Synchronization in complex networks of phase oscillators: A survey <s> Synchronization in Complete Networks <s> Abstract The impressive variety of biological rhythms leaves no doubt that autonomously periodic processes contribute to the coordination of life-processes. The question here raised is, “What modes of temporal organization—if any—could result from weak interactions in a population of innately oscillatory devices (e.g. electronic oscillators, secretory cells, spontaneously active neurons, or individual animals)?” By mathematical analysis, electronic experiments, and digital computer simulation, answers are found as functions of the nature of the periodic processes and the kind of interactions involved. For populations of “generalized relaxation oscillators”, threshold conditions are discovered for mutual synchronization in any of a variety of modes. It is proposed that self-entraining communities of this sort may exist within individual metazoan animals and plants as the basis of the observed diurnal coordination of their physiological process. <s> BIB001 </s> Synchronization in complex networks of phase oscillators: A survey <s> Synchronization in Complete Networks <s> Improved mobility control in secondary and tertiary recovery processes is obtained by injecting at least two slugs into the reservoir, one slug designed to effect a permeability reduction and the other slug containing an agent which more effectively increases the fluid viscosity. For example, an aqueous slug containing a partially hydrolyzed, high molecular weight polyacrylamide can be injected (this slug efficiently reduces the relative permeability to the flow of the aqueous slug through the reservoir), followed by an aqueous slug containing a biopolymer, e.g., a linear polysaccharide having a molecular weight of about one million (this agent more effectively increases the viscosity of the water and has a lesser effect on permeability reduction). This combination of slugs gives more efficient and economical mobility control in flooding subterranean oil-bearing formations to recover oil therefrom. <s> BIB002
In this section, we study heterogeneous oscillators coupled in a complete graph with uniform weights a ij = K/n. In this case, the coupled oscillator model (1) reduces to the celebrated Kuramoto model given in (3). The Kuramoto model will reach synchronization provided that the coupling gain K is larger than a critical value, which depends on the dissimilarity among the natural frequencies ω. Starting from Winfree's and Kuramoto's pioneering work BIB001 BIB002  , this trade-off has been characterized by parametric inequalities. In what follows, we review various estimates of the critical coupling strength to characterize the on-set of synchronization as well as the ultimate stage of synchronization. We consider both infinite-dimensional as well as finite-dimensional Kuramoto models.