Mutual synchronization in a system of differential equations

Abstract

The perturbation analysis showed that for the given system of N interacting units, a totally synchronized state was possible over an unbounded N-dimensional volume in the parameter space spanned by A 1, A 2,…, A N . If one of the A n is fixed, the corresponding synchronization region in the remaining ( N − 1)-dimensional parameter space has finite non-zero measure. This synchronized state in principle requires corresponding initial relative phase differences; however numerical simulation reveals insensitivity to the initial phase differences, even though the ultimate average phase differences will be correct. When the initial phase differences are too far from those suggested by the perturbation theory for total synchronization, the system will, on occasion, apparently lock into a state where subsets are synchronized, but the individual subsets slip with respect to each other. In any practical system this situation might not occur as small fluctuations in parameter values could cause the eventual requisite alignment of phases prior to total locking.