  Saving energy and enhancing performance are secular preoccupations shared by
both nature and human beings. In animal locomotion, flapping flyers or swimmers
rely on the flexibility of their wings or body to passively increase their
efficiency using an appropriate cycle of storing and releasing elastic energy.
Despite the convergence of many observations pointing out this feature, the
underlying mechanisms explaining how the elastic nature of the wings is related
to propulsive efficiency remain unclear. Here we use an experiment with a
self-propelled simplified insect model allowing to show how wing compliance
governs the performance of flapping flyers. Reducing the description of the
flapping wing to a forced oscillator model, we pinpoint different nonlinear
effects that can account for the observed behavior ---in particular a set of
cubic nonlinearities coming from the clamped-free beam equation used to model
the wing and a quadratic damping term representing the fluid drag associated to
the fast flapping motion. In contrast to what has been repeatedly suggested in
the literature, we show that flapping flyers optimize their performance not by
especially looking for resonance to achieve larger flapping amplitudes with
less effort, but by tuning the temporal evolution of the wing shape (i.e. the
phase dynamics in the oscillator model) to optimize the aerodynamics.
