Supra-resonant wing oscillations in insect flight

We study a minimal dynamical model of the mechanism generating wing beats in insects based on a coupled wing–thorax–muscle oscillator with delayed muscle activation. The system exhibits self-sustained oscillations arising from an instability due to the stretch-activated muscle forcing. We identify two distinct dynamical regimes depending on the relative magnitude of the muscle response rate and the thoracic resonance frequency, corresponding to slow- and fast-muscle dynamics. In the slow-muscle regime, the instability is unconditional with respect to the muscle forcing and oscillations emerge at the natural mechanical frequency. In the fast-muscle regime, oscillations arise only above a finite threshold and are inherently supra-resonant already at their onset. Using the method of harmonic balance, we derive an analytical relation between oscillation amplitude and frequency that works beyond the weakly nonlinear regime. The analytical predictions are validated against numerical simulations, showing good agreement in the biologically relevant parameter range.