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We present the results of a numerical investigation of the emergence of chaos in the orbital dynamics of droplets walking on a vertically vibrating fluid bath and acted upon by one of the three different external forces, specifically, Coriolis, Coulomb, or linear spring forces. As the vibrational forcing of the bath is increased progressively, circular orbits destabilize into wobbling orbits and eventually chaotic trajectories. We demonstrate that the route to chaos depends on the form of the external force. When acted upon by Coriolis or Coulomb forces, the droplet's orbital motion becomes chaotic through a period-doubling cascade. In the presence of a central harmonic potential, the transition to chaos follows a path reminiscent of the Ruelle-Takens-Newhouse scenario.

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... They concluded that the walking state is stable to in-line perturbations and neutrally stable to lateral perturbations in the parameter regime accessible in the laboratory. The stroboscopic model has been used to study the walker's complex nonlinear dynamics in a number of settings [17,27,31,32,34,36,45,46,48,205,[219][220][221] to be detailed in sections 5-8. ...

... Tambasco et al [221] characterized theoretically the onset of chaos through the destabilization of circular orbits in orbital pilot-wave dynamics using the stroboscopic model (section 4.3). The authors considered the dynamics of walking droplets acted upon by external forces, specifically Coriolis, Coulomb, and linear spring forces. ...

... Rahman and Blackmore [256] review the walking-droplet system from the perspective of the dynamical-systems-theory community. It is noteworthy that attempts to characterize transitions to chaos experimentally face the difficulty that the transition typically happens abruptly; for example, in the case of a walker in a simple harmonic potential, over a span of Δγ/γ F ∼ 0.004 [221]. Perrard and Labousse [34] drew a distinction between the chaotic dynamics observed just at the onset of chaos, and the intermittency observed in the high-memory limit. ...

The walking droplet system discovered by Yves Couder and Emmanuel Fort presents an example of a vibrating particle self-propelling through a resonant interaction with its own wave field. It provides a means of visualizing a particle as an excitation of a field, a common notion in quantum field theory. Moreover, it represents the first macroscopic realization of a form of dynamics proposed for quantum particles by Louis de Broglie in the 1920s. The fact that this hydrodynamic pilot-wave system exhibits many features typically associated with the microscopic, quantum realm raises a number of intriguing questions. At a minimum, it extends the range of classical systems to include quantum-like statistics in a number of settings. A more optimistic stance is that it suggests the manner in which quantum mechanics might be completed through a theoretical description of particle trajectories. We here review the experimental studies of the walker system, and the hierarchy of theoretical models developed to rationalize its behavior. Particular attention is given to enumerating the dynamical mechanisms responsible for the emergence of robust, structured statistical behavior. Another focus is demonstrating how the temporal nonlocality of the droplet dynamics, as results from the persistence of its pilot wave field, may give rise to behavior that appears to be spatially nonlocal. Finally, we describe recent explorations of a generalized theoretical framework that provides a mathematical bridge between the hydrodynamic pilot-wave system and various realist models of quantum dynamics.

... The specificity of walker dynamics is that it exhibits wave-like behaviours and quantised sets of attractors both in the deterministic regime [2][3][4] or in the highly disordered regime [5,6]. The study of the transient from one regime to another has been less studied, but it has now been reported experimentally [7] and numerically [8]. Other studies have also reported the arising of more complex trajectories [9] that are located in this intermediate regime. ...

... However the use of standard dynamical system tools (bifurcation diagram, first return map, period doubling research) have drawn a transition to low-dimensional chaos which implies a loss of determinism by increasing lack of predictability. Tambasco et al. [8] in particular have compared three confinement configurations (Coriolis force, central force and Coulomb potential) to show that several scenarii of transition to chaos can be observed, depending on the confinement type. ...

... We provide evidence of a transition to chaos from periodic orbits to unpredictable dynamics of two types. One type follows the previous observation of Perrard et al. [7] and the previous detailed analysis of Tambasco et al. [8]. The transition is in particular now thoroughly characterised and consistent between authors for small extension orbits. ...

A walker is the association of a sub-millimetric bouncing drop moving along with a co-evolving Faraday wave. When confined in a harmonic potential, its stable trajectories are periodic and quantised both in extension and mean angular momentum. In this article, we present the rest of the story, specifically the chaotic paths. They are chaotic and show intermittent behaviors between an unstable quantised set of attractors. First, we present the two possible situations we find experimentally. Then, we emphasise theoretically two mechanisms that lead to unstable situations. It corresponds either to noise-driven chaos or low-dimensional deterministic chaos. Finally, we characterise experimentally each of these distinct situations. This article aims at presenting a comprehensive investigation of the unstable paths in order to complete the picture of walkers in a two dimensional harmonic potential.

... Through a detailed analysis, Oza et al. 12 reached an integro-differential equation of motion for the bouncing droplets and coined the term "pilot-wave hydrodynamics". Numerical investigations of this model with confining central-potential terms (Coriolis, harmonic, or Coulomb type) demonstrated different routes to chaos 13 and rich subsequent dynamics. Similar to their experimental counterparts, these systems also exhibit rotation symmetry and are at the focus of the current paper. ...

... With the wave field given as in (28), the trajectory equation (27) is a delay-differential system, which is very hard to study both analytically and numerically. Therefore, Oza et al. 12 and several subsequent studies 13,27,28 approximated the sum in (28) by the integral ...

... The model parameters (48) are determined by the experimental conditions as explained by Oza et al. 12 and Perrard et al. 10 . In the following numerical work, we are going to adopt the numbers reported by Tambasco et al. 13 and set m = 0.25 × 10 −6 kg, f = 80Hz, D = 2.0 × 10 −6 kg/s, A = 3.5 × 10 −6 m, and k = 3.2 × 10 −6 N/m. These choices yield the nondimensional system parameters ...

We consider the motion of a droplet bouncing on a vibrating bath of the same fluid in the presence of a central potential. We formulate a rotation symmetry-reduced description of this system, which allows for the straightforward application of dynamical systems theory tools. As an illustration of the utility of the symmetry reduction, we apply it to a model of the pilot-wave system with a central harmonic force. We begin our analysis by identifying local bifurcations and the onset of chaos. We then describe the emergence of chaotic regions and their merging bifurcations, which lead to the formation of a global attractor. In this final regime, the droplet's angular momentum spontaneously changes its sign as observed in the experiments of Perrard et al. [Phys. Rev. Lett. 113(10), 104101 (2014)].

... Through a detailed analysis, Oza et al. 12 reached an integro-differential equation of motion for the bouncing droplets and coined the term "pilot-wave hydrodynamics". Numerical investigations of this model with confining central-potential terms (Coriolis, harmonic, or Coulomb type) demonstrated different routes to chaos 13 and rich subsequent dynamics. Similar to their experimental counterparts, these systems also exhibit rotation symmetry and are at the focus of the current paper. ...

... With the wave field given as in (28), the trajectory equation (27) is a delay-differential system, which is very hard to study both analytically and numerically. Therefore, Oza et al. 12 and several subsequent studies 13,27,28 approximated the sum in (28) by the integral ...

... The model parameters (48) are determined by the experimental conditions as explained by Oza et al. 12 and Perrard et al. 10 . In the following numerical work, we are going to adopt the numbers reported by Tambasco et al. 13 and set m = 0.25 × 10 −6 kg, f = 80Hz, D = 2.0 × 10 −6 kg/s, A = 3.5 × 10 −6 m, and k = 3.2 × 10 −6 N/m. These choices yield the nondimensional system parameters ...

We consider the motion of a droplet bouncing on a vibrating bath of the same fluid in the presence of a central potential. We formulate a rotation symmetry-reduced description of this system, which allows for the straightforward application of dynamical systems theory tools. As an illustration of the utility of the symmetry reduction, we apply it to a model of the pilot-wave system with a central harmonic force. We begin our analysis by identifying local bifurcations and the onset of chaos. We then describe the emergence of chaotic regions and their merging bifurcations, which lead to the formation of a global attractor. In this final regime, the droplet's angular momentum spontaneously changes its sign as observed in the experiments of Perrard et al. (Phys. Rev. Lett., 113(10):104101, 2014).

... Under this perspective, it forms a natural candidate to address the questions above. Previous experiments [19,20] have shown that in the case of long memory, the walker's dynamics can exhibit intermittent chaotic behaviors. We wish to endeavor the very high memory regimes. ...

... It was shown that a positional information can be stored in an oscillating wave field [17,18,25,26]. This dynamical storage of information can lead to selforganization [23,[26][27][28][29], can trigger different types of transition to chaos [19,20,30]. As the basics of a Turing machine [31], the memory can be written, stored, read and erased. ...

