In this thesis, convergence of time averages near statistical attractors of continuous flows are investigated. A relation between statistical attractor and essential Ω-limit set is proved, and using this a general definition for statistical attractor is given. Sufficient conditions are given for an observable to admit a convergent time average along the orbits of the flow. The general results are applied to flows on a torus, and in particular to systems of coupled phase oscillators that admit attracting heteroclinic networks in their phase space. A particular heteroclinic network that we call heteroclinic ratchet is observed and analysed in detail. Heteroclinic ratchets give rise to a novel phenomenon, unidirectional desynchronization of oscillators (ratcheting). The results obtained about the convergence of time averages near statistical attractors implies that heteroclinic ratchets induce, besides its other interesting consequences, frequency synchronization without phase synchronization. Different coupling structures that can give rise to ratcheting of oscillators are also investigated.

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