On hysteresis and retardation

Hysteresis and retardation are two kinds of popular phenomena in natural sciences, engineering sciences and social sciences. However, they are often confused in both academic and technical circles. This paper, starting from the definition and the nature of two kinds of phenomena, presents their general and individual characteristics, as well as their relations. The illustrative examples in the paper show that the hysteresis implies the phase delay of two processes varying periodically with a physical parameter, while the retardation reflects the time delay of two dynamic processes varying arbitrarily in the time domain. In the case of both linear hysteresis and harmonically time-varying input, they look identical. The nonlinear hysteresis, however, will reduce their relevance even the harmonically time-varying input remains unchanged. In general, they are two kinds of quite different phenomena by nature. In the aspect of memory, for example, the hysteresis and retardation characterize local memory and global memory, respectively. As for their transfer property, a hysteretic system corresