Bouncing states of a droplet on a liquid surface under generalized forcing

Droplets can exhibit complex dynamics when vertically and sinusoidally forced by a liquid surface from which they remain separated by a thin air cushion. Here we extend previous studies to include a family of periodic forcing functions that vary smoothly from sinusoidal to square wave by changing a single parameter. Through analytical and numerical work we ﬁnd that the dynamics of the droplets and transitions between regular and chaotic regimes are effectively controlled by the impulse imparted on the droplets over a half-period. We also ﬁnd that having nonsinusoidal forcing lowers the threshold amplitudes for most of the dynamical regimes. This is explained on the basis of a correlation between impulse increases and subsequent energy increases.


I. INTRODUCTION
Since the pioneering work of Couder et al. [1] demonstrating that a droplet is able to bounce indefinitely without coalescing on the surface of a vertically vibrated liquid bath, the rebounding of such droplets has attracted a great deal of attention, partly because controlled droplet bouncing may be a promising way to manipulate fluids for microfluidic operations [2], and partly due to its reminiscence of the wave-particle duality [3,4]. The bouncing dynamics of drops atop a liquid bath vibrated sinusoidally at a given frequency has been previously studied [5][6][7][8][9], in the case where the vibration frequency corresponds to a given excited wavelength according to the dispersion relation of Faraday waves, while a simple theoretical model of the bouncing states has been developed [8,10] for the bouncing dynamics on a soap film vertically and sinusoidally vibrated. Also, the vertical dynamics of a bouncing droplet on a bath subjected to forcing signals (FSs) composed of a combination of two commensurable frequencies has been experimentally studied in a recent work [11]. In all these cases diverse bouncing states were found, including periodic and multiperiodic states as well as chaos, as the amplitude of the harmonic FSs is varied. Beyond a critical amplitude threshold the stationary bouncing state becomes unstable due to the underlying wave field, giving rise to a walking regime in which the drops walk across the liquid surface.
In such previous works, however, the robustness of the bouncing scenario (BS) against diversity in the vertical vibrations of the liquid surface frame was not studied since harmonic vibrations have been systematically used because of their simplicity. Clearly, this mathematically convenient choice imposes a drastic and unnecessary restriction in the BS, limiting thus its possible implications and applications [12]. Thus, to fully explore and exploit the physics of the BS, it seems appropriate to consider FSs exhibiting general features of periodic signals which are the output of generic (nonlinear) systems, therefore being appropriately represented by Fourier series-not just by a single harmonic term. Since there are infinitely many different waveforms, an important question, both scientifically and technologically, is how can one explain in physical terms-providing in turn a quantitative characterization-the effect of the FS's waveform on the BS once its amplitude and period are fixed.
Here we theoretically and numerically show that for a generic FS f (t ) having equidistant zeros, the impulse transmitted by the signal over a half-period (hereafter referred to simply as the signal's impulse), is a relevant quantity that characterizes the effectiveness of such FS. Here T is the period and I a quantity integrating the conjoint effects of the signal's amplitude, period, and waveform. In the accelerating reference frame of the liquid surface, f (t ) introduces a fictitious inertial force .. f (t ) having the same period T and an impulse given by (2) i.e., I is given by the (absolute) difference between the slopes of the FS at two of its consecutive zeros. The relevance of the signal's impulse has been observed previously in quite different physical contexts, such as adiabatically ac driven periodic Hamiltonian systems [13], chaotic dynamics of a pumpmodulation Nd:YVO 4 laser [14], ratchet transport [15][16][17][18], discrete breathers in nonlinear oscillator networks [19], topological amplification effects in scale-free networks of signaling devices [20], driven two-level systems and periodically curved waveguide arrays [21], and control of chaos in damped driven systems by secondary periodic excitations [22]. The effect of the FS's impulse will be studied below by generalizing the sinusoidal FS to a family of periodic functions, related to Jacobian elliptic functions, which vary smoothly from sinusoidal to square wave by varying a single parameter (m). This family of signals will be introduced in Sec. II, along with a model that describes the behavior of droplets on a soap film in terms of a second-order ODE with two parameters, the previously introduced m and an amplitude parameter, . That section also describes the analytical and numerical tools used to address this problem, which include energy-based analysis, numerical integration, extraction of power spectra, and Lyapunov exponents (LEs) from the generated time series. Section III presents our results. We conclude the paper with Sec. IV, a general conclusion. Figure 1 shows a schematic representation of a droplet bouncing on a soap film vibrated with a vertical generic Tperiodic displacement f (t ).

