| This thesis concerns a three-parameter family of discrete time models of N coupled oscillators, N being an arbitrary integer. The couplings are all-to-all and piecewise linear. The three parameters are omega, representing the rotation in the uncoupled oscillators, alpha, representing the coupling strength, and epsilon, on which the geometry of the coupling function depends. After reviewing some previous work including the well known Kuramoto model, definitions of phase-locking and synchronization will be given. These are the objects of interest in this thesis. The phase space of the model is the N-torus. It will be shown that synchronized and phase-locked states correspond to invariant circles on the torus, and can be viewed as fixed points in a quotient dynamical system. The parameter epsilon in [0, 1/2], is related to the portion of phase space on which the map expands distances. At epsilon = 0, the dynamical system reduces to a piecewise linear contraction with discontinuity. At epsilon= 1/2, the system bears nontrivial similarity to the Kuramoto model. The case epsilon = 0 is explored first, and it is used as a basis for exploring systems corresponding to epsilon > 0. It will be shown analytically that for epsilon = 0 there are finitely many phase-locked solutions and that they obey certain structural conditions. The idea of clustering will be developed, and theorems will be proved concerning the possible number of clusters a phase-locked solution can have. It will also be argued that there is a signicant difference between odd and even N. Numerical results on the basins of attraction will also be given. These simulations suggest that when the basins are grouped by cluster number they obey simple properties, in particular monotonicty and a 1/sqrt(N) scaling law. Rigorous results for positive epsilon, where the map no longer reduces to a piecewise contraction, will be given. It will be shown analytically that the phase-locked solutions that persist are tied to the multiplicative factors of N. The effects that the parameters have on synchronization in this case are also discussed. |
