Horizontal stability of a bouncing ball

A ball bouncing on a vibrating platform is a classical example of a simple dynamical system that exhibits a rich variety of nonlinear behaviors, in particular, a period-doubling cascade towards chaos.¹ This system has remained of significant interest over the past several decades in research and instruction, in part due to its relative simplicity of conception and its application to vibrated granular gases.^2–11 Experimental realizations of the bouncing ball system require that the ball remains on a finite plate surface despite ambient perturbations. Confinement is often achieved by use of a guiding tube;^12–14 however, the tube may introduce an additional unwanted source of friction into the system. A favorable alternative is the use of an underlying concave surface,^15–18 typically a concave parabolic lens. Despite repeated impacts on an inwardly curved surface, it has been noted that the ball does not always tend to the vertex but may actually achieve persistent horizontal motion. In this work, we explore the simplest possible model of a partially elastic ball bouncing on a vibrated parabolic surface and arrive at the striking conclusion that a concave impact surface can destabilize periodic vertical bouncing at the vertex such that horizontal motion of the ball would be observed in experiment.

Despite its relevance to experimental realizations of this rich dynamical system, the problem of the bouncing ball's horizontal motion has only received minimal attention. Wright, Swift, and King first investigated the problem of a partially elastic ball bouncing on a concave parabolic surface numerically but focused on relatively large driving amplitudes.¹⁹ They also include a continuous application of noise, which they describe as a model for the surface roughness inherent to an actual experiment. Curiously, they discovered that the maximum horizontal range of the ball does not simply decrease monotonically with the curvature of the surface as one might intuitively expect. Instead after a critical value of surface curvature, the ball intermittently makes large horizontal excursions across the surface. More recently, Chastaing et al. investigated this system experimentally and showed that for the range of parameters tested, the ball readily leaves the vertex and moves laterally over the surface.¹⁸ They rationalize their experimental observations with a simple 2D model that captures some of the qualitative behaviors such as the possibility of a pendulum-like oscillation in the ball's horizontal position. However, their model predicts unbounded exponential growth, whereas bounded horizontal motion is observed in their experiments. Furthermore, they assume that the vertical motion remains unchanged despite any horizontal displacement, which will not always be guaranteed when impacting a curved surface.

Despite these recent works, many outstanding questions remain. Most significantly, does a concave parabolic impact surface necessarily result in horizontal displacement away from the vertex? In this work, we focus on the parameter regime where in the absence of surface topography, the ball bounces periodically on a vibrated flat platform, impacting the plate exactly once per vibration cycle, which we define as simple vertical bouncing. We show that a critical parabolic coefficient exists beyond which simple vertical bouncing is unstable and bounded horizontal motion ensues. Furthermore, the resulting horizontal and vertical motion can be periodic or even chaotic.

We model the motion of a partially elastic ball bouncing on a platform in two dimensions with surface height

where K is the dimensional parabolic coefficient that defines the static shape of the platform. As in Ref. 4, we will work in dimensionless units, by non-dimensionalizing time by the time scale $2πω$ and space by the length scale $2π2gω2$⁠. This results in the non-dimensional surface height

where $Γ=γω2πg, κ=2π2gKω2$⁠, and g is the acceleration due to gravity. After leaving the platform at a time τ₀ from a horizontal position ξ₀ with relative horizontal and vertical take-off velocities u₀ and v₀, respectively, the absolute height of the ball prior to the next impact is given by

and its horizontal position by

Thus, the height of the ball relative to the platform below can be written as

In order to find the time of the next impact, we find the first zero of the relative height $Δ(τ1)=0$ such that $τ1>τ0$⁠. Note that it is possible for multiple zeros to exist that satisfy $Δ(τ)=0$ and $τ>τ0$⁠, so care must be taken to locate the first zero.²⁰ Knowing the time of next impact allows us to update the horizontal position of impact

the relative vertical velocity into the impact

and the horizontal velocity into the impact

We then update the velocities exiting the impact according to the following equation:

where M₁ is a rotation matrix that rotates the frame of reference to be aligned with the surface's local normal vector at the impact location ξ₁

with

and A is our matrix of restitution coefficients

where α_T is the tangential coefficient of restitution and α_N is the normal coefficient of restitution. Equations (2)–(12) complete an implicit map $(τ0,v0,u0,ξ0)→(τ1,v1,u1,ξ1)$⁠. This procedure is then repeated for each successive bounce.

