamically Asymmetric Sphere, Regul. Chaotic Dyn., 2011, vol. 16, no. 5, pp. 465­483. 4. Borisov, A. V., Kilin, A. A., and Mamaev, I. S., Generalized Chaplygin's Transformation and Explicit Integration of a System with a Spherical Support, Regul. Chaotic Dyn., 2012, vol. 17, no. 2, pp. 170­190. 5. Duistermaat, J. J., Chaplygin's Sphere, arXiv:math/0409019v1 [math.DS] 1 Sep 2004.
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6. Kilin, A. A., The Dynamics of Chaplygin Ball: The Qualitative and Computer Analysis, Regul. Chaotic Dyn., 2001, vol. 6, no. 3, pp. 291­306. 7. Bolsinov, A. V., Borisov, A. V., and Mamaev, I. S., Topology and Stability of Integrable Systems, Uspekhi Mat. Nauk, 2010, vol. 65, no. 2, pp. 71­132 [Russian Math. Surveys, 2010, vol. 65, no. 2, pp. 259­318]. 8. Bolsinov, A. V. and Fomenko, A. T., Integrable Hamiltonian Systems: Geometry, Topology and Classification, Bo ca Raton, FL: CRC Press, 2004. 9. Borisov, A. V., Mamaev, I. S., and Marikhin, V.G., Explicit Integration of one Problem in Nonholonomic Mechanics, Dokl. Akad. Nauk, 2008, vol. 422, no. 4, pp. 475­478 [Dokl. Phys., 2008, vol. 53, no. 10, pp. 525­528]. 10. Borisov, A. V. and Fedorov, Yu. N., On Two Mo dified Integrable Problems of Dynamics, Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., 1995, no. 6, pp. 102­105 (Russian); see also: http://ics.org.ru/do c?pdf=384&dir=e (English transl.). 11. Borisov, A. V. and Tsygvintsev, A. V., Kowalewski Exponents and Integrable Systems of Classic Dynamics: 1, 2, Regul. Chaotic Dyn., 1996, vol. 1, no. 1, pp. 15­37 (Russian). 12. The Clebsch System: Separation of Variables and Explicit Integration?: Col lected Papers, A. V. Borisov, A. V. Tsiganov (Eds.), Moscow­Izhevsk: R&C Dynamics, Institute of Computer Science, 2009, pp. 7­20 (Russian). 13. Fundamental and Applied Problems in the Theory of Vortices, A. V. Borisov, I. S. Mamaev, M. A. Sokolovskiy (Eds.), Moscow­Izhevsk: R&C Dynamics, Institute of Computer Science, 2003 (Russian). 14. Kharlamov, M. P., Topological Analysis of Integrable Problems of Rigid Body Dynamics, Leningrad: Leningr. Gos. Univ., 1988. 15. Tsyganov, A. V., One Invariant Measure and Different Poisson Brackets for Two Non-Holonomic Systems, Regul. Chaotic Dyn., 2012, vol. 17, no. 1, pp. 72­96. 16. Chaplygin, S. A., On a Ball's Rolling on a Horizontal Plane, Math. Sb., 1903, vol. 24, no. 1, pp. 139­168 [Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 131­148]. 17. Bolsinov, A. V., Borisov, A. V., and Mamaev, I. S., Bifurcation Analysis and the Conley Index in Mechanics, Regul. Chaotic Dyn., 2012, vol. 17, no. 5 , pp. 451­478. 18. Bolsinov, A. V., Borisov, A. V., and Mamaev, I. S., Rolling of a Ball without Spinning on a Plane: The Absence of an Invariant Measure in a System with a Complete Set of Integrals, Regul. Chaotic Dyn., 2012, vol. 17, no. 6, pp. 571­579.

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