||In natural science and technology, there is a universality class of dynamical systems that show complex chaotic behaviors. These chaotic behaviors are strongly related to the nonlinear characteristics of each dynamical system. As a function of control parameter, periodic oscillation bifurcates to nonperiodic and irregular motion, namely "deterministic chaos". Nonper-iodicity or chaos stems from deterministic property hidden behind nonlinear characteristics of dynamical systems. It is surprising that deterministic chaos can be seen in numerous dynamical systems: biological systems, chemical reaction systems (Belousov-Zhabotinski reaction), lasers, semiconductor systems, fluid flow systems, and so on. Many different nonlinear systems show universal rules of bifurcation routes to chaos.
Very often, spatially-extended dynamical systems give rise to a pattern formation as a result of nonequilibrium phase transition, where the spatial pattern will be static or evolve to an oscillatory mode (periodic or chaotic). While a deep understanding about the pattern formation process in a specified physical system or chemical system is necessary for a full comprehension of spatiotemporal behaviors, it is quite interesting to note that the temporal evolutions are governed by universality rules of the chaos theory which arise essentially from the nonlinear character of the dynamical systems.
In semiconductors physics, studies on nonlinear carrier transport have been ongoing since the 1960s. At the early stage in semiconductor physics, many authors studied the nonlinear carrier transport associated with the impact ionization of neutral impurities in Ge and GaAs, the Gunn oscillations in GaAs and InP, and the acousto-electric effects in CdS. Many devices have been proposed in order to develop new, modern electronic devices of oscillators and switching devices such as the "oscillistor", the "cryosar", and the Gunn diode. However, many of these proposed devices, except for the Gunn diode, were found not to be on the application base due to unstable oscillations and unreliable switching. The failures could have been partly attributed to the unavoidable occurrence of chaos. Two decades later, chaos in semiconductor systems was discovered by the author in 1982. After this discovery, many researchers have studied the complex chaotic behaviors in GaAs, InSb, InP, Si, and Ge. The