Rhythmic states and first-order phase transitions in adaptive coupled three-dimensional limit-cycle oscillators.

This paper reports a phase transition in coupled three-dimensional limit-cycle oscillators with an adaptive coupling. We reveal that the multiple-cluster rhythmic states emerge when natural frequencies of oscillators follow uniform distribution (case I), but disappear for Gaussian distribution (case II). Furthermore, as the coupling strength K increases, two first-order phase transitions occur sequentially. When K→0^{+}, an inevitable and nonhysteretic discontinuous phase transition occurs. For K>0, another discontinuous phase transition with a hysteresis loop emerges, and its occurrence depends on the width of the natural frequency distribution. From the microscopic perspective of the system, there are double switches between suppression and revival of oscillations as K varies in case I, but only one switch occurs in case II. Theoretical analyses of the incoherent states and the fixed points are given, which can be accurately verified by numerical simulations. This paper provides insights into our understanding of high-dimensional oscillators and their variants.