Ball bouncing down rounded edge stairs: chaotic but tricky

Each step consists of a horizontal and a vertical part, connected smoothly with an arc formed by a quarter of a circle of radius R. Relations (5) and (6) of the previous section imply that it is worth measuring all lengths in the unit of the step tread L. In this representation the steps are of unit length, and of rise m = M/L, while the radius of the arc is r = R/L, so that the horizontal part of a step is of length 1 − r, as figure 1 indicates. We shall be interested in the range 0 < r ⩽ 0.2. The figure also contains the schematic trajectory of a ball, and the quantities needed for a unique description of the bouncing dynamics are also given.

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Our goal is to find a relation between the location and velocity data of the nth and the n + 1st bounces. Velocities are convenient to measure in units of the initial horizontal velocity u₀, as e.g., (4) suggests. From now on, the rebound velocities u_n and v_n , right after the nth bounce, are considered to be dimensionless. The only quantity in which the dimensional u₀ appears is the dynamical parameter H given by (6). From the dimensionless data x_n , u_n , v_n of the nth bounce, we would like to know those of the next bounce, i.e. we determine the mapping

$$\begin{equation}\left({x}_{n},{u}_{n},{v}_{n}\right)\to \left({x}_{n+1},{u}_{n+1},{v}_{n+1}\right).\end{equation} \tag{ 7 }$$

The y coordinates do not appear here since on the staircase this is never an independent quantity, it follows from the x coordinate and the shape of the stairs. Initial data of the entire motion are represented by n = 0. An important auxiliary quantity is the jump number, N_n , expressing how many steps are jumped over between the nth and the n + 1st collision. Relation (7) concentrates only on the bounces but, if desirable, the continuous-time motion can be reconstructed from these data in the form of parabola arcs.

To keep things simple, we place the origin of our coordinate system to the left edge of the step on which the collision occurs. This means that we always shift the coordinate x of the collision back to the unit interval, and the y coordinate is either zero (if the bounce is on the horizontal part of the step), or negative. As in the rectangular case, air drag is assumed to be negligible.

To simplify the monitoring process, we cut the trajectory into two pieces: the first one lasts until the ball hits an imagined rectangular step at some coordinate ${\tilde {x}}_{n+1}$ (see figure 2), because the impact data here can be determined analytically, and the rest, until the real bounce with the step at some x_n+1 occurs, to which numerical methods should be applied ⁵ .

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Let the motion, right after the nth bounce, start on a step at dimensionless horizontal coordinate 0 < x_n < 1. The corresponding y coordinate is ${y}_{n}=\sqrt{{r}^{2}-{\left({x}_{n}-\left(1-r\right)\right)}^{2}}-r{< }0$ if the bounce has been on the curvature of the step (x_n > 1 − r), and y_n = 0 otherwise. After the start with horizontal and vertical velocity components u_n and v_n , respectively, we consider the stairs to be rectangular, and assume that the number of steps, ${\tilde {N}}_{n}$, flown over until the next collision with this imagined staircase is known. Thus the collision will happen at a depth $-m{\tilde {N}}_{n}$, and all the quantities characterising this collision can easily be determined from the laws of oblique projection. The results are marked by a tilde to indicate that these are not necessarily the data for the collision with the rounded stairs. For the x coordinate one obtains

$$\begin{equation}{\tilde {x}}_{n+1}={x}_{n}+\frac{{u}_{n}}{H}\left({v}_{n}+\sqrt{{v}_{n}^{2}+2H\left(m{\tilde {N}}_{n}+{y}_{n}\right)}\right)-{\tilde {N}}_{n}.\end{equation} \tag{ 8 }$$

where

$$\begin{equation}{\tilde {N}}_{n}=\left[{x}_{n}+\frac{{u}_{n}}{H}\left({v}_{n}+\sqrt{{v}_{n}^{2}+2H\left(m{\tilde {N}}_{n}+{y}_{n}\right)}\right)\right],\end{equation} \tag{ 9 }$$

and the square brackets represent the integer part. The jump number ${\tilde {N}}_{n}$ is the smallest integer for which equations (8) and (9) possess a solution. The impact velocities (marked with commas) with this ${\tilde {N}}_{n}$ can be expressed as

$$\begin{equation}{\tilde {u}}_{n}^{\prime }={u}_{n},\enspace \enspace \enspace {\tilde {v}}_{n}^{\prime }=-\sqrt{{v}_{n}^{2}+2H\left(m{\tilde {N}}_{n}+{y}_{n}\right)}.\end{equation} \tag{ 10 }$$

