Topological synchronization of coupled nonlinear oscillators

Synchronization of coupled oscillators is a ubiquitous phenomenon found throughout nature. Its robust realization crucial to our understanding various nonlinear systems, ranging from biological functions electrical engineering. On another front, in condensed matter physics, topology utilized realize properties like topological edge modes, as demonstrated by celebrated insulators. Here, we integrate these two research avenues and propose phenomenon, namely synchronization, where only the synchronize while bulk ones exhibit chaotic dynamics. We analyze concrete prototypical models demonstrate presence positive Lyapunov exponents vectors localized along edge. As unique characteristic find that unconventional extra boundary modes appear at emerging effective boundaries. Furthermore, proposal shows promise for spatially controlling such on-demand pattern designing defect detection. The synchronization can ubiquitously thus provide guiding principle robust, geometrical, flexible way.