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ISSN 1560-3547, Regular and Chaotic Dynamics, 2014, Vol. 19, No. 6, pp. 718­733. c Pleiades Publishing, Ltd., 2014.

The Reversal and Chaotic Attractor in the Nonholonomic Mo del of Chaplygin's Top
Alexey V. Borisov1* , Alexey O. Kazakov2** , and Igor R. Sataev3
1 Udmurt State University, ul. Universitetskaya 1, Izhevsk, 426034 Russia; Moscow Institute of Physics and Technology, Inststitutskii per. 9, Dolgoprudnyi, 141700 Russia 2 Lobachevsky State University of Nizhny Novgorod, pr. Gagarina 23, Nizhny Novgorod, 603950 Russia 3

***

Saratov Branch of Kotelnikov's Institute of Radio-Engineering and Electronics of RAS ul. Zelenaya 38, Saratov, 410019 Russia
Received August 26, 2014; accepted Septemb er 8, 2014

Abstract--In this paper we consider the motion of a dynamically asymmetric unbalanced ball on a plane in a gravitational field. The point of contact of the ball with the plane is sub ject to a nonholonomic constraint which forbids slipping. The motion of the ball is governed by the nonholonomic reversible system of 6 differential equations. In the case of arbitrary displacement of the center of mass of the ball the system under consideration is a nonintegrable system without an invariant measure. Using qualitative and quantitative analysis we show that the unbalanced ball exhibits reversal (the phenomenon of reversal of the direction of rotation) for some parameter values. Moreover, by constructing charts of Lyaponov exponents we find a few types of strange attractors in the system, including the so-called figure-eight attractor which belongs to the genuine strange attractors of pseudohyperbolic type. MSC2010 numbers: 37J60, 37N15, 37G35, 70E18, 70F25, 70H45 DOI: 10.1134/S1560354714060094 Keywords: rolling without slipping, reversibility, involution, integrability, reversal, chart of Lyapunov exponents, strange attractor.

1. INTRODUCTION The problem of motion of a p erfectly rigid b ody on a plane in a gravitational field can b e considered in the context of two mathematical models: a nonholonomic model and a model with friction. In the former case, the b ody moves on a (p erfectly rough) plane without slipping. The absence of slipping is provided by the force of friction, which, however, does not p erform any work. In the latter case, slipping is p ossible and, therefore, the forces of friction are dissipative. An advantage of nonholonomic models is that these models are, as a rule, simpler than models with dissipative friction and, therefore, help to explain the nature of dynamical phenomena in many problems. For example, the nonholonomic model of a Celtic stone helps to explain the nature of the reversal characterized by the change of the direction of rotation of the stone to the opp osite when it rotates ab out the vertical axis in the "inappropriate" direction. This phenomenon has b een known for a long time and evidently was first describ ed in the work of G. Walker [1], while the explanation of this phenomenon was recently given by I. S. Astap ov, A. V. Karap etyan and A. P. Markeev [2­4] within the framework of the nonholonomic model. However, some phenomena of rigid b ody dynamics cannot b e explained within the nonholonomic model. The flip-over of the
* ** ***

E-mail: borisov@rcd.ru E-mail: kazakovdz@yandex.ru E-mail: sataevir@rambler.ru

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tipp e top is one of the most illustrative examples. This flip-over is caused by the dissipative force of friction [5, 6] and so cannot b e explained within the nonholonomic model. In this pap er we consider the nonholonomic model of motion on a plane of a dynamically asymmetric ball whose center of mass does not lie on any of the principal planes of inertia. In [7], problems of controllability of this ball were investigated and the term Chaplygin's top was introduced to refer to the ball. This term will b e used in this pap er. We note that Chaplygin's ball is the most complicated ball with resp ect to mass distribution. There are several sp ecial cases dep ending on the typ e of displacement of the center of mass and on the mass distribution inside the ball. When the center of mass of the ball coincides with its geometrical center, such a ball is called Chaplygin's bal l. The nonholonomic model of Chaplygin's ball was studied in the early 20th century by S. A. Chaplygin [8], who proved integrability of this problem and found the first (area, angular momentum and total energy) integrals and an invariant measure. The motion of Chaplygin's ball in absolute space is studied in [9, 10], where the equations of the contact p oint of the ball are describ ed and the conditions for finite and infinite motions of this ball are determined. In a generic case, for a ball with a displaced center of mass, the first integrals found by S. A. Chaplygin disapp ear [11]. An exception is the so-called Routh bal l -- a ball whose center of mass is displaced along one of the axes of inertia, while the principal momenta of inertia corresp onding to the other two axes are equal. Such a ball was first explored in [12], where the integrability of the nonholonomic model was proved and the first (total energy, Jellett's and Routh's) integrals were found. Generally the ball whose center of mass is displaced along one axis of inertia is called rock'n'rol ler. This term was introduced in [13], where the dynamical phenomenon of recession -- reversal of precession -- was investigated. Figure 1 shows the hierarchy of balls of different typ es dep ending on the typ e of dynamical asymmetry and displacement of the center of mass. In another nonholonomic approach where the p oint of contact of a ball with a plane is sub ject to a "non-spin" constraint1) , in addition to the non-slip Fig. 1. Hierarchy of the balls. constraint, the problem of motion of Chaplygin's top is generally also nonintegrable [15]. Moreover, in [16, 17] it is shown that the dynamics of this problem significantly dep ends on the typ e of involutions which admits a corresp onding nonholonomic system. The numb er of involutions is determined by the typ e of displacement of the center of mass. For different typ es of displacement, strange attractors with weak dissipation and another typ e of dynamical chaos -- the so-called mixed dynamics [18­20] -- were discovered in this problem. From a dynamical p oint of view, the nonholonomic model of a Celtic stone, in which Hamiltonianlike structures (for example, mixed dynamics) [21] as well as different limit regimes, including strange attractors [23, 24], were found, is the most similar to the model of Chaplygin's top. Of sp ecial note is the discovery of the Lorenz-like attractor in the nonholonomic model of a Celtic stone [25, 26]. This typ e of strange attractors b elongs to the class of genuine attractors and has not b een observed in any problems of rigid b ody dynamics b efore. In this pap er we explore the dynamics of Chaplygin's top on a plane in the framework of a nonholonomic model. A more complex model which takes into account dissipative forces of friction will b e investigated in our future work. The pap er is organized as follows. In Section 2 the nonholonomic model of Chaplygin's top is introduced: equations of motion, first integrals and involutions are presented, and the procedure for constructing a Poincar´ map is describ ed. In Section 3 we investigate the reversal dynamics e of Chaplygin's ball associated with the existence of asymptotically stable and completely unstable equilibria which corresp ond to p ermanent rotations ab out the vertical axis in opp osite directions.
1)