We present a wave-memory driven system that exhibits a macroscopic diffusive-like behavior emerging from a deterministic set of microscopic rules. This diffusive-like motion originates from a self-scattering process that the wave-particle coupling generates spontaneously. We show that the stochastic aspect of this self-scattering process derives from a Shil'nikov type chaos. The chaotic nature induces a bimodal statistics analogue to a run-and-tumble processes usually observed in living systems of much higher complexity. This is the first evidence of controlled and tunable diffusive-like motion of a single particle ruled by deterministic dynamics. We show that the resulting diffusive properties are determined by the duration of the evanescent wave-memory.

... Under this perspective, it forms a natural candidate to address the questions above. Previous experiments [19,20] have shown that in the case of long memory, the walker's dynamics can exhibit intermittent chaotic behaviors. We wish to endeavor the very high memory regimes. ...

... It was shown that a positional information can be stored in an oscillating wave field [17,18,25,26]. This dynamical storage of information can lead to selforganization [23,[26][27][28][29], can trigger different types of transition to chaos [19,20,30]. As the basics of a Turing machine [31], the memory can be written, stored, read and erased. ...

We present a wave-memory driven system that exhibits a macroscopic diffusive-like behavior emerging from a deterministic set of microscopic rules. This diffusive-like motion originates from a self-scattering process that the wave-particle coupling generates spontaneously. We show that the stochastic aspect of this self-scattering process derives from a Shil'nikov type chaos. The chaotic nature induces a bimodal statistics analogue to a run-and-tumble processes usually observed in living systems of much higher complexity. This is the first evidence of controlled and tunable diffusive-like motion of a single particle ruled by deterministic dynamics. We show that the resulting diffusive properties are determined by the duration of the evanescent wave-memory.

... According to this scenario, from a fixed point, three bifurcations induce additional incommensurate frequencies into the spectrum, after which it is likely (but not guaranteed) that a strange attractor appears in the phase space. 49 Following the methodology of Tambasco et al., 48 we fix κ = 0.03 and initialize a simulation for a value of where the periodic motion is stable, as indicated by the linear stability analysis. The simulation runs for N 0 + 2 p impacts, where the first N 0 impacts are discarded to remove transient effects. ...

... This evolution invokes a qualitative change in the statistics, with several peaks emerging in the droplet position stationary distribution [ Fig. 10(b)]. Unlike the route to chaos of circular orbits, 48 we do not observe any frequency locking between f 1 and f 2 . ...

A millimetric droplet may bounce and self-propel on the surface of a vertically vibrating bath, where its horizontal “walking” motion is induced by repeated impacts with its accompanying Faraday wave field. For ergodic long-time dynamics, we derive the relationship between the droplet’s stationary statistical distribution and its mean wave field in a very general setting. We then focus on the case of a droplet subjected to a harmonic potential with its motion confined to a line. By analyzing the system’s periodic states, we reveal a number of dynamical regimes, including those characterized by stationary bouncing droplets trapped by the harmonic potential, periodic quantized oscillations, chaotic motion and wavelike statistics, and periodic wave-trapped droplet motion that may persist even in the absence of a central force. We demonstrate that as the vibrational forcing is increased progressively, the periodic oscillations become chaotic via the Ruelle-Takens-Newhouse route. We rationalize the role of the local pilot-wave structure on the resulting droplet motion, which is akin to a random walk. We characterize the emergence of wavelike statistics influenced by the effective potential that is induced by the mean Faraday wave field.

... Here the wave persistence defines a memory time during which the positional information is stored and can be read or erased, similarly to a Turing machine [23]. Besides wave-particle-inspired dynamics [19,20,24,25,[28][29][30][31][32][33][34][35][36][37][38][39][40][41][42], walking droplets exhibit cascades of bifurcation to chaos in Coriolis and Coulomb force field [43] as well as intermittency in harmonic potential [36,[43][44][45][46][47][48]. Nonsteady propulsions have been reported in asynchronous bouncing modes [49,50] and speed limit cycle and chaotic behavior for the free particle [51] have been investigated for synchronous bouncing modes. ...

... Here the wave persistence defines a memory time during which the positional information is stored and can be read or erased, similarly to a Turing machine [23]. Besides wave-particle-inspired dynamics [19,20,24,25,[28][29][30][31][32][33][34][35][36][37][38][39][40][41][42], walking droplets exhibit cascades of bifurcation to chaos in Coriolis and Coulomb force field [43] as well as intermittency in harmonic potential [36,[43][44][45][46][47][48]. Nonsteady propulsions have been reported in asynchronous bouncing modes [49,50] and speed limit cycle and chaotic behavior for the free particle [51] have been investigated for synchronous bouncing modes. ...

We present a wave-memory-driven system that exhibits intermittent switching between two propulsion modes in free space. The model is based on a pointlike particle emitting periodically cylindrical standing waves. Submitted to a force related to the local wave-field gradient, the particle is propelled, while the wave field stores positional information on the particle trajectory. For long memory, the linear motion is unstable and we observe erratic switches between two propulsive modes: linear motion and diffusive motion. We show that the bimodal propulsion and the stochastic aspect of the dynamics at long time are generated by a Shil'nikov chaos. The memory of the system controls the fraction of time spent in each phase. The resulting bimodal dynamics shows analogies with intermittent search strategies usually observed in living systems of much higher complexity.

... In that critical case the drop stops and turns back which triggers a chaotic regime ( = 1 in Fig. 3(a) and a strange attractor Fig. 3(b)). Although, chaotic behaviors for synchronous bouncing states have been observed in confined situations [30][31][32][33] or in the particular cases mediated by complex bouncing modes [23,24], it is the first time that we observe chaotic free regimes of walking droplets which intrinsically rely on their horizontal dynamics. The boundaries separating the three distinct regimes mainly depend on the damping parameter and very little on the memory parameter. ...

... We expect that the existence of a chaotic free walking regimes itself does not depend on the dimension of the motion but that exact nature of the chaos does [29]. We also note that the transition to chaos is here very different from the chaotic paths observed in confining potentials [30][31][32][33]. The two free states are attractors and stable. ...

We investigate the dynamics of a deterministic self-propelled particle endowed with coherent memory. We evidence experimentally and numerically that it exhibits several stable free states. The system is composed of a self-propelled drop bouncing on a vibrated liquid driven by the waves it emits at each bounce. This object possesses a propulsion memory resulting from the coherent interference of the waves accumulated along its path. We investigate here the transitory regime of the buildup of the dynamics which leads to velocity modulations. Experiments and numerical simulations enable us to explore unchartered areas of the phase space and reveal the existence of a self-sustained oscillatory regime. Finally, we show the coexistence of several free states. This feature emerges both from the spatiotemporal nonlocality of this path memory dynamics as well as the wave nature of the driving mechanism.

... As the memory is increased progressively, the quantized orbital states go unstable via one of the classic routes to chaos. 35,[51][52][53] At high memory, the walker switches between unstable periodic orbits, resulting in multimodal, quantum-like statistics. 27,32 While some of this behavior is captured by the stroboscopic models, discrepancies suggest the significance of variable vertical dynamics. ...

... The first mechanism is an intermittency dynamics typical of low-dimensional chaos. As such transitions have been captured theoretically with stroboscopic models, 42,45,53 one may surmise that variability in the vertical dynamics is not an essential ingredient for them. Conversely, the second mechanism, noise-driven chaos, relies explicitly on the influence of noise in the vertical dynamics. ...

Hydrodynamic quantum analogs is a nascent field initiated in 2005 by the discovery of a hydrodynamic pilot-wave system [Y. Couder, S. Protière, E. Fort, and A. Boudaoud, Nature 437, 208 (2005)]. The system consists of a millimetric droplet self-propeling along the surface of a vibrating bath through a resonant interaction with its own wave field [J. W. M. Bush, Annu. Rev. Fluid Mech. 47, 269–292 (2015)]. There are three critical ingredients for the quantum like-behavior. The first is “path memory” [A. Eddi, E. Sultan, J. Moukhtar, E. Fort, M. Rossi, and Y. Couder, J. Fluid Mech. 675, 433–463 (2011)], which renders the system non-Markovian: the instantaneous wave force acting on the droplet depends explicitly on its past. The second is the resonance condition between droplet and wave that ensures a highly structured monochromatic pilot wave field that imposes an effective potential on the walking droplet, resulting in preferred, quantized states. The third ingredient is chaos, which in several systems is characterized by unpredictable switching between unstable periodic orbits. This focus issue is devoted to recent studies of and relating to pilot-wave hydrodynamics, a field that attempts to answer the following simple but provocative question: Might deterministic chaotic pilot-wave dynamics underlie quantum statistics?

... This instability cascade has been characterized experimentally in two systems: mode-locked lasers [33] and RDEs [34,35,36]. The hydrodynamic quantum analog has been experimentally observed [40,41] to produce the same bifurcating cascade as these systems, showing that that the particle-wave interaction dynamics is a manifestation of canonical damped-driven dynamics in spatially extended systems. The objective of this work is to construct an approximate theoretical framework to model the energy balance in the hydrodynamic quantum analog system. ...

