Self-sustained musical instruments are nonlinear dynam- ical systems that can produce a wealth of sound regimes. These can be periodic but also quasiperiodic (oscillating with a strong beating component), and their character- istics depend sensitively on both the design and control parameters. We investigate here the emergence of non periodic sound regimes in two types of flute-like instru- ments: a recorder and a pre-hispanic Chilean flute. The specific geometry of the Chilean flutes, whose resonator are made of two cylinders of similar length but different cross-sections, has been shown to favor a strong inhar- monicity between the acoustic resonance frequencies. It has been demonstrated recently that both instruments can produce quasiperiodic sounds, which are either avoided or played on purpose. Here, we adopt a dynamical sys- tem point of view to investigate the emergence of these quasiperiodic sound regimes. A bifurcation analysis of a physical model of the instruments unveils the key role played by both a design parameter - the resonator inhar- monicity - and a control parameter - the pressure in the musician’s mouth - on the emergence of stable quasiperi- odic sound regimes.