Business cycles tend to comove across countries. However, standard models that attribute comovement to propagation of exogenous shocks struggle to generate a level of comovement that is as high as in the data. In this paper, we consider models that produce business cycles endogenously, through some form of non-linear dynamics -- limit cycles or chaos. These models generate stronger comovement, because they combine shock propagation with synchronization of endogenous dynamics. In particular, we study a demand-driven reduced-form model in which business cycles emerge from strategic complementarities within countries, synchronizing their oscillations through international trade linkages. We develop an eigendecomposition that explores the interplay between non-linear dynamics, shock propagation and network structure, and use this theory to understand the mechanisms of synchronization. Next, we calibrate the model to data on 24 countries and show that the empirical level of comovement can only be matched by combining endogenous business cycles with exogenous shocks. Despite the limitations of using a stylized model, our results support the hypothesis that business cycles are at least in part caused by underlying non-linear dynamics.