In a system of coupled oscillators, synchronization occurs when the oscillators spontaneously lock to a common frequency or phase. We study a system of n ≫ 1 phase oscillators placed on a circle with random initial positions and sinusoidal coupling with their k nearest neighbors on each side. When all of the oscillators are identical, the final state of the system is either full phase-locking (in which all oscillators have the same relative phase) or a splay state characterized by a winding

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... er q with the oscillators uniformly spread apart in phase. However, when the internal frequencies of the oscillators are uniformly distributed on a small interval, we demonstrate that they settle into an "approximate" splay state, and their phases are no longer uniformly spread. For k = 1, we examine the system's final state as a map and show that phase-locking synchronization can never occur for nonidentical oscillators. We also derive a sufficient and necessary condition for the existence of cycles. The study of synchronization has been extremely prominent for over two decades. The phenomenon is present in many systems in physics, biology, and engineering. The Kuramoto model, one popular system that models synchronization, describes n ≫ 1 phase oscillators on a line. In this paper, we investigate a variant which was first proposed by Wiley, Strogatz, and Girvan [1]. This variant differs from the Kuramoto model by describing n oscillators on a ring rather than a line topology. Our system generalizes theirs in that the oscillators are not necessarily identical. Furthermore, to better understand the final states that arise, we include an examination of the final state as a map between oscillators rather than a set of coupled differential equations.