## Abstract

Following part III in the series, the linear stability of previously identified steady states is analysed for a general convex axisymmetric body spinning on the horizontal plane in order to determine spin orientations leading to the ‘rising egg’ phenomenon. The viscous friction law is assumed between the body and the plane, which is linear in the velocity of the point of contact and allows for analytical treatment of the problem. In the analysis, the emphasis is put on the relationship between the geometrical structure of interconnected structures of non-isolated fixed-points, representing the steady-spin states in the system phase space, and their stability properties. It is shown that the rising egg phenomenon, discussed initially in part I for the flip-symmetric geometry of a uniform spheroid, occurs in a much broader class of spinning axisymmetric bodies. It is also shown that for some geometries, the steady spin configurations of minimum potential energy are always stable, contrary to the flip-symmetric case, so that even a rapid spin does not cause the centre-of-mass to rise. Particular attention is focused on a spheroid with displaced centre-of-mass and the tippe-top.

## 1. Introduction

It was shown in part I of this series (Moffatt *et al*. 2004) that the paradoxical ‘rising egg’ phenomenon, characterized by a rise in the centre-of-mass of a rapidly spinning axisymmetric body, could be successfully explained for the case of a uniform spheroid by studying the geometry and stability properties of a structure of non-isolated fixed-points (i.e. states of steady spin) in the phase space of the corresponding dynamical system. However, the strong assumption of flip-symmetry used in part I resulted in a highly symmetric fixed-point structure which was shown in part III (Branicki *et al*. 2006) to be structurally unstable (i.e. any departure from flip-symmetry destroys such a fixed-point structure). It therefore remained unclear whether the approach presented in part I could also explain the rising phenomenon observed in many real-life, non-flip-symmetric objects like a hard boiled egg or the well-known tippe-top, which generally rise to spin on one end but not on the other.

The analysis of part I was further generalized in part III of this series (Branicki *et al*. 2006), where the structure of steady states was studied in the dynamics of a convex axisymmetric body spinning on the horizontal surface in the presence of friction (figure 1). In such a general case, four elementary categories of non-isolated fixed-points (i.e. dynamic steady states) were identified in the six-dimensional phase space of the governing dynamical system. It was shown that these elementary fixed-point branches formed a complex global structure in the phase space which, depending on geometry of the body and its inertia tensor, could be differently interconnected. For the prototype problem of a spheroid with displaced centre-of-mass, nine topologically distinct classes of fixed-point structures were identified.

In the present paper, we continue the analysis of part III and determine the linear stability of the previously identified fixed-point structures; the viscous friction law is assumed which makes the problem amenable to analytical treatment. We look in particular for steady spin orientations, which become unstable and lead to the rising phenomenon in this general case. The stability of each elementary fixed-point class is analysed separately with the emphasis on the relationship between the topological configuration of the overall fixed-point structure and stability properties of its elementary branches. This approach enables determination of important stability properties directly from the geometry of the underlying fixed-point structure. We show, in particular, that the stability of spin in the vertical orientation can only be altered (by varying the spin value) if, for a given geometry and a sufficiently small neighbourhood of the considered vertical orientation, there exists a continuum of steady spin states with the axis of symmetry inclined to the vertical. This observation leads in turn to the identification of situations when the vertical spin cannot be destabilized, regardless of the spin value, hence prohibiting the rising phenomenon. The general treatment is followed by detailed examples for the spheroid with displaced centre-of-mass or the tippe-top. In particular, it is shown that the nine classes of fixed-point structures, derived for the spheroid in part III, have distinct stability properties and are thus characterized by different dynamical behaviour of the spinning body.

Part III is a prerequisite for understanding the following analysis; here we simply recapitulate the fundamental equations derived in the previous publications (parts I–III).