A. Forcing functions
Specifically, we study the effect of the FS's impulse by generalizing the sinusoidal FS to the case Similar to the case of a sinusoidal FS, writing the force balance in the accelerating reference frame introduces the (dimensionless) periodic forcing term −F (τ ) with where ≡  (4), respectively] and using standard tables of integrals [24], one can obtain the dependence on the shape parameter m of their respective impulses I and I [see where the coefficients b n and a n (m) are given below. Note that the functions I (m), I (m), a 0 (m) monotonically increase as the shape parameter m is increased from zero, while keeping constant B and T , which allows us to expect impulse-induced dynamical changes in the BS, as shown below. After Taylor expanding the function arcsin [sn ( t; m)] in Eq. (3), one straightforwardly obtains the impulse I (m) given by Eq.
where the integrals in Eq. (8) can be obtained in closed analytical form by using Mathematica: Similarly, after applying standard tables of integrals [24], one straightforwardly obtains the impulse I (m) given by Eq. (6). We next provide the Fourier series of the FS f (t ) [Eq. (3)] and its second derivative and using the Fourier series of the Jacobian elliptic function [23] cn (·; m), one obtains where 0 ≡ 2π/T . After taking the second derivative of the series in Eq. (13), one obtains Note that b n (m) = (2n + 1) 2 a n (m), and hence i.e., the dependence of the respective (normalized) Fourier coefficients of the FS and its second time derivative are the same, as expected, which allows us to understand the common dependence on the shape parameter of their associated (normalized) impulse functions [cf. Fig. 2(c)]. Figure 3 shows the relative size of the second and third Fourier coefficients with respect to the first one as a function of the shape parameter, illustrating thus the extremely rapid convergence of the Fourier series even for values of m very close to 1.

B. Model system
Introducing the FS just described into a model of bouncing states [8,10] one obtains where all parameters and variables are dimensionless, . y ≡ dy/dτ , U (y) ≡ y + y 2 H (−y)/2 + 1/2 is the effective potential associated with the forces exerted by the gravitational field and by the soap film [see Fig. 1(d)], H (·) is the Heaviside function, is an experimentally adjustable parameter, while y ≡ −kz/(m d g) provides a measure of the vertical position of the droplets (see Fig. 1). This model will be analyzed in the next section.

A. Numerical methods
The differential system (16) was integrated by using a fourth-order Runge-Kutta method with time steps chosen in the range τ = 0.001-0.005. The simplest way to visualize the effect of the FS's impulse is to study the evolution of the attracting orbits as the shape parameter m (or the normalized amplitude ) is varied. To provide different ways of dealing with the possibility of impulse-induced multiple attractors, we considered two options for the initial conditions for each of the parameter values of the bifurcation diagrams. In one case, we initialized on the last state found for the previous, slightly smaller parameter value (i.e., "following the attractor"), while we initialized on the same, fixed initial condition for each new parameter value in the other case. The numerical integration of Eq. (16) yields pseudo-orbits of the system in the form of time series y(τ ), . y(τ ); a standard fast Fourier transform then yields the power spectrum S(ω) = |a(ω)| 2 . Usual averaging procedures were used to improve its quality [25]. Examples of time series and corresponding power spectra are shown below. We calculated LEs by using a version of the algorithm introduced in Ref. [26], with integration typically up to 10 4 drive cycles for each fixed set of parameters. To construct the LE diagrams we followed two steps. First, the maximal LE, λ + , was calculated for each point on a N × N grid with amplitude factor and shape parameter m along the horizontal and vertical axes. Second, a diagram was constructed by only plotting points on the grid according to a color code. The temporal average energy E was typically calculated over 10 5 forcing cycles after the transients had been removed.

