Fig. 3: Transitions between synchronous and asynchronous modes in simulation and robotics.

a, A unified biophysical model combines hawkmoth body mechanics (equation (3)) with time-periodic, neurogenic (synchronous) and delayed stretch activation (asynchronous) forcing. Stretch activation is implemented as a feedback filter (or convolution) of wing angle (ϕ) converted to muscle strain rate (\(\dot{\varepsilon }\)) and scaled to wingstroke conditions (µF_a) (equations (10) and (11)). The parameter K_r interpolates between the two sources of muscle force (equation (1)). b, K_r and stretch activation time-to-peak normalized to the mechanical natural frequency (t₀/T_n) define a parameter space. High-power flapping occurs at both extremes, but intermediate modes only generate appreciable power along a bridge where the rate of stretch activation approximately matches the synchronous drive (25 Hz). M. sexta is plotted on the basis of estimates of t₀/T_n and \({\widetilde{K}}_{r}\) from quasi-static data. c, Emergent wingbeat frequencies (f) normalized by the synchronous drive frequency (f_s). Dark blue indicates regions where the emergent wingbeat frequency is entrained to the synchronous driving frequency (f = f_s). The red regions indicate where the asynchronous dynamics dominate (f ≠ f_s). The grey line indicates the boundary between synchronous- and asynchronous-dominant dynamics. d, A robophysical system (roboflapper) implementing both types of actuation, plus real-world fluid physics and friction. e,f, Results from the setup in d are qualitatively similar to the simulations in b,c, but with a region of no wingstrokes due to system friction with low K_r and high t₀/T_n. g, A centimetre-scale robotic wing modelled after the Harvard robobee¹², consisting of (1) a wing; (2) a transmission; (3) a carbon fibre frame; (4) a piezoelectric bending actuator; and (5) a wing displacement sensor. h, A single hybrid robobee transitioning from synchronous (K_r = 1, blue) to asynchronous (K_r = 0, red) in real time. Transitions are smooth when synchronous and asynchronous frequencies are approximately equal (blue and red markers, respectively). i, When the frequencies differ, interference causes frequency and amplitude fluctuations in the transition regime.