  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 concrete prototypical models
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 designing 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.
