Multiple Coupling in Chains of Oscillators

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Abstract
Chains of oscillators with coupling to more neighbors than the nearest ones are considered. The equations for the phase-locked solutions of an infinite chain of such type may be considered as a one-parameter family of $(2m - 1)$st-order discrete dynamical systems, whose independent variable is position along the chain, whose dependent variable is the phase between successive oscillators, and where m is the number of neighbors connected to each side. It is shown that for each value of the parameter in some range, the $(2m - 1)$st-order system has a one-dimensional hyperbolic global center manifold. This is done by using the theory of exponential dichotomies to show that the system “shadows” a simple one-dimensional system. The exponential dichotomy is constructed by exploiting an algebraic structure imposed by the geometry of the multiple coupling. For a finite chain, the dynamical system is constrained by manifolds of boundary conditions. It is shown that for open sets of such conditions, the solution to the equation for phase-locking in long chains stays close to the center manifold except near the boundaries. This is used to show that a multiply coupled system behaves, except near the boundaries, as a modified nearest-neighbor system. The properties of the nearest-neighbor and multiply coupled systems are then compared.

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