For y_n = 0 these relations provide the exact description of the bouncing on rectangular stairs, as given in [2] where tildes are not used as the results always describe the state at the moment of a real collision.

Problem 3. Derive relations (8)–(10).

In the case of ${\tilde {x}}_{n+1}{ >}1-r$, numeric calculations are required. We consider the moment when the ball reaches the imagined rectangular stair, to be at time t = 0 for the subsequent treatment. This way, the x and y coordinates of the ball can be determined as functions of time: x(t), y(t). The y coordinate of the stair is a function of x, but when calculated at the time dependent x(t) coordinate of the ball, it will practically become a function of time, telling us the height of the stair at the point directly underneath the ball at any given moment. Subtracting this from the ball's height, y(t), we get the vertical distance Δy(t) between the ball and the stair.

Problem 4. Determine the function Δy(t).

The numeric algorithm checks this quantity after every small time increment Δt, at t = 0, Δt, 2Δt, ... to see its sign. When Δy(t) becomes negative, it indicates that the collision has happened prior to that moment. Hence, we return to the last instance t of time that gave a positive value and start using a smaller increment Δt' that we conveniently choose to be one tenth of Δt. The process continues, now checking after every increment Δt' until we run into another negative value for the vertical distance of the ball and curvature. An even smaller increment can be chosen, and the same steps can be executed several times, to achieve a sufficiently accurate determination of the time t* that passes between the ball being on the imagined rectangular stair and colliding with the actual one. We have the error, the shortest increment used, be below 10⁻¹⁴. ⁶ We have experienced with other increments, too. By choosing this error to be ten times smaller, i.e. 10⁻¹⁵, the deviation in the values of x_n , u_n , v_n was found to be typically of order 10⁻¹¹ or less, even at collision numbers n above 100. We, therefore, decided to stick to 10⁻¹⁴. Another way to show the consistency, was to check that the results of [2] are recovered for r = 0. It is worth mentioning here that chaotic systems are error-magnifiers [7], thus errors unavoidably grow in the long-term dynamics, and the well-known chaos quantifier, Lyapunov exponent, is just the rate of this growth (see more about this in sections 4.4, 4.5 and 6). For this reason all long-term quantities should be given as averages, that do not depend on individual details. The Lyapunov exponent, too, is itself a kind of average.

This done, the ball's position, ${x}_{n+1}={\tilde {x}}_{n+1}+{\tilde {u}}_{n}^{\prime }{t}^{{\ast}}$, and impact velocities as ${u}_{n}^{\prime }={\tilde {u}}_{n}^{\prime }={u}_{n}$, ${v}_{n}^{\prime }={\tilde {v}}_{n}^{\prime }-H{t}^{{\ast}}$ are also known.

In the simplest scenario, the ball bounces on the same step as it would if the stairs was rectangular, just lands on the curvature instead of the horizontal surface (see figure 2). Occasionally, the ball can 'skip' a step, when the value of ${\tilde {N}}_{n}$ derived via the analytic calculations is not the actual value of N_n . In this case, the ball misses the curvature of the step that it would have hit if it were rectangular. Since the trajectory of the ball does not intersect with the curvature, the numeric algorithm will not find a solution. This problem can be eliminated by constantly checking the inequality x(t) < 1, and whenever it becomes incorrect, it is certain that the ball has 'skipped' the step. Then, we return to the analytic calculations, substituting ${\tilde {N}}_{n}+1$ as the new value of ${\tilde {N}}_{n}$. Although plausible, seldom does the ball 'skip' two or more stairs this way, nevertheless, the same process of switching between analytic and numeric calculations can then be performed multiple times.

