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