Stability and persistence of synchronized manifold of diffusively coupled oscillators
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The study of Synchronization, Stability and Robustness of a system of oscillators has attracted great interest because of its application in many fields such as Neurobiology and Biological Systems [5, 6], Communication Systems , Mechanical and Electrical Systems , Stabilization of Unstable Periodic Orbits  and many others. In this paper, we study the condition for stability and persistence of synchronized manifold of diffusively coupled oscillators of linear and planar simple Bravais lattices. We considered "" d-dimensional oscillators each with an asymptotically stable limit cycle coupled by a near neighbor linear diffusive like path. We will state and prove a theorem that gives the conditions for stability and persistence of the synchronized manifold. The invariant manifold theory and Lyapunov exponents enabled us to establish the range of coupling strength for stability and robustness of the synchronized state. The comparison of the trajectories of oscillators in the manifolds was by comparing the amplitudes of graphed trajectories generated using ode45 Matlab solver.