This book presents an important condition for stability and persistence of synchronized manifold of diffusively coupled oscillators of linear and planar simple Bravais lattices. This is done by considering n ( ? 2), d-dimensional oscillators each with an assymptoticaly stable limit cycle coupled by a nearest neighbour linear diffusive like path. In chapter two we review what has been done in relation to the three aspects; namely synchronization, stability and persistence of the synchronized manifold. In Chapter three, we state and prove a theorem that gives the conditions for stability and persistence of the synchronized state. Here we give the equations that describe the nature of dynamics of coupled oscillators and a detailed analysis where Invariant manifold Theory and Lyapunov exponents are used to establish the range of coupling strength for stability and robustness of synchronized state. The book is of great value to the fields of both Applied Mathematics and Statistics.