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We show that the existence of these equilibria is the main reason for reversal. In Section 4 the focus is on the scenario of the app earance of the so-called figure-eight attractor. This is a new typ e of attractor which, to our knowledge, has not yet b een observed in any physical models. In App endix 1 we give a full list of involutions for balls with different typ es of displacement of the center of mass. Finally, in App endix 2 the procedure of searching for equilibria in the nonholonomic system is describ ed. 2. THE NONHOLONOMIC MODEL OF CHAPLYGIN'S BALL 2.1. Equations of Motion and First Integrals Let us introduce a coordinate system Cxy z attached to the center of mass of a ball (see Fig. 2) and chosen in such a way that the inertia tensor of Chaplygin's ball is a diagonal matrix I = diag(I1 ,I2 ,I3 ), where Ii are the principal moments of inertia. In the nonholonomic model the contact p oint of the ball with a plane is sub ject to the nonholonomic constraint which forbids slipping of the ball. If we denote the velocity of the center of mass and the angular velocity of the ball by v and , resp ectively, and the radius vector connecting the center of mass of the ball with the point of contact P by r , then the condition of absence of slipping can b e represented as
Fig. 2. Chaplygin's top on a plane.

v + â r = 0.

(2.1)

It is well known (see, for example, [27]) that the equations of motion of a rigid b ody in the variables M and , where M is the angular momentum relative to the p oint of contact and is the vertical unit vector, can b e written as M = M â + mr â ( â r )+ mgr â = â . (2.2)

Here m is the mass of the ball and g is the acceleration of gravity. In our case, since the b ody is a ball with a displaced center of mass (see Fig. 2), we have r = -R - a , (2.3) where the vector a = (a1 ,a2 ,a3 ) sp ecifies the displacement of the center of mass. As we know (see, for example, [11]) the vectors M and are related by M = I + mr â ( â r ). (2.4) If we express the vectors r , r and using (2.3) and (2.4), we can get the system of ordinary differential equations (M , ) = F (M , ,), (2.5) which dep ends on the parameters characterizing the physical and dynamical prop erties of Chaplygin's ball. In a generic case the system (2.2) admits only two integrals 1 (M , ) - mg(r , ), G = ( , ). (2.6) 2 The former is an energy integral and the latter is a geometrical integral. Due to normalization the geometrical integral is fixed by 1 (G = 1). Thus, for the system to b e integrable by the Euler ­ Jacobi theorem, we need two additional integrals and an invariant measure. From [11] it follows that the system under consideration is integrable only in two sp ecial cases: E= · Chaplygin's ball (a = 0); · Routh's ball (I1 = I2 ,a1 = a2 = 0).
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In the other cases the system (2.2) admits neither an additional integral nor an invariant measure, and hence, on the common level set of two integrals (2.6), the system can exhibit dynamical b ehavior typical of dissipative systems. Therefore, for the nonholonomic model of Chaplygin's top one can exp ect the existence of asymptotically stable regimes such as equilibria, limit cycles, invariant tori and even strange attractors. ´ 2.2. THE POINCARE MAP On the common level set of two integrals (2.6) the phase volume of the system is a fourdimensional manifold M4 = {(M , ) : G ( ) = 1, E (M , ) = E0 = const}, (2.7) which is homeomorphic to S3 â S3 . For the parameterization of this manifold it is very convenient to use the Andoyer ­ Deprit variables (see, for example, [27]) which are related to (M , ) by M1 = 1 = 2 = 3 = G2 - L2 sin l, M2 = H G H G HL - G2 1- 1- L2 L + 2 G G L2 L + G2 G L2 G2 1- 1- G2 - L2 cos l, M3 = L, H2 cos g sin l + G2 H2 cos g cos l - G2 1- 1- H2 sin g cos l G2 H2 sin g sin l G2 (2.8)

1-

1-

H2 cos g. G2

Note that in the new variables the condition ( , ) = 1 holds automatically. Thus, (2.8) defines one-to-one corresp ondence b etween the variables ((M , ) : ( , ) = 1) and (L, H, G, g, l) everywhere except on four planes L/G = ±1 and H/G = ±1 on which l and g resp ectively are undefined. In the Andoyer ­ Deprit variables the plane g = g0 can b e considered as a secant of the flow defined by (2.2). Then on this secant we can determine a Poincar´ map. To do so, we choose e L H the coordinates l (which is 2 -p eriodic), G [-1, 1] and G [-1, 1] on the g = g0 . Thus, we can investigate the dynamics of Chaplygin's ball using the three-dimensional Poincar´ map e x = Fg0 (x), x = ¯ l, LH , GG , (2.9)