The output of the numerical part is the position x_n+1 of the n + 1st collision, and the components u'_n , v'_n of the impact velocity.

The collision rule can be described analytically. If the collision occurs on the horizontal surface, the vertical impact velocity v'_n < 0 becomes multiplied by COR k < 1, and changes sign. The horizontal velocity remains unchanged, the collision rule is thus

$$\begin{equation}{v}_{n+1}=-k{v}_{n}^{\prime },\qquad {u}_{n+1}={u}_{n}^{\prime }.\end{equation} \tag{ 11 }$$

On the curvature, the collision rule is naturally formulated in normal/radial and tangential components denoted by v_r and v_t, respectively. Here, k is applied for the normal component, thus we call it the normal COR. As will be seen later, it is worth considering the effect of another COR, the tangential COR, whose value will be denoted by j ⩽ 1. In terms of physics, this is a consequence of the collision not being instantaneous, thus forces are present while being in contact. The relevance of such a COR has been shown experimentally [18, 19]. We are here offering an elementary argument for its existence, which is based on the presence of a friction force during the short duration of contact. Let F_f,n (t) denote the magnitude of this force at the nth collision which always points against the tangential component of the velocity (see inset to figure 3), and is thus negative. Its time-dependence is not known, but thanks to Newton's second law this is not even needed to determine the momentum loss. The latter is given by

$$\begin{equation*}\int {F}_{f,n}\left(t\right)\mathrm{d}t={\Delta}{p}_{\mathrm{t},n}\end{equation*}$$

where the integral is taken over the time of contact, p_t is the tangential momentum, and the momentum difference Δp_t,n is negative. We consider the impact velocity v'_t,n to be the one right before the contact, and v_t,n+1 the one at the end of it. The letter is the rebound velocity with which the flight of the ball starts. For a point of mass m* (not to be confused with m, the slope of the staircase), the tangential momentum difference is

$$\begin{equation*}{\Delta}{p}_{\mathrm{t}}={m}^{{\ast}}\left({v}_{\mathrm{t},n+1}-{v}_{\mathrm{t},n}^{\prime }\right)={m}^{{\ast}}\left(j-1\right){v}_{\mathrm{t},n}^{\prime }.\end{equation*}$$

The tangential COR j < 1 is thus set by the time integral of the friction force (and the mass), but, for simplicity, we will consider it a constant. For completeness, we mention that in models taking into account the internal structure of the ball, the tangential COR also depends on the internal details [20, 21]. ⁷ The tangential COR is applied only on the curvature in order to recover the dynamics of [2] with no curvature, r = 0.

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All in all, the collision rule on the curvature is that for impact velocity components v'_r,n < 0, v'_t,n , the same components right after the collision become (see figure 3)

$$\begin{equation}{v}_{\mathrm{r},n+1}=-k{v}_{\mathrm{r},n}^{\prime },\qquad {v}_{\mathrm{t},n+1}=j{v}_{\mathrm{t},n}^{\prime }.\end{equation} \tag{ 12 }$$

The numerical part provides, however, the impact velocity in rectangular components as u'_n = u_n , v'_n with the coordinate x_n+1 also given. If the bounce is on the curvature, x_n+1 > 1 − r, the angle α_n+1 under which the collision point is seen from the centre of the circle defining the rounding is obtained from (see figure 3)

$$\begin{equation}\enspace \mathrm{cos}\enspace {\alpha }_{n+1}=\frac{{x}_{n+1}-1+r}{r}.\end{equation} \tag{ 13 }$$

The α_n+1 is called the impact angle.