B. Energy-based analysis
By analyzing the variation of the droplets' energy, one can rationalize both the impulse-induced emergence of complexity (multiperiodicity) and the impulse-induced emergence of chaos. Indeed, Eq. (16) has the associated energy equation where E(τ ) ≡ where the remaining integral in Eq. (19) represents the dissipation work, τ * ∈ [nT , nT + T /2], while is the impulse transmitted per unit of period and unit of amplitude [cf. Eq. (6)]. One can understand the occurrence of impulse-induced onset of chaos, by analyzing Eq. (19): a simple heuristic argument is given next. Let us consider fixing the parameters ( , , T ) for the droplet's motion to present a periodic orbit near the bottom of the potential well at m = 0. Since the initial state is a steady (periodic) state, τ * will depend solely on the FS but not on n. In this situation, one increases m while holding the remaining parameters constant. For values m > 0 such that the state of the bouncing droplet is still a periodic orbit (which will be relatively near the initial periodic orbit in phase space), one expects that . y(τ * ) and the remaining integral in Eq. (19) will maintain approximately their initial values (at m = 0) while the impulse I (m) | =T =1 will rise from its initial value. This means that, in some case depending upon the remaining parameters and the sign of chaotic attractor, i.e., the oscillation amplitude threshold that will allow escape from the basin of attraction of the initial periodic orbit to be able to reach the basin of attraction of a different periodic orbit or that of a chaotic attractor. Clearly the probability of this event increases as the shape parameter m is increased from zero, i.e., as the impulse transmitted is increased from the value corresponding to a sinusoidal FS.  Fig. 5(b)], such that only the peaks corresponding to odd subharmonics, ω = 3ω 0 , 5ω 0 , 7ω 0 , . . ., are higher than those associated with the power spectrum of the period-2 solution existing for m = 0 (compare the two lowermost lines in Fig. 6). Further increase of the impulse may give rise to chaos, as for m = 0.95 [cf. Fig. 5(c)], which is reflected in the visible enhancement of the power background in the corresponding power spectrum (second line from the top in Fig. 6). This further effect is ever higher as m → 1, i.e., as the FS approximates a square wave function [see Fig. 2(a)], as for  Fig. 6]. The power of the four spectra shown in Fig. 6 is exactly the same at the fundamental (driving) frequency ω = ω 0 ≡ 2π/T , as expected.