which is defined everywhere except on the ab ove-mentioned four planes. 2.3. Reversibility and Involutions In [16] it was shown that the dynamics of Chaplygin's top moving on a plane without slipping and spinning (rubb er-b ody) significantly dep ends on the numb er of involutions in the system, which is defined by the numb er of zero comp onents of the vector a. When all ai = 0, the system admits only one involution. In this case the dynamics of the system exhibit strange attractors. If one, or two, of the comp onents of the vector a is zero, additional involutions app ear. There are no strange attractors in this case, but one can observe another typ e of dynamical chaos ­ the socalled mixed dynamics, which is similar to Hamiltonian chaos but differs from it by the existence of asymptotically stable and unstable orbits of large p eriods [16, 17]. In this pap er we consider the dynamics of Chaplygin's ball moving on a plane only without slipping (spinning is p ossible) and obtain in a sense similar results. For arbitrary parameter values the system (2.2) is reversible with resp ect to the trivial involution R0 : M -M , ,t -t, (2.10) which is resp onsible for the reversal of the angular momentum (and, therefore, the angular velocities ). The involution R0 significantly affects the dynamics of the system. Due to R0 , for
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each stable dynamical regime there exists a symmetric unstable one with reversed angular velocities. Thus, for each attractor there is a rep eller in the system. The system (2.2) can admit, along with R0 , additional involutions whose numb er equals the numb er of zero comp onents of the vector a. So, Chaplygin's ball has the maximal numb er of involutions. A full list of the involutions is describ ed in App endix A. In this work we are interested only in the case of arbitrary displacement of the center of mass, since only in this case can we find interesting features of the dissipative dynamics2) . Note that each involution of the flow system gives rise to an involution for the Poincar´ map if e a manifold invariant under the involution is chosen as a section. If one takes g0 = 0 as the section, e then the trivial involution R0 gives rise to an involution r0 for the Poincar´ map (2.9) LH H L - , - ,l l + . (2.11) G GG G Thus, if a set of tra jectories A b elongs to some attractor on the Poicar´ map, then r0 (A) b elongs e to the rep eller. r0 : 3. THE REVERSAL The first thing that attracted our attention in the nonholonomic model of Chaplygin's ball is the reversal -- the prop erty of reversing the direction of rotation (the sing of vector ) of the ball when it rotates ab out the vertical axis in an "inappropriate" direction. It has b een known for a long time that the reversal occurs in real physical exp eriments with Celtic stones, which are tops of a different typ e [1, 28]. However, the explanation of this phenomenon in the framework of the nonholonomic model was given quite recently in [2­4]. In this model, asymptotically stable equilibrium corresp onds to rotations ab out the vertical axis in "appropriate" directions (for example, clockwise) and asymptotically unstable equilibrium corresp onds to rotations ab out the same vertical axis in "inappropriate" directions (counterclockwise). Thus, if we spin the stone fast enough ab out this vertical axis counterclockwise, the stone starts, after a few rotations, to rock and oscillate, and then these oscillations cause it to spin clockwise. Unlike a Celtic stone, Chaplygin's ball is a completly geometrically symmetric b ody. Therefore, the existence of asymptotically stable and unstable states of equilibrium3) differing only in the direction of rotation ab out the same vertical axis is, in our view, a surprising and remarkable prop erty. Further, we note that even for a Celtic stone the reversal can b e of different nature dep ending on the geometrical and physical prop erties of the stone and also on how fast it is spun. In some cases the unstable vertical rotations can turn into stable rotations with a small precession. In other cases this precession can b e strong enough. We call the reversal of such typ e periodic reversal and note that the explanation of such a motion can also b e given in the framework of the nonholonomic model, in which (for some parameter values) the stable equilibrium undergoes the Andronov ­ Hopf bifurcation, b ecoming unstable, and a stable limit cycle arises in its neighb orhood. In this case, after the stone is spun in an "inappropriate" direction, its final motion reaches the limit cycle, and moving in this cycle, the stone rotates in the opp osite direction with small precession. For some parameter values there coexist 2 stable limit cycles in the system and the dynamics in the second cycle corresp ond to the motion with large precession. In this pap er we are interested in a genuine reversal which implies that Chaplygin's top, spun fast enough ab out a certain vertical axis, reverses the angular velocities and continues stable rotations in the opp osite direction4) . For the existence of the genuine reversal of Chaplygin's ball we require that the following two conditions b e satisfied.
2)

3)

4)

When, for example, a1 = 0, a2 = 0 and a3 = 0, preliminary investigations show that the phase space of the system is foliated by invariant tori and no signs of the dissipative dynamics are observed. Here and in the sequel the equilibrium state of Chaplygin's ball is understood to mean a state of equilibrium of the system (2.2). Note that in the case at hand only the direction of rotation of Chaplygin's ball reverses, while the axis of rotation of the ball, as opp osed to that of a "sup erCeltic stone" [29], remains unchanged. REGULAR AND CHAOTIC DYNAMICS Vol. 19 No. 6 2014


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· The stable rotation of the top ab out a constant vertical axis with constant angular velocity is p ossible only in a stable equilibrium of the system (2.2). Therefore, the system must admit a stable equilibrium. If this condition holds, then due to the involution R0 the system will also admit an asymptotically unstable equilibrium corresp onding to unstable rotations (in the opp osite direction) ab out the same axis. · There must b e initial p oints for which the tra jectories of the system (2.2) reach a stable equilibrium in forward time and a neighb orhood of an unstable equilibrium in backward time. According to the ab ove conditions, we describ e the algorithm for investigating the p ossibility of a genuine reversal. First we analyze the equilibria of the system (2.2) and their stability. In a state of equilibrium the vectors M = M and = satisfy the following system of equations: â = 0, M â + mr â ( â r )+ mgr â = 0. (3.1)

It follows from the first equation that in a state of equilibrium the vectors and are collinear, i.e., = ( = 0, otherwise the ball does not move). Substituting this dep endence into the second equation of the system and using (2.3) and (2.4), we can obtain the relation 2 I - m[2 (R +(a, )) + g] a â = 0, which leads to the equation relating the vectors and a through the coefficients and : (2 I - E) = m[2 (R +(a, )) + g]a.