We thus have to transform the impact velocity into polar components in a frame of polar angle α_n+1. Then (12) is applied with these components, and the new velocity components are transformed back in rectangular coordinates. For these rebound values u_n+1, v_n+1 we obtain

$$\begin{equation}{u}_{n+1}={u}_{n}^{\prime }\left(j\enspace {\mathrm{sin}}^{2}\enspace {\alpha }_{n+1}-k\enspace {\mathrm{cos}}^{2}\enspace {\alpha }_{n+1}\right)-{v}_{n}^{\prime }\left(k+j\right)\mathrm{sin}\enspace {\alpha }_{n+1}\enspace \mathrm{cos}\enspace {\alpha }_{n+1},\end{equation} \tag{ 14 }$$

$$\begin{equation}{v}_{n+1}=-{u}_{n}^{\prime }\left(k+j\right)\enspace \mathrm{sin}\enspace {\alpha }_{n+1}\enspace \mathrm{cos}\enspace {\alpha }_{n+1}+{v}_{n}^{\prime }\left(j\enspace {\mathrm{cos}}^{2}\enspace {\alpha }_{n+1}-k\enspace {\mathrm{sin}}^{2}\enspace {\alpha }_{n+1}\right).\end{equation} \tag{ 15 }$$

Problem 5. Derive relations (14) and (15).

Problem 6. Determine the smallest impact angle α_c permitted in (14) and (15) for given impact velocity components u_n ' and v_n '.

Relation (14) clearly shows that, in sharp contrast to perpendicular stairs, the horizontal velocity component does not remain constant on the rounded stairs.

By the end of these calculations, we obtain the location of the n + 1st bounce, and the velocity components right after the collision, i.e. map (7) is completed. From here on the algorithm can be iteratively repeated an arbitrary number of times.

In this special case, collisions with the curvature are rather rare, the motion is mostly dominated by long sequences of the ball bouncing only on the horizontal part of the stairs. The numerical simulation on such a stair of rounding radius r = 0.01 also illustrates (figure 4) that without an energy loss in the tangential velocity, j = 1, the horizontal velocity, u always increases. This tricky feature indicates a strong deviation from the rectangular case.

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After a bounce on the curvature, there is a jump in the u-component, while in v_n a transient increase can be observed over about 10 bounces, after which a plateau follows.

Problem 7. At collisions with the curvature, why do we see red dots below the level of the previous plateau?

From here on the motion is like on the rectangular stairs. Since long sequences exist without the ball visiting the curvature, we can interpret the findings by considering u_n+1 right after the first hitting of the curvature as a new initial velocity u'₀. In view of the definition (6) of parameter H, there is a new, smaller value ${H}^{\prime }=H{\left({u}_{0}/{u}_{0}^{\prime }\right)}^{2}$ governing the motion leading, in view of (4), to average vertical velocity ${\bar{v}}^{\prime }=\bar{v}\left({u}_{0}^{\prime }/{u}_{0}\right){ >}\bar{v}$. The plateau after the first bounce corresponds to the attractor ⁸ , described in [2], belonging to parameter H' on the rectangular stair. The fine structure in the red lines of figure 4 indicates that the motion is quasi-periodic, although the stationary motion on the rectangular stair with H = 4, k = 0.6 (panel (a)) is periodic. The first plateau is, however, of finite length, after some time a bounce occurs on the curvature again, leading to a new initial velocity u''₀ > u'₀ for the forthcoming motion with an even smaller H''. The sequence of u₀, u'₀, u''₀, ..., is increasing, indicating that the velocities are ever increasing, and larger and larger jumps occur. There is a tendency for an inherent instability in the system.

As the parameter H changes, the spectrum of the discrete k values that result in periodic motion changes with it, thus making it statistically impossible for periodicity to occur after a bounce on a curvature. This also implies, that motions for k = 0.6 are no longer fundamentally different from those for k = 0.75. So much so, that it should not be treated as a separate case, as all of our results for k = 0.75 similarly hold for k = 0.6. It is more practical to analyse motions that start with parameters inducing quasi-periodicity because then the ball will eventually bounce on the curvature under every initial condition (in contrast to motions with parameters of periodic motions, when only a portion of the initial conditions do so, as it can only happen before the periodicity sets in). For this reason we decide to keep only the COR value k = 0.75 in what follows.

