Active Soft-Impact Oscillator: Dynamics of a Walking Droplet in a Non-Smooth Potential

TL;DR

The paper addresses how a memory‑driven wave–particle entity (a walking droplet) behaves in a non‑smooth confinement. It develops a minimal Lorenz‑like model with a piecewise‑smooth soft‑impact potential, yielding a 4D dynamical system that captures memory effects and impact interactions. A key contribution is the analytic Hopf stability boundary $R = \frac{1}{M^2} + \frac{k}{M+1}$, along with a rich set of dynamical regimes including extended weak chaos, grazing‑induced crises, multistability, fractal basins, and invisible attractor switching, all examined through parameter‑space and spectral analyses. The results offer testable predictions for walking droplets in engineered non‑smooth potentials and provide a bridge between active matter dynamics and hydrodynamic quantum analogs in non‑smooth environments.

Abstract

Walking droplets are millimetric fluid drops that propel themselves across a vibrated liquid bath through interaction with their self-generated waves. They constitute classical active wave-particle entities and exhibit a range of hydrodynamic quantum analogs. We investigate an \emph{active soft-impact oscillator} as a minimal model for a walking droplet moving within a piecewise-smooth external potential, analogous to classical mass-spring soft-impact oscillators and recently explored quantum soft-impact oscillators. Our active soft-impact oscillator model couples a non-smooth soft-impact force to the Lorenz-like dynamics arising from the wave-particle entity. Theoretical and numerical exploration of the full parameter space reveals a wide variety of nonlinear behaviors and bifurcations driven by impact and grazing events. These include grazing-induced and impact-induced transitions between periodic and chaotic motion, as well as grazing-mediated attractor switching and impact-free (invisible) attractor switching. The active soft-impact oscillator thus provides a versatile platform for probing nonlinear impact dynamics in active systems and exploring hydrodynamic quantum analogs in non-smooth potentials.

Figures (7)

• Figure 1: Schematic of the setup showing a one-dimensional active wave-particle entity (blue) in a non-smooth harmonic potential (red). A particle (blue circle) of unit dimensionless mass located at $x_d$ and moving horizontally with velocity $\dot{x}_d$ experiences a wave-memory force from its self-generated wave field (blue filled area), and an effective drag force. The underlying wave field is a superposition of the individual waves of spatial form $W(x)=\cos(x)$, which are generated by the particle continuously along its trajectory and decay exponentially in time (black and gray curves). An external piecewise-smooth harmonic potential exerts an additional force on the particle.
• Figure 2: Different types of WPE dynamical behaviors observed in the system with increasing memory parameter $M$. Phase-space trajectory in the $(x_d,X)$ projection showing (a) $M = 0.5$, stable fixed point, (b) $M = 2$, a stable periodic orbit, (c) $M=10$, a chaotic attractor. The vertical dashed line denotes $x=x_{\text{wall}}$. Other parameters are fixed to $k=1$, $A=5$, $x_{\text{wall}}=1.0$ and $R=1.5$ and initial condition $(x_d(0),X(0),Y(0),Z(0))=(1.0,0.1,0,0)$.
• Figure 3: Dynamics in the $R$-- $M$ parameter space for three values of the stiffness parameter $A$, where the color denotes the maximum Lyapunov exponent (MLE). The black dashed curve indicates the analytical stability boundary given by Eq. \ref{['eq: stability_curve']}. Regions with negative, zero, and positive MLE correspond to fixed-point, periodic/quasiperiodic dynamics, and chaotic dynamics, respectively: (a) $A = 0$, (b) $A = 5$, and (c) $A = 100$. Other parameters are fixed to $k=1$, $x_{\text{wall}}=1.0$, and initial conditions are fixed to $(x_d(0),X(0),Y(0),Z(0))=(1.0,0.1,0,0)$.
• Figure 4: Dynamics in the $x_{\text{wall}}$--$M$ parameter space for different values of the stiffness parameter $A$, where the color denotes the maximum Lyapunov exponent (MLE). Three values of the stiffness parameter $A$ are considered: (a) $A = 0$, (b) $A = 5$, and (c) $A = 100$. The color scale indicates the dynamical regime: dark blue and yellow correspond to negative MLE (stable points), white to near-zero MLE (periodic/quasiperiodic dynamics), and light blue and red to positive MLE (chaotic motion). Other parameters are fixed to $k = 1$, $R = 1.0$, and initial conditions are fixed to $(x_d(0),X(0),Y(0),Z(0)) = (1.0,0.1,0.0,0.0)$.
• Figure 5: Bifurcation diagram illustrating multistability as a function of the memory parameter $M$ for $x_{\text{wall}} = 1.0$, $R = 1.0$, $k = 1$, $A = 0$ computed over initial droplet positions $x_d(0) \in [-2.5,\, 2.0]$ and initial velocity $X = 1.0$, initial wave memory force $Y = 0.0$, initial wave-field height $Z = 0.0$. The bifurcation diagram is constructed by treating $M$ as the control parameter and recording the particle position $x^n_d$ at the Poincaré section defined by $X = 0$.