Quasi phase reduction of all-to-all strongly coupled coupled λ − ω oscillators near incoherent states

Abstract

The dynamics of an ensemble of 𝑁 weakly coupled limit-cycle oscillators can be captured by their 𝑁 phases using standard phase reduction techniques. However, it is a phenomenological fact that all-to-all strongly coupled limit-cycle oscillators may behave as “quasiphase oscillators,” evidencing the need of novel reduction strategies. We introduce, here, quasi phase reduction (QPR), a scheme suited for identical oscillators with polar symmetry (𝜆−𝜔 systems). By applying QPR, we achieve a reduction to 𝑁+2 degrees of freedom: 𝑁 phase oscillators interacting through one independent complex variable. This “quasi phase model” is asymptotically valid in the neighborhood of incoherent states, irrespective of the coupling strength. The effectiveness of QPR is illustrated in a particular case, an ensemble of Stuart-Landau oscillators, obtaining exact stability boundaries of uniform and nonuniform incoherent states for a variety of couplings. An extension of QPR beyond the neighborhood of incoherence is also explored. Finally, a general QPR model with 𝑁+2⁢𝑀 degrees of freedom is obtained for coupling through the first 𝑀 harmonics.