The well-known property of an absolutely elastic ball bouncing down on stairs, namely that its jumps are ever increasing (since the potential energy is decreasing on average, but the total energy is constant), appears to remain valid on rounded stairs even in the presence of a normal COR. This COR is unable to lead to a sufficiently strong energy loss to oblige the kinetic energy to stay on average. The tendency of increase follows from relation (14) without tangential energy dissipation (j = 1).

Problem 8. Show, based on relation (14), that with j = 1, the horizontal velocity can never decrease: u_n+1 ⩾ u'_n = u_n .

The inclusion of a tangential COR j < 1 provides a stronger dissipation, and there is a chance for having smaller horizontal rebound velocities then impact ones. A consequence of (14) with j < 1 is that u_n+1/u'_n ⩾ j.

Problem 9. Show, based on relation (14), that its maximum is taken for an angle α* = π/4 + α_c /2, i.e. for the average of the permitted angles in (α_c , π/2). The minimum is achieved at the two ends α_c and π/2, where u_n+1 = ju_n .

When analysing the velocity ratio u_n+1/u'_n a difficulty is that it contains v'_n /u'_n , a quantity not known analytically. In the limit of small r, however, there is a possibility for simplification: there are long stretches of bounces occurring only on the horizontal parts of the steps, and therefore, between two collisions with the curvature the results of section 2 apply. It is sure that after an impact with v'_n , the rebound velocity is −k v'_n which, after the transients, takes one of the values for a typically quasi-periodic bouncing. Their average is given by (4). Therefore, −k v'_n can be approximated by $\bar{v}$ in which u₀ should be taken as the horizontal velocity after the last collision with the curved surface, i.e. u'_n in the current notation. We can thus write

$$\begin{equation}\frac{{v}_{n}^{\prime }}{{u}_{n}^{\prime }}=-\frac{\bar{v}}{k{u}_{n}^{\prime }}=-\frac{2m}{1-k}.\end{equation} \tag{ 16 }$$

Substituting this relation into (14) we obtain

$$\begin{equation}\frac{{u}_{n+1}}{{u}_{n}^{\prime }}=j-\left(k+j\right)\enspace \mathrm{cos}\enspace {\alpha }_{n+1}\left(\mathrm{cos}\enspace {\alpha }_{n+1}-\frac{2m}{1-k}\enspace \mathrm{sin}\enspace {\alpha }_{n+1}\right).\end{equation} \tag{ 17 }$$

The advantage of this approximate relation is that the velocity ratio appears as a one-variable function of the impact angle α_n+1.

Problem 10. Based on (16), determine the critical angle α_c as a function of the parameters m and k. What is this angle for m = 1/2 with our baseline COR value k = 0.75?

In figure 5 we plot the velocity ratio (17) as a function of the impact angle, for different values of the tangential COR j. We see that the velocity ratio can be both smaller and larger than unity, and for sufficiently low values of j the average can be close to unity. For k = 0.75 such a value is j = 0.2. A balance between increased and decreased rebound u values can only be ensured with rather strong tangential momentum loss. We shall use the mentioned j in what follows.

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Even if a tangential COR is applied at which the average velocity ratio u_n+1/u_n ' is close to unity (like e.g. the j value represented with the bold line in figure 5), there always exist initial conditions which lead, sooner or later, to an ever increasing sequence of u. For large velocities, however, the air drag cannot be neglected, although our reasoning is based on the rules of oblique projection. We are thus running out of the validity of our model if the velocity is too large. To avoid this, we stop considering the results reliable if the collision data exceed a threshold value. For practical reasons, we prescribe a threshold in the jump number, N_th = 100.

Whenever N_n is larger than this, simulation is stopped and we say that the ball has flown away. As an arbitrary long monitoring of the motion is not possible within the model, chaos, if present, can only appear as a transient. The theory of transient chaos [7, 15] requires the existence of long lasting motion, we shall, therefore, be after trajectories remaining well defined over several hundreds in n at least.

Problem 11. The threshold in the jump number implies a threshold in the velocities, too. Based on (5) and N_th = 100, show that for k = 0.75 u_th ≈ 7.5.