(3.2)

Thus, the vectors , , and hence (by (2.4)) also M , are defined in terms of the system parameters and the coefficients and . To determine the unknowns and , we use two first integrals (2.6). Assuming E = E0 = const, we obtain the system of two equations G ( ( , )) = 1, E ( ( , ), ( , )) = E0 . (3.3)

The procedure of finding the unknowns and is describ ed in App endix B. Unfortunately, it is imp ossible to solve this system analytically. However, we have succeeded in estimating the numb er of its solutions for any parameter values and hence in determining the numb er of equilibrium states. According to (3.2), quadratically dep ends on , and hence each solution of the system (3.3) corresp onds to two equilibrium states: O+ = ( , ) and O- = (- , ). Due to reversibility, if the equilibrium state O+ has the eigenvalues + = = ( ,... , ), then O- will have the 1 6 eigenvalues - = - = (- ,... , - ). Thus, for stability analysis it is sufficient to investigate 1 6 one of the pair of equilibrium states corresp onding to a sp ecific solution of the system (3.3). The characteristic equation of the linearized system has the form 2 (4 + a3 + b2 + c + d) = 0, (3.4) where a, b, c and d are some coefficients dep ending on the system parameters. We note that the characteristic equation has two zero roots resp onsible for the existence of two integrals (2.6). On the level set of these integrals the equilibrium of the system is asymptotically stable if all nonzero roots of the characteristic equation (3.4) have a negative real part. Below we present numerical results for the equilibria of the system (2.2) on the parameter plane Q = (E0 ,a3 ): E0 [200, 800], a3 [0.1, 2.0]; the other parameters can b e fixed as follows: I1 = 2, I2 = 6, I3 = 7, m = 1, g = 100, R = 3, a1 = 1, a2 = 1.5. (3.5) To find the unknowns (, ) of the system (3.3) and to linearize the system in a neighb orhood of the equilibrium states, we have used the software package MAPLE. As the parameters (E0 ,a3 ) change in the region Q, the numb er of real solutions (corresp onding to the equilibrium states) of the system (3.3) varies from one to two. In Fig. 3a the grey area indicates the range of parameters for which there exists a unique solution of the system (1 ,1 ) and the dark grey area corresp onds
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+ - to the region in which the second solution (2 ,2 ) app ears. Let O1 and O1 denote the equilibrium + - states corresp onding to the solution (1 ,1 ) and let O2 and O2 denote the equilibrium states for the solution (2 ,2 ). Numerical calculations show that in the region Q all equilibrium states p ossess + two pairs of nonzero complex-conjugate roots. Figure 3b shows a stability diagram for O1 . In the shaded region Q1 all nonzero eigenvalues have a negative real part, and hence the equilibrium state - is asymptotically stable (accordingly, O1 is completely unstable). When the parameter values pass + from Q1 to Q2 , one pair of complex-conjugate roots of the equilibrium state O1 passes into the right half-plane, the equilibrium state b ecomes a saddle-focus, and an asymptotically stable limit cycle is b orn in its neighb orhood. This limit cycle corresp onds to a motion of the top around some axis with a small precession (see Fig. 4)5) .

Fig. 3. On the parameter plane Q one can see: (a) regions of parameters for which there exist one (light grey) + and two (dark grey) equilibrium states, (b) stability diagram for the equilibrium state O1 corresp onding to the first solution of the system (3.3).

Fig. 4. (a) Visualization of the top's motion after the loss of stability of the equilibrium state through the Andronov ­ Hopf bifurcation for the parameters E0 = 500, a3 = 1.4. (b) A limit cycle is b orn around the equilibrium state that has b ecome unstable. The dynamics on this limit cycle corresp onds to the motion of the stone inside the "ring" with some precession.
5)

We note that the equilibrium state corresp onding to the vertical rotation of the Celtic stone loses stability in a similar fashion, when the initial energy decreases b elow some critical level [3]. REGULAR AND CHAOTIC DYNAMICS Vol. 19 No. 6 2014


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+ - Similar stability analyses have b een carried out for the equilibrium states O2 and O2 . Calculations show that for any parameter values these states are saddle­foci and hence cannot b e stable. Thus, for the parameter values under consideration the system can p ossess the only asymptotically stable equilibrium and the only completely unstable equilibrium. However, in some regions of parameters from Q1 the dynamics of Chaplygin's top exhibit a developed multistability -- the existence of various stable limit regimes dep ending on the initial conditions. For example, Fig. 5 shows three attractors (in addition to the stable equilibrium state + O1 ) coexisting for the same parameter values: a torus and two 2-round tori (the figure shows the attractors in the Poincar´ section, where the torus corresp onds to an invariant closed curve and the e 2-round torus corresp onds to an invariant curve of p eriod 2). Thus, for the same parameter values at least four attractors coexist in the system! So, it can happ en that the tra jectories run away from the neighb orhood of the unstable equilibrium state to one of the tori, and no genuine reversal is observed.

Fig. 5. Coexistence of the equilibrium state and another three attractors -- tori for the same values of parameters.

To find and classify the limit regimes, we have constructed charts of Lyapunov exp onents in forward and backward time on the parameter plane Q devided into 600 â 600 nodes. A tra jectory was launched from the same initial p oint6) P0 of each node. To preclude a transient process, the system was integrated for T = 105 time units, and then the Lyapunov exp onents were estimated on the interval T = 104 by the Benettin method [30]. Numerical solutions of the equations of motion were obtained with the help of the Dormand ­ Prince integrator [31], which uses the Runge ­ Kutta method of order 8 with an automatically variable size of the integration step. The local accuracy of integration for calculating the exp onents was chosen to b e 10-12 . At the last p oints, the right-hand side of the system (2.2) was also estimated to sp ecify the instant when the tra jectory enters into the neighb orhood of the equilibrium state. As a result of the calculations, the pixels on the chart have the following colors: black indicates an equilibrium state (all Lyapunov exp onents are negative or the norm of the right-hand sides of equations (2.2) b ecomes smaller than some threshold), dark blue corresp onds to the limit cycle (the largest exp onent vanishes), light blue is a torus (two largest exp onents vanish), and red corresp onds to the chaotic regime (the largest exp onent value is p ositive). Figures 6a and 6b show the charts of Lyapunov exp onents. The charts have b een constructed in forward and backward time from the initial p oint P0 . Analysis of the charts allows the conclusion that the parameter region for which a stable equilibrium state is a limit regime from the given p oint is much smaller than the parameter region from which an unstable equilibrium state is attainable
6)

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in backward time7) . This is illustrated in Fig. 6c, where the parameter regions for which a stable equilibrium state is attainable in forward time and an unstable equilibrium state is attainable in backward time are overlapp ed. In the intersection of these two regions, the nonholonomic model of Chaplygin's top demonstrates the phenomenon of a genuine reversal, which can b e clearly observed in numerical exp eriments. Figure 7 represents the time dep endencies of the angular velocities of Chaplygin's top for the parameters E0 = 500 and a3 = 1. The p oint P0 is chosen as the initial p oint, and the tra jectory is launched in b oth forward and backward time. Thus, according to the nonholonomic model, if the system parameters and the initial p oint are given in accordance with Fig. 6 and Chaplygin's top is spun around the vertical axis = in an "inappropriate" direction, then, after a transient process (which is rather long for some parameters) accompanied by oscillations and rolling motions, the top reverses the direction of rotation and rotates ab out the same axis in the opp osite direction.