Note that our model treats the ball as a point mass, which is valid when the radius of the ball R is much smaller than the minimum radius of curvature of the surface, specifically when $KR≪1$⁠. This simplification also allows us to safely neglect any influence of the ball's rotation.¹⁹ Furthermore, we have also made the typical assumptions that the contact of the ball occurs instantaneously and that the influence of air resistance during flight is negligible.

For κ = 0, the model equations reduce to the standard one-dimensional scenario that has been studied extensively. Luck and Mehta⁴ showed that the following relations are satisfied for the simple vertical bouncing solution (⁠$τ1=τ0+1$⁠, v₁ = v₀):

where $τ*≡τn (mod 1)$ and $v*=vn$⁠. Furthermore, by linearizing the governing equations about solutions of this form, Luck and Mehta⁴ demonstrated that the 1D system allows for a stable simple vertical bouncing state provided that the forcing amplitude Γ lies within the following range:

Just above this range, period-doubling in the vertical motion begins. Clearly for $κ>0$⁠, the simple vertical bouncing solution will still exist for the case when $ξn=0$ and u_n = 0, i.e., when the ball bounces repeatedly at the vertex of the parabola. The question remains, however, whether such motion is stable to horizontal perturbation. To address this question, we apply small perturbations to each of the variables (⁠$τ0=τ*+δτ0, v0=v*+δvo, u0=0+δu0$⁠, and $ξ0=0+δξ0$⁠) and assess whether the perturbation amplitude grows or decays. Following one bounce, to linear order, we arrive at the following system whose eigenvalues dictate the stability of the simple vertical bouncing state:

where

and

The block diagonal structure of the linearization matrix in (16) shows that the horizontal and vertical perturbations are decoupled to linear order. In fact, the 2 × 2 block in the upper-left corner is identical to that obtained by Luck and Mehta,⁴ which is independent of κ. The first two eigenvalues, λ₁ and λ₂, both have magnitude less than 1 for the simple vertical bouncing state under consideration. The horizontal stability is thus dictated by the eigenvalues λ₃ and λ₄ associated with the 2 × 2 block in the lower-right corner of (16), which depends explicitly on κ. Below a critical value of $κ<κc$⁠, the motion is linearly stable to horizontal perturbations and thus relaxes back to the simple vertical bouncing state centered at ξ = 0. However, above this critical value of $κ>κc$⁠, the motion is linearly unstable and thus will not return to the simple vertical bouncing state. The value of κ_c is only weakly dependent on α_N and α_T as is shown in Figure 1 and occurs around $κc≈1$ over a wide range of parameters. We also note that any other higher periodic vertical bouncing modes (of positive-integer period m such that $τ1=τ0+m$⁠) become unstable at even lower values of κ, specifically when $κ>κc/m2$ (see supplementary material for further discussion), thus precluding the possibility of the ball transitioning to such a periodic vertical bouncing state at ξ = 0.

After simple vertical bouncing becomes laterally unstable, what is the resultant motion? We will explore the evolution of the motion as κ is increased with $αN=0.975, αT=0.6$⁠, and $Γ=0.1$⁠. For these parameters, a stable vertical bouncing solution exists for the 1D (flat-plate) dynamics, which can be verified by Equation (15). However, according to Equation (16), we expect this state to become unstable due to the topography if κ is increased beyond its critical value, for this case when $κ>κc=1.00318…$⁠.

We begin by numerically solving and iterating the 2D model equations and recording the final 64 horizontal positions of impact ξ_n after letting any transient behavior decay, the result of which is shown in Figure 2(a). As is expected from the linear stability analysis, for $κ<κc$ the ball finds the simple vertical bouncing state with $ξn=0$⁠. However, at $κ=κc$⁠, the simple vertical bouncing state loses its stability in a supercritical flip bifurcation, beyond which horizontal motion is observed. At the point of bifurcation, a period-2 lateral bouncing state emerges which corresponds to symmetrically bouncing back and forth over the vertex, as shown in the lower inset of Figure 2(a).