Geometry of the spinning body determines the height function *h* of the centre-of-mass G above the table. Smooth convex shapes are considered so that and are continuous; in particular we assume at , *π*. Six (dimensionless) variables are needed to define the state of motion of a spinning axisymmetric body (see figure 1),(1.1)where are the *X*- and *Y*- components of velocity of the centre-of-mass G, is the rate of precession of G*z* about , *θ* is the angle between and , , and *n* is the spin (i.e. the component of angular velocity about ). The governing evolution equations can be cast in the form of a sixth-order nonlinear dynamical system with components(1.2a)(1.2b)(1.2c)(1.2d)(1.2e)(1.2f)where and are the components of the frictional force at P and are the (dimensionless) principal moments of inertia at G. Evolution of the spinning body from the initial state is then represented by the trajectory in the phase space of the system (1.2). In the following analysis, we assume the ‘viscous’ law ,1 where is the velocity of the point of contact with components given by(1.3a)(1.3b)(1.3c)

Finally, the normal reaction *R* is given by(1.4)We restrict the following analysis to the situation when so that the body remains in contact with the plane throughout the evolution.

## 2. Stability of the steady states

As shown in part III, each of the four elementary categories of non-isolated fixed-points (i.e. dynamic steady states of the spinning body) can be represented as(2.1)where *s* is a parameter along the particular fixed-point branch.

Suppose now that one such steady state (i.e. for fixed *s*) is perturbed to(2.2)Then, the corresponding linearization of the dynamical system (1.2) can be written as(2.3)where is the matrix . The linear system (2.3) has solutions of the form where *p* satisfies a determinantal condition that can be reduced to the polynomial(2.4)In general, the coefficients in (2.4) depend on the fixed-point (via *s*) and the system parameters. Since all the fixed-points are non-isolated, there is always one ‘neutral’ direction corresponding to the perturbation along the fixed-point line; this is reflected in the structure of (2.4) by the existence of the neutral eigenvalue . We may disregard such neutral perturbations in what follows.

The following analysis is a generalization of results presented in part I for the flip-symmetric spheroid. We first determine the stability of each elementary category of the steady states for a general height function and later discuss some specific examples. In general, two types of instabilities can be identified in the system. In the frictionless situation (*μ*=0), when the system is non-dissipative, all the fixed-points are either (neutrally stable) centres or (unstable) saddles. When weak dissipation is introduced , the (structurally unstable) centres are obviously destroyed and may be either stabilized by the friction or give rise to slow dissipation-induced instabilities with the growth rate proportional to *μ* (see Bloch *et al*. 1994; Bou-Rabee *et al*. 2004). The saddles, on the other hand, are not substantially affected by the friction and the fast instabilities, associated with the unstable manifolds, survive in the system with the growth rate in the limit . We will show below that if the rising phenomenon from a given steady spin orientation occurs, it always corresponds to the slow dissipation-induced instability which is triggered for sufficiently large spin (or precession) value. This instability fuels the Coriolis effect, against the action of gravity, by transferring the kinetic energy of spin about the axis of symmetry into the energy of precession about the vertical (or vice versa).

As shown in part III, there are many possible configurations of the elementary fixed-point branches depending on the actual geometry of the spinning body. As we proceed, we point out how geometry of the resulting global fixed-point structure affects stability properties of the individual branches.

### (a) Stability of vertical spin

Consider an axisymmetric body spinning on the horizontal surface with its axis of symmetry in the vertical. We show here that such a configuration is stable provided that it either corresponds to a slow spin in the minimum (local or global) of the potential energy or to a fast spin in the maximum (local or global) of the potential energy. By analysing the dynamical system (1.2), we show that the change of stability of the vertical spin, which for some geometries leads to the rising phenomenon, always occurs for the spin value *n* at which the intermediate states branch out from the vertical spin states. If such a branching does not occur, the stability of the spin in a given vertical orientation remains independent of *n* (i.e. either always stable or always unstable) thus inhibiting the rising phenomenon.

