In a most surprising manner, the development of modern nonlinear dynamics (bifurcation and chaos theory) has revealed a number of misinterpretations in our understanding of the laws of physics. In particular, the conception of predictability for deterministic systems is incorrect and based on a few simple examples where the equations of motion can be solved analytically. Conservative systems in generel display regions of parameter space where the slightest change of the initial conditions after a while will produce major deviations in the trajectory. Problems of this type are of interest in discussions, for instance, of the development of our solar system. More important, however, is the fact that the theory of conservative systems forms the basis for both quantum mechanics and classical statistical mechanics.
Dissipative systems (i.e., systems with friction and other forms of damping) can also show complex dynamical phenomena, provided that their motion is maintained by a sufficient supply of energy and/or nutrients. This is obviously also a criterion for life, and living systems depend on a wide variety of nonlinear dynamic phenomena both for the regulation of their normal biological processes and for the development of physiological structures. Oscillations, synchronization, wave propagation, pattern formation, etc., are manifest at all levels of the animate world from regulation of intracellular metabolic processes over cellular communication and the control of various functional units to macrophysiological processes such as respiration and the beating of the heart.
Our aim is to contribute to a deeper understanding of nonlinear dynamic processes such as synchronization and clustering of chaotic oscillators, border-collision bifurcations in piecewise-smooth dynamical systems, torus destruction, and pattern formation in chemical reaction-diffusion systems. At the same time we try to illustrate how the concepts and methods of nonlinear dynamics can be applied to solve practical problems in engineering and other fields of science. We have demonstrated, for instance, how transonic flutter in modern aircraft wings arises through a subcritical Hopf bifurcation and we have analyzed the nonlinear dynamic phenomena associated with application of thrust vectoring in the maneuvering of an airplane.
In recent years, much effort has gone into the analysis of the bifurcation structure in DC/DC electrical power converters with relay control or pulse-width modulation. By virtue of their lower weight and costs such converters are gradually displacing the traditional transformer systems, particularly in spaceships, aircrafts, trains, etc.
The figure illustrates the intermingled basins of attraction in a coupled map system that displays two alternative states of complete chaotic synchronization.