Low dimensional behavior of large systems of globally coupled oscillators

This follows by noting that the Cauchy–Riemann conditions imply that the complex function $α(ωr+iωi,0)$ satisfies Laplace’s equation in $ω$⁠, $∇ω2α(ωr+iωi,0)=0$⁠, where $∇ω2=∂2∕∂ωr2+∂2∕∂ωi2$⁠. Setting $α=∣α∣e−iψ$⁠, this gives $∇ω2∣α∣=∣α∣[(∂ψ∕∂ωr)2+(∂ψ∕∂ωi)2]$⁠. Since $∇ω2∣α∣≥0$ in $ωi<0$⁠, $∣α∣$ cannot assume a maximum anywhere interior to the lower-half complex $ω$-plane. Hence the maximum value of $∣α∣$ in $ωi≤0$ must occur on $ωi=0$⁠, i.e., on the real $ω$-axis.

While the time-asymptotic behavior of the order-parameter obtained from the dynamics of the full system is seen to correspond to the attractors of the reduced systems (10) and (13) [where Eq. (13) is treated in Ref. 10], we emphasize that $M$ does not need to be attracting for the microscopic state $f(ω,θ,t)$⁠. This can be simply seen in the case of the Kuramoto problem (4), for which $f(ω,θ,t)=[g(ω)∕2π]{1+∑mβm(ω)exp[im(θ−ωt)]}$ and $r(t)=0$ for all $t$ is a solution for $β±1(ω)=0$ and any choice of perturbations $βm(ω)$⁠, $∣m∣≥2$⁠.