### Synchronization in a network of Oscillators with Delayed Coupling

#### Abstract

The study of coupled oscillators with time lag can get its applications in; Neurobiology,Laser arrays, Microwave devices, Communications satellites and electronic circuits, justto mention but few. That is why we studied a population of n oscillators each with anasymptotically stable limit cycle coupled all-to-all by a linear diffusive like path with atime lag, t . The system of equations was inbuilt with symmetries which we exploitedto get an analytical understanding of the dynamics of the system. The symmetries thenhelped us get two n-dimensional invariant manifolds: the diagonal manifold and theother orthogonal manifold. We exploited the symmetries in the coupling terms toestablish the range of time delay t for stability of synchronized state.We did a rigorous study of the condition of stability and persistence of the synchronizedmanifold of diffusively coupled oscillators of linear and planar simple Bravais Latticesby considering n (n ³ 2) , d-dimensional oscillators each with an asymptotically stablelimit cycle coupled all-to-all by a nearest neighbor linear diffusive like path. We usedthe invariant Manifold Theory and Lyapunov exponents to establish the range ofcoupling strength for stability and robustness of the synchronized manifold. The 4th and5th order Runge-Kutta method, together with ode-45 and dde-23 Mat lab solvers werethe numerical methods we used to get the numerical solution of our problem. Weestablished the estimate for bound of t for which the synchronized manifold remainsstable when the oscillators are coupled in an all-to-all configuration. The synchronizedstate is seen to be stable when t < 9. Even for significant time delays, a stablesynchronized state exists at a very low coupling strength.From the study we realized that if synchronization exists for a certain couplingconfiguration, then there exist a k0 > 0 such that for all k0 > k , synchronizationmanifold is stable and persist under perturbation.

#### References

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