Manuscript submitted to Website: http://AIMsciences.org AIMS’ Journals Volume X, Number 0X, XX 200X pp. X–XX STABILITY OF SYNCHRONIZATION IN A SHIFT-INVARIANT RING OF MUTUALLY COUPLED OSCILLATORS R. Yamapi Department of Physics, Faculty of Science University of DOUALA, PO. Box 24157 DOUALA, Cameroon R.S. Mackay Mathematics Institute University of Warwick, Coventry CV4 7AL, U.K. (Communicated by the associate editor ) Abstract. This paper treats synchronization dynamics in a shift-invariant ring of N mutually coupled self-sustained electrical units. Via some qualita- tive theory for the Lyapunov exponents, we derive the regimes of coupling parameters for which synchronized oscillation is stable or unstable in the ring. 1. Introduction. Synchronisation of dynamical systems is a phenomenon of widespread technolog- ical and natural importance, ranging from the operation of electricity distribution networks to the neural generation of breathing rhythm, for example. It is of inter- est to determine the parameter regimes in which stable synchronisation can occur. There are many works on this, e.g. [1]- [15] and it would fill a book to review the literature. Here we restrict attention to the simplest context for synchronisation, a shift- invariant ring of identical nearest neighbour coupled second order oscillators, and the strongest notion of synchronisation, identical time dependence. Such a system has been used to model a parallel operating system of microwave oscillators [16, 17], and the interest in synchronisation is to produce high power. This allows us to derive strong general results on the parameter regime for stable synchronisation. The key new ingredient is qualitative analysis of the parameter dependence of the time-dependent flow on projective space of an associated two-dimensional linearised system about a synchronous solution. The paper is developed around a specific electrical example, but the theory applies more generally. The paper is organised as follows. In Section 2 the electrical system is described. In Section 3, basic results on linear stability of synchronous solutions of such a system are explained. In Section 4, the Pr¨ ufer transformation is used to develop extensive qualitative theory for a generalised class of linear equations; parts are deferred to appendices. In Section 5, the theory is compared to the results of numerics on the electrical system, with good agreement. Section 6 summarises the conclusions, gives a simple extension of the results, and suggests some directions for further work. 2000 Mathematics Subject Classification. Primary: 58F15, 58F17; Secondary: 53C35. Key words and phrases. Synchronization, Master stability function, Lyapunov exponents. 1