... Regardless, we are able to construct a simple model that highlights the energy balance in the system. Moreover, these simple energy considerations are sufficient to describe the observed bifurcation sequence of period-doubling to chaos [40,41]. More broadly, such model reductions reflect the observation of Robert May, who was highly influential in popularizing the logistic map, that simple systems can produce quite complicated behavior [42]. ...

We consider the dynamics of a droplet on a vibrating fluid bath. This hydrodynamic quantum analog system is shown to elicit the canonical behavior of damped-driven systems, including a period doubling route to chaos. By approximating the system as a compositional map between the gain and loss dynamics, the underlying nonlinear dynamics can be shown to be driven by energy balances in the systems. The gain-loss iterative mapping is similar to a normal form encoding for the pattern forming instabilities generated in such spatially-extended system. Similar to mode-locked lasers and rotating detonation engines, the underlying bifurcations persist for general forms of the loss and gain, both of which admit explicit representations in our approximation. Moreover, the resulting geometrical description of the particle-wave interaction completely characterizes the instabilities observed in experiments.

... Our experiments consist of a single droplet bouncing on a bath with variable topography that introduces a radially confining force, which is known to enable chaotic dynamics in hydrodynamic pilot-wave systems. 15,27,28 In this setup, we slowly change the control parameter to uncover the series of bifurcations that lead to the formation of the system's chaotic attractor. In particular, we observe crisis bifurcations, i.e., discontinuous changes of the system's attractor upon small changes of the control parameter. ...

The theory of chaos has been developed predominantly in the context of low-dimensional systems, well known examples being the three-body problem in classical mechanics and the Lorenz model in dissipative systems. In contrast, many real-world chaotic phenomena, e.g. weather, arise in systems with many degrees of freedom rendering most of the tools of chaos theory inapplicable to these systems in practice. In the present work, we demonstrate that the hydrodynamic pilot-wave systems offer a bridge between low- and high-dimensional chaotic phenomena by allowing for a systematic study of how the former connects to the latter. Specifically, we present experimental results which show the formation of low-dimensional chaotic attractors upon destabilization of regular dynamics and a final transition to high-dimensional chaos via the merging of distinct chaotic regions through a crisis bifurcation. Moreover, we show that the post-crisis dynamics of the system can be rationalized as consecutive scatterings from the nonattracting chaotic sets with lifetimes following exponential distributions.

... The convergence of this method for the parallel walking solution is shown in Fig. 15. Using our method, we have been able to reproduce the exotic trajectories of a single walker in a rotating frame by Tambasco et al. 23 and Oza et al. 10 Fig. 14 shows the comparison with different timesteps of the closed circular trajectory at κ = 0.6 and β = 4 where the pair of walkers are in a lopsided mode and the right-angled discrete turning walkers at κ − 0.5 and β = 6. Simulating trajectories at this parameter value with timesteps ∆t = 2 −6 , 2 −8 and 2 −10 with noise in initial conditions confirm that these exotic behaviours are robust. ...

A droplet bouncing on the surface of a vibrating liquid bath can move horizontally guided by the wave it produces on impacting the bath. The wave itself is modified by the environment, and thus, the interactions of the moving droplet with the surroundings are mediated through the wave. This forms an example of a pilot-wave system. Taking the Oza-Rosales-Bush description for walking droplets as a theoretical pilot-wave model, we investigate the dynamics of two interacting identical, in-phase bouncing droplets theoretically and numerically. A remarkably rich range of behaviors is encountered as a function of the two system parameters, the ratio of inertia to drag,
κ
, and the ratio of wave forcing to drag,
β
. The droplets typically travel together in a tightly bound pair, although they unbind when the wave forcing is large and inertia is small or inertia is moderately large and wave forcing is moderately small. Bound pairs can exhibit a range of trajectories depending on parameter values, including straight lines, sub-diffusive random walks, and closed loops. The droplets themselves may maintain their relative positions, oscillate toward and away from one another, or interchange positions regularly or chaotically as they travel. We explore these regimes and others and the bifurcations between them through analytic and numerical linear stability analyses and through fully nonlinear numerical simulation.

... In a series of configurations recently summarised by Bush (2015), the behaviour of walkers is strongly reminiscent of that of quantum particles. Quantisation of bound states is observed in a central force field (Perrard et al. 2014a,b;Labousse et al. 2014Labousse et al. , 2016Tambasco et al. 2016), in a rotating frame (Fort et al. 2010;Eddi et al. 2012;Harris & Bush 2014;Oza et al. 2014a,b), or with two-body interactions (Protiere, Boudaoud & Couder 2006;Protière, Bohn & Couder 2008;Borghesi et al. 2014;Filoux, Hubert & Vandewalle 2015;Durey & Milewski 2017;Oza et al. 2017). Walkers interact with submerged boundaries through non-specular reflection (Pucci et al. 2016) and tunnelling (Eddi et al. 2009;Carmigniani et al. 2014;Nachbin, Milewski & Bush 2017). ...

A walker is a fluid entity comprising a bouncing droplet coupled to the waves that it generates at the surface of a vibrated bath. Thanks to this coupling, walkers exhibit a series of wave-particle features formerly thought to be exclusive to the quantum realm. In this paper, we derive a model of the Faraday surface waves generated by an impact upon a vertically vibrated liquid surface. We then particularise this theoretical framework to the case of forcing slightly below the Faraday instability threshold. Among others, this theory yields a rationale for the dependence of the wave amplitude to the phase of impact, as well as the characteristic timescale and length scale of viscous damping. The theory is validated with experiments of bead impact on a vibrated bath. We finally discuss implications of these results for the analogy between walkers and quantum particles.

... The convergence of this method for the parallel walking solution is shown in Fig. 15. Using our method, we have been able to reproduce the exotic trajectories of a single walker in a rotating frame by Tambasco et al. 23 and Oza et al. 10 Fig. 14 shows the comparison with different timesteps of the closed circular trajectory at κ = 0.6 and β = 4 where the pair of walkers are in a lopsided mode and the right-angled discrete turning walkers at κ − 0.5 and β = 6. Simulating trajectories at this parameter value with timesteps ∆t = 2 −6 , 2 −8 and 2 −10 with noise in initial conditions confirm that these exotic behaviours are robust. ...

A droplet bouncing on the surface of a vibrating liquid bath can move horizontally guided by the wave it produces on impacting the bath. The wave itself is modified by the environment, and thus the interactions of the moving droplet with the surroundings are mediated through the wave. This forms an example of a pilot-wave system. Taking the Oza Rosales Bush description for walking droplets as a theoretical pilot-wave model, we investigate the dynamics of two interacting identical, in-phase bouncing droplets theoretically and numerically. A remarkably rich range of behaviors is encountered as a function of the two system parameters, the ratio of inertia to drag, \k{appa}, and the ratio of wave forcing to drag, \b{eta}. The droplets typically travel together in a tightly bound pair, although they unbind when the wave forcing is large and inertia is small or inertia is moderately large and wave forcing is moderately small. Bound pairs can exhibit a range of trajectories depending on parameter values, including straight lines, sub-diffusive random walks, and closed loops. The droplets themselves may maintain their relative positions, oscillate towards and away from one another, or interchange positions regularly or chaotically as they travel. We explore these regimes and others and the bifurcations between them through analytic and numerical linear stability analyses and through fully nonlinear numerical simulation.

... 27,32,[36][37][38] Finally, the stroboscopic model was adopted in the first theoretical study of the transitions to chaos in orbital pilot-wave dynamics. 39 The success of the stroboscopic model has been limited in rationalizing the stability of configurations with multiple droplets. Oza et al. 13 have recently studied the interactions of two identical droplets above the walking threshold γ W , where the droplets may settle into orbits with quantized radii, as observed by Couder et al. 10 and Protière et al. 6,40,42 Two additional refinements to the stroboscopic model were required in order to achieve a match between experimental results and theoretical predictions. ...

The walking droplet system has extended the range of classical systems to include several features previously thought to be exclusive to quantum systems. We review the hierarchy of analytic models that have been developed, on the basis of various simplifying assumptions, to describe droplets walking on a vibrating fluid bath. Particular attention is given to detailing their successes and failures in various settings. Finally, we present a theoretical model that may be adopted to explore a more generalized pilot-wave framework capable of further extending the phenomenological range of classical pilot-wave systems beyond that achievable in the laboratory.

... (9)] do not depend on the exact shape of the repulsive potential, it is tantalizing to apply our model to this situation. In the experiments, the erratic crossing events originate from the interaction between the propelling waves and memory effects, which are known to trigger a transition to chaos as soon as the drop interacts with external potential [27,28]. The chaotic regime generates an effective distribution of incident angles. ...

