Enlarged Kuramoto model: Secondary instability and transition to collective chaos

Cluster states and π-transition in the Kuramoto model with higher order interactions

“…Recently, the analysis of an enlarged Kuramoto model encompassing a first harmonic pair-wise interaction plus symmetric and anti-symmetric three body interactions has revelead the emergence of collective chaos [32]. In particular, in [32], the authors have obtained mean-field results for unimodal Gaussian frequency distributions by employing a truncated Hermite-Fourier decomposition of the oscillator density.…”

“…Recently, the analysis of an enlarged Kuramoto model encompassing a first harmonic pair-wise interaction plus symmetric and anti-symmetric three body interactions has revelead the emergence of collective chaos [32]. In particular, in [32], the authors have obtained mean-field results for unimodal Gaussian frequency distributions by employing a truncated Hermite-Fourier decomposition of the oscillator density. It would be worth performing an analysis to interpret the observed transition to collective chaos in terms of our mean-field potential landscape V (ϕ) and to compare the results of the self-consistent approach with those reported in [32].…”

Heteroclinic cycles and chaos in a system of four identical phase oscillators with global biharmonic coupling

“…The study of the behavior of systems of coupled phase oscillators occupies a special place in this research field. A phase model was first introduced by Winfree [Win67,Win80] as a phenomenological explanation [BYMS21,LP22] for an emergence of synchronisation in populations of weakly coupled limit cycle oscillators. Subsequently, the model proposed by Kuramoto opened the way for a large number of both analytical and numerical studies of various phenomena of collective dynamics [ABV + 05, PR15].…”

Solvable Dynamics of Coupled High-Dimensional Generalized Limit-Cycle Oscillators

“…For D = 2, (1) reduces to the model addressed in [42][43][44][45][46], where the single uncoupled unit (coined as the Stuart-Landau oscillator) represents a canonical form near a supercritical Hopf bifurcation [1]. For K → 0, the first-order phase reduction to (1) with D = 2 results in the classic Kuramoto model [8]; the second-order phase-reduction approach leads to the enlarged Kuramoto model [47,48].…”

“…To conclude, we have introduced and studied a new model of globally coupled D-dimensional generalized limit-cycle oscillators with amplitude dynamics. Under the weak coupling limit K → 0, our model reduces to the D-dimensional Kuramoto phase model, which is akin to a similar classic construction of the seminal Kuramoto phase model from weakly coupled two-dimensional limit-cycle oscillators [47,48]. In this sense, our work puts the recent studies regarding the D-dimensional Kuramoto model [17][18][19][20][21][22][23][24][25] on a stronger footing by providing a much more general framework to consider the previous results, owing to no longer being constrained by fixed amplitude dynamics.…”