# Mathematical modeling for the synchronization of two interacting active rotors

## TL;DR

This paper develops a mathematical model to understand how two active rotors synchronize their rotation through chemical concentration fields, revealing conditions for in-phase and anti-phase synchronization.

## Contribution

It introduces a combined numerical and theoretical framework to analyze the synchronization of active rotors influenced by surface-active chemical fields.

## Key findings

- In-phase and anti-phase synchronization depend on rotor distance.
- Theoretical predictions match numerical simulation results.
- Stability of synchronization modes is analyzed via phase reduction.

## Abstract

We investigate the synchronization of active rotors. A rotor is composed of a free-rotating arm with a particle that releases a surface-active chemical compound. It exhibits self-rotation due to the surface tension gradient originating from the concentration field of the surface-active compound released from the rotor. In a system with two active rotors, they should interact through the concentration field. Thus, the interaction between them does not depend only on the instantaneous positions but also on the dynamics of the concentration field. By numerical simulations, we show that in-phase and anti-phase synchronizations occur depending on the distance between the two rotors. The stability of the synchronization mode is analyzed based on phase reduction theorem through the calculation of the concentration field in the co-rotating frame with the active rotor. We also confirm that the numerical results meet the prediction by theoretical analyses.