Abstract:
Equations
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ABSTRACT
The study of coupled oscillators with time lag can get its applications in; Neurobiology,
Laser arrays, Microwave devices, Communications satellites and electronic circuits, just
to mention but few. That is why we studied a population of n oscillators each with an
asymptotically stable limit cycle coupled all-to-all by a linear diffusive like path with a
time lag, t . The system of equations was inbuilt with symmetries which we exploited
to get an analytical understanding of the dynamics of the system. The symmetries then
helped us get two n-dimensional invariant manifolds: the diagonal manifold and the
other orthogonal manifold. We exploited the symmetries in the coupling terms to
establish 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 synchronized
manifold of diffusively coupled oscillators of linear and planar simple Bravais Lattices
by considering n (n ³ 2) , d-dimensional oscillators each with an asymptotically stable
limit cycle coupled all-to-all by a nearest neighbor linear diffusive like path. We used
the invariant Manifold Theory and Lyapunov exponents to establish the range of
coupling strength for stability and robustness of the synchronized manifold. The 4th and
5th order Runge-Kutta method, together with ode-45 and dde-23 Mat lab solvers were
the numerical methods we used to get the numerical solution of our problem. We
established the estimate for bound of t for which the synchronized manifold remains
stable when the oscillators are coupled in an all-to-all configuration. The synchronized
state is seen to be stable when t < 9. Even for significant time delays, a stable
synchronized state exists at a very low coupling strength.
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From the study we realized that if synchronization exists for a certain coupling
configuration, then there exist a k0 > 0 such that for all k0 > k , synchronization
manifold is stable and persist under perturbation.