C. Bouncing scenario
As already mentioned above, Fig. 3 shows the relative dependence on the shape parameter of the second, a 1 (m), and third, a 2 (m), Fourier coefficients of the FS [cf. Eqs. (13) and (14)] with respect to the first one, a 0 (m), indicating the extremely rapid convergence of the Fourier series even for values of m very close to 1. Thus, regarding experiments, this property is relevant in the sense that it allows one to effectively approximate the elliptic FS by solely retaining the first two harmonics of its Fourier expansion over a sufficiently large range of values of the shape parameter (0 ≤ m 0.99; see Fig. 3): where 0 ≡ 2π/T [cf. Eq. (13)]. The second-order nonautonomous differential equation (16) may be solved subject to the initial conditions y(0) = 0, . y(0) = −V at impact. Here we are mainly interested in investigating the structural stability of the BS under changes in the FS's impulse (i.e., in the shape parameter m), while keeping constant the remaining parameters. For the sake of comparison with previous studies, we will assume parameter values which were used in the simple case of a soap film vertically and sinusoidally vibrated [8,10]. Our numerical simulations show that an effect of increasing the impulse transmitted by the FS is to diminish the value of its amplitude factor at which the bouncing droplet undergoes a period-doubling transition both in the case of initial conditions "following the attractor" and in the case of fixed initial conditions (see Fig. 4). Another effect of increasing the impulse is to diminish the threshold value of the amplitude factor at which the chaotic behavior appears, chaos = chaos (m), again in the two cases of initial conditions (see Fig. 4). The same aforementioned impulse-induced dynamical features can be seen in the bifurcation diagrams for the variable dy/dτ with the shape parameter m in the range 0 ≤ m ≤ 1−10 −14 , which are shown in Fig. 7.
Additional information of the bouncing motions appearing at different values of the shape parameter can be obtained by analyzing the spectral properties of the corresponding solutions (see Fig. 6). The comparison of such power spectra indicates that only the power associated with the peaks corresponding to odd subharmonics increases as the impulse transmitted is increased (i.e., as m is increased from zero), which is a consequence of the increase of the relative weight of such subharmonics in the Fourier series of both f (t ) and F (τ ) [Eqs. (13) and (14), respectively] as m → 1. Furthermore, our numerical simulations indicate that the extension of the region in phase space which is visited by the corresponding chaotic attractors existing at > chaos (m) increases as the shape parameter m is increased, i.e., as the impulse transmitted is increased. We systematically studied the chaotic regime by calculating Lyapunov exponents (LEs), as in the example shown in Fig. 8. One sees again that the effect of increasing the impulse (i.e., the shape parameter m) is to diminish the threshold value of the amplitude factor at which the Remarkably, our computations of the temporal average energy, E , confirmed the predictions from the energy-based analysis (see Sec. III B). Indeed, we found an excellent correlation among the temporal average energy E , the impulse I [Eq. (6)], and the maximal LE λ + as functions of the shape parameter m over the complete range of periodic solutions (compare Figs. 2 and 9), while we found an approximate correlation over the complete range of chaotic solutions (see Fig. 9, bottom panels). The asymptotic behavior of E and λ + as m → 1 is ultimately due to the functional dependence [24] of K (m) on m.
When the coefficient [Eq. (16)] is sufficiently small, one can expect the appearance of the phenomenon of multistability [28]. We have indeed detected the coexistence of different attractors when the amplitude parameter is varied for a given set of the remaining parameters. Figure 10 shows an example for four randomly chosen initial conditions and m = 0, = 0.01. We found that multistability occurs over an intermediate range of values between two ranges of monostability (a period-1 attractor for sufficiently small amplitudes and a chaotic attractor for sufficiently large amplitudes). Remarkably, we also observed multistability when the shape parameter m is varied while the remaining parameters are held constant. Figure 11 shows an example for four randomly chosen initial conditions and = 0.04, = 0.01. One sees the coexistence of period-1, period-2, and period-3 attractors over distinct ranges of m values. Thus reshaping a FS appears as an alternative procedure to control multistability [28].

IV. CONCLUSIONS
Our theoretical and numerical results show that the impulse transmitted by the forcing signal which yields vertical vibrations of a liquid surface frame is a fundamental quantity for the reliable control of the scenario associated with the stationary bouncing regime of droplets. Specifically, we have shown through a single model that the values of the forcing signal's amplitude at which different dynamical bifurcations occur, such as transitions from simple periodic states to complex periodic, as well as transitions from periodic states to chaotic states can be suitably diminished by increasing the impulse transmitted, while keeping constant the remaining parameters. We have analytically shown that this effect of the impulse may be understood on the basis of a correlation between increases of the impulse and subsequent increases of the energy transmitted by the vertical acceleration of the liquid surface. We have also found that multistability can be effectively controlled by the impulse imparted on the droplets.
Moreover, the present bouncing droplets scenario contains and extends the prevalent one, since the latter essentially arises from the systematic application of sinusoidal forcing signals and hence its control depends upon two quantities that have systematically been assumed to be independent of each other: the amplitude and the frequency of the sinusoidal signal. In contrast, the present bouncing droplets scenario identifies a single quantity-the impulse transmitted by the forcing signal per unit of period and unit of amplitudewhich is controlled by a single parameter while providing reliable control of the droplets' dynamic irrespective of the forcing signal waveform. It is worth mentioning that two FSs transmitting the same impulse but with different period and amplitude can yield different BSs because of the possible existence of resonances with underlying periodic solutions.
More importantly, we have shown that an effective experimental realization of the impulse-based control of the droplets' dynamic can be readily achieved by solely implementing the two first harmonics of the Fourier series of our generalized forcing signal, which facilitates its application in microfluidics. In addition, our work could shed some light on our understanding and control of the hydrodynamics of pilot wave systems.