We investigate the crossing of an energy barrier by a self-propelled particle described by a Rayleigh friction term. We reveal the existence of a sharp transition in the external force field whereby the amplitude dramatically increases. This corresponds to a saddle point transition in the velocity flow phase space, as would be expected for any type of repulsive force field. We use this approach to rationalize the results obtained by Eddi et al. [Phys. Rev. Lett. 102, 240401 (2009)] who studied the interaction between a drop propelled by its accompanying wave field and a submarine obstacle. This wave particle entity can overcome potential barrier, suggesting the existence of a “macroscopic tunneling effect.” We show that the effect of self-propulsion is sufficiently strong to generate crossing of the high-energy barrier. By assuming a random distribution of initial angles, we define a probability distribution to cross the potential barrier that matches with the data of Eddi et al. This probability is similar to the one encountered in statistical physics for Hamiltonian systems, i.e., a Boltzmann exponential law.

... Most of the quantum-like phenomena described above were observed for a memory larger than 20. The mechanisms of transition to chaos for the confined walkers were rationalized in Shirokoff, 44 Gilet, 19 and Tambasco et al. 45 The wave-mediated interaction of multiple walkers has received significantly less attention. Protière et al. 39 reported that two identical walkers launched at one another could either scatter or bind to each other. ...

Walkers are dual objects comprising a bouncing droplet dynamically coupled to an underlying Faraday wave at the surface of a vibrated bath. In this paper, we study the wave-mediated interaction of two walkers launched at one another, both experimentally and theoretically. Different outcomes are observed in which either the walkers scatter or they bind to each other in orbits or promenade-like motions. The outcome is highly sensitive to initial conditions, which is a signature of chaos, though the time during which perturbations are amplified is finite. The vertical bouncing dynamics, periodic for a single walker, is also strongly perturbed during the interaction, owing to the superposition of the wave contributions of each droplet. Thanks to a model based on inelastic balls coupled to the Faraday waves, we show that this perturbed vertical dynamics is the source of horizontal chaos in such a system.

... During the past four decades, there are significantly developments in computational power, advances in technology, and mathematical theory that have facilitated formulation of nonlinear approaches for intricate systems [1,2] since many physical, socioeconomic, and natural systems are intrinsically nonlinear, so these systems show large range of characteristics. During the same time, chaos has been also noted in several experimental works and it redesigns many researches in different fields of engineering and science [3][4][5]; also, the study of chaotic behaviour in dynamical systems has been the interest of many scientists, engineers, and mathematicians. Applications of it can be extensively found in variety of disciplines such as modeling [6,7], optimization [8], stock market [9], photovoltaic plant [10], fashion cycle model [11], and other [5]. ...

The focus of this research work is to obtain the chaotic behaviour and bifurcation in the real dynamics of a newly proposed family of functions f λ , a x = x + 1 − λ x ln a x ; x > 0 , depending on two parameters in one dimension, where assume that λ is a continuous positive real parameter and a is a discrete positive real parameter. This proposed family of functions is different from the existing families of functions in previous works which exhibits chaotic behaviour. Further, the dynamical properties of this family are analyzed theoretically and numerically as well as graphically. The real fixed points of functions f λ , a x are theoretically simulated, and the real periodic points are numerically computed. The stability of these fixed points and periodic points is discussed. By varying parameter values, the plots of bifurcation diagrams for the real dynamics of f λ , a x are shown. The existence of chaos in the dynamics of f λ , a x is explored by looking period-doubling in the bifurcation diagram, and chaos is to be quantified by determining positive Lyapunov exponents.

... More recently, Tambasco et al. [87] used and Adams-Bashforth method to solve the trajectory equation of (4) with a Coriolis, Coulomb or simple harmonic potential force. They found that increasing γ (or M e) in the cases of Coriolis and Coulomb forces produced a period-doubling cascade to chaos, while for a simple harmonic force the transition to chaos resembled the mechanism described in Newhouse et al. [88]. ...

Over the past decade the study of fluidic droplets bouncing and skipping (or "walking") on a vibrating fluid bath has gone from an interesting experiment to a vibrant research field. The field exhibits challenging fluids problems, potential connections with quantum mechanics, and complex nonlinear dynamics. We detail advancements in the field of walking droplets through the lens of Dynamical Systems Theory, and outline questions that can be answered using dynamical systems analysis. The article begins by discussing the history of the fluidic experiments and their resemblance to quantum experiments. With this physics backdrop, we paint a portrait of the complex nonlinear dynamics present in physical models of various walking droplet systems. Naturally, these investigations lead to even more questions, and some unsolved problems that are bound to benefit from rigorous Dynamical Systems Analysis are outlined.

... Different types of trajectory such as stable circular orbit, quasiperiodic orbit with a significantly different mean radius, wobbling orbits characterized by a radial oscillation, wobble-and-leap orbits, complex periodic or quasiperiodic orbits, and chaotic trajectories were numerically predicted. The transition to chaos was analytically studied by Tambasco et al. [183] for droplets walking in the presence of Coriolis acceleration, or in the presence of linear spring, or Coulomb forces. In the presence of Coriolis or Coulomb forces, the droplet's orbital motion was found to reach chaotic state through a period-doubling cascade. ...

This article considers additional phenomena that complement the earlier topics addressed by Ibrahim [(Liquid Sloshing Dynamics: Theory and Applications. Cambridge University Press, Cambridge, 2005), (ASME J Fluids Eng 137(9):090801, 2015)]. The first phenomenon is the localized Faraday waves known as oscillons, which were observed in granular materials and liquid layers subjected to parametric excitation. Extreme waves, known as rogue, generated in the Faraday surface ripples, are related to the increase in the horizontal mobility of oscillating solitons (oscillons), and their horizontal motion is random over a limited range of excitation acceleration amplitude. Parametric excitation of water in a Hele–Shaw cell and the associated localized standing surface waves of large amplitude will be discussed. The surface wave pattern exhibited a certain similarity with the three-dimensional axisymmetric oscillon. Faraday waves in superfluid Fermi–Bose mixtures and their wave function will be addressed in terms of position and time as described by the Schrödinger equation with time-dependent parabolic potential. The phenomenon of walking fluid droplets on Faraday waves constitutes the majority portion of this article. Different regimes of droplet motion in terms of droplet physical properties, the fluid bath excitation acceleration amplitude and frequency will be discussed. The droplet trajectory diffraction, when passes through a slit, shares the same random features of electron diffraction. The duality of the droplet-wave field together with the path-memory-driven nonlocality and other related topics will be assessed. This article is complemented with the fascinating phenomenon of the stone and bombs skipping/ricochet over water surface.

... This stroboscopic model averages over the droplet's vertical periodic bouncing motion and provides a trajectory equation for its two-dimensional horizontal walking motion by taking into account two key horizontal forces acting on the walker: (i) the horizontal wave force proportional to the gradient of the underlying wave field generated by the walker, and (ii) an effective horizontal drag force composed of aerodynamic drag and momentum loss during impact with the fluid surface. This stroboscopic model rationalizes several hydrodynamic quantum analogs 5,6,16,22,23,[32][33][34][35] and also results in rich dynamical behaviors for walkers [36][37][38][39][40][41][42] . ...

Vertically vibrating a liquid bath can give rise to a self-propelled wave-particle entity on its free surface. The horizontal walking dynamics of this wave-particle entity can be described adequately by an integro-differential trajectory equation. By transforming this integro-differential equation of motion for a one-dimensional wave-particle entity into a system of ordinary differential equations (ODEs), we show the emergence of Lorenz-like dynamical systems for various spatial wave forms of the entity. Specifically, we present and give examples of Lorenz-like dynamical systems that emerge when the wave form gradient is (i) a solution of a linear homogeneous constant coefficient ODE, (ii) a polynomial and (iii) a periodic function. Understanding the dynamics of the wave-particle entity in terms of Lorenz-like systems may provide to be useful in rationalizing emergent statistical behavior from underlying chaotic dynamics in hydrodynamic quantum analogs of walking droplets. Moreover, the results presented here provide an alternative physical interpretation of various Lorenz-like dynamical systems in terms of the walking dynamics of a wave-particle entity.

... This stroboscopic model averages over the droplet's vertical periodic bouncing motion and provides a trajectory equation for its two-dimensional horizontal walking motion by taking into account two key horizontal forces acting on the walker: (i) the horizontal wave force proportional to the gradient of the underlying wave field generated by the walker, and (ii) an effective horizontal drag force composed of aerodynamic drag and momentum loss during impact with the fluid surface. This stroboscopic model rationalizes several hydrodynamic quantum analogs 5,6,16,22,23,[32][33][34][35] and also results in rich dynamical behaviors for walkers [36][37][38][39][40][41][42] . ...

