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ISSN 1560-3547, Regular and Chaotic Dynamics, 2013, Vol. 18, No. 5, pp. 508­520. c Pleiades Publishing, Ltd., 2013.

Strange Attractors and Mixed Dynamics in the Problem of an Unbalanced Rubb er Ball Rolling on a Plane
Alexey O. Kazakov*
Institute of computer science ul. Universitetskaya 1, Izhevsk, 426034, Russia The Research Institute of Applied Mathematics and Cybernetics, Nizhny Novgorod State University, pr. Gagarina 23, Nizhny Novgorod, 603950, Russia
Received May 30, 2013; accepted Septemb er 3, 2013

Abstract--We consider the dynamics of an unbalanced rubber ball rolling on a rough plane. The term rubber means that the vertical spinning of the ball is impossible. The roughness of the plane means that the ball moves without slipping. The motions of the ball are described by a nonholonomic system reversible with respect to several involutions whose number depends on the type of displacement of the center of mass. This system admits a set of first integrals, which helps to reduce its dimension. Thus, the use of an appropriate two-dimensional Poincar´ ap is em enough to describe the dynamics of our system. We demonstrate for this system the existence of complex chaotic dynamics such as strange attractors and mixed dynamics. The type of chaotic behavior depends on the type of reversibility. In this paper we describe the development of a strange attractor and then its basic properties. After that we show the existence of another interesting type of chaos -- the so-called mixed dynamics. In numerical experiments, a set of criteria by which the mixed dynamics may be distinguished from other types of dynamical chaos in two-dimensional maps is given. MSC2010 numbers: 37J60, 37N15, 37G35 DOI: 10.1134/S1560354713050043 Keywords: mixed dynamics, strange attractor, unbalanced ball, rubber rolling, reversibility, twodimensional Poincar´ map, bifurcation, fo cus, saddle, invariant manifolds, homo clinic tangency, e Lyapunov's exponents.

On the occasion of the 60th birthday of my advisor Sergey V. Gonchenko, a prominent scientist and an outstanding man! 1. INTRODUCTION This pap er complements a series of works devoted to the study of a new, p oorly known typ e of motion -- rubber rol ling and describ es the motions of a dynamically asymmetric ball with a displaced center of mass on a plane. The term rubber was first prop osed with resp ect to motions in [1, 2], but this typ e of motions had b een considered more than a century ago by J. Hadamard [3]. For more information see the recently published pap er [4], which presents the results of investigations of the integrability of a system governing the motion of a rubb er b ody (a ball or an ellipsoid) on a plane or a sphere. These problems are also dealt with in [5], where sp ecial cases of integrability of the problem of the motion of a rubb er ellipsoid on a plane and a sphere are considered by combining analytical and numerical approaches. There are two reasons for the increased interest in rubb er b ody dynamics. On the one hand, such motions are to investigate numerically than the motions with spinning. Systems with rubb er rolling admit an additional integral, which helps to reduce the analysis of such problems to the
*

E-mail: kazakovdz@yandex.ru

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investigation of the two-dimensional Poincar´ map. On the other hand, such an idealization can b e e achieved in real exp eriments. To do this, the b ody should have a rubb er surface. However, even the dynamics in such systems turns out to b e very rich and complex. An integrable case was p ointed out only in [6], when two principal moments of inertia are equal and the center of mass of the ball is displaced only along one principal axis of inertia. Moreover, in [7] it is proved that a system governing the motion of an unbalanced rubb er ball (with one non-zero comp onent of displacement) on a plane has no an invariant measure even in the case of a full set of first integrals (when the acceleration of gravity is zero). If the acceleration of gravity is non-zero, the additional integral disapp ears and the absence of an invariant measure b ecomes obvious from the analysis of the Poincar´ map. e The main goal of the pap er is to demonstrate the richness and complexity of the dynamics of an unbalanced rubb er ball on a plane rather than investigating all dynamical prop erties in detail. 2. EQUATIONS OF A MOTION AND FIRST INTEGRALS Consider the motion of an unbalanced rubb er ball on a plane. There is no slipping and spinning at the contact p oint of the ball with the plane. These conditions are governed by nonholonomic constraints represented as: v + â r = 0, (, ) = 0, (2.1) where r is the radius vector connecting the center of mass with the contact p oint P , v and are the velocity of the center of mass and the angular velocity of the ball, resp ectively, and is the normal unit vector of the plane at the contact p oint (see Fig. 1). Note that r , v , and n are pro jected onto the moving axes Cx, C y and Cz attached to the ball. Fig. 1. An unbalanced ball The equations governing the evolution of and in the gravity on a plane. field can b e represented as: ~ = (~) â - mr â ( â r )+ mag ( â a)+ 0 (ag ) I I = â , (2.2)