For other initial conditions, the ball, after a period of bounces on different steps, might undergo an infinite number of collisions on a single step. Parallel to this, the sequence of v_n tends to zero. This indicates the turning of the bouncing motion into a different kind of motion: sliding. Such cases provide another kind of escaping process, and we say the ball has stuck down. Numerically, simulation is stopped when either of the velocity components become less then 0.01 after a bounce on a horizontal part of a step. Sticking is a well-known phenomenon for bouncing balls (see e.g. [23–26]) but was found in [2] to exists only for k < 0.4. It is the presence of another source of dissipation, described by j, which leads to sliding even for larger k values in our cases.

Problem 12. The lowest graph of figure 5 represents parameter sets where practically no increase would occur in the horizontal velocity. Why do not we choose such COR values to prohibit balls flying away?

We are thus interested in long lasting sequences of bounces before flying away or turning to sliding. For small r there is a wide range of such initial conditions. An interesting pattern arises if we examine the total number of bounces τ before any kind of escape takes place, and plot this lifetime as a function of the initial position along the step. Red and blue columns mark motions which end in flying away and in sliding, respectively. Figure 6 exhibits an intricate pattern: the heights, the lifetimes before escape, are irregularly distributed, and the change of colouring is also irregular.

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To gain more insight, we select a tiny, hardly visible interval around x = 0.36, zoom in on it, and plot the lifetimes on it in panel (b). The structure is very similar to that of in panel (a), no regularity occurs on fine scales. Statistical features, like e.g. the ratio of flying away and sliding remain the same (roughly 1 to 2).

The irregularity of the distribution both in height and colour provides an example of sensitivity to initial conditions, a general feature of chaos, while the fractal-like pattern in the lifetime distribution is a property characterising transient chaos specifically [7, 15].

The presence of a curved surface act as a convex mirror and scatter the beam of balls falling on it along parallel paths. This tendency for divergence serves as the source of chaos in the motion of the balls. A measure of the strength of this sensitivity to initial conditions is the so-called Lyapunov exponent (see [7, 22]). This also quantifies the growth of small initial errors as time progresses. An elementary argument for its estimation is as follows.

Consider two balls hitting the curvature of radius r along parallel paths and with approximately the same velocity. The impact points are close to each other and the corresponding impact angles α_n+1 differ by some small Δα. The rebound directions will be different, characterised by a relative angle Δφ (see figure 7). It is intuitively clear that these two small angles are proportional to each other:

$$\begin{equation}{\Delta}\varphi =a{\Delta}\alpha \end{equation} \tag{ 18 }$$

where a is a proportionality constant.

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Problem 13. Determine coefficient a in (18) for collisions with CORs k and j.

Because of the divergence in their path, the distance Δx between the balls increases in time. As an estimation, we can write this distance as Δx = Δφ ⋅ s after a length s along the paths. We take s to be of order unity, i.e., corresponding to the length of a single step. Rearranging this as Δx = Δφ ⋅ r/r = Δα ⋅ r ⋅ a/r, and using that Δα ⋅ r = Δx₀ is approximately the initial distance between the balls, we find Δx = Δx₀ a/r. This distance increases by a factor of a/r. The logarithm of this, ln(a/r) can be considered the Lyapunov exponent belonging to a bounce on the curvature. Hitting this part is, however, rather rare (at least for small r), and the corresponding probability is expected to be proportional to r (as the step tread is chosen to be unity). We thus can write the average Lyapunov exponent as

$$\begin{equation}\lambda =cr\enspace \mathrm{ln}\left(a/r\right)\end{equation} \tag{ 19 }$$

where c > 0 is a proportionality constant. For r << 1, the logarithm ln(a/r) = ln a − ln r is dominated by ln r and the expression simplifies to

$$\begin{equation}\lambda =cr\enspace \mathrm{ln}\left(1/r\right).\end{equation} \tag{ 20 }$$

This result indicates that the knowledge of constant a is not necessary in this limit. For r → 0 the Lyapunov exponent vanishes in harmony with finding no chaos on the rectangular stairs in [2]. The illuminating feature of (20) is that it provides a positive value for any r > 0. There is thus a potential for chaos on any rounded stairs, no matter how small r is [27].

