The aim of this work is to find the behaviour of the mean field of a
population of coupled dynamical systems with diversity (microscopic noise or
parameter mismatch). The population is under two opposite effects: the
coupling (an all-to-all, diffusive term), that pushes the system toward a full
synchronized state, and the presence of noise or parameter mismatch, that
pushes the system toward incoherence. Here we focus on the case of strong
coupling. In the (trivial) case of no diversity, all the elements eventually
synchronize, and the mean field inherits exactly the dynamics of each
population element. However, when diversity is introduced, new properties
appear. The mean field heads toward attractors that the uncoupled elements do
not have, and complex perturbation responses arise. All these phenomena can be
explained quantitatively showing that new macroscopic degrees of freedom,
coupled to the mean field, appear when diversity is introduced. Using an
expansion in order parameters, the equations of motion for all the revant
macroscopic variables can be obtained esplicitly. Macroscopic bifurcations and
perturbation responses are then studied through a standard bifurcation
analysis. The method requires that the population is close to the case of
perfect synchronization. The method works for any kind of population (e.g.,
coupled limit cycles, chaotic oscillators, or maps), with parameter mismatch
or noise, and for any population size. Potential and quantitative applications
that will be discussed are multicellular behaviours (especially in connection
with yeast glycolysis) and reaction-advection-diffusion systems.
References:
S. De Monte, F. d\\\'Ovidio, Europhys. Lett. 58, 21-27 (2002)
S. De Monte, F. d\\\'Ovidio, E. Mosekilde, Phys. Rev. Lett. 90, 054102 (2003)