where ~ = I + m(r , r ) · E - mr · r T is the tensor of inertia relative to the p oint of contact, m is I the mass of the ball and ag is the acceleration of gravity, I = diag(I1 ,I2 ,I3 ), where I1 ,I2 ,I3 are the principal moments of inertia relative to the moving axes Cx, C y and Cz . For the unbalanced ball the vectors r and are related by r = -R - a, where a is the displacement of the center of mass with the comp onents (a1 ,a2 ,a3 ). The undetermined multiplier 0 is resp onsible for the rubb er constraints ( , ) = 0 and can b e represented as 0 (ag ) = - ~-1 , (~) â - mr â ( â r )+ mag ( â a) I I ( , ~-1 ) I . (2.3)

Equations (2.2) admit three first integrals: 1 I (2.4) E = (, ~) - mag (r , ), ( , ) = 1, (, ) = 0. 2 The energy and geometric integrals are common for all nonholonomic problems. The third integral is sp ecified by rubb er constraints (in what follows we will call it rubber integral). Thus, by the Euler­Jacobi theorem for integrability of the system (2.2) the existence of an invariant measure and an additional fourth integral is necessary. Such additional invariants exist in the case I1 = I2 ,a1 = a2 = 0. Moreover, in [6] it is proved that this system can b e represented in a conformally Hamiltonian form. An additional integral also exists in the case of arbitrary parameters I , a, when ag = 0. But in this case the system (2.2) is not integrable due to the absence of an invariant measure (see [7]). For arbitrary values of the parameters the system (2.2) admits neither an additional integral nor an invariant measure. Due to these facts the dynamics of the system turns out to b e very rich and complex.
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´ 3. THE POINCARE MAP As in most problems of nonholonomic mechanics, it is very convenient to use the Poincar´ crosse sections for numerical investigation of the system under consideration. On the common level set of three integrals (2.4) our system can b e restricted to the three-dimensional manifold M3 = {(, ) : ( , ) = 1, ( , ) = 0,E (, ) = const}. (3.1) For parametrization of this manifold we use sp ecial variables (L, G, l, g), which can b e represented as: 1 = G2 - L2 sin l, 2 = 1 = G2 - L2 cos l, 3 = L,

L cos g sin l +sin g cos l, (3.2) G L cos g cos l - sin g sin l, 2 = G L2 cos g. 3 = - 1 - G Remark 1. Similar variables are used for many problems of nonholonomic and rigid b ody dynamics (see, for example, [8]), where they are called the Andoyer­Deprit variables. Note that in the new variables the geometric and rubb er integrals are conserved automatically. Hence, Eqs. (3.2) sp ecify one-to-one transformations everywhere except for L/G = ±1. As a secant for this flow we choose a manifold given by g = g0 = const. The cross-section of the three-dimensional manifold M3 formed by the intersection with the e secant g = g0 forms the two-dimensional Poincar´ map Fg0 : M20 M20 , g g M20 = {x M3 |g(x) = g0 }, g which is defined on S2 . For parametrization of M20 we use the variables l mod 2 and g L e Thus, the pair (l, G ) defines a p oint in S2 on which the Poincar´ map Fg0 is applied.
L G

(3.3) [-1, 1].

Remark 2. In the problems of "classical" rolling with spinning the Poincar´ cross-section is threee dimensional [9, 10] and hence more difficult to analyze. 4. REVERSIBILITY OF THE FLOW AND THE MAP Our investigation shows that the dynamics of the system under consideration significantly dep ends on the typ e of reversibility. We represent the equations of motion (2.2) as X = v (X ), where X = (1 ,2 ,3 ,1 ,2 ,3 ). Recall that the map of the phase space R(X ) : X X is called involution for the flow v (X ) if dR(X ) = -v (R(X )),R R = id. (4.1) dt In this case the flow v (X ) is called reversible with resp ect to the involution R(X ). In the system (2.2) there exists the trivial involution (4.2) R0 : - , ,t -t, which reverses the angular velocities of the system. The set of fixed p oints of this involution is the subspace of zero angular velocities (1 = 2 = 3 = 0). In addition to the trivial involution, the system admits additional involutions, whose numb er is defined by the numb er of non-zero comp onents of the displacement of the center of mass.When the ball is balanced (a1 = a2 = a3 = 0), the system has the maximal numb er of involutions. Dep ending on the typ e of transformation of the evolution variables ( , ), all additional involutions can b e divided into two classes:
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· involutions corresp onding to the rotation of the ball by the angle ab out one of three axes attached to this ball (-1 , -2 ,3 ), t -t 1 : (1 ,2 , -3 ), (-1 ,2 , -3 ), t -t (4.3) 2 : (1 , -2 ,3 ), 3 : (-1 ,2 ,3 ), (1 , -2 , -3 ), t -t · involutions corresp onding to the reflection passing through a pair of the axes attached 1 : (1 ,2 , -3 ), 2 : (1 , -2 ,3 ), 3 : (-1 ,2 ,3 ), of to the the ball with resp ect to one of three planes ball (1 ,2 , -3 ), t -t (1 , -2 ,3 ), t -t (4.4) (-1 ,2 ,3 ), t -t

If the center of mass of the ball is displaced only along one axis, then the system (2.2) admits three involutions (in addition to R0 ): the involution corresp onding to the rotation of the ball by the angle along the axis of displacement (one of i , i = 1,... , 3) and two involutions corresp onding to the reflection of the ball with resp ect to the planes passing through the axis of displacement and another axis (two from i , i = 1,... , 3). If the center of mass of the ball is displaced along two axes, then the system (2.2) admits only one additional involution corresp onding to the reflection of the ball with resp ect to the plane passing through these axes. In the case of arbitrary displacement of the center of mass (when all comp onents a1 ,a2 and a3 are non-zero) the system under consideration does not admit additional involutions. We now recall the definitions of reversibility and involution for maps. The transformation r (x) : x x is called involution for the map (3.3) if
- Fg0 r = r Fg01 .

(4.5)

The map Fg0 (x) is called reversible with resp ect to the involution r (x). It is clear that every involution for a flow system can b e reduced to the involution of its Poincar´ e map if a manifold invariant under this involution is chosen as a secant. In what follows, such involutions reduced from the flow system to the Poincar´ cross-section (3.3) e will b e called reduced involutions. For the Poincar´ map (3.3) the reduced involutions can b e e represented as · Reduced R0 : r0 : L L - , G G 3 - L , G l l, l - l, l -l, g -g g -g g -g (4.7) l l + , g -g (4.6)

· Reduced 1 , 2 and L 1 : G L 2 : G L 3 : G · Reduced 1 , 2 and L 1 : G L 2 : G L 3 : G

L , G L , G
3

-

L , G

l l, l - l, l -l,
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L , G L , G

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For each involution (except for r0 ) the set of fixed p oints is represented as lines in the variables (L, G, l, g). Hereafter we will call such lines the lines of fixed points or fixed lines. For the involution r0 the set of fixed p oints is undefined b ecause of uncertainty of the variables (L, G, l, g), when e the angular velocities are zero. Due to the involution r0 the phase p ortraits in the Poincar´ map preserve an obvious symmetry, according to which each tra jectory in the map has a symmetrical analog. Thus, for each stable tra jectory the system under consideration has a symmetrical unstable tra jectory. It is well known from [11, 12] that the b ehavior of reversible systems in a neighb orhood of the intersection of the images of the Poincar´ map for fixed lines (hereafter images of fixed lines) is e close to a conservative, i.e, area-preserving b ehavior. This prop erty helps to search for regions with p ossible dissipation by analyzing the b ehavior of images of fixed lines. The intersections of various images of fixed lines contain conservative p eriodic p oints (elliptic or saddle p oints with unit saddle values). Hence, the system near such p oints is mostly areapreserving [13]. The regions without such p oints can contain sinks and sources such as fixed or p eriodic p oints, invariant curves and even strange attractors. When the Poincar´ map is reversible with resp ect to several involutions with fixed lines, the e network of the images of fixed lines can b e very dense. Thus, the dynamical b ehavior in the system (2.2) b ecomes more complex from the case of the maximal numb er of involutions (balanced ball) to the case with only trivial involution r0 (completely unbalanced ball, when the displacement of the center of mass has three non-zero comp onents). In this pap er we consider two cases in detail. · The case of a completely unbalanced ball: the system (Poincar´ map) reversible with resp ect e to only one involution R0 (r0 ). · The case of a partially unbalanced ball (two of three comp onents of the displacement are non-zero and the third comp onent is zero): the system (Poincar´ map) is reversible with e resp ect to two involutions R0 (r0 ) and one of i (i ), i = 1 ... 3. 5. STRANGE ATTRACTORS The problem of an unbalanced rubb er ball rolling on a plane is not the first nonholonomic problem where strange attractors were found. Not long ago strange attractors were found in the nonholonomic model of a Celtic stone [14, 15]. The pap ers [16, 17] are also devoted to investigations of strange attractors in this model. In [17] the spiral strange attractor of Shilnikov was discovered in the nonholonomic model of a Celtic stone. The scenario of development of this strange attractor was also presented in [18]. Some more strange attractors were found in the ab ove-mentioned model in [16], where they were investigated using numerical calculations. Quite recently the "famous" strange attractor of Lorenz type has b een discovered in the same model of the Celtic stone [18]. Remarkably, this model is the first model from applications where the Lorenz-like attractor was found. 5.1. Development of a Strange Attractor In the case of a completely unbalanced ball, strange attractors can exist in the system (2.2). Fig. 2 shows one of such attractors for the following values of parameters: E = 50,R = 3,m = 1, I1 = 1,I2 = 2,I3 = 3, a1 = 1,a2 = 1.5,a3 = 0.5, g = 0. Remark 3. Note that the triangle inequality for the principal moments of in does not hold strictly. But this does not matter in our case, since the main goal demonstrate the richness and complexity of the dynamics of the system (2.2). attractors were also found for a wide range of parameters of the system (when the holds strictly), but for the values (5.1) the attractor looks particularly clear.
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(5.1)

ertia I1 ,I2 and I3 of this pap er is to Moreover, strange triangle inequality

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Fig. 2. The Poincar´ map of a strange attractor and a strange repel ler. The chaotic set on the left is a strange e attractor and its symmetric (with resp ect to the involution r0 ) analog is a strange rep eller on the right. Almost all tra jectories starting from the rep eller evolve onto the attractor with a small numb er of iterations.

