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Despite their simplicity, networks of coupled phase oscillators can give riseto intriguing collective dynamical phenomena. However, the symmetries ofglobally and identically coupled identical units do not allow solutions wheredistinct oscillators are frequency-unlocked -- a necessary condition for theemergence of chimeras. Thus, forced symmetry breaking is necessary to observechimera-type solutions. Here, we consider the bifurcations that arise when fullpermutational symmetry is broken for the network to consist of coupledpopulations. We consider the smallest possible network composed of four phaseoscillators and elucidate the phase space structure, (partial) integrabilityfor some parameter values, and how the bifurcations away from full symmetrylead to frequency-unlocked weak chimera solutions. Since such solutions windaround a torus they must arise in a global bifurcation scenario. Moreover,periodic weak chimeras undergo a period doubling cascade leading to chaos. Theresulting chaotic dynamics with distinct frequencies do not rely on amplitudevariation and arise in the smallest networks that support chaos.

Symmetry breaking yields chimeras in two small populations of Kuramoto-type oscillators