Fig. 6. Charts of Lyapunov exp onents in forward and backward time, and their intersection on the plane Q.

Fig. 7. Time dep endences of the comp onents of the angular velocity for the genuine reversal at (E0 = 500, a3 = 1).

4. CHAOTIC DYNAMICS We now turn our attention to chaotic dynamics in the nonholonomic model of Chaplygin's top. As in the analysis of reversal, we shall consider the dynamics of the ball for the values of the parameters E0 and a3 from the region Q. The values of the other parameters are given according to (3.5). Analysis of the chart of Lyapunov exp onents has help ed to detect strange attractors and to
7)

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explore the scenarios of their app earance 8) . A detailed description of the algorithm for constructing the chart of Lyapunov exp onents is given at the end of the previous section. Here we only note that to accelerate the convergence to the steady-state dynamical regime and due to multistability of the system, the initial conditions in the internal nodes of the grid were chosen by using an inheritance scheme which implies that the state obtained by applying the algorithm in the previous node was used as the initial p oint in each subsequent node of the grid. The constructed chart of Lyapunov exp onents is presented in Fig. 8. We note that the construction of the chart was essentially influenced by the develop ed multistability, due to which the chart had to b e "glued together" from several pieces constructed by various typ es of scanning of the initial conditions. For convenience, we comment once again on the rules of coding of the regimes shown in Fig. 8. Periodic regimes (limit cycles), quasip eriodic regimes (tori) and chaotic regimes are shown in dark blue, light blue and red, resp ectively. The pixels colored black corresp ond to the states in which the tra jectories reach the equilibrium. The chart of regimes was constructed with inheritance, as describ ed ab ove, except that if the tra jectory reaches the stable equilibrium, the initial conditions at the next p oint were chosen to b e the same as those at the edge of the chart. This was done to reveal regimes differing from the stable equilibrium state.

Fig. 8. Chart of Lyapunov exp onents on the parameter plane Q.

Typical attractors for the Poincar´ map of the system under consideration are fixed and p eriodic e p oints, invariant curves and tori-chaos, which arise as a result of destruction of invariant curves according to the Afraimovich ­ Shilnikov scenario [33]. The white arrows on the chart of dynamical regimes indicate the paths along which one can observe sequences of bifurcations leading to the birth of strange attractors. A typical feature of the system is a visible absence of p eriod doublings for stable p eriodic p oints9) . The b oundaries of stability regions of these p oints are formed by bifurcation lines of saddle-node bifurcations and Neimark ­ Sacker bifurcations. Thus, the scenarios of transition to chaos in our system are mainly associated with the destruction of invariant curves. We consider one of the most interesting scenarios of chaos development, which is associated with the app earance of a figure-eight attractor. 4.1. Figure-eight Attractor The p ossibility of app earance of figure-eight (and Lorentz typ e) attractors in three-dimensional maps due to simple bifurcation scenarios was established in [35] (see also [36]). In these scenarios it
8)

9)

Generally sp eaking, it is convenient to use the charts of Lyapunov exp onents for the investigation of dissipative chaotic dynamics in various models [32]. This phenomenon is evidently typical of three-dimensional maps whose Jacobian is not too close to zero. For example, in [34] it was discovered that in the case of three-dimensional H´ on maps with the constant Jacobian en B > 1 no second p eriod doubling is observed for a stable fixed p oint. 3 Vol. 19 No. 6 2014

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was assumed that at first a fixed p oint is an attractor in the map. According to one of the scenarios, as the parameter changes, this p oint then undergoes a p eriod-doubling bifurcation, b ecoming a saddle with a one-dimensional unstable manifold, and a stable cycle of p eriod 2 is b orn in its neighb orhood. Then this cycle loses stability (in principle, it does not matter how exactly), and the stable and unstable manifolds of the saddle p oint b egin to intersect. The resulting "homoclinic attractor" is, dep ending on the multipliers of the saddle p oint, either a Lorentz typ e or a figure-eight attractor (see Fig. 9).

Fig. 9. The simplest scenarios of the birth of two typ es of homoclinic attractors.

In our case, a homoclinic figure-eight attractor is b orn, and at the initial stage the scenario of its birth is somewhat different from that describ ed in [35]. We now consider the stages of app earance of this attractor. Let us fix a3 = 1.9 and analyze the bifurcations arising on the route A in the chart of Lyapunov exp onents (Fig. 8). At first, when 417.5 E1 < E < E2 455.60, the attractor is the p oint (o1 ,o2 ) of p eriod 2, which is b orn as a result of saddle-node bifurcation together with the saddle p oint (s1 ,s2 ). The map (2.9) also has the fixed saddle p oint S1 , which is located b etween o1 and o2 (see Fig. 10a). The p oint S1 is a saddle-focus up to E E3 = 456.162, whereup on its unstable complex-conjugate multipliers b ecome real negative. At E E4 = 456.30 a p eriod-2 p oint (s1 ,s2 ) merges into the saddle S1 (as a result of the sub critical doubling bifurcation) and the saddle itself changes its typ e from (1, 2) to (2, 1) (its unstable manifold b ecomes one-dimensional). After this bifurcation the saddle S1 has multipliers 1 , 2 and such that < -1 < 2 < 0 < 1 < 1. Now the unstable manifold is coiled around the invariant curve (l1 ,l2 ) of p eriod 2 (see Fig. 10b). This curve arises due to the Neimark ­ Sacker bifurcation from the stable cycle (o1 ,o2 ) at E 455.60. With further increase of the parameter E the invariant curve L undergoes a series of "torus-doubling" bifurcations (see Figs. 10c and 10d) and then decays to form a torus-chaos (Fig. 10e). Soon after that, the unstable manifold of the saddle S1 b egins to intersect with the stable manifold, and a strange attractor which is visually similar to the figure-eight attractor is formed. Figure 10f shows a p ortrait of the detected attractor for E = 457.913. The p ortrait has b een obtained by iterating a p oint launched from the neighb orhood of the saddle S1 with coordinates l = 0.514231, L/G = -0.700259, H/G = -0.930815.
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(a) E = 455

(b) E = 457

(c) E = 457.904

(d) E = 457.910

(e) E = 457.911

(f ) E = 457.913

Fig. 10. The main stages of the app earance of the figure-eight attractor.

