We furnish necessary and sufficient conditions for the occurrence of local Hopf bifurcation in a notably significant fluid–structure problem, where a Navier–Stokes liquid interacts with a rigid body that is subject to an undamped elastic restoring force. The motion of the coupled system is driven by a uniform flow at spatial infinity, with constant dimensionless velocity \(\lambda >0\) . The study is particularly challenging since 0 is in the essential spectrum of the relevant linearized operator, for any value of \(\lambda \) , which makes classical bifurcation theories inapplicable. To successfully address this situation, we build upon the method introduced by Galdi (Arch Ration Mech Anal 222:285–315, 2016) that overcomes the problem of the absence of a spectral gap. The most remarkable feature of our result is that no restriction is imposed on the frequency \(\omega \) of the bifurcating solution, which may thus coincide with one of the natural structural frequencies \(\omega _\textsf{n}\) of the body. Therefore, resonance cannot occur as a result of this bifurcation. However, when \(\omega \rightarrow \omega _\textsf{n}\) , the amplitude of oscillations may become very large when the fluid density is negligible compared to the mass of the body. To our knowledge, our result is the first rigorous investigation of the existence of a Hopf bifurcation in a fluid–structure interaction problem.
