External Dynamics Model in Active Matter

• External Dynamics Model is a framework that decomposes system behavior into internal mechanisms and external forces using coupled nonlinear equations.
• It demonstrates how gravitational-like and electric-like forces induce diverse transitions, including straight, circular, and zigzag motions through bifurcation analysis.
• The model informs applications in active matter, cell migration, and soft robotics by linking self-propulsion, deformation, and feedback mechanisms.

An external dynamics model characterizes how the behavior of a system—composed of one or more interacting agents, particles, or subsystems—responds to forces, fields, or influences originating outside the intrinsic or internal dynamics of the system. These models formalize the interplay between internal mechanisms (self-organization, passive or active motion, feedbacks) and external drivers (constant or time-varying forces, fields, signals), allowing researchers to analyze and predict the resulting complex behaviors, including transitions between qualitatively distinct dynamical regimes.

1. Mathematical Foundations of External Dynamics Models

External dynamics models are typically formulated as sets of coupled, often nonlinear, differential or difference equations. The core principle involves decomposing the total dynamics into internal (autonomous) contributions and explicit external input terms, which may act additively or multiplicatively on the system variables.

For example, in the context of active matter, the dynamics of a deformable self-propelled particle under external forcing is governed by coupled equations for the center-of-mass velocity $\mathbf{v}$ and a deformation tensor $S_{\alpha\beta}$:

$$\begin{aligned} \frac{d v_\alpha}{dt} &= \gamma v_\alpha - |\mathbf{v}|^2 v_\alpha - a S_{\alpha\beta} v_\beta + g_\alpha \ \frac{d S_{\alpha\beta}}{dt} &= -\kappa S_{\alpha\beta} + b\left(v_\alpha v_\beta - \frac{1}{2} |\mathbf{v}|^2 \delta_{\alpha\beta}\right) + Q_{\alpha\beta} \end{aligned}$$

Here, $g_\alpha$ is a gravitational-like force acting on the center of mass, and $Q_{\alpha\beta}$ is an electric-like force modifying particle shape. The mathematical structure permits the analysis of both stationary and time-dependent external forces, and the resulting model encompasses a rich variety of dynamical regimes for different force strengths and particle parameters.

2. Types of External Forces and Their Coupling

The model supports two distinct kinds of external forces:

• Gravitational-like force: Enters additively into the velocity equation and is represented as a vector $\mathbf{g}$. This force acts uniformly in a fixed direction and magnitude, analogous to gravity or a constant electric field on a charged particle.
• Electric-like force: Enters additively into the deformation tensor equation as a second-rank tensor $Q_{\alpha\beta}$. This models, for example, the response of a deformable particle to a uniform electric field, inducing a dipole moment and thereby changing the particle's shape.

Mathematically, the gravitational-like force is of the form $g_\alpha = (0, -g)$, and the electric-like force assumes $$Q_{\alpha\beta} = h ( E_\alpha E_\beta - \frac{1}{2}|E|^2 \delta_{\alpha\beta} )$$, where $h$ parametrizes the field strength, and $\mathbf{E}$ is the external field direction (chosen as $(1, 0)$ in prototype scenarios).

The deformation response is encoded by a symmetric, traceless tensor: $$S_{\alpha\beta} = s(n_\alpha n_\beta - \frac{1}{2} \delta_{\alpha\beta})$$ with $\mathbf{n}$ specifying the orientation and $s$ the deformation magnitude.

3. Bifurcations and Richness of Dynamics

Changing the magnitude of external forces and the degree of self-propulsion, as parameterized by $\gamma$, induces bifurcations between qualitatively distinct dynamic states:

Major dynamical behaviors under gravitational-like forcing ($g \neq 0, h = 0$):

• Straight-falling motion: A passive particle moves directly along the external field.
• Circular motion: For supercritical activity ($\gamma > \gamma_c$), unforced particles move in circles.
• Circular-drift motion: The particle moves along a drifting circular path whose center advances almost perpendicularly to the external force, reflecting complex feedbacks between shape and propulsion.
• Zigzag motions: Oscillatory states (zigzag-1 and zigzag-2) emerge with further increases in force or activity, showing rich time dependence.

Under electric-like forcing ($h \neq 0, g = 0$):

• Motionless and straight motion: The external field can both suppress and orient motion, even flipping the alignment of velocity and deformation (parallel or perpendicular) across sharp thresholds.
• Circular and zigzag motions: Oscillatory and periodic trajectories result from the interaction of self-propulsion, deformation, and the applied field.

Transitions between regimes (e.g., straight to zigzag motion) are mediated by bifurcations such as Hopf and pitchfork bifurcations, which are analytically characterized in the model.

Forcing Type	Dynamics Observed	Key Bifurcations
Gravitational-like	Straight-falling, circular-drift, zigzag	Hopf, saddle-homoclinic
Electric-like	Motionless, straight, zigzag, circular	Hopf, pitchfork

4. Physical Interpretations and Phase Diagrams

This model demonstrates that the interplay between deformability and self-propulsion fundamentally alters how particles respond to external forces. Notably, certain emergent behaviors, such as drift motion perpendicular to the applied field or intricate multi-component oscillations, do not occur in rigid or non-self-propelled particles. Force strength, self-propulsion, and shape feedback mechanisms combine nonlinearly, giving rise to phase diagrams with boundaries demarcating the regions of different dynamical regimes.

Key quantitative results include:

• Drift bifurcation threshold:

$\gamma_c = \frac{\kappa^2}{a b} + \frac{\kappa}{2}$

• Hopf bifurcation (onset of zigzag-1 motion):

$$g^* = \frac{2B(\gamma - \gamma_c)}{1-B} \left( \frac{\gamma - \kappa}{1-B} \right)^{1/2}$$

with $B = \frac{ab}{2\kappa}$.

Phase diagrams map out the parameter space of $(\gamma, g)$ or $(\gamma, h)$, showing the domains of each dynamical regime and the nature of their transitions.

5. Broader Significance and Theoretical Implications

The external dynamics model for deformable self-propelled particles demonstrates that:

• External forcing can both lock and diversify particle motion, encompassing not just trivial alignment but also emergent classes of periodic and aperiodic behaviors.
• The feedback between self-propulsion and deformation is essential for capturing the full range of observed behaviors; neglecting such feedback would miss significant dynamical regimes, such as drift perpendicular to the force or zigzag-2 motion.
• Rich phase behavior arises even in a minimal, symmetry-based model, underscoring the importance of incorporating both internal structural degrees of freedom and external drivers in theoretical descriptions of soft active systems.

Such models have implications for biological and synthetic systems—the migration of deformable cells in external fields, the response of soft robotic agents, or the design of active materials subject to environmental cues.

6. Analytical Methods and References

Key equations, bifurcation conditions, and phase diagrams are derived explicitly:

• Core model equations: Eqs. (1.1), (1.2)
• Scalar/angular variable system: Eqs. (1.3)–(1.6)
• Bifurcation conditions and phase boundaries: Eq. (B), Eq. (N1.5), etc.

Phase diagrams and time series data (detailed in figure references within the paper) provide visual mapping of dynamic regimes and underscore the sensitivity to force type and parameter variation.

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In summary, the external dynamics model detailed here provides a general framework for understanding how the coupling between internal degrees of freedom (deformation, self-propulsion) and external forces orchestrates a broad spectrum of dynamic states. The force type, strength, and particle parameters conspire to produce transitions and behaviors that are analytically tractable, yet phenomenologically rich, establishing a foundation for more complex modeling in diverse physical and biological systems.