Vertically vibrating a liquid bath can give rise to a self-propelled wave–particle entity on its free surface. The horizontal walking dynamics of this wave–particle entity can be described adequately by an integro-differential trajectory equation. By transforming this integro-differential equation of motion for a one-dimensional wave–particle entity into a system of ordinary differential equations (ODEs), we show the emergence of Lorenz-like dynamical systems for various spatial wave forms of the entity. Specifically, we present and give examples of Lorenz-like dynamical systems that emerge when the wave form gradient is (i) a solution of a linear homogeneous constant coefficient ODE, (ii) a polynomial, and (iii) a periodic function. Understanding the dynamics of the wave–particle entity in terms of Lorenz-like systems may prove to be useful in rationalizing emergent statistical behavior from underlying chaotic dynamics in hydrodynamic quantum analogs of walking droplets. Moreover, the results presented here provide an alternative physical interpretation of various Lorenz-like dynamical systems in terms of the walking dynamics of a wave–particle entity.

We explore the effects of an imposed potential with both oscillatory and quadratic components on the dynamics of walking droplets. We first conduct an experimental investigation of droplets walking on a bath with a central circular well. The well acts as a source of Faraday waves, which may trap walking droplets on circular orbits. The observed orbits are stable and quantized, with preferred radii aligning with the extrema of the well-induced Faraday wave pattern. We use the stroboscopic model of Oza et al. [J. Fluid Mech. 737, 552–570 (2013)] with an added potential to examine the interaction of the droplet with the underlying well-induced wavefield. We show that all quantized orbits are stable for low vibrational accelerations. Smaller orbits may become unstable at higher forcing accelerations and transition to chaos through a path reminiscent of the Ruelle-Takens-Newhouse scenario. We proceed by considering a generalized pilot-wave system in which the relative magnitudes of the pilot-wave force and drop inertia may be tuned. When the drop inertia is dominated by the pilot-wave force, all circular orbits may become unstable, with the drop chaotically switching between them. In this chaotic regime, the statistically stationary probability distribution of the drop’s position reflects the relative instability of the unstable circular orbits. We compute the mean wavefield from a chaotic trajectory and confirm its predicted relationship with the particle’s probability density function.

A millimetric droplet may bounce and self-propel on the surface of a vertically vibrating fluid bath, guided by its self-generated wave field. This hydrodynamic pilot-wave system exhibits a vast range of dynamics, including behavior previously thought to be exclusive to the quantum realm. We present the results of a theoretical investigation of an idealized pilot-wave model, in which a particle is guided by a one-dimensional wave that is equipped with the salient features of the hydrodynamic system. The evolution of this reduced pilot-wave system may be simplified by projecting onto a three-dimensional dynamical system describing the evolution of the particle velocity, the local wave amplitude, and the local wave slope. As the resultant dynamical system is remarkably similar in form to the Lorenz system, we utilize established properties of the Lorenz equations as a guide for identifying and elucidating several pilot-wave phenomena, including the onset and characterization of chaos.

We present the results of an integrated experimental and theoretical investigation of the promenade mode, a bound state formed by a pair of droplets walking side by side on the surface of a vibrating fluid bath. Particular attention is given to characterizing the dependence of the promenading behavior on the vibrational forcing for drops of a given size. We also enumerate the different instabilities that may arise, including transitions to smaller promenade modes or orbiting pairs. Our theoretical developments highlight the importance of the vertical bouncing dynamics on the stability characteristics. Specifically, quantitative comparison between experiment and theory prompts further refinement of the stroboscopic model [A. U. Oza et al., J. Fluid Mech. 737, 552 (2013)] through inclusion of phase adaptation and reveals the critical role that impact phase variations play in the stability of the promenading pairs.

We present the results of a numerical investigation of droplets walking in a harmonic potential on a vibrating fluid bath. The droplet's trajectory is described by an integro-differential equation, which is simulated numerically in various parameter regimes. We produce a regime diagram that summarizes the dependence of the walker's behavior on the system parameters for a droplet of fixed size. At relatively low vibrational forcing, a number of periodic and quasiperiodic trajectories emerge. In the limit of large vibrational forcing, the walker's trajectory becomes chaotic, but the resulting trajectories can be decomposed into portions of unstable quasiperiodic states.

A millimetric liquid droplet may walk across the surface of a vibrating liquid bath through a resonant interaction with its self-generated wavefield. Such walking droplets, or “walkers,” have attracted considerable recent interest because they exhibit certain features previously believed to be exclusive to the microscopic, quantum realm. In particular, the intricate motion of a walker confined to a closed geometry is known to give rise to a coherent wave-like statistical behavior similar to that of electrons confined to quantum corrals. Here, we examine experimentally the dynamics of a walker inside a circular corral. We first illustrate the emergence of a variety of stable dynamical states for relatively low vibrational accelerations, which lead to a double quantisation in angular momentum and orbital radius. We then characterise the system’s transition to chaos for increasing vibrational acceleration and illustrate the resulting breakdown of the double quantisation. Finally, we discuss the similarities and differences between the dynamics and statistics of a walker inside a circular corral and that of a walker subject to a simple harmonic potential.

A deterministic low-dimensional iterated map is proposed here to describe the interaction between a bouncing droplet and Faraday waves confined to a circular cavity. Its solutions are investigated theoretically and numerically. The horizontal trajectory of the droplet can be chaotic: it then corresponds to a random walk of average step size equal to half the Faraday wavelength. An analogy is made between the diffusion coefficient of this random walk and the action per unit mass h/m of a quantum particle. The statistics of droplet position and speed are shaped by the cavity eigenmodes, in remarkable agreement with the solution of Schrödinger equation for a quantum particle in a similar potential well.

Small drops bouncing across a vibrating liquid bath display many features reminiscent of quantum systems.

A millimetric droplet bouncing on the surface of a vibrating fluid bath can self-propel by virtue of a resonant interaction with its own wave field. This system represents the first known example of a pilot-wave system of the form envisaged by Louis de Broglie in his double-solution pilot-wave theory. We here develop a fluid model of pilot-wave hydrodynamics by coupling recent models of the droplet's bouncing dynamics with a more realistic model of weakly viscous quasi-potential wave generation and evolution. The resulting model is the first to capture a number of features reported in experiment, including the rapid transient wave generated during impact, the Doppler effect and walker–walker interactions.

Yves Couder, Emmanuel Fort, and coworkers recently discovered that a millimetric droplet sustained on the surface of a vibrating fluid bath may self-propel through a resonant interaction with its own wave field. This article reviews experimental evidence indicating that the walking droplets exhibit certain features previously thought to be exclusive to the microscopic, quantum realm. It then reviews theoretical descriptions of this hydrodynamic pilot-wave system that yield insight into the origins of its quantumlike behavior. Quantization arises from the dynamic constraint imposed on the droplet by its pilot-wave field, and multimodal statistics appear to be a feature of chaotic pilot-wave dynamics. I attempt to assess the potential and limitations of this hydrodynamic system as a quantum analog. This fluid system is compared to quantum pilot-wave theories, shown to be markedly different from Bohmian mechanics and more closely related to de Broglie’s original conception of quantum dynamics, his double-solution theory, and its relatively recent extensions through researchers in stochastic electrodynamics.

A walker is a droplet bouncing on a liquid surface and propelled by the waves that it generates. This macroscopic wave-particle association exhibits behaviors reminiscent of quantum particles. This article presents a toy model of the coupling between a particle and a confined standing wave. The resulting two-dimensional iterated map captures many features of the walker dynamics observed in different configurations of confinement. These features include the time decomposition of the chaotic trajectory in quantized eigenstates and the particle statistics being shaped by the wave. It shows that deterministic wave-particle coupling expressed in its simplest form can account for some quantumlike behaviors.

The transmission of information can couple two entities of very different nature, one of them serving as a memory for the other. Here we study the situation in which information is stored in a wave field and serves as a memory that pilots the dynamics of a particle. Such a system can be implemented by a bouncing drop generating surface waves sustained by a parametric forcing. The motion of the resulting "walker" when confined in a harmonic potential well is generally disordered. Here we show that these trajectories correspond to chaotic regimes characterized by intermittent transitions between a discrete set of states. At any given time, the system is in one of these states characterized by a double quantization of size and angular momentum. A low dimensional intermittency determines their respective probabilities. They thus form an eigenstate basis of decomposition for what would be observed as a superposition of states if all measurements were intrusive.

A bouncing droplet on a vibrated bath can couple to the waves it generates, so that it becomes a propagative walker. Its propulsion at constant velocity means that a balance exists between the permanent input of energy provided by the vibration and the dissipation. Here we seek a simple theoretical description of the resulting non-Hamiltonian dynamics with a walker immersed in a harmonic potential well. We demonstrate that the interaction with the recently emitted waves can be modeled by a Rayleigh-type friction. The Rayleigh oscillator has well defined attractors. The convergence toward them and their stability is investigated through an energetic approach and a linear stability analysis. These theoretical results provide a description of the dynamics in excellent agreement with the experimental data. It is thus a basic framework for further investigations of wave-particle interactions when memory effects are included.

