Frequency entrainment in long chains of oscillators with random natural frequencies in the weak coupling limit.

We study oscillator chains of the form phi; (k) = omega(k) +K[Gamma( phi(k-1) - phi(k) )+Gamma( phi(k+1) - phi(k) )], where phi(k) epsilon[0,2pi) is the phase of oscillator k. In the thermodynamic limit where the number of oscillators goes to infinity, for suitable choices of Gamma(x), we prove that there is a critical coupling strength K(c), above which a stable frequency-entrained state exists, but below which the probability is zero to have such a state. It is assumed that the natural frequencies are random with finite bandwidth. A crucial condition on Gamma(x) is that it is nonodd, i.e., mid R: Gamma(x)+Gamma(-x) mid R: not equal 0. The interest in the results comes from the fact that any chain of limit-cycle oscillators can be described by equations of the above form in the limits of weak coupling and narrow distribution of natural frequencies.