RESUMEN
A discrete and periodic complex Ginzburg-Landau equation, coupled to a mean equation, is systematically derived from a driven and dissipative lattice oscillator model, close to the onset of a supercritical Andronov-Hopf bifurcation. The oscillator model is inspired by recent experiments exploring active vibrations of quasi-one-dimensional lattices of self-propelled millimetric droplets bouncing on a vertically vibrating fluid bath. Our systematic derivation provides a direct link between the constitutive properties of the lattice system and the coefficients of the resultant amplitude equations, paving the way to compare the emergent nonlinear dynamics-namely, the onset and formation of discrete dark solitons, breathers, and traveling waves-against experiments. The framework presented herein is expected to be applicable to a wider class of oscillators characterized by the presence of a dynamic coupling potential between particles. More broadly, our results point to deeper connections between nonlinear oscillators and the physics of active and driven matter.
RESUMEN
We present the results of a theoretical investigation into the dynamics of a vibrating particle propelled by its self-induced wave field. Inspired by the hydrodynamic pilot-wave system discovered by Yves Couder and Emmanuel Fort, the idealized pilot-wave system considered here consists of a particle guided by the slope of its quasi-monochromatic "pilot" wave, which encodes the history of the particle motion. We characterize this idealized pilot-wave system in terms of two dimensionless groups that prescribe the relative importance of particle inertia, drag and wave forcing. Prior work has delineated regimes in which self-propulsion of the free particle leads to steady or oscillatory rectilinear motion; it has further revealed parameter regimes in which the particle executes a stable circular orbit, confined by its pilot wave. We here report a number of new dynamical states in which the free particle executes self-induced wobbling and precessing orbital motion. We also explore the statistics of the chaotic regime arising when the time scale of the wave decay is long relative to that of particle motion and characterize the diffusive and rotational nature of the resultant particle dynamics. We thus present a detailed characterization of free-particle motion in this rich two-parameter family of dynamical systems.
RESUMEN
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.
RESUMEN
We present the results of a theoretical investigation of a dynamical system consisting of a particle self-propelling through a resonant interaction with its own quasi-monochromatic pilot-wave field. We rationalize two distinct mechanisms, arising in different regions of parameter space, that may lead to a wavelike statistical signature with the pilot-wavelength. First, resonant speed oscillations with the wavelength of the guiding wave may arise when the particle is perturbed from its steady self-propelling state. Second, a random-walk-like motion may set in when the decay rate of the pilot-wave field is sufficiently small. The implications for the emergent statistics in classical pilot-wave systems are discussed.
RESUMEN
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.