A growing number of dynamical situations involve the coupling of particles or singularities with physical waves. In principle these situations are very far from the wave particle duality at quantum scale where the wave is probabilistic by nature. Yet some dual characteristics were observed in a system where a macroscopic droplet is guided by a pilot wave it generates. Here we investigate the behaviour of these entities when confined in a two-dimensional harmonic potential well. A discrete set of stable orbits is observed, in the shape of successive generalized Cassinian-like curves (circles, ovals, lemniscates, trefoils and so on). Along these specific trajectories, the droplet motion is characterized by a double quantization of the orbit spatial extent and of the angular momentum. We show that these trajectories are intertwined with the dynamical build-up of central wave-field modes. These dual self-organized modes form a basis of eigenstates on which more complex motions are naturally decomposed.

We present the results of a combined experimental and theoretical investigation of millimetric droplets bouncing on a vertically vibrating fluid bath. We first characterize the system experimentally, deducing the dependence of the droplet dynamics on the system parameters, specifically the drop size, driving acceleration and driving frequency. As the driving acceleration is increased, depending on drop size, we observe the transition from coalescing to vibrating or bouncing states, then period-doubling events that may culminate in either walking drops or chaotic bouncing states. The drop’s vertical dynamics depends critically on the ratio of the forcing frequency to the drop’s natural oscillation frequency. For example, when the data describing the coalescence–bouncing threshold and period-doubling thresholds are described in terms of this ratio, they collapse onto a single curve. We observe and rationalize the coexistence of two non-coalescing states, bouncing and vibrating, for identical system parameters. In the former state, the contact time is prescribed by the drop dynamics; in the latter, by the driving frequency. The bouncing states are described by theoretical models of increasing complexity whose predictions are tested against experimental data. We first model the drop–bath interaction in terms of a linear spring, then develop a logarithmic spring model that better captures the drop dynamics over a wider range of parameter space. While the linear spring model provides a faster, less accurate option, the logarithmic spring model is found to be more accurate and consistent with all existing data.

We present the results of a combined experimental and theoretical investigation of droplets walking on a vertically vibrating fluid bath. Several walking states are reported, including pure resonant walkers that bounce with precisely half the driving frequency, limping states, wherein a short contact occurs between two longer ones, and irregular chaotic walking. It is possible for several states to arise for the same parameter combination, including high- and low-energy resonant walking states. The extent of the walking regime is shown to be crucially dependent on the stability of the bouncing states. In order to estimate the resistive forces acting on the drop during impact, we measure the tangential coefficient of restitution of drops impacting a quiescent bath. We then analyse the spatio-temporal evolution of the standing waves created by the drop impact and obtain approximations to their form in the small-drop and long-time limits. By combining theoretical descriptions of the horizontal and vertical drop dynamics and the associated wave field, we develop a theoretical model for the walking drops that allows us to rationalize the limited extent of the walking regimes. The critical requirement for walking is that the drop achieves resonance with its guiding wave field. We also rationalize the observed dependence of the walking speed on system parameters: while the walking speed is generally an increasing function of the driving acceleration, exceptions arise due to possible switching between different vertical bouncing modes. Special focus is given to elucidating the critical role of impact phase on the walking dynamics. The model predictions are shown to compare favourably with previous and new experimental data. Our results form the basis of the first rational hydrodynamic pilot-wave theory.

Bouncing droplets can self-propel laterally along the surface of a vibrated fluid bath by virtue of a resonant interaction with their own wave field. The resulting walking droplets exhibit features reminiscent of microscopic quantum particles. Here we present the results of an experimental investigation of droplets walking in a circular corral. We demonstrate that a coherent wavelike statistical behavior emerges from the complex underlying dynamics and that the probability distribution is prescribed by the Faraday wave mode of the corral. The statistical behavior of the walking droplets is demonstrated to be analogous to that of electrons in quantum corrals.

Three scenarios leading to turbulence in theory and experiment are outlined. The respective mathematical theories are explained and compared.

The linear stability of the plane free surface of a viscous liquid on a horizontal plate under vertical sinusoidal oscillation is analysed theoretically. The free surface of a laterally unbounded liquid of any depth h may always be excited to standing waves if the external acceleration is raised above a critical value ac. For a fixed external frequency omega , solutions are possible only within certain bands of wave numbers k for a given forcing amplitude above ac, that is, within tongue-like stability zones in the a-k plane. The analysis for a shallow layer of viscous fluids shows new qualitative behaviours compared to the nearly inviscid theory. It predicts a series of bicritical points, where both harmonic and subharmonic solutions exist for the same forcing amplitude and forcing frequency. This makes harmonic solutions possible at the onset in a laterally large container, which is qualitatively different from the results of nearly inviscid theory. For a low viscosity fluid of small depths, the damping coefficient may be considered proportional to (nu omega )1/2/h in contrast to nu kappa 2 predicted by the nearly inviscid theory. An approximate analytic expression is derived for the lower part of the lowest marginal curve in cases when the depth of the liquid is much larger than the thickness of the viscous boundary layer formed at the bottom plate. This approximate threshold agrees well with that of recent experiments with viscous liquids.

Using automated laser-Doppler methods we have identified four distinct sequences of instabilities leading to turbulent convection at low Prandtl number (2·5–5·0), in fluid layers of small horizontal extent. Contour maps of the structure of the time-averaged velocity field, in conjunction with high-resolution power spectral analysis, demonstrate that several mean flows are stable over a wide range in the Rayleigh number R, and that the sequence of time-dependent instabilities depends on the mean flow. A number of routes to non-periodic motion have been identified by varying the geometrical aspect ratio, Prandtl number, and mean flow. Quasi-periodic motion at two frequencies leads to phase locking or entrainment, as identified by a step in a graph of the ratio of the two frequencies. The onset of non-periodicity in this case is associated with the loss of entrainment as R is increased. Another route to turbulence involves successive subharmonic (or period doubling) bifurcations of a periodic flow. A third route contains a well-defined regime with three generally incommensurate frequencies and no broadband noise. The spectral analysis used to demonstrate the presence of three frequencies has a precision of about one part in 104 to 105. Finally, we observe a process of intermittent non-periodicity first identified by Libchaber & Maurer at lower Prandtl number. In this case the fluid alternates between quasi-periodic and non-periodic states over a finite range in R. Several of these processes are also manifested by rather simple mathematical models, but the complicated dependence on geometrical parameters, Prandtl number, and mean flow structure has not been explained.

On a vertically vibrating fluid interface, a droplet can remain bouncing indefinitely. When approaching the Faraday instability onset, the droplet couples to the wave it generates and starts propagating horizontally. The resulting wave–particle association, called a walker, was shown previously to have remarkable dynamical properties, reminiscent of quantum behaviours. In the present article, the nature of a walker's wave field is investigated experimentally, numerically and theoretically. It is shown to result from the superposition of waves emitted by the droplet collisions with the interface. A single impact is studied experimentally and in a fluid mechanics theoretical approach. It is shown that each shock emits a radial travelling wave, leaving behind a localized mode of slowly decaying Faraday standing waves. As it moves, the walker keeps generating waves and the global structure of the wave field results from the linear superposition of the waves generated along the recent trajectory. For rectilinear trajectories, this results in a Fresnel interference pattern of the global wave field. Since the droplet moves due to its interaction with the distorted interface, this means that it is guided by a pilot wave that contains a path memory. Through this wave-mediated memory, the past as well as the environment determines the walker's present motion.

A droplet bouncing on a liquid bath can self-propel due to its interaction with the waves it generates. The resulting "walker" is a dynamical association where, at a macroscopic scale, a particle (the droplet) is driven by a pilot-wave field. A specificity of this system is that the wave field itself results from the superposition of the waves generated at the points of space recently visited by the particle. It thus contains a memory of the past trajectory of the particle. Here, we investigate the response of this object to forces orthogonal to its motion. We find that the resulting closed orbits present a spontaneous quantization. This is observed only when the memory of the system is long enough for the particle to interact with the wave sources distributed along the whole orbit. An additional force then limits the possible orbits to a discrete set. The wave-sustained path memory is thus demonstrated to generate a quantization of angular momentum. Because a quantum-like uncertainty was also observed recently in these systems, the nonlocality generated by path memory opens new perspectives.

A droplet bouncing on a vibrated bath becomes a "walker" moving at constant velocity on the interface when it couples to the surface wave it generates. Here the motion of a walker is investigated when it collides with barriers of various thicknesses. Surprisingly, it undergoes a form of tunneling: the reflection or transmission of a given incident walker is unpredictable. However, the crossing probability decreases exponentially with increasing barrier width. This shows that this wave-particle association has a nonlocality sufficient to generate a quantumlike tunneling at a macroscopic scale.