We present a numb er of quantitative and qualitative characteristics of the detected attractor. The multipliers of the saddle S1 for E = 457.913, a3 = 1.9 are given b elow: 1 0.98885, 2 -0.99732, (4.1) -1.00907. Thus, |1 | < |2 | < 1 < | |. The multiplier 1 resp onsible for strong compression is p ositive, which is characteristic of the figure-eight attractors and distinguishes them from the Lorentz typ e attractors (see Fig. 9). The condition |2 || | > 1 implies the extension of areas that are transversal to the direction of strong compression, and is indicative of pseudohyp erb olicity of the detected attractor. The Lyapunov exp onents of the tra jectory randomly chosen on the attractor take the following values: 1 0.00063, 2 -0.00003, (4.2) 3 -0.00492. The condition 1 +2 > 0 is also indicative of pseudohyp erb olicity, and the negative sum of all 3 exp onents p oints to a volume compression typical of the genuine strange attractors. The analysis made allows one to classify the detected attractor as a pseudohyp erb olic figure-eight attractor. 5. CONCLUSION In this pap er we have investigated the nonholonomic model of Chaplygin's top, within which we have detected the phenomenon of reversal. It is well known that the nonholonomic model is an idealization in which the friction force does not p erform any work. However, the analysis of such a model help ed to explain the cause of the app earance of reversal for Celtic stones. Whereas in the case with Celtic stones the phenomenon of reversal was at first noticed in a real exp eriment and only much later was explored on the basis of a mathematical model, the situation with Chaplygin's top was the opp osite. In the latter case, we have managed so far to detect the phenomenon of reversal only in the nonholonomic model. In our further research we plan to construct a more realistic model taking into account dissipative friction forces. In the case of success, attempts will b e made to design a top and to simulate the phenomenon of reversal in a real exp eriment.
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In addition to revealing the reversal, the nonholonomic model turned out to b e very interesting for the investigation of chaotic dynamics. Analysis of the chart of Lyapunov exp onents allowed us to detect several typ es of strange attractors. In this pap er we considered the scenario of the onset of a figure-eight attractor (which is, in our opinion, the most interesting of the attractors detected so far). In our future work we plan to carry out investigations of other scenarios (typical of the model considered) of onset of strange attractors. APPENDIX A. ON INVOLUTIONS We recall the definition of reversibility and involution for a flow system. Supp ose the equations of motion (2.2) have the form X = v (X ), where X = (M1 ,M2 ,M3 ,1 ,2 ,3 ). The map of the phase space R(X ): X X is said to b e an involution for the flow v (X ) if d (R(X )) = -v (R(X )), R R = id. (5.1) dt In this case, the flow v (X ) is called reversible with resp ect to the involution R(X ). Dep ending on the typ e of transformation of the variables (M , ), all additional (to R0 ) involutions can b e divided into two classes: · involutions resp onsib the angle , 1 : 2 : 3 : le for the rotation of the ball ab out one of the axes of inertia through M (M1 ,M2 , -M3 ), (-1 , -2 ,3 ), t -t, M (M1 , -M2 ,M3 ), (-1 ,2 , -3 ), t -t, M (-M1 ,M2 ,M3 ), (1 , -2 , -3 ), t -t,

(5.2)

· involutions resp onsible for the reflection of the ball with resp ect through a pair of the axes of inertia of the ball, 1 : M (M1 ,M2 , -M3 ), (1 ,2 , -3 2 : M (M1 , -M2 ,M3 ), (1 , -2 ,3 3 : M (-M1 ,M2 ,M3 ), (-1 ,2 ,3

to one of three planes passing ), t -t, ), t -t, ), t -t.

(5.3)

If the center of mass of the ball is displaced only along one axis, then the system (2.2) admits three additional (to R0 ) involutions: involution resp onsible for the rotation of the ball through the angle along the axis of displacement (one of i , i = 1,... , 3), and two involutions resp onsible for the reflection of the ball with resp ect to the planes passing through the axis of displacement and one of the two axes of inertia (two of i , i = 1,... , 3). If the center of mass of the ball is displaced along the two axes, the system has the only additional involution resp onsible for the reflection of the ball with resp ect to the plane passing through the axes of displacement (one of i , i = 1,... , 3). In the case of arbitrary displacement of the center of mass of the ball (when all comp onents a1 , a2 and a3 are nonzero) the system does not p ossess any additional involutions. We recall the definition of reversibility and involution for maps. A transformation r (x): x x is called an involution for the map (2.9) if
- Fg0 r = r Fg01 .

