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Cyclops states are intriguing cluster patterns observed in oscillatornetworks, including neuronal ensembles. The concept of cyclops states formed bytwo distinct, coherent clusters and a solitary oscillator was introduced in[Munyayev {\it et al.}, Phys. Rev. Lett. 130, 107021 (2023)], where we exploredthe surprising prevalence of such states in repulsive Kuramoto networks ofrotators with higher-mode harmonics in the coupling. This paper extends ouranalysis to understand the mechanisms responsible for destroying the cyclops'states and inducing new dynamical patterns called breathing and switchingcyclops' states. We first analytically study the existence and stability ofcyclops states in the Kuramoto-Sakaguchi networks of two-dimensionaloscillators with inertia as a function of the second coupling harmonic. We thendescribe two bifurcation scenarios that give birth to breathing and switchingcyclops states. We demonstrate that these states and their hybrids areprevalent across a wide coupling range and are robust against a relativelylarge intrinsic frequency detuning. Beyond the Kuramoto networks, breathing andswitching cyclops states promise to strongly manifest in other physical andbiological networks, including coupled theta-neurons.

Breathing and switching cyclops states in Kuramoto networks with higher-mode coupling