In the neighbourhood of the vertical spin state (parameterized by *n*), the perturbation is(2.5)We disregard here the perturbation of *n* since it corresponds to the neutral direction along the branch of vertical spin states. Consequently, the linearized system (1.2) is(2.6a)(2.6b)(2.6c)(2.6d)(2.6e)(2.6f)where and with either or . Note that the spin *n* can be regarded as a parameter in the linear approximation since is quadratic in the small quantities ; this is self-consistent with the perturbation (2.5). We may now drop the primes and rewrite the system (2.6) using and . Then, (2.6*a*) and (2.6*b*) transform to(2.7)and (2.6*c*) and (2.6*d*) become(2.8)Following part I, if we now let and , and combining this transformation with (2.7) and (2.8), we obtain(2.9)

The linear system (2.9) admits solutions of the form(2.10)provided that *p* satisfies the cubic characteristic equation(2.11)

Note first that when *μ*=0,2 the roots of (2.11) are(2.12)with(2.13)For stability in the frictionless case, we thus require that(2.14)a condition that is clearly satisfied for any *n* if the height function *h* has a local minimum at . Then, the corresponding fixed-point line of the vertical spin states is composed entirely of centres (with the neutral direction along the line). If, however, *h* has a local maximum at , there is a bifurcation at(2.15)and the corresponding vertical spin state becomes a saddle and gives rise to the fast instability discussed at the beginning of this section. For , the spin at is insufficient to overcome the effect of gravity and such orientation is unstable.

Suppose now that . Provided , the roots (2.12) are perturbed at order *μ* to(2.16)It is clear that is damped by friction. If the vertical orientation corresponds to the local maximum of the height function , the root is responsible for the fast instability for . For , stability of both oscillatory modes requires that(2.17)which reduces to(2.18)As shown in appendix A, we have . If (2.18) is satisfied, the vertical spin at is stable. Otherwise, the vertical spin is destabilized through the dissipation-induced instability. It is important to realize here that the critical *n* in (2.18), at which the stability of a vertical spin state is altered, corresponds to the intersection point between a branch of intermediate fixed-points and the vertical spin states (compare (3.28), (3.29) of part III with (2.18)). Conversely, if for a given fixed-point structure (corresponding to a particular geometry of the spinning body) there is no such intersection, the stability of the vertical spin states at remains invariant (i.e. independent of *n*). This observation indicates that, contrary to the flip-symmetric case of part I, there exist geometries for which a rapid spin in the minimum-potential-energy orientation does not lead to the rising of the centre-of-mass (see examples given below). The rising from a vertical spin state, corresponding to a local minimum of the height function, is only possible when a branch of intermediate states intersects the vertical spin states. Moreover, a vertical spin in the maximum-potential-energy orientation can only be stabilized by sufficiently large spin if the respective vertical spin states are intersected by a branch of intermediate fixed-points. Note finally that such situations can be identified directly from the geometry of the underlying fixed-point structure.

Consider now two examples that illustrate in detail the above results.

#### (i) Spheroid with displaced centre-of-mass

In this case, the height function has the form(2.19)where , and we obtain(2.20)For stability in the frictionless case we require that (see (2.14))(2.21)

If , the condition (2.18) for stability of the oscillatory modes can be written as(2.22)where(2.23)Thus, a number of cases is possible for each of the two orientations :

and , i.e. : the vertical spin is stable if ,

and , i.e. : the vertical spin is stable if ,

and , i.e. : the vertical spin is always stable,

and , i.e. : the vertical spin is always unstable.

The above classification can be rewritten in terms of the four parameters of the system which, as illustrated in figure 2, leads to the identification of different stability regions of the vertical spin states in the parameter space. In a generic situation, when , there are three distinct regions of stability for each orientation with the boundaries determined by (thick solid curve) and (thin solid curve).

Comparison of fig. 6 (part III) and figure 3 shows that there exists a correspondence between the classes of the intermediate states and the ‘joint-stability classes’ (non-risen state+risen state) of the vertical spin states. In fact, the curve is equivalent to *Χ*=*a* (see (4.13) in part III) and the curve is equivalent to ; similarly, the condition corresponds to (see (4.10) in part III). All these curves are plotted in figure 3 for three distinct cases *A*<*C*, *A*=*C* and *A*>*C*.

We may now characterize the stability of the vertical spin states according to the classes of the fixed-point structures which were identified in part III (see also figure 3).

