Collective actuation in active solids in the presence of a polarizing field: a systematic analysis of the dynamical regimes

TL;DR

The paper provides a systematic analysis of collective actuation in active solids subjected to an external polarizing field, distinguishing the impact of degenerate versus non-degenerate mode spectra. It employs a triad of approaches—a microscopic model, a reduced single-particle description, and a coarse-grained mean-field model—to map the dynamical regimes and transitions, deriving analytic results near exceptional points and Hopf bifurcations while relying on numerics where needed. Key findings include the emergence of new oscillatory states such as Windscreen Wiper dynamics, a reentrance transition that can either promote or suppress collective actuation depending on field strength, and a rigorous link between fixed-point stability and rotating/oscillatory solutions across levels of description. The results illuminate how external fields can serve as precise controls for actuation, with potential relevance to biological systems and engineered active materials, and open questions about non-degenerate regime complexity and spatial coexistence in large systems.

Abstract

Collective actuation in active solids, the spontaneous condensation of the dynamics on a few elastic modes, takes place whenever the deformations of the structure reorient the forces exerted by the active units composing, or embedded in, the solid. In a companion paper, we show through a combination of model experiments, numerical simulations, and theoretical analysis that adding an external field that polarizes the active forces strongly affects the dynamical transition to collective actuation. A new oscillatory regime emerges, and a reentrance transition to collective actuation takes place. Depending on the degeneracy of the modes on which the dynamics condensates, and on the orientation of the field with respect to the stiff direction of the solid, several new dynamical regimes can be observed. The purpose of the present paper is to review these dynamical regimes in a comprehensive way, both for the single-particle dynamics and for the coarse-grained one. Whenever possible the dynamical regimes and the transition between them are described analytically, otherwise numerically.

Figures (9)

• Figure 1: Single particle dynamics in a degenerate harmonic potential, in the absence of external field: (a) The drift-pitchfork bifurcation; the white cone describes the set of marginal fixed points for increasing $\Pi$. For $\Pi=\omega_0^2$ all fixed points turn unstable and leave place to an orbiting solution, the oscillation frequency of which, $\Omega$ increases from zero at the transition. (b) Destabilization mechanism; when $\Pi<\omega_0^2$, the displacement vector (black) catches up the orientation one (red) and the system restabilizes on a new fixed point; when $\Pi>\omega_0^2$, the displacement vector chases the orientation one indefinitely, leading to the oscillating solution.
• Figure 2: Single particle dynamics in a degenerate harmonic potential, in the presence of an external field: (a) Phase diagram; the color codes for the value of $\Omega_y/\Omega_x$ -- the ratio of the oscillation frequency of the displacement along the $x$ and $y$ directions -- as indicated in the legend (green : $\Omega_y/\Omega_x=1$, light blue : $\Omega_y/\Omega_x=2/3$, red: $\Omega_y/\Omega_x=2$); each point in the diagram is the result from an independent simulation, with random initial condition. The solid and dashed black lines indicate $\Pi_c = \omega_{0}^{2} + h$ and $\Pi^{\star} = \omega_{0}^{2} + 3h$, respectively. (b-c-d)-left: dynamics of the displacements in the WW$_y$, WW$_y^2$, and CO regimes; the trajectory is plotted during one period of oscillation, and colored with time running from dark blue to red; the dark arrows are snapshots of the orientation of the active force $\hat{\boldsymbol{n}}$. (b-c-d)-right: corresponding dynamics of the phases $(\theta,\varphi)$ with the same color code.
• Figure 3: Single particle in a degenerate harmonic potential: mapping with the physical pendulum close to the exceptional point. (a) Energy drift $\delta E$ as a function of $E/\tilde{H}$, for different values of $\tilde{H}$ (App. \ref{['app:pendulum']}). Stable orbits are highlighted with a green marker. (b) Phase portrait of two transient regimes at $\tilde{H} = 1/3$, with initial energies slightly above and below the heteroclinic orbit's energy (black dashed line); the heteroclinic orbit is the unstable fixed point in panel (a) for $\tilde{H} = 1/3$, and the separatrix of transients. The stationary regimes obtained correspond to the stable orbits shown in panels (a) for $\tilde{H} = 1/3$. (c) Rescaled fundamental frequency of oscillation of $\varphi$ as a function of the rescaled distance to the threshold, for a small field $H = 10^{-4}$. Colored markers represent numerical simulations (green: CO, red: WW$_y$), and the solid black lines (resp. solid grey line) are the stable (resp. unstable) solutions of the pendulum equations.
• Figure 4: Linear stability of the fixed points of the dynamics for a single particle in a non-degenerate harmonic potential with $\omega^2_x < \omega^2_y$, in the absence of an external field: The fixed points, represented in the ($u_x$,$u_y$)-plane, are distributed along an ellipse, as given by Eq. (\ref{['eq:ellipse']}), and the stability threshold $\Pi_c$ depends on the orientation $\theta_0$ (which is also the angle with respect to the $x$-axis of this fixed point). The red overlay indicates the marginally stable fixed points; when absent the fixed points are linearly unstable.
• Figure 5: Single particle dynamics in a non-degenerate harmonic potential $(\omega_x^2 = 1, \omega_y^2 = 2)$ , in the presence of an external field: (a) Phase diagram with the field in the stiff direction $\boldsymbol{h} = h_y \boldsymbol{\hat{e}}_y$; (b) phase diagram with the field in the soft direction $\boldsymbol{h} = h_x \boldsymbol{\hat{e}}_x$; the color codes for the value of $\Omega_y/\Omega_x$ -- the ratio of the oscillation frequency of the displacement along the $x$ and $y$ directions -- as indicated in the legend. Each point in the diagram is the result of an independent simulation, with a random initial condition. As such, areas of parameter space where two colors appear intermingled, indicate zones of coexistence between different dynamical regimes. The solid black lines indicate $\Pi_c(h_{x/y})$, as obtained from linear stability analysis. (c-g)-left: dynamics of the displacements in the dynamical regimes as named; the trajectory is plotted during one period of oscillation, and colored with time running from dark blue to red; the dark arrows are snapshots of the orientation of the active force $\hat{\boldsymbol{n}}$. (c-g)-right: corresponding dynamics of the phases $(\theta,\varphi)$ with the same color code. Nota bene: the WW$_y$ and the CO$^2$ regimes share the same color code because $\Omega_y/\Omega_x=2$ for both of them; they can however not be confused because WW$_y$ is a bounded phases regime, while the phases are unbounded in the CO$^2$ regime.