Let us describ e the development of the strange attractor from Fig. 2. We consider the acceleration of gravity ag as a bifurcational parameter. Note that the energy parameter E can b e used as a bifuractional parameter instead of ag , since the increase in the value of ag is equal to the decrease in the total energy of the system. The development of the strange attractor is a rather complicated process and is associated with a series of local and global bifurcations. By the local bifurcations we mean (as usual) bifurcations of fixed and p eriodic p oints and bifurcations of invariant curves on the Poincar´ map. Among the e global bifurcations we will take into account those resulting in qualitative changes of the regions of attraction (repulsion) of invariant sets (fixed or p eriodic p oints or invariant curves), which are related to evolutions of the invariant manifolds of saddle p oints. The main stages of development of the strange attractor are describ ed b elow (for more details see [19]). e · When ag = 0, the phase p ortrait in the Poincar´ map looks area-preserving. The invariant curves surround the elliptic p oints Fl , Fr , Fd and Ft separated by two saddle p oints Sl and Sr (see Fig. 3a). When the parameter ag b ecomes p ositive, the elliptic p oints b ecome foci and most of invariant curves are destroyed. When ag > 7.58, the region of attraction of the focus Fl (hereafter referred as the region L) are b ounded by the unstable invariant manifold of the saddle Sr . Due to the involution r0 the region R = r0 (L) is b ounded by the stable invariant manifold of the saddle Sl (see Fig. 3b).

(a) ag = 0

(b) ag = 8

Fig. 3. Poincar´ maps. a) The phase p ortrait is foliated into invariant curves which surround the elliptic e points Fl , Fr , Fd and Ft separated by two saddle p oints Sl and Sr . b) Elliptic p oints b ecome foci. Fl and Ft are stable foci, while Fr and Ft are unstable foci. A stable invariant manifold of the saddle Sr is coiled around the focus Fl , while an unstable invariant manifold of Sl is coiled around Fr .

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· When ag 8.65, saddle-node bifurcations occur. The stable node p oint Fn and the saddle point Sn are b orn in L, and (due to the involution r0 ) an unstable node and a saddle are born in R (see Fig. 4a). The node Fn very soon b ecomes a focus. When ag > 8.85, stable and unstable manifolds of the saddle Sn form a homoclinic figure-eight [20]. Thus, b oth stable manifolds of Sn are intersected by its unstable manifold, while the second unstable manifold is coiled around the focus Fl (see Fig. 4c). Such intersections lead to the onset of chaos in a neighb orhoods of Sn .

(a) ag = 8.7

(b) ag = 8.8

(c) ag = 9.0

Fig. 4. Poincar´ maps. a) Saddle-node bifurcation for ag 8.65. The stable focus Fn and the saddle Sn are e b orn in the region L. b-c) Behavior of stable (black) and unstable (gray) invariant manifolds of the saddle Sn . The region containing Fl , Sn and Fn is b ounded by the unstable invariant manifold of Sr .

· As ag increases, the saddle Sn approaches the focus Fl and, when ag to a saddle-node bifurcation. But the chaos in the region L exists as sadd le node with a transversal homoclinic vanishes [21]. Thereafter, the fixed p oint in the region L. Starting with ag 8.94 Fn loses stability local bifurcations, after which only saddle p oints remain in the region L interesting and will b e describ ed in Section 5.1.1.

9.14, disapp ears due b efore, b ecause the focus Fn is only one due to a cascade of . This cascade seems

· When ag 9.77, the final evolution of the invariant manifolds of Sl and Sr occurs (see Fig. 5), after which almost all tra jectories evolve from the region R into the region L with p ositive iterations of the Poincar´ map. Thus, the attractive set in L is a strange attractor and the e symmetric repulsion set in R is a strange repel ler.

Fig. 5. ag = 9.77. The final p osition of the invariant manifolds in the Poincar´ map. e

5.1.1. A Cascade of Local Bifurcations in a Neighborhood of Fn

When ag 9.14, the focus Fl vanishes due to saddle-node bifurcation, and the stability in the region L b ecomes associated with bifurcations of the focus Fn . In a neighb orhood of Fn a transition to chaos is associated with a cascade of local bifurcations of the focus Fn . This process is
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rather complicated and interesting, therefore we describ e it b elow in detail. By the cascade of local bifurcations we mean an infinite sequence of local bifurcations for a fixed or p eriodic p oint of focus typ e. In such cascades period-doubling bifurcations can generally alternate with the Neimark­Sacker bifurcations. When ag 8.94, Fn loses stability due to the Neimark­Sacker bifurcation and b ecomes an unstable focus. After that the stable invariant curve Cn born from Fn attracts most tra jectories in the neighb orhood of Fn (see Fig. 6a).

