It is shown, under weak conditions, that the dynamical evolution of large systems of globally coupled phase oscillators with Lorentzian distributed oscillation frequencies is, in an appropriate physical sense, time-asymptotically attracted toward a reduced manifold of the system states. This manifold was previously known and used to facilitate the discovery of attractors and bifurcations of such systems. The result of this paper establishes that attractors for the order parameter dynamics obtained by restriction to this reduced manifold are, in fact, the only such attractors of the full system. Thus all long time dynamical behaviors of the order parameters of these systems can be obtained by restriction to the reduced manifold.
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Equation (24), together with the final condition , can be viewed as generating a conformal mapping from the complex -plane to the complex -plane. Since Eq. (24) with given by Eq. (20) is a Riccati equation, this mapping is a Möbius transformation, , where the coefficients and depend on . [See, for example, Ref. 21, Eq. (4.1.6).] Thus the unit disk in (i.e., ) is mapped into a disk in , and since, by Eq. (25), at for , this disk is contained within the region . Our result, as , implies that the radius of this -disk shrinks to zero as .
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.© 2009 American Institute of Physics.
2009
American Institute of Physics
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