  The synchronization of coupled chaotic systems represents a fundamental
example of self organization and collective behavior. This well-studied
phenomenon is classically characterized in terms of macroscopic parameters,
such as Lyapunov exponents, that help predict the systems transitions into
globally organized states. However, the local, microscopic, description of this
emergent process continues to elude us. Here we show that at the microscopic
level, synchronization is captured through a gradual process of topological
adjustment in phase space, in which the strange attractors of the two coupled
systems continuously converge, taking similar form, until complete topological
synchronization ensues. We observe the local nucleation of topological
synchronization in specific regions of the systems attractor, providing early
signals of synchrony, that appear significantly before the onset of complete
synchronization. This local synchronization initiates at the regions of the
attractor characterized by lower expansion rates, in which the chaotic
trajectories are least sensitive to slight changes in initial conditions. Our
findings offer an alternative description of synchronization in chaotic
systems, exposing its local embryonic stages that are overlooked by the
currently established global analysis. Such local topological synchronization
enables the identification of configurations where prediction of the state of
one system is possible from measurements on that of the other, even in the
absence of global synchronization.
