Data-driven exploration of swarmalators with second-order harmonics.

We explore the dynamics of a swarmalator population comprising second-order harmonics in phase interaction. A key observation in our study is the emergence of the active asynchronous state in swarmalators with second-order harmonics, mirroring findings in the one-dimensional analog of the model, accompanied by the formation of clustered states. Particularly, we observe a transition from the static asynchronous state to the active phase wave state via the active asynchronous state. We have successfully delineated and quantified the stability boundary of the active asynchronous state through a completely data-driven method. This was achieved by utilizing the enhanced image processing capabilities of convolutional neural networks, specifically, the U-Net architecture. Complementing this data-driven analysis, our study also incorporates an analytical stability of the clustered states, providing a multifaceted perspective on the system's behavior. Our investigation not only sheds light on the nuanced behavior of swarmalators under second-order harmonics, but also demonstrates the efficacy of convolutional neural networks in analyzing complex dynamical systems.