Systems of N identical globally coupled phase oscillators can demonstrate a multitude of complex behaviours. Such systems can have chaotic dynamics for N > 4 when a coupling function is biharmonic. The case N = 4 does not possess chaotic attractors when the coupling is biharmonic, but has them when the coupling includes nonpairwise interactions of phases. Previous studies showed that some of chaotic attractors in this system are organized by heteroclinic networks. In present paper we discuss which heteroclinic cycles are forbidden and which are supported by this particular system. We also discuss some of the cases regarding homoclinic trajectories to saddle-foci equilibria.1 This contrasts the case of non-identical oscillators where chaotic attractors can emerge even in the simplest Kuramoto model due to strong detuning of oscillators' natural frequencies [PMT05].

We argue that a discrete Shilnikov attractor exists in the system of five identical globally coupled phase oscillators with biharmonic coupling. We explain the scenario that leads to birth of this kind of attractor and numerically illustrate the sequence of bifurcations that supports our statement.1 Note that there are also other ways to get chaotic attractors in systems of identical globally coupled phase oscillators. This could be done by using coupling function with higher order harmonics [BTP + 11] or by taking into account nonpairwise interactions between phases [BAR16].2 This is in contrast with a case of non-identical oscillators where chaotic attractors can emerge even in the simplest Kuramoto model due to strong detuning of oscillators' natural frequencies [PMT05].3 An attractor is called homoclinic if it contains a saddle fixed point[GG16].

We study chaotic dynamics in a system of four differential equations describing the interaction of five identical phase oscillators coupled via biharmonic function. We show that this system exhibits strange spiral attractors (Shilnikov attractors) with two zero (indistinguishable from zero in numerics) Lyapunov exponents in a wide region of the parameter space. We explain this phenomenon by means of bifurcation analysis of a three-dimensional Poincaré map for the system under consideration. We show that chaotic dynamics develop here near a codimension three bifurcation, when a periodic orbit (fixed point of the Poincaré map) has the triplet of multipliers [Formula: see text]. As it is known, the flow normal form for such bifurcation is the well-known three-dimensional Arneodó–Coullet–Spiegel–Tresser (ACST) system, which exhibits spiral attractors. According to this, we conclude that the additional zero Lyapunov exponent for orbits in the observed attractors appears due to the fact that the corresponding three-dimensional Poincaré map is very close to the time-shift map of the ACST-system.

Phase response curve is an important tool in studies of stable self-sustained oscillations; it describes a phase shift under action of an external perturbation. We consider multistable oscillators with several stable limit cycles. Under a perturbation, transitions from one oscillating mode to another one may occur. We define phase transfer curves to describe the phase shifts at such transitions. This allows for a construction of one-dimensional maps that characterize periodically kicked multistable oscillators.We show, that these maps are good approximations of the full dynamics for large periods of forcing.

We study a system of four identical globally coupled phase oscillators with biharmonic coupling function. Its dimension and the type of coupling make it the minimal system of Kuramoto-type (both in the sense of the phase space's dimension and the number of harmonics) that supports chaotic dynamics. However, to the best of our knowledge, there is still no numerical evidence for the existence of chaos in this system. The dynamics of such systems is tightly connected with the action of the symmetry group on its phase space. The presence of symmetries might lead to an emergence of chaos due to scenarios involving specific heteroclinic cycles. We suggest an approach for searching such heteroclinic cycles and showcase first examples of chaos in this system found by using this approach.