*Class*:

: The non-risen state (global minimum of

*h*) is stable for and the risen state (global maximum of*h*) is stable if . Evolution from the non-risen state corresponds to the slow rise of the centre-of-mass.: The non-risen state (global minimum of

*h*) is stable for and the risen state (local minimum of*h*) is always stable. (Global maximum of*h*corresponds to a static equilibrium located at intermediate ; such configuration exists only for*a*<1.) Only the non-risen states, which are intersected by the intermediate fixed-point branch, can be destabilized for sufficiently large spin, leading to the subsequent rising phenomenon.: The non-risen state (local maximum of

*h*) is always unstable and the risen state (global maximum of*h*) is stable for . (Exists only for*a*>1.) The initial stage of the unstable evolution from the non-risen state, which might end in one of the intermediate steady states, corresponds to lowering of the centre-of-mass. If spun sufficiently fast in the non-risen state, the body will rise to the risen state.: The non-risen state (global minimum of

*h*) is always stable and the risen state (global maximum of*h*) is stable for . The rising phenomenon from a vertical spin orientation does not occur. The centre-of-mass does rise if the body is spun sufficiently fast at a sufficiently large inclination angle (see §2*c*).: The non-risen state (global minimum of

*h*) is stable for and the risen state (global maximum of*h*) is always unstable. Rising from the non-risen state is possible for sufficiently large initial spin. Evolution from the risen state always corresponds to decrease of potential energy.Two cases must be considered separately:

*a*>1, : The non-risen state (local maximum of*h*) and the risen state (global maximum of*h*) are both unstable. The body evolves towards a spin in an intermediate state which has lower potential energy than the risen state but not necessarily lower than the non-risen state.*a*<1, : The non-risen (global minimum of*h*) and the risen (local minimum of*h*) states are both stable. Initial spin in any of these states does not lead to the rising phenomenon.

Two cases must be considered separately:

*a*>1, : The non-risen state (local maximum of*h*) is stable for and the risen state (global maximum of*h*) is stable for . Insufficient spin in any of these states results in a (fast) evolution towards a lower-potential-energy state (i.e. one of the intermediate spin states or a horizontal precession state).*a*<1, : The non-risen state (global minimum of*h*) is stable for and the risen state (local minimum of*h*) is stable for . For sufficiently large spin, evolution from any of these states results in the slow rising of the centre-of-mass.

Note finally that the cases (6) and (9) describe in particular the stability properties of the vertical spin in the flip-symmetric case of part I.

#### (ii) Tippe-top

It can be easily seen from figure 3 that if we model the tippe-top as a sphere with displaced centre-of-mass , then three different stability scenarios are possible for the spinning body, depending on which fixed-point structure exists in the system phase space. As shown in part III, these structures can be classified for the tippe-top using the parameter . If the geometry of the tippe-top is such that (class (1)), its centre-of-mass will rise when the top is spun in the non-risen state with , possibly reaching the risen state (the flipping behaviour) for sufficiently large initial spin. If the tippe-top is characterized by *Χ*>1 (class (4)) and it is spun initially in the non-risen vertical orientation, its centre-of-mass will not rise at all, regardless of the spin value. Finally, if , the centre-of-mass of the tippe-top will rise3 from the minimum non-risen orientation for sufficiently large initial spin. These analytical predictions are confirmed numerically in figure 4 where the roots of the characteristic polynomial (2.11) are plotted for three different examples of tippe-tops that are representative of the three different stability classes.

### (b) Horizontal precession states

As shown in part III, these states exist only if ; note that the body need not be flip-symmetric in such a situation. We show here that, similar to the degenerate flip-symmetric geometry considered in part I, if the horizontal precession becomes unstable, the corresponding unstable manifold is one-dimensional (two-dimensional if the whole line of non-isolated horizontal precession fixed-points is considered). The instability, which is triggered by an intersection of the horizontal precession states and the intermediate states, leads to the rising phenomenon if and if the precession is sufficiently large. Otherwise, when , this instability leads to lowering of the centre-of-mass if the body spins sufficiently slowly.

Similar to the case of previously considered vertical spin states, the system (1.2) can be linearized in the neighbourhood of a horizontal precession state(2.24)characterized by a steady precession with axis of symmetry parallel to horizontal plane. The exact form of the linearized system, which can be obtained using standard techniques, is shown in the electronic supplementary material. We obtain from the linearization that is quadratic in small quantities (similar to in (2.6*f*)), which is self-consistent with the form of the perturbation (2.24). The dimension of the linearized system is thus reduced from 6 to 5 (the neutrally stable mode corresponding to perturbation along the fixed-point line being suppressed). This fifth-order system admits solutions with , where *p* satisfies the quintic characteristic polynomial(2.25)where(2.26)Note that is quadratic in *μ* and has the same symmetry, , as in the flip-symmetric case of part I.

