We numerically and analytically analyze transitions between different synchronous states in a network of globally coupled phase oscillators with attractive and repulsive interactions. The elements within the attractive or repulsive group are identical, but natural frequencies of the groups differ. In addition to a synchronous two-cluster state, the system exhibits a solitary state, when a single oscillator leaves the cluster of repulsive elements, as well as partially synchronous quasiperiodic dynamics. We demonstrate how the transitions between these states occur when the repulsion starts to prevail over attraction.PACS numbers: 05.45.Xt Synchronization; coupled oscillators Networks of coupled oscillators are a popular model for many engineered or natural systems. The main effect -emergence of a collective mode via synchronization -is now well-understood and therefore focus of research shifted recently to analysis of different complex states. These states include chimeras, when a population of identical units splits into a synchronous and asynchronous part, quasiperiodic partially synchronous states, characterized by the difference of frequencies of individual units and of the collective mode, and clusters and heteroclinic cycles, to name just a few. Of particular interest are ensembles where some elements have only attractive connections while others have only repulsive ones. This model is motivated by studies of neuronal networks that are built from excitatory and inhibitory neurons. In this paper we analyze how the state of such a setup changes with the interplay of attraction and repulsion. We demonstrate that if the frequency mismatch between attractive and repulsive units is smaller than some critical value then desynchronization occurs via appearance of the solitary state. With the further increase of repulsion the system undergoes a transition to quasiperiodic partial synchrony. In the latter state the attractive units remain synchronized, while the repulsive group settles between synchrony and asynchrony so that the mean fields of both groups remain locked, but the frequency of the repulsive elements is larger than that of their mean field. For a large frequency mismatch of attractive and repulsive groups desynchronization immediately leads to partial synchrony.

We explore the phase reduction in networks of coupled oscillators in the higher orders of the coupling parameter. For coupled Stuart–Landau oscillators, where the phase can be introduced explicitly, we develop an analytic perturbation procedure to explicitly obtain the higher-order approximation. We demonstrate this by deriving the second-order phase equations for a network of three Stuart–Landau oscillators. For systems where explicit expressions of the phase are not available, we present a numerical procedure that constructs the phase dynamics equations for a small network of coupled units. We apply this approach to a network of three van der Pol oscillators and reveal components in the coupling with different scaling in the interaction strength.

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We explore the phase reduction in networks of coupled oscillators in the higher orders of the coupling parameter. For coupled Stuart-Landau oscillators, where the phase can be introduced explicitly, we develop an analytic perturbation procedure to allow for the obtaining of the higher-order approximation explicitly. We demonstrate this by deriving the second-order phase equations for a network of three Stuart-Landau oscillators. For systems where explicit expressions of the phase are not available, we present a numerical procedure that constructs the phase dynamics equations for a small network of coupled units. We apply this approach to a network of three van der Pol oscillators and reveal components in the coupling with different scaling in the interaction strength.

Partial synchronous states appear between full synchrony and asynchrony and exhibit many interesting properties. Most frequently, these states are studied within the framework of phase approximation. The latter is used ubiquitously to analyze coupled oscillatory systems. Typically, the phase dynamics description is obtained in the weak coupling limit, i.e., in the first-order in the coupling strength. The extension beyond the first-order represents an unsolved problem and is an active area of research. In this paper, three partially synchronous states are investigated and presented in order of increasing complexity. First, the usage of the phase response curve for the description of macroscopic oscillators is analyzed. To achieve this, the response of the mean-field oscillations in a model of all-to-all coupled limit-cycle oscillators to pulse stimulation is measured. The next part treats a two-group Kuramoto model, where the interaction of one attractive and one repulsive group results in an interesting solitary state, situated between full synchrony and self-consistent partial synchrony. In the last part, the phase dynamics of a relatively simple system of three Stuart-Landau oscillators are extended beyond the weak coupling limit. The resulting model contains triplet terms in the high-order phase approximation, though the structural connections are only pairwise. Finally, the scaling of the new terms with the coupling is analyzed.