Stable chimera states: A geometric singular perturbation approach

Over the past decades, chimera states have attracted considerable attention given their unexpected symmetry-breaking spatiotemporal nature and simultaneously exhibiting synchronous and incoherent behaviors under specific conditions. Despite relevant precursory results of such unforeseen states for diverse physical and topological configurations, there remain structures and mechanisms yet to be unveiled. In this work, using mean-field techniques, we analyze a multilayer network composed of two populations of heterogeneous Kuramoto phase oscillators with coevolutive coupling strengths. Moreover, we employ the geometric singular perturbation theory through the inclusion of a time-scale separation between the dynamics of the network elements and the adaptive coupling strength connecting them, gaining a better insight into the behavior of the system from a fast-slow dynamics perspective. Consequently, we derive the necessary and sufficient condition to produce stable chimera states when considering a coevolutionary intercoupling strength. Additionally, under the aforementioned constraint and with a suitable adaptive law election, it is possible to generate intriguing patterns, such as persistent breathing chimera states. Thereafter, we analyze the geometric properties of the mean-field system with a coevolutionary intracoupling strength and demonstrate the production of stable chimera states. Next, we give arguments for the presence of such patterns in the associated network under specific conditions. Finally, relaxation oscillations and canard cycles, seemingly related to breathing chimeras, are numerically produced under identified conditions due to the geometry of our system.