When *μ*=0, the five roots of (2.25) are(2.27)If the height function has a local minimum in this orientation , all the roots (2.27) have zero real part and the corresponding horizontal precession states are (linearly) neutrally stable. If, on the other hand, *h* has a local maximum at , there is a bifurcation at(2.28)Then, the horizontal precession is neutrally stable for . If however , the roots become real and can be rewritten as(2.29)where the mode corresponding to the root is clearly unstable.

When , the roots (2.27) are slightly perturbed, the perturbation being regular if , but singular at the bifurcation point if . It is necessary to consider these cases separately.

#### (i) Horizontal precession at a local minimum of the height function

Here, it is sufficient to consider a regular perturbation of each of the roots to order *μ*.

The root

Suppose this root is to be perturbed to ; substituting it in (2.25) and retaining only terms of order

*μ*leads to(2.30)where , and(2.31)If , i.e. when no intermediate states branch intersects the horizontal precession states (part III), this mode is damped by friction (*q*<0). If, on the other hand, , stability of the horizontal precession states is altered at the point of intersection with the intermediate states branch located at(2.32)For , the horizontal precession is stable, and for , the corresponding mode is subject to slow dissipation-induced instability with growth rate .The roots

These are similarly perturbed to, say, . Since , we have , . Hence, we find from (2.25) after some algebra that and consequently(2.33)i.e. both modes are damped by the frictional dissipation.

The roots

We write similarly . In this case, , , and (2.25) gives in this case(2.34)Since , as shown in appendix A,

*q*is real and negative and these modes are always damped by the frictional effect, whatever the value of .

In summary, if *h* has a local minimum at , a single non-oscillatory mode becomes unstable for the horizontal precession states when . There are also four oscillatory modes (two complex conjugate pairs) which are all damped by friction in the linear stability analysis. Consequently, the unstable manifold of any such horizontal precession fixed-point is one-dimensional and it corresponds to the slow rising, at least in the initial stage. (Note that if the *line* of non-isolated fixed-points is considered, the corresponding unstable manifold becomes two-dimensional.)

#### (ii) Horizontal precession at a local maximum of the height function

In this case, the above type of perturbation analysis may be carried out provided is not near the bifurcation point (given by (2.28)). We may briefly summarize the results as follows:

If , the non-oscillatory mode gives rise to the fast instability if ; the horizontal precession is stable if . For , we see from (2.31) and (2.28) that . There is a slow instability for and fast instability for in this case.

For arbitrary values of *μ* or in the vicinity of , the roots of (2.25) have to be determined numerically. The stability of the horizontal precession states for the flip-symmetric spheroid was presented in part I and we do not discuss any further examples here. Note that the prolate and oblate geometries, discussed in part I, are representative of, respectively, the local minimum and the local maximum of the height function at .

### (c) Stability of intermediate states

We showed in §2*a*,*b* that the change of stability of the vertical spin states and of the horizontal precession states was triggered by the intersection with a branch of intermediate states which represent a steady spin with axis of symmetry inclined to the vertical at . We show here that the intermediate fixed-point branches, which are generally stable at these intersection points, might have two-dimensional unstable manifolds in other regions of the phase space. These manifolds exist either near unstable intermediate static equilibria (i.e. , ) or originate from branches that extend into the gyroscopic region of the phase space (i.e. ). The unstable manifolds existing in the gyroscopic region are responsible for the ‘slow’ rising phenomenon from the neighbourhood of the intermediate fixed-point branches. The unstable manifolds located near the unstable static equilibria correspond to ‘fast’ evolution towards a state of lower potential energy.

We may write a perturbation of any intermediate fixed-point as(2.35)where and are related by (see part III)(2.36)and(2.37) belongs to intervals where the right-hand side of (2.36) is positive, and , , .

