Elsewhere we have reported our simulations of general hovering aerodynamics as practiced, for example, by hummingbirds and dragonflies. These studies explain rather satisfactorily from the point of view of partial differential equations (Navier-Stokes) how such high lift is generated in such motions. Excellent match with laboratory findings is achieved. In this paper, I will first bring up-to-date these studies. New results include a study of the qualitative features of these motions, treated as a nonlinear dynamical system. Lift phase portraits and power spectra reveal a progression up the bifurcation ladder through periodic and increasingly complicated aperiodic motions toward chaos. A connection to the inertial manifold framework is established. A new structural stability theory is postulated for such dynamical systems.
Secondly, I will bring together here for the first time a synthesis of other key dynamical subsystems which enter critically into the success of the total dynamical system description of hummingbird, dragonfly and similar hovering species. These include thermodynamical, structural dynamical, neurodynamical systems. Of necessity my treatment of these will be brief.