Small drops can bounce indefinitely on a bath of the same liquid if the container is oscillated vertically at a sufficiently high acceleration. Here we show that bouncing droplets can be made to 'walk' at constant horizontal velocity on the liquid surface by increasing this acceleration. This transition yields a new type of localized state with particle-wave duality: surface capillary waves emanate from a bouncing drop, which self-propels by interaction with its own wave and becomes a walker. When two walkers come close, they interact through their waves and this 'collision' may cause the two walkers to orbit around each other.

A droplet bouncing on a vertically vibrated bath can become coupled to the surface wave it generates. It thus becomes a "walker" moving at constant velocity on the interface. Here the motion of these walkers is investigated when they pass through one or two slits limiting the transverse extent of their wave. In both cases a given single walker seems randomly scattered. However, diffraction or interference patterns are recovered in the histogram of the deviations of many successive walkers. The similarities and differences of these results with those obtained with single particles at the quantum scale are discussed.

A decade ago, Couder and Fort [Phys. Rev. Lett. 97, 154101 (2006)] discovered that a millimetric droplet sustained on the surface of a vibrating fluid bath may self-propel through a resonant interaction with its own wave field. We here present the results of a combined experimental and theoretical investigation of the interactions of such walking droplets. Specifically, we delimit experimentally the different regimes for an orbiting pair of identical walkers and extend the theoretical model of Oza et al. [J. Fluid Mech. 737, 552 (2013)] in order to rationalize our observations. A quantitative comparison between experiment and theory highlights the importance of spatial damping of the wave field. Our results also indicate that walkers adapt their impact phase according to the local wave height, an effect that stabilizes orbiting bound states.

We present the results of a theoretical investigation of the dynamics of a droplet walking on a vibrating fluid bath under the influence of a harmonic potential. The walking droplet's horizontal motion is described by an integro-differential trajectory equation, which is found to admit steady orbital solutions. Predictions for the dependence of the orbital radius and frequency on the strength of the radial harmonic force field agree favorably with experimental data. The orbital quantization is rationalized through an analysis of the orbital solutions. The predicted dependence of the orbital stability on system parameters is compared with experimental data and the limitations of the model are discussed.

Direct numerical simulations of the transition process from laminar to chaotic flow in converging-diverging channels are presented. The chaotic flow regime is reached after a sequence of successive supercritical Hopf bifurcations to periodic, quasiperiodic, and chaotic self-sustained flow regimes. The numerical experiments reveal three distinct bifurcations as the Reynolds number is increased, each adding a new fundamental frequency to the velocity spectrum. In addition, frequency-locked periodic solutions with independent but synchronized periodic functions are obtained. A scenario similar to the Ruelle-Takens-Newhouse scenario of the onset of chaos is verified in this forced convective open system flow. The results are illustrated for different Reynolds numbers using time-velocity histories, Fourier power spectra, and phase space trajectories. The global structure of the self-sustained oscillatory flow for a periodic regime is also discussed.

We present a first-principles model of drops bouncing on a liquid reservoir. We consider a nearly inviscid liquid reservoir and track the waves that develop in a bounded domain. Bouncing drops are modeled as vertical linear springs. We obtain an expression for the contact force between drop and liquid surface and a model where the only adjustable parameter is an effective viscosity used to describe the waves on the reservoir’s surface. With no adjustable parameters associated to the drop, we recover experimental bouncing times and restitution coefficients. We use our model to describe the effect of the Bond, Ohnesorge, and Weber numbers on drops bouncing on a stationary reservoir. We also use our model to describe drops bouncing on an oscillated reservoir, describing various bouncing modes and a walking threshold.

A droplet bouncing on a vertically vibrated liquid bath can be self-propelled by the surface waves it generates. Theses Faraday waves are sustained by the vertical bath vibration for a memory time which can be tuned experimentally. The wave field thus contains in its interference pattern a memory of the past-trajectory. The resulting entity called a walker is characterized by the interaction between the drop and its surrounding waves through this path-memory.
This thesis is devoted to an experimental and theoretical investigation of such a wave-mediated path-memory. For this purpose a bouncing drop is magnetically loaded with a droplet of ferrofluid and can then be trapped in an harmonic well. The drop is thus forced to interact with its own path. The confinement induces a self-organization process between the particle and its wave packet, leading to wave-type behavior for a particle. Notions such quantization or probability of measuring an eigenstate can thus be used for the walker dynamics description. These features originate from the temporal coherence of the walker’s dynamics. In that sense, the walker is an entity extended in time, we cannot reduce to a point-like approximation. It reminds us, in another context, the pilot wave theory developped by de Broglie at the beginning of the XXst century.

In a thought-provoking paper, Couder and Fort [Phys. Rev. Lett. 97, 154101 (2006)10.1103/PhysRevLett.97.154101] describe a version of the famous double-slit experiment performed with droplets bouncing on a vertically vibrated fluid surface. In the experiment, an interference pattern in the single-particle statistics is found even though it is possible to determine unambiguously which slit the walking droplet passes. Here we argue, however, that the single-particle statistics in such an experiment will be fundamentally different from the single-particle statistics of quantum mechanics. Quantum mechanical interference takes place between different classical paths with precise amplitude and phase relations. In the double-slit experiment with walking droplets, these relations are lost since one of the paths is singled out by the droplet. To support our conclusions, we have carried out our own double-slit experiment, and our results, in particular the long and variable slit passage times of the droplets, cast strong doubt on the feasibility of the interference claimed by Couder and Fort. To understand theoretically the limitations of wave-driven particle systems as analogs to quantum mechanics, we introduce a Schrödinger equation with a source term originating from a localized particle that generates a wave while being simultaneously guided by it. We show that the ensuing particle-wave dynamics can capture some characteristics of quantum mechanics such as orbital quantization. However, the particle-wave dynamics can not reproduce quantum mechanics in general, and we show that the single-particle statistics for our model in a double-slit experiment with an additional splitter plate differs qualitatively from that of quantum mechanics.

Bouncing droplets on a vibrating fluid bath can exhibit wave-particle behavior, such as being propelled by interacting with its own wave field. These droplets seem to walk across the bath, and thus are dubbed walkers. Experiments have shown that walkers can exhibit exotic dynamical behavior indicative of chaos. While the integro-differential models developed for these systems agree well with the experiments, they are difficult to analyze mathematically. In recent years, simpler discrete dynamical models have been derived and studied numerically. The numerical simulations of these models show evidence of exotic dynamics such as period doubling bifurcations, Neimark--Sacker (N--S) bifurcations, and even chaos. For example, in [Gilet, PRE 2014], based on simulations Gilet conjectured the existence of a supercritical N-S bifurcation as the damping factor in his one-dimensional path model. We prove Gilet's conjecture and more; in fact, both supercritical and subcritical (N-S) bifurcations are produced by separately varying the damping factor and wave-particle coupling for all eigenmode shapes. Then we compare our theoretical results with some previous and new numerical simulations, and find complete qualitative agreement. Furthermore, evidence of chaos is shown by numerically studying a global bifurcation.
many of the interesting dynamical properties have yet to be proven. In recent
years discrete dynamical models have been derived and studied numerically. We
prove the existence of a Neimark--Sacker bifurcation for a variety of eigenmode
shapes of the Faraday wave field from one such model. Then we reproduce
numerical simulations and produce new numerical simulations and apply our
theorem to the test functions used for that model in addition to new test
functions. Evidence of chaos is shown by numerically studying a global
bifurcation.

We present the results of a numerical investigation of droplets walking on a rotating vibrating fluid bath. The drop's trajectory is described by an integro-differential equation, which is simulated numerically in various parameter regimes. As the forcing acceleration is progressively increased, stable circular orbits give way to wobbling orbits, which are succeeded in turn by instabilities of the orbital center characterized by steady drifting then discrete leaping. In the limit of large vibrational forcing, the walker's trajectory becomes chaotic, but its statistical behavior reflects the influence of the unstable orbital solutions. The study results in a complete regime diagram that summarizes the dependence of the walker's behavior on the system parameters. Our predictions compare favorably to the experimental observations of Harris and Bush ["Droplets walking in a rotating frame: from quantized orbits to multimodal statistics," J. Fluid Mech. 739, 444-464 (2014)]. (c) 2014 AIP Publishing LLC.

Electrodynamic shakers are widely used in experimental investigations of vibrated fluids and granular materials. However, they are plagued by undesirable internal resonances that can significantly impact the quality of vibration. In this work, we measure the performance of a typical shaker and characterize the influence that a payload has on its performance. We present the details of an improved vibration system based on a concept developed by Goldman (2002) [1] which consists of a typical electrodynamic shaker with an external linear air bearing to more effectively constrain the vibration to a single axis. The principal components and design criteria for such a system are discussed. Measurements characterizing the performance of the system demonstrate considerable improvement over the unmodified test shaker. In particular, the maximum inhomogeneity of the vertical vibration amplitude is reduced from approximately 10 percent to 0.1 percent; moreover, transverse vibrations were effectively eliminated.

We present the results of an experimental investigation of a droplet walking on the surface of a vibrating rotating fluid bath. Particular attention is given to demonstrating that the stable quantized orbits reported by Fort et al. (Proc. Natl Acad. Sci., vol. 107, 2010, pp. 17515-17520) arise only for a finite range of vibrational forcing, above which complex trajectories with multimodal statistics arise. We first present a detailed characterization of the emergence of orbital quantization, and then examine the system behaviour at higher driving amplitudes. As the vibrational forcing is increased progressively, stable circular orbits are succeeded by wobbling orbits with, in turn, stationary and drifting orbital centres. Subsequently, there is a transition to wobble-and-leap dynamics, in which wobbling of increasing amplitude about a stationary centre is punctuated by the orbital centre leaping approximately half a Faraday wavelength. Finally, in the limit of high vibrational forcing, irregular trajectories emerge, characterized by a multimodal probability distribution that reflects the persistent dynamic influence of the unstable orbital states.

We present the results of a theoretical investigation of droplets walking on a rotating vibrating fluid bath. The droplet's trajectory is described in terms of an integro-differential equation that incorporates the influence of its propulsive wave force. Predictions for the dependence of the orbital radius on the bath's rotation rate compare favourably with experimental data and capture the progression from continuous to quantized orbits as the vibrational acceleration is increased. The orbital quantization is rationalized by assessing the stability of the orbital solutions, and may be understood as resulting directly from the dynamic constraint imposed on the drop by its monochromatic guiding wave. The stability analysis also predicts the existence of wobbling orbital states reported in recent experiments, and the absence of stable orbits in the limit of large vibrational forcing.

A millimetric droplet can be induced to bounce on the surface of a fluid bath by vibrating the bath near the droplet's resonant frequency (Figure 1(a) ). [superscript 1–3] The localized field of Faraday waves excited by the bouncing droplet can cause it to propel itself laterally across the surface, moving in resonance with its guiding wave field (Figure 1(b) ). [superscript 4,5] These walking droplets, or “walkers,” generally move in a straight line at constant speed; however, they can be diverted through interaction with boundaries or external forces. This hydrodynamic system represents a macroscopic realization of the pilot-wave theory of quantum dynamics proposed by Louis de Broglie, according to which microscopic particles are propelled through a resonant interaction with a wave field generated by the particle's internal vibration. [superscript 6] Coincidentally, it exhibits many behaviors once thought to be exclusive to the microscopic quantum realm, including single-particle diffraction, [superscript 7] tunneling, [superscript 8] quantized orbits, [superscript 9] and orbital-level splitting. [superscript 10]

We present the results of a theoretical investigation of droplets bouncing on a vertically vibrating fluid bath. An integro-differential equation describing the horizontal motion of the drop is developed by approximating the drop as a continuous moving source of standing waves. Our model indicates that, as the forcing acceleration is increased, the bouncing state destabilizes into steady horizontal motion along a straight line, a walking state, via a supercritical pitchfork bifurcation. Predictions for the dependence of the walking threshold and drop speed on the system parameters compare favourably with experimental data. By considering the stability of the walking state, we show that the drop is stable to perturbations in the direction of motion and neutrally stable to lateral perturbations. This result lends insight into the possibility of chaotic dynamics emerging when droplets walk in complex geometries.

A small liquid drop can be kept bouncing on the surface of a bath of the same fluid for an unlimited time when this substrate oscillates vertically. With fluids of low viscosity the repeated collisions generate a surface wave at the bouncing frequency. The various dynamical regimes of the association of the drop with its wave are investigated first. The drop, usually a simple , undergoes a drift bifurcation when the forcing amplitude is increased. It thus becomes a propagating at a constant velocity on the interface. This transition occurs just below the Faraday instability threshold, when the drop becomes a local emitter of a parametrically forced wave. A model of the particle–wave interaction accounts for this drift bifurcation. The self-organization of several identical bouncers is also investigated. At low forcing, bouncers form bound states or crystal-like aggregates. At larger forcing, the collisions between walkers reveal that their interaction can be either repulsive or attractive, depending on their distance apart. The attraction leads to the spontaneous formation of orbiting pairs, the possible orbit diameters forming a discrete set. A theoretical model of the non-local interaction resulting from the interference of the waves is given. The nature of the interaction is thus clarified and the various types of self-organization recovered.

A vessel containing a heavy liquid vibrates vertically with constant frequency and amplitude. It has been observed that for some combinations of frequency and amplitude standing waves are formed at the free surface of the liquid, while for other combinations the free surface remains plane. In this paper the stability of the plane free surface is investigated theoretically when the vessel is a vertical cylinder with a horizontal base, and the liquid is an ideal frictionless fluid making a constant angle of contact of 90 degrees with the walls of the vessel. When the cross-section of the cylinder and the frequency and amplitude of vibration of the vessel are prescribed, the theory predicts that the mth mode will be excited when the corresponding pair of parameters (pm, qm) lies in an unstable region of the stability chart; the surface is stable if none of the modes is excited. (The corresponding frequencies are also shown on the chart.) The theory explains the disagreement between the experiments of Faraday and Rayleigh on the one hand, and of Matthiessen on the other. An experiment was made to check the application of the theory to a real fluid (water). The agreement was satisfactory; the small discrepancy is ascribed to wetting effects for which no theoretical estimate could be given.

Light-scattering measurements of the time-dependent local radial velocity in a rotating fluid reveal three distinct transitions as the Reynolds number is increased, each of which adds a new frequency to the velocity spectrum. At a higher, sharply defined Reynolds number all discrete spectral peaks suddenly disappear. Our observations disagree with the Landau picture of the onset of turbulence, but are perhaps consistent with proposals of Ruelle and Takens.

Direct numerical simulations of the transition process from laminar to chaotic flow in converging–diverging channels are presented. The chaotic flow regime is reached after a sequence of successive supercritical Hopf bifurcations to periodic, quasiperiodic, and chaotic self-sustained flow regimes. The numerical experiments reveal three distinct bifurcations as the Reynolds number is increased, each adding a new fundamental frequency to the velocity spectrum. In addition, frequency-locked periodic solutions with independent but synchronized periodic functions are obtained. A scenario similar to the Ruelle–Takens–Newhouse scenario of the onset of chaos is verified in this forced convective open system flow. The results are illustrated for different Reynolds numbers using time-velocity histories, Fourier power spectra, and phase space trajectories. The global structure of the self-sustained oscillatory flow for a periodic regime is also discussed.

The relation between the solutions of the time‐independent Schrödinger equation and the periodic orbits of the corresponding classical system is examined in the case where neither can be found by the separation of variables. If the quasiclassical approximation for the Green's function is integrated over the coordinates, a response function for the system is obtained which depends only on the energy and whose singularities give the approximate eigenvalues of the energy. This response function is written as a sum over all periodic orbits where each term has a phase factor containing the action integral and the number of conjugate points, as well as an amplitude factor containing the period and the stability exponent of the orbit. In terms of the approximate density of states per unit interval of energy, each stable periodic orbit is shown to yield a series of δ functions whose locations are given by a simple quantum condition: The action integral differs from an integer multiple of h by half the stability angle times ℏ. Unstable periodic orbits give a series of broadened peaks whose half‐width equals the stability exponent times ℏ, whereas the location of the maxima is given again by a simple quantum condition. These results are applied to the anisotropic Kepler problem, i.e., an electron with an anisotropic mass tensor moving in a (spherically symmetric) Coulomb field. A class of simply closed, periodic orbits is found by a Fourier expansion method as in Hill's theory of the moon. They are shown to yield a well‐defined set of levels, whose energy is compared with recent variational calculations of Faulkner on shallow bound states of donor impurities in semiconductors. The agreement is good for silicon, but only fair for the more anisotropic germanium.

It is shown that by a smallC
2 (resp.C
) perturbation of a quasiperiodic flow on the 3-torus (resp. them-torus,m>3), one can produce strange AxiomA attractors. Ancillary results and physical interpretation are also discussed.

A mechanism for the generation of turbulence and related phenomena in dissipative systems is proposed.

Etude d'une dynamique a m emoire de chemin: une exp erimentation th eorique

- M Labousse

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103107-10 Tambasco et al

103107-10
Tambasco et al.
Chaos 26, 103107 (2016)

- Tambasco

Tambasco et al.
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The pilot-wave dynamics of walking droplets in confinement Massachusetts Institute of Technology Chaos driven by interfering memory

- D M Harris
- M Labousse
- E Fort
- Y Couder

29
D. M. Harris, " The pilot-wave dynamics of walking droplets in confinement, " Ph.D. thesis, Massachusetts Institute of Technology,
Department of Mathematics, 2015.
30
S. Perrard, M. Labousse, E. Fort, and Y. Couder, " Chaos driven by interfering memory, " Phys. Rev. Lett. 113, 104101 (2014).

liquid in vertical periodic motion

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Drops of liquid can be made to float on the liquid. What enables them to do so?

- J Walker

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