(b) ag = 9.5 (a) ag = 8.94
Fig. 6. The Poincar´ maps in a neighb orhood of Fn . a) The focus Fn loses stability due to the Neimark­Sacker e bifurcation. b) The birth of the unstable 1 : 3 resonance in a neighb orhood of Fn .

When ag 9.375, a resonance of a large p eriod occurs on Cn , and soon Cn disapp ears according to the Afraimovich­Shilnikov scenario [22]. When ag = 9.49, the unstable orbit of p eriod 3 is b orn in the neighb orhood of the unstable Fn (due to occurring of the strong resonance 1 : 3) (see Fig. 6b). After that Fn remains an unstable focus. With further increase in the value of ag the unstable p eriodic focus of p eriod 3 b ecomes a stable p eriodic focus (due to the Neimark­Sacker bifurcation), and when ag 9.72, it loses stability due to a cascade of local bifurcations. The unstable focus Fn bifurcates as well. Since ag 9.634, two p eriod-doubling bifurcations occur, after which an unstable focus of p eriod 4 is b orn in the neighb orhood of Fn . With further increase in ag , the last focus of p eriod 4 b ecomes stable due to the Neimark­Sacker bifurcation. Then the stable focus of p eriod 4 bifurcates due to a cascade of local bifurcations, after which only the focus of p eriod 3 attracts the tra jectories in the region L (see Fig. 7b). When ag 9.7715, the cascade of local bifurcations for the stable focus of p eriod 3 terminates, and our numerical investigation shows that the region L does not contain stable fixed or p eriodic (of not large p eriod) p oints. This region is entirely filled with chaotic orbits. Now we describ e bifurcations of the focus Fn in more detail. It is well known that the transition to chaos due to a cascade of period-doubling bifurcations (also known as the Feigenbaum scenario) is typical of b oth conservative and dissipative two-dimensional maps continuously dep ending on a parameter ([23, 24]). The sequence of bifurcational values of parameters converges in this case, and the sp eed of such convergence (also called the Feigenbaum constant) is asymptotically constant and universal for conservative and dissipative maps. In our case the b eginning of the bifurcations for Fn is the following: 2 p eriod-doubling bifurcations, a Neimark­Sacker bifurcation and a series of 5 p eriod-doubling bifurcations (we could not find bifurcations after the last p eriod-doubling due to numerical miscalculations). We supp osed that the subsequence of bifurcations occurring after the Neimark­Sacker bifurcations is sub ject to
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(a)

(b)

Fig. 7. The Poincar´ maps in a neighb orhood of Fn (a) b efore and (b) after the focus Fn bifurcates due to a e cascade of local bifurcations.