To explore this bifurcation in further detail, we solve for all solutions of our model equations that satisfy $τ1=τ0+1$⁠, v₁ = v₀, $u1=−u0$⁠, and $ξ1=−ξ0$⁠, which captures both the simple vertical bouncing solutions and the period-2 lateral bouncing solutions. Note that this type of solution always appears as a pair of which at most one is stable for a given κ. The phase of impact for these solutions is plotted in Figure 2(b) and the line types of the curves signify their linear stability (see supplementary material for expanded details on the stability analysis). Consistent with the numerical results, at $κ=κc$⁠, a stable period-2 lateral bouncing solution emerges. As κ is increased further, this solution persists but is not always stable. The period-2 lateral bouncing solutions eventually vanish when the pair of solutions collides at θ = 0. Beyond this point, the simple vertical bouncing state remains unstable, and even more complex horizontal and vertical dynamics arise.

We next investigate the dynamics further by gradually increasing κ over an extended range. We again plot the final 64 horizontal positions of impact for each value of κ in Figure 3(a), after having allowed any initial transient behavior to die out. As can be seen, following the initial bifurcation, horizontal motion continues to persist as κ is increased. We also numerically estimate the largest Lyapunov exponent (LLE) associated with the trajectories,²¹ the results of which are presented in Figure 3(b). The motion varies between periodic, quasi-periodic, or chaotic, depending on κ. For this set of parameters, we observe chaotic motion over several ranges of κ, a direct consequence of the underlying topography. Several sample sequences of horizontal impact positions are shown in Figures 4(a)–4(d) for different values of κ. Finally, in Figure 4(e), we show the projection of a chaotic attractor observed for $κ=2.3$ with each point representing a single impact with horizontal position ξ_n and impact phase θ_n.

In all of the simulations appearing in this work, “chattering,” wherein the ball bounces an infinite number of times in a finite amount of time,^4,22 was not observed. Although we only explored one set of parameters in such detail, the qualitative sequence of dynamics appears generic as κ is increased beyond κ_c.

Until recently, it has been assumed that a concave surface will prevent or at least minimize the horizontal motion of a ball bouncing on a vibrating platform. However, in our analysis, we have shown that a persistent bounded horizontal motion of the ball can actually be a direct consequence of the surface's concavity alone. Specifically, we have shown that simple vertical bouncing of a ball at the vertex of a vibrated parabolic surface is unstable when the surface's parabolic coefficient exceeds a critical value. The simple vertical bouncing solution loses its stability in a supercritical flip bifurcation and complex horizontal motion emerges. Due to the simplicity of the model, we were able to isolate the critical role that surface topography plays in this classical dynamical system.

While the parabolic shape may seem like a restrictive class of topography, a smooth local minimum is, in general, locally parabolic, that is, when the leading (non-constant) term in the Taylor expansion is quadratic. Thus, our analysis will also apply to vertical bouncing at the local minima of significantly more general surfaces.

Future work will extend the current model to three dimensions and include the influence of ball rotation, eventually allowing for quantitative comparison with the recent 3D experiments.¹⁸ We also aim to explore surfaces with periodic topography, where net particle transport²³ and diffusive behavior^24,25 are possible. Furthermore, we plan to consider the motion of particles on vibrating surfaces with time-dependent topography, which may lead to simple models for the motion of small granules on a resonating plate²⁶ or the motion of small fluid droplets bouncing on standing Faraday waves.^27,28

See supplementary material for additional details on the numerical methods, stability analysis, and movies of some possible bouncing states.

The authors would like to thank the Joint Applied Math and Marine Sciences Fluids Lab at the University of North Carolina at Chapel Hill for support and engaging discussions. D.M.H. would also like to acknowledge the financial support of the National Science Foundation (Grant No. RTG DMS-0943851).