We divide the unit interval in 10⁶ pieces and initiate from all these x₀ coordinates a trajectory (with (dimensionless) velocity v₀ = 3). The distance Δx_n between originally neighbouring pairs of initial distance Δx₀ = 10⁻⁶ is numerically followed as a function of the number of bounces, n. In order to keep only pairs with long lasting motion, we keep only those which possess at least 1000 bounces. The logarithm ln Δx_n of the distance is taken from each of the 76 674 pairs kept. The average of these, $\bar{\mathrm{ln}\enspace {\Delta}{x}_{n}}$, is evaluated and plotted as a function of n in figure 8. The growth of this quantity (indicating an approximately exponential increase) is a clean sign of sensitivity to initial condition, i.e. of chaos. The slope (after discarding the initial transient of 30) is the Lyapunov exponent. Its value is obtained as λ = 0.176 ± 0.004. Although (20) need not hold since r = 0.01 is small but not very small, a comparison is worth while: r ln(1/r) = 0.046 for r = 0.01, not the same as the numerical value but certainly on the same order of magnitude. (If (20) is taken seriously, the proportionality constant is c ≈ 4.)

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Chaotic motions are accompanied with intricate patterns in the phase space. In our model the phase space is three-dimensional with variables as appearing in mapping (7). We shall investigate two projections, the (u_n , x_n ), (v_n , x_n ) phase planes. We saw that there is a convergence time on the order of 10 bounces to reach a kind of stationary motion, therefore we cut the first 30 iterations of long trajectories. The last 30 iterations are also cut in order to avoid points characterising the escape process. One example of such trajectories is shown in figure 9.

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Both patterns are rather complex, fractal-like. The most dramatic effect is perhaps that the permitted u values extend over an interval of about [0.4, 4], while in the rectangular case u is constant and the corresponding phase plane pattern belongs to horizontal line u = 1 (upper right inset in panel (b)). The (v, x) pattern is also more complicated since the quasi-periodic motion of the rectangular case would practically appear as the union of only four horizontal intervals (upper left inset in panel (b)).

A large number of short horizontal intervals can be seen in both panels of figure 9 indicating that the chaotic motion at this small r value consists of a mixture of quasi-periodic sections separated by bounces on the curved surface, and irregularity afterwards indicated by all the points scattered around randomly. An additional novelty in the (v, x) plane is that points close to the end of the step might exhibit negative v values. This is a consequence of nearly tangential collisions with the curvature. The fact that the set of points in both panels appear to be subjected to an upper cut-off is the consequence of the application of the threshold value N_th = 100 and the fact that the last 30 points before reaching this value are not plotted.

It is important to note that other long trajectories produce practically the same patterns as seen in figure 9. The fact that these are independent of the initial pair (x₀, v₀) illustrates that there exists a chaotic set in the phase space underlying the motion. (In the terminology of transient chaos this is called a non-attracting chaotic set or a chaotic saddle [7, 15, 22]).

Let us, finally, show the motion in real space. The path of the ball (built up from appropriate parabola arcs) corresponding to the points shown in figure 9 are exhibited in panel (b) of figure 10. For comparison, the quasi-periodic motion belonging to the insets of figure 9 is shown in panel (a). The difference is enormous. Arcs in panel (b) are of different heights (below a maximum) mostly irregularly arranged. The existence of nearly repetitive patterns is a consequence of the presence of quasi-periodic epochs in the motion (horizontal segments in figure 9). The full motion is a mixture of irregular periods and such epochs, each of the latter, if lasting infinitely long, would be similar to the path in panel (a).

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We have thus demonstrated all the main characteristics of chaotic motions already for a small r: irregularity in time, sensitivity to initial conditions, and complex, fractal-like patterns in phase space. The motion keeps, however, its character for finite times only: chaos is transient.