TABLE OF CONTENTS As illustrated for the R ssler system in Fig. 1.6, nonlinear dynamic phenomena are often initiated by a Hopf bifurcation [6, 7, 8, 9] where the stationary equilibrium state of a dynamical system becomes unstable and yields to a state of self-sustained oscillations of finite amplitude. Since the first application of radiowaves (G.Marconi, 1901), we have made use of active devices to produce oscillations and waves of a broad spectrum of frequencies, ranging from ultrasound oscillations for crack detection in aircraft wings and scanning of pregnant women over radio- and microwaves to laser, X-ray and synchrotron radiation as used in optical communication and modern materials science.
As we strive for cheaper, faster, and more efficient solutions to a variety of technical problems we will be forced to operate closer and closer to the stability threshold. In certain systems, crossing of the threshold may be acceptable. In other systems, the emergence of self-sustained oscillations could represent a severe threat to safety or to the durability of the structure. In all cases, it will be important to understand precisely how the oscillations arise, and what happens as we cross into the region of instability. This is precisely the area of interest for nonlinear dynamics.
Transonic flutter in aircraft wings represents a typical example of a problem in contemporary engineering [37, 38]. Flutter denotes a characteristic form of self-excited oscillations that can arise through the interaction of an aerodynamic flow and the elastic modes of a...