As in the case of the previously considered fixed-point branches, the system (1.2) can be linearized in the neighbourhood of an intermediate fixed-point (which is a tedious but a well-defined procedure; see electronic supplementary material for details). The linearized system admits solutions , where *p* satisfies a determinantal condition in the form(2.38)The coefficients , obtained with the help of Maple, are given in appendix B. Note that are even, and are odd (actually linear) in *μ*. Thus, similar to , satisfies the symmetry condition . The factor *p* in (2.38) corresponds to neutrally stable perturbations in phase space in the direction tangent to the continuous curve of intermediate states and we disregard such perturbations in what follows.

Consider first the situation when *μ*=0. Then, (2.38) gives(2.39)where(2.40)the function is presented in appendix B. The roots of (2.39) at the fixed-point are(2.41)If then *α*<0, the corresponding intermediate state is unstable, since . If however , the intermediate state is (linearly) stable, since . Note also that if(2.42)there is a resonance between the and the modes as in the flip-symmetric case discussed in part I.

For arbitrary *μ*, the roots of (2.38) have to be calculated numerically. We illustrate this procedure in figure 5 for the spheroid with displaced centre-of-mass where nine examples, representative of distinct topological configuration of the intermediate fixed-point branches, are analysed in detail. Each example, corresponding to particular values of the system parameters , is composed of two figures: one presents a projection of the (non-isolated) fixed-point branches onto the subspace of the system phase space and the other shows the computed values of along these branches as a function of .

It is important to note here that, while the exact values of vary with the system parameters, certain stability properties of the intermediate branches remain invariant within the fixed-point classes, as shown in figure 6. In particular, the fixed-points in branches that belong to classes (3), (5), (9) and (6) (*a*>1) are (linearly) stable except a possible narrow band in the neighbourhood of the resonant angle (2.42); the (very) slow instabilities corresponding to these unstable bands can occur in any class. Branch configurations of classes (2), (4), (7), (8) and (6) (*a*<1) always have a two-dimensional unstable manifold which corresponds to an unstable non-oscillatory mode (see grey area in figure 6). This mode leads to the fast instability near the unstable static equilibria (classes (2) and (6)–(8) when *a*<1). The dynamics associated with the fast instability always leads to a decrease of the potential energy of the body. In classes (4), (7) and (8) when *a*>1 the unstable manifold gives rise to the slow instability associated with the rising phenomenon. In this case, the manifold is located in the gyroscopic region of the phase space (i.e. rapid spin). The fixed-points in class (1) are generally all stable except the borderline region when *A*<*C* (figure 6*a*).

### (d) Stability of pure rolling states

The perturbation in the neighbourhood of any rolling state can be written as(2.43)where corresponds to extremal points of the height function , and .

The linearization of the system (1.2) in the neighbourhood of a rolling state4 and seeking eigensolutions in the form leads to the characteristic polynomial in the form(2.44)where are given in appendix C.

Note that (2.44) has a double root . As in the flip-symmetric case of part I, the first of these roots corresponds to neutrally stable perturbations along the line (in phase space) of pure rolling states. The second root corresponds to perturbations in a direction orthogonal to this line and is neutrally stable only in linear analysis. The nonlinear analysis on the centre manifold corresponding to these roots becomes extremely complex in the general case and we do not discuss it here. In the following, we focus on the linear stability of modes corresponding to the roots , bearing in mind that the ‘zero’ root leads to a weak nonlinear instability which is observed in numerical simulations of the system (1.2).

When *μ*=0, the roots of the in (2.44) are(2.45)It is then clear that provided (local maximum), there is a fast (frictionless) instability if(2.46)when the non-oscillatory mode, corresponding to the (purely real) positive root , dominates over the nonlinear effects.

When , the perturbed roots can be obtained to order *μ* as in the previous sections provided we are not near the bifurcation point . Due to a complicated form of these expressions, we present them in appendix C. As shown in part III, the rolling states are always connected either to a branch of intermediate states or to the horizontal states through a static equilibrium located at(2.47)where the inclination angle of the rolling body is determined by . Consequently, the roots determined from (2.38) coincide with the roots of (2.44) at the intersection point (2.47) and their signs are invariant within each class. From (C 5)–(C 9) one can see that, at the intersection point, these roots are(2.48)(2.49)and(2.50)It is then clear that the roots are always real and negative at a static equilibrium. The fast instability of the rolling states, when is positive, is associated with unstable static equilibria and corresponds to evolution towards a lower-potential-energy state.

