Topological states and continuum model for swarmalators without force reciprocity

Abstract: Swarmalators are systems of agents which are both self-propelled particles and oscillators. Each particle is endowed with a phase which modulates its interaction force with the other particles. In return, relative positions modulate phase synchronization between interacting particles. In the present model, there is no force reciprocity: when a particle attracts another one, the latter repels the former. This results in a pursuit behavior. In this paper, we derive a hydrodynamic model of this swarmalator system and show that it has explicit doubly-periodic travelling-wave solutions in two space dimensions. These special solutions enjoy non-trivial topology quantified by the index of the phase vector along a period in either dimension. Stability of these solutions is studied by investigating the conditions for hyperbolicity of the model. Numerical solutions of both the particle and hydrodynamic models are shown. They confirm the consistency of the hydrodynamic model with the particle one for small times or large phase-noise but also reveal the emergence of intriguing patterns in the case of small phase-noise.

• The paper introduces a novel hydrodynamic model for swarmalators showing how non-reciprocal forces produce topologically protected travelling-wave solutions.
• It derives a continuum model from a kinetic framework, linking velocity alignment and phase synchronization through hyperbolicity conditions.
• Numerical simulations validate the model by demonstrating coherent segregation and stable topological states with implications for swarm robotics.

Topological States and Continuum Model for Swarmalators Without Force Reciprocity

Introduction

The paper "Topological states and continuum model for swarmalators without force reciprocity" (2205.15739) introduces a novel model for swarmalators, systems of self-propelled particles that also act as oscillators, and explores the absence of force reciprocity. In this setup, when one particle exerts an attractive force, the counterpart exerts a repellant force, resulting in pursuit dynamics. A hydrodynamic model of this swarmalator system is derived, showcasing specific travelling-wave solutions with non-trivial topological characteristics.

Hydrodynamic Model Derivation

The study establishes a hydrodynamic limit from a kinetic description. Notably, it highlights the interaction between velocity alignment and phase synchronization, defining an explicit doubly-periodic travelling-wave solution in two-dimensional space. The absence of reciprocity in interaction forces leads to a compelling pursuit behavior, contrasting with classical symmetrical interaction models. This non-reciprocity is critical to the observed topology of the solutions.

Figure 1: Pursuit dynamics between two particles located at $X_1$ and $X_2$. The forces acting on these particles illustrate the pursuit behavior.

Stability and Hyperbolicity Analysis

The paper examines the stability of these travelling-wave solutions by investigating hyperbolicity conditions. The analysis reveals that stability is contingent upon the relationship between interaction parameters and configurations, particularly the angle between orientation and phase gradients. It emphasizes that under certain conditions, perturbations may lead to loss of hyperbolicity, thereby affecting solution stability.

Figure 2: Geometric setting of the hyperbolicity analysis, highlighting the spatial configuration crucial for understanding the model's stability.

Numerical Simulations

Numerical simulations are used to validate the hydrodynamic model and illustrate the formation of topologically protected states. These simulations demonstrate that the swarmalators maintain coherent topological states over time, influenced by the level of noise in phase dynamics. The particles exhibit segregation into distinct regions based on phase, separated by low-density interfaces.

Figure 3: Plot of the phase field $\alpha(x,0)$ (when $\alpha_0 = 0$) for specific travelling-wave solutions showcasing non-trivial topology.

Implications and Future Directions

The implications of this research are significant, providing insights into the collective dynamics of systems where interaction forces deviate from classical symmetrical laws. The introduction of non-reciprocal interactions opens new avenues in designing artificial systems with emergent behaviors that mimic biological organisms. Future work may explore topological characteristics, leveraging the robust topological structures identified in these systems for applications in swarm robotics and collective algorithm design.

Conclusion

This paper provides a comprehensive analysis of a novel swarmalator model that eschews traditional force reciprocity, revealing new topological states in collective dynamics. By deriving and numerically validating a continuum model, it lays the groundwork for future explorations into non-reciprocal interactions and their role in natural and artificial systems.