the Feigenbaum scenario and tried to evaluate the sp eed of convergence by the series of 5 p erioddoubling bifurcations. However, we could not find a convergence to one of the universal Feigenbaum constants. We assume that this phenomenon is associated with a complex alternating divergency of the system (2.2), due to which p eriod-doubling bifurcations comp ete with the Neimark­Sacker bifurcations. Therefore, in addition to moving in a unit circle through -1, the multipliers of fixed p oints can also leave this circle through ei . Due to this comp etition the Neimark­Sacker bifurcation can occur after a series of p eriod-doubling bifurcations, which will eventually affect the asymptotical Feigenbaum constant. 5.2. Prop erties of the Strange Attractor In this Section we will describ e qualitative and quantitative characteristics of the chaotic set in the region L. The attractive region of the strange attractor is formed by uniting L and R. The location of the stable invariant manifolds of saddle p oints helps to construct the b oundary of the strange attractor. On the left the attractor is b ounded by the stable invariant manifold (light blue) of the saddle Sr , and on the right the attractor is b ounded by the stable invariant manifold (dark blue) of Sl (see Fig. 5). In the upp er and lower parts of the Poincar´ map these manifolds approach each other as e if to close the region of the strange attractor. Thus, almost all tra jectories starting from R evolve into L and wander ab out in the strange attractor. In what follows we analyze the Lyapunov exponents of the strange attractors for the flow system (2.2). For our system, 3 of 6 Lyapunov exp onents are zero due to the existence of 3 first integrals (2.4). The fourth Lyapunov exp onent is resp onsible for a shift in time and hence is zero, too. Thus, the tra jectories of the system generally admit two non-zero Lyapunov exp onents. We use the well-known Binettin algorithm [25] to calculate a full sp ectrum of the Lyapunov exp onents. To evaluate the numerical miscalculations, we compute the sp ectrum of the Lyap onov exp onents many times from different p oints of the attractor. Then we consider the calculated set as a sampling of a random variable, for which we evaluate the exp ectation value and sample variance of the Lyapunov exp onents. The non-zero Lyapunov exp onents with evaluation of the numerical miscalculations and their sum are shown b elow: 1 = 0.083368 ± 0.000422, 2 = -0.084553 ± 0.000423, (5.2) 1 +2 = -0.001184 ± 0.000011.
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The p ositive Lyapunov exp onent proves the chaotic nature of the attractor, while the negative sum of the Lyapunov exp onents p oints to the compression of volume, which is characteristic of the typical attractors. Note that this compression in our strange attractor is very weak due to small values of the sum of the Lyapunov exp onents. Now we calculate the Kaplan­Yorke dimension [16] of the strange attractor: D =1+ 1 |2 | 1.986. (5.3)

This formula shows that the Kaplan­Yorke dimension is close to 2, which explains why the area of the attractor is comparable with the area of the phase space of (3.3). The describ ed prop erties imply that the discovered stable chaotic set in the region L is a genuine strange attractor. However, this attractor is not related to one of the well-known attractors, such as the attractors of Lorenz or Henon typ e. Due to the weak dissipation the attractor may b e classified as a weak strange attractor. The attractor of this typ e is presented in [26]. 6. MIXED DYNAMICS As was mentioned ab ove, when the center of mass of the ball is displaced along two axes, the system (2.2) admits an additional involution corresp onding to the reflection of the ball with resp ect to the plane passing through the axes of displacement. For the Poincar´ map this involution e is reduced to one of three involutions of the set (4.8). In this case we have found another interesting typ e of chaotic b ehavior -- the so-called mixed dynamics [27, 28] (see also [18] where this phenomenon is p ointed out for the nonholonomic model of Celtic stone). Here we give this definition only for two-dimensional reversible maps. In this case, mixed dynamics is a closed chaotic set of orbits with the following prop erties: · This set contains a countable set of asymptotically stable, asymptotically unstable, saddle and symmetric elliptic orbits. · The closure of the sets of orbits of different typ es has a non-empty intersection. The latter prop erty means that attractors and rep ellers can intersect each other. Moreover, in the numerical exp eriments an attractor and a rep eller can form a single whole. Following [29], by an attractor we will mean a set of images of a fixed line of the involution iterated in forward time and, accordingly, by a repel ler we will mean a set of such images iterated in backward time. The exp erimental evidence of the fact that a closed chaotic set has a mixed nature is that an attractor and a repel ler differ due to the asymmetry of the asymptotically stable orbits b elonging to the attractor and the asymptotically unstable orbits b elonging to the repel ler. Since we iterate a fixed line of an involution, symmetric p eriodic (elliptic and saddle) orbits also b elong to this intersection [11]. Moreover, asymptotically stable and unstable p eriodic orbits should generically present in any neighb orhood of this intersection. However, it is imp ossible to find all these orbits in numerical investigations. Hence, instead orbits of large p eriod we tried to find p eriodic orbits of any p eriod. In addition, we have replaced the requirement of an intersection of the closures of stable and unstable orbits with the requirement of vicinity of these orbits. Figures 8a and 8b show, resp ectively, the attractor and the rep eller obtained by iterating the fixed line forward and backward 1) for the following values of parameters: E = 50,ag = 9.7715, R = 3,m = 1, I1 = 1,I2 = 2,I3 = 3, a1 = 1,a2 = 1.5,a3 = 0,g = /2.