Figure 7 shows examples of exact computations for a spheroid with displaced centre-of-mass where the roots of (2.44), parameterized by *n*_{*}, are calculated numerically along the rolling fixed-point line. The inclination angle of steady rolling is in this case given by eqn (4.9) in part III and the condition for existence of the fast instability (2.46) becomes(2.51)where(2.52)As can be seen from (2.51), this fast ‘rolling’ instability can occur only for the oblate geometry (*a*<1) and is therefore always present for the rolling states that belong to classes (2) and (7)–(9) when *a*<1 (figure 7). The ‘slow’ instabilities, visible, for example, in figure 7((2), (3), (6)), can be deduced from (C 5)–(C 9) and are not invariant within any fixed-point class.

## 3. Discussion and conclusions

We studied the linear stability of steady states for a general convex axisymmetric body spinning on a horizontal surface in the presence of viscous friction. Stability of four elementary categories of steady states, corresponding to branches of non-isolated fixed-points, was analysed in order to determine situations which lead to the rising phenomenon in such a general setting. It was shown that the stability of any fixed-point in these branches depends on five parameters , i.e. the value of the height function and its first two derivatives at the fixed-point and the two principal moments of inertia. In the analysis, we emphasized the relationship between the stability properties of these elementary branches and the geometrical configuration of the resulting global structure. In particular, it was shown that the stability of vertical spin states is altered by an intersection with a branch of intermediate fixed-points and remains unchanged, regardless of the spin value, if there is no such intersection. Similarly, it could be seen that the fast instabilities, associated with evolution towards a state of lower potential energy, occur only in the neighbourhood of unstable static equilibria; hence they are confined to the ‘low-spin’ region of the phase space where the gravity dominates over the Coriolis effect. The slow instabilities, which were shown to occur for sufficiently large spin (or precession) values, are usually (but not exclusively) associated with the rising phenomenon. These instabilities are always induced by the presence of friction between the body and the plane, which breaks the classical frictionless invariants and fuels the Coriolis effect by transferring the kinetic energy of spin about the axis of symmetry into the energy of precession about the vertical (or vice versa).

In the case of a spheroid with displaced centre-of-mass (and, in particular, the tippe-top), such a geometrical approach enabled classification of the stability of the vertical spin states according to the classes of the fixed-point structures derived in part III.

The analysis of the degenerate case of the flip-symmetric spheroid, discussed in part I, revealed existence of the two-dimensional unstable manifold associated with the presence of an unstable non-oscillatory mode in the horizontal precession states. Analysis of the geometry of this manifold proved crucial for understanding the nonlinear phase of the ‘rising phenomenon’ under the ‘gyroscopic balance’ for a rapidly spinning spheroid. The stability analysis of the intermediate states for the spheroid with displaced centre-of-mass showed that, when the symmetry is broken, the unstable manifold survives on one of the intermediate fixed-point branches. However, the dynamics on the unstable manifold is in general much more complicated than in the flip-symmetric case of part I. This problem will be addressed in future communications.

## Acknowledgements

The authors would like to thank Prof. H. K. Moffatt for helpful comments and discussions. M.B. is supported by a scholarship of the Gates Cambridge Trust. Y.S. acknowledges the support of the Keio Gijyuku Academic Development Fund.

## Footnotes

The electronic supplementary material is available at http://dx.doi.org/10.1098/rspa.2006.1727 or via http://www.journals.royalsoc.ac.uk.

↵Note that we neglect the rolling friction here.

↵Note that in such a case

*n*becomes a constant of motion, as seen from (1.2*f*).↵The flipping behaviour, when the top evolves from a spin in the non-risen to the risen vertical states, does not occur for tippe-tops of class (5). The unstable evolution from the non-risen state terminates on one of the (stable) intermediate spin states.

↵The full linearized system is shown in the electronic supplementary material.

- Received August 11, 2005.
- Accepted March 29, 2006.

- © 2006 The Royal Society