(5.4)

In this case, the map Fg0 (x) is called reversible with resp ect to the involution r (x). After an appropriate choice of a secant the involutions of the flow system (5.2) and (5.3) generate involutions for the Poincar´ map (2.9): e · those generated from 1 , 2 an L 1 : G L 2 : G L 3 : G d 3 - L , G H H - , G G H H - , G G H H - , G G l l, l - l, l -l.
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L , G L , G

(5.5)

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· those generated from 1 , 2 and 1 :

3

L L - , G G L L , 2 : G G L L , 3 : G G

H H , G G H H , G G H H , G G

l l, l - l, l -l. (5.6)

APPENDIX B. ON THE SEARCH FOR EQUILIBRIUM STATES To find equilibrium states of the system (2.2), it is necessary to define the unknowns and using the expressions for first integrals and the relations (3.1)­(3.2). The relation (3.2) can b e rewritten as g (5.7) I - 2 = m R +(a, )+ 2 a, whence, on introducing the notation = we obtain the relation = m R +(a, )+ g Ia. We p erform a scalar multiplication of the last relation by a and express (a, ) and : (a, ) = = m(R + g)(Ia, a) 1 - m(Ia, a) 1 - m(Ia, a) . , (5.8) 1 , 2 = , 2 I= I- 2
-1

,

m(R + g)(Ia, a)

Further, using the normalization condition (the geometrical integral is fixed on the unit level set) and denoting R + g by , we write the first equation of the system (3.3) as m2 2 (Ia, Ia) (1 - m(Ia, a))2 - 1 = 0. (5.9)

Now we transform the expression for the energy integral (2.6). In a state of equilibrium the vector M , expressed by the formula (2.4), can b e represented as M = [I + m(R(a, ) + a2 - Ra - (a, )a)]. Then the expression for the energy integral b ecomes 1 E0 = 2 [(I , )+ m(a2 - (a, )2 )] + mg((a, )+ R). 2 Performing a scalar multiplication of (5.7) by , we obtain an expression for (I , ): (I , ) = + m(R +(a, )+ g)(a, ). Substituting this expression into (5.11) and using (5.8), we obtain an expression for the second equation of the system (3.3): 3m2 (Ia, a) 1 - m(Ia, a) 2 + 2mR - 2mR2 (Ia, a) 2E0 2E0 R + ma 2 + + - - 2mR2 = 0. g g 1 - m(Ia, a)
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(5.10)

(5.11)

(5.12)

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Thus, the system (3.3) is formed by Eqs. (5.9) and (5.12). Subtracting from Eq. (5.12) Eq. (5.9) multiplied by 3(Ia, a)(1 - m(Ia, a)) (Ia, Ia) we obtain an equation relating to : 2mR2 - = 2E0 R 3(1 - m(Ia, a))(Ia, a) - - ma 2 - g (Ia, Ia) 2mR 1 - m(Ia, a) 1 - m(Ia, a) 2E0 - g ,

.

Substituting into (5.9), we obtain for an irrational equation of high degree. For a numerical solution of this equation with sp ecific parameter values we have applied the software package Maple. ACKNOWLEDGMENTS The authors wish to acknowledge the valuable advice and consultation given by A. A. Kilin, S. P. Kuznetsov, I. S. Mamaev, and I. A. Bizyaev. Sp ecial thanks are due to S. V. Gonchenko for his invaluable help in writing the manuscript of the pap er. The work of A. V. Borisov was supp orted by the Ministry of Education and Science of the Russian Federation within the framework of the basic part of the state assignment to institutions of higher education. The work of A. O. Kazakov on Section 3 was supp orted by the grant of the Russian Scientific Foundation No 14-12-00811, the work of Section 4.1 was partially supp orted by the grant of the Russian Scientific Foundation 14-41-00044 and by the grant of the President of the Russian Federation for supp ort of young doctors of science MD-2324.2013.1. The remaining part of the work of A. O. Kazakov was supp orted by the Ministry of Education and Science (pro ject No 2000). The work of I. R. Sataev was supp orted by the grant of the President of the Russian Federation for supp ort of leading scientific schools NSh-1726.2014.2. REFERENCES
1. Walker, G. T., On a Curious Dynamical Property of Celts, Proc. Cambridge Phil. Soc., 1895, vol. 8, pt. 5, pp. 305­306. 2. Astapov, I. S., On Rotation Stability of Celtic Stone, Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., 1980, no. 2, pp. 97­100 (Russian). 3. Karapetyan, A. V., On Realizing Nonholonomic Constraints by Viscous Friction Forces and Celtic Stones Stability, J. Appl. Math. Mech., 1981, vol. 45, no. 1, pp. 30­36; see also: Prikl. Mat. Mekh., 1981, vol. 45, no. 1, pp. 42­51. 4. Markeev, A. P., The Dynamics of a Rigid Bo dy on an Absolutely Rough Plane, J. Appl. Math. Mech., 1983, vol. 47, no. 4, pp. 473­478; see also: Prikl. Mat. Mekh., 1983, vol. 47, no. 4, pp. 575­582. 5. Kane, T. R. and Levinson, D. A., A Realistic Solution of the Symmetric Top Problem, J. Appl. Mech., 1978, vol. 45, no. 4, pp. 903­909. 6. Aleshkevich, V. A., Dedenko, L. G., and Karavaev, V. A., Lectures on Solid Mechanics, Moscow: Mosk. Gos. Univ., 1997 (Russian). 7. Shen, J., Schneider, D. A., and Blo ch, A. M., Controllability and Motion Planning of a Multibody Chaplygin's Sphere and Chaplygin's Top, Int. J. Robust Nonlinear Control, 2008, vol. 18, no. 9, pp. 905­ 945. 8. Chaplygin, S. A., On a Ball's Rolling on a Horizontal Plane, Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 131­148; see also: Math. Sb., 1903, vol. 24, no. 1, pp. 139­168. 9. Kilin, A. A., The Dynamics of Chaplygin Ball: The Qualitative and Computer Analysis, Regul. Chaotic Dyn., 2001, vol. 6, no. 3, pp. 291­306. 10. Borisov, A. V., Kilin, A. A., and Mamaev, I. S., The Problem of Drift and Recurrence for the Rolling Chaplygin Ball, Regul. Chaotic Dyn., 2013, vol. 18, no. 6, pp. 832­859. 11. Borisov, A. V., Mamaev, I. S., and Kilin, A. A., The Rolling Motion of a Ball on a Surface: New Integrals and Hierarchy of Dynamics, Regul. Chaotic Dyn., 2002, vol. 7, no. 2, pp. 201­219. 12. Routh, E. J., A Treatise on the Dynamics of a System of Rigid Bodies: P. 2. The Advanced Part, 6th ed., New York: Macmillan, 1905; see also: New York: Dover, 1955 (reprint).
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13. Lynch, P. and Bustamante, M. D., Precession and Recession of the Ro ck'n'Roller, J. Phys. A, 2009, vol. 42, no. 42, 425203, 25 pp. 14. Koiller, J. and Ehlers, K. M., Rubber Rolling over a Sphere, Regul. Chaotic Dyn., 2007, vol. 12, no. 2, pp. 127­152. 15. Borisov, A. V., Mamaev, I. S., and Bizyaev, I. A., The Hierarchy of Dynamics of a Rigid Bo dy Rolling without Slipping and Spinning on a Plane and a Sphere, Regul. Chaotic Dyn., 2013, vol. 18, no. 3, pp. 277­328. 16. Kazakov, A. O., Strange Attractors and Mixed Dynamics in the Problem of an Unbalanced Rubber Ball Rolling on a Plane, Regul. Chaotic Dyn., 2013, vol. 18, no. 5, pp. 508­520. 17. Kazakov, A. O., On the Chaotic Dynamics of a Rubber Ball with Three Internal Rotors, Nonlinear Dynamics & Mobile Robotics, 2014, vol. 2, no. 1, pp. 73­97. 18. Gonchenko, S. V., Turaev, D. V., and Shilnikov, L. P., On Newhouse Domains of Two-Dimensional Diffeomorphisms That Are Close to a Diffeomorphism with a Structurally Unstable Hetero clinic Contour, Proc. Steklov Inst. Math., 1997, vol. 216, pp. 70­118; see also: Tr. Mat. Inst. Steklova, 1997, vol. 216, pp. 76­125. 19. Lamb, J. S. W. and Stenkin, O. V., Newhouse Regions for Reversible Systems with Infinitely Many Stable, Unstable and Elliptic Periodic Orbits, Nonlinearity, 2004, vol. 17, no. 4, pp. 1217­1244. 20. Delshams, A., Gonchenko, S. V., Gonchenko, A. S., L´zaro, J. T., and Sten'kin, O., Abundance of a Attracting, Repelling and Elliptic Perio dic Orbits in Two-Dimensional Reversible Maps, Nonlinearity, 2013, vol. 26, no. 1, pp. 1­33. 21. Gonchenko, A. S., Gonchenko, S. V., and Kazakov, A. O., On Some New Aspects of Celtic Stone Chaotic Dynamics, Rus. J. Nonlin. Dinam., 2012, vol. 8, no. 3, pp. 507­518 (Russian). 22. Gonchenko, A. S., Gonchenko, S. V., and Kazakov, A. O., Richness of Chaotic Dynamics in the Nonholonomic Mo del of Celtic Stone, Regul. Chaotic Dyn., 2013, vol. 18, no. 5, pp. 521­538. 23. Borisov, A. V. and Mamaev, I. S., Strange Attractors in Rattleback Dynamics, Physics­Uspekhi, 2003, vol. 46, no. 4, pp. 393­403; see also: Uspekhi Fiz. Nauk, 2003, vol. 173, no. 4, pp. 408­418. 24. Borisov, A. V., Jalnine, A. Yu., Kuznetsov, S. P., Sataev, I. R., and Sedova, J. V., Dynamical Phenomena Occurring due to Phase Volume Compression in Nonholonomic Mo del of the Rattleback, Regul. Chaotic Dyn., 2012, vol. 17, no. 6, pp. 512­532. 25. Gonchenko, A. S. and Gonchenko, A. S., On Existence of Lorenz-Like Attractors in a Nonholonomic Mo del of Celtic Stones, Rus. J. Nonlin. Dinam., 2013, vol. 9, no. 1, pp. 77­89 (Russian). 26. Gonchenko, A. S., Lorenz-Like Attractors in Nonholonomic Mo dels of Celtic Stone, Nonlinearity, 2015 (to appear). 27. Borisov, A. V. and Mamaev, I. S., Rigid Body Dynamics: Hamiltonian Methods, Integrability, Chaos, Izhevsk: R&C Dynamics, Institute of Computer Science, 2005 (Russian). 28. Walker, J., The Amateur Scientist: The Mysterious "Rattleback": A stone That Spins in One Direction and Then Reverses, Sci. Am., 1979, vol. 241, pp. 172­184. 29. Borisov, A. V., Kilin, A. A., and Mamaev, I. S., New Effects in Dynamics of Rattlebacks, Dokl. Phys., 2006, vol. 51, no. 5, pp. 272­275; see also: Dokl. Akad. Nauk, 2006, vol. 408, no. 2, pp. 192­195. 30. Benettin, G., Galgani, L., Giorgilli, A., and Strelcyn, J.-M., Lyapunov Characteristic Exponents for Smo oth Dynamical Systems and for Hamiltonian Systems: A Metho d for Computing All of Them: P. 1: Theory; P. 2: Numerical Application, Meccanica, 1980, vol. 15, pp. 9­30. 31. Hairer, E., Norsett. S. P., and Wanner, G., Solving Ordinary Differential Equations: 1. Nonstiff Problems, Berlin: Springer, 1987. 32. Kuznetsov, S. P., Dynamical Chaos, 2nd ed., Moscow: Fizmatlit, 2006 (Russian). 33. Afraimovich, V. S. and Shilnikov, L. P., Invariant Two-Dimensional Tori, Their Breakdown and Sto chasticity, in Methods of Qualitative Theory of Differential Equations, E. A. Leontovich-Andronova (Ed.), Gorky: Gorky Gos. Univ., 1983, pp. 3­26 (Russian). 34. Gonchenko, S. V., Ovsyannikov, I. I., Sim´, C., and Turaev, D., Three-Dimensional H´ on-Like Maps o en and Wild Lorenz-Like Attractors, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2005, vol. 15, no. 11, pp. 3493­3508. 35. Gonchenko, A. S., Gonchenko, S. V., and Shilnikov, L. P., Towards Scenarios of Chaos Appearance in Three-Dimensional Maps, Rus. J. Nonlin. Dyn., 2012, vol. 8, no. 1, pp. 3­28 (Russian). 36. Gonchenko, A. S., Gonchenko, S. V., Kazakov, A. O., and Turaev, D. V., Simple Scenarios of Onset of Chaos in Three-Dimensional Maps, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2014, vol. 24, no. 8, 1440005, 25 pp.

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