(6.1)

For convenience, we divide each of the figures (8a and 8b) into two parts. Two large "white" regions b elong to the first part. These "white" regions are associated with dissipative dynamics.
1)

Figures 8a and 8b show, resp ectively, 100 forward and backward iterations for p oints of the line L/G = 0 with the step 0.001. Vol. 18 No. 5 2013

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(a)

(b)

(d)

(c)

(e)

Fig. 8. The Poincar´ maps. a) An attractor with 100 forward iterations of the Poincar´ map for p oints of the e e line L/G = 0 with the step 0.001. b) A repel ler with 100 backward iterations of the Poincar´ map for p oints of e the line L/G = 0 with the step 0.001. c) A zoomed region (indicated with a rectangle from Figs. (a) and (b)) containing orbits of b oth the attractor and the repel ler. d) A zoomed region of the unstable p oints of p eriod 19 (in the center of the figure) and 133 (surrounding the foci of p eriod 19). e) A zoomed region of the stable p oints of p eriod 19 (in the center of the figure) and 133 (surrounding the foci of p eriod 19).

Tra jectories starting from the lower part of these regions (from the neighb orhood of unstable foci) pass to the stable foci of the upp er part of these regions. Moreover, these two regions are invariant in the sense that the orbits cannot leave them and cannot evolve into them from outside. The second part contains a region with chaotic dynamics.It can b e seen that the chaos in the second part of Fig. 8a differs from the chaos in the second part of Fig. 8b. However, these two parts have a non-empty intersection. In addition, we find stable and unstable p eriodic p oints of p eriod 19 and 133 inside this chaos. Moreover, the stable p eriodic p oints are close to the unstable ones (see Figs. 8c­8e). Figure 8c shows a zoomed region (indicated with a rectangle in Figs. 8a and 8b) containing orbits of b oth the attractor and the repel ler. Figures 8d and 8e show a zoomed region, resp ectively, of the unstable and stable p oints of p eriod 19 (in the center of the figures) and 133 (surrounding the foci of p eriod 19). Remark 4. Small "white" regions in Figs. 8a and 8b and large "white" regions in Fig. 8c contain p eriodic elliptic orbits. Why do the images of fixed lines not evolve into these regions? We supp ose that the answer is quite simple. We iterate only the fixed line of the involution 1 from the set (4.8). But it is well known [11] that if a map (or a system) is reversible with resp ect to the involution , - then this map is also reversible with resp ect to the involution Fg01 . But it is difficult to find and iterate the fixed line of this involution, so we have not dealt with it. Thus, we assume that the ab ove-mentioned "white" regions will b e filled with such iterations of the fixed line of the - involution Fg01 . All these facts confirm that the chaotic dynamics under consideration is mixed dynamics.
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ACKNOWLEDGMENTS The author thanks A. V. Borisov, S. V. Gonchenko, S. P. Kuznetsov, I. S. Mamaev and I. R. Sataev for fruitful discussions and comments. This work was supp orted by the RFBR grants No. 13-01-00589 and 13-01-97028-p ovolzhye, the Federal Target Program "Personnel" No.14.B37.21.0361, and by the Federal Target Program "Scientific and Scientific-Pedagogical Personnel of Innovative Russia" (Contract No. 14.B37.21.0863). REFERENCES
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