Tunable bimodal explorations of space from memory-driven deterministic dynamics.

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.

Multistable Free States of an Active Particle from a Coherent Memory Dynamics.

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.

Introduction to focus issue on hydrodynamic quantum analogs.

Hydrodynamic quantum analogs is a nascent field initiated in 2005 by the discovery of a hydrodynamic pilot-wave system [Y. Couder, S. Protière, E.

Self-attraction into spinning eigenstates of a mobile wave source by its emission back-reaction.

The back-reaction of a radiated wave on the emitting source is a general problem. In the most general case, back-reaction on moving wave sources depends on their whole history. Here we study a model system in which a pointlike source is piloted by its own memory-endowed wave field.

Wave-Based Turing Machine: Time Reversal and Information Erasing.

The investigation of dynamical systems has revealed a deep-rooted difference between waves and objects regarding temporal reversibility and particlelike objects. In nondissipative chaos, the dynamic of waves always remains time reversible, unlike that of particles. Here, we explore the dynamics of a wave-particle entity.

Interaction of two walkers: wave-mediated energy and force.

A bouncing droplet, self-propelled by its interaction with the waves it generates, forms a classical wave-particle association called a "walker." Previous works have demonstrated that the dynamics of a single walker is driven by its global surface wave field that retains information on its past trajectory. Here we investigate the energy stored in this wave field for two coupled walkers and how it conveys an interaction between them.

Chaos driven by interfering memory.

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.

Self-organization into quantized eigenstates of a classical wave-driven particle.

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.

Wavelike statistics from pilot-wave dynamics in a circular corral.

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.

Level splitting at macroscopic scale.

A walker is a classical self-propelled wave particle association moving on a fluid interface. Two walkers can interact via their waves and form orbiting bound states with quantized diameters. Here we probe the behavior of these bound states when setting the underlying bath in rotation.

Shape transition in artificial tumors: from smooth buckles to singular creases.

Using swelling hydrogels, we study the evolution of a thin circular artificial tumor whose growth is confined at the periphery. When the volume of the outer proliferative ring increases, the tumor loses its initial symmetry and bifurcates towards an oscillatory shape. Depending on the geometrical and elastic parameters, we observe either a smooth large-wavelength undulation of the swelling layer or the formation of sharp creases at the free boundary.

Mutual adaptation of a Faraday instability pattern with its flexible boundaries in floating fluid drops.

Hydrodynamic instabilities are usually investigated in confined geometries where the resulting spatiotemporal pattern is constrained by the boundary conditions. Here we study the Faraday instability in domains with flexible boundaries. This is implemented by triggering this instability in floating fluid drops.

Unpredictable tunneling of a classical wave-particle association.

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.

Turning a plant tissue into a living cell froth through isotropic growth.

The forms resulting from growth processes are highly sensitive to the nature of the driving impetus, and to the local properties of the medium, in particular, its isotropy or anisotropy. In turn, these local properties can be organized by growth. Here, we consider a growing plant tissue, the shoot apical meristem of Arabidopsis thaliana.

Developmental patterning by mechanical signals in Arabidopsis.

A central question in developmental biology is whether and how mechanical forces serve as cues for cellular behavior and thereby regulate morphogenesis. We found that morphogenesis at the Arabidopsis shoot apex depends on the microtubule cytoskeleton, which in turn is regulated by mechanical stress. A combination of experiments and modeling shows that a feedback loop encompassing tissue morphology, stress patterns, and microtubule-mediated cellular properties is sufficient to account for the coordinated patterns of microtubule arrays observed in epidermal cells, as well as for patterns of apical morphogenesis.

Exotic orbits of two interacting wave sources.

As shown recently, it is possible to create, on a vibrating fluid interface, mobile emitters of Faraday waves [Y. Couder, S. Protière, E.

Spiral patterns in the packing of flexible structures.

Spiral patterns are found to be a generic feature in close-packed elastic structures. We describe model experiments of compaction of quasi-1D sheets into quasi-2D containers that allow simultaneous quantitative measurements of mechanical forces and observation of folded configurations. Our theoretical approach shows how the interplay between elasticity and geometry leads to a succession of bifurcations responsible for the emergence of such patterns.

Single-particle diffraction and interference at a macroscopic scale.

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.

Dynamical phenomena: walking and orbiting droplets.

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.

Side-branch growth in two-dimensional dendrites. II. Phase-field model.

The development of side-branching in solidifying dendrites in a regime of large values of the Peclet number is studied by means of a phase-field model. We have compared our numerical results with experiments of the preceding paper and we obtain good qualitative agreement. The growth rate of each side branch shows a power-law behavior from the early stages of its life.

From bouncing to floating: noncoalescence of drops on a fluid bath.

When a drop of a viscous fluid is deposited on a bath of the same fluid, it is shown that its coalescence with this substrate is inhibited if the system oscillates vertically. Small drops lift off when the peak acceleration of the surface is larger than g. This leads to a steady regime where a drop can be kept bouncing for any length of time.

Hierarchical crack pattern as formed by successive domain divisions. II. From disordered to deterministic behavior.

Hierarchical crack patterns, such as those formed in the glaze of ceramics or in desiccated layers of mud or gel, can be understood as a successive division of two-dimensional domains. We present an experimental study of the division of a single rectangular domain in drying starch and show that the dividing fracture essentially depends on the domain size, rescaled by the thickness of the cracking layer e. Utilizing basic assumptions regarding the conditions of crack nucleation, we show that the experimental results can be directly inferred from the equations of linear elasticity.

Hierarchical crack pattern as formed by successive domain divisions. I. Temporal and geometrical hierarchy.

Crack patterns, as they can be observed in the glaze of ceramics or in desiccated mud layers, are formed by successive fractures and divide the two-dimensional plane into distinct domains. On the basis of experimental observation, we develop a description of the geometrical structure of these hierarchical networks. In particular, we show that the essential feature of such a structure can be represented by a genealogical tree of successive domain divisions.

Side-branch growth in two-dimensional dendrites. I. Experiments.

The dynamics of growth of dendrites' side branches is investigated experimentally during the crystallization of solutions of ammonium bromide in a quasi-two-dimensional cell. Two regimes are observed. At small values of the Peclet number a self-affine fractal forms.

Four sided domains in hierarchical space dividing patterns.

The cracks observed in the glaze of ceramics form networks, which divide the 2D plane into domains. It is shown that, on the average, the number of sides of these domains is four. This contrasts with the usual 2D space divisions observed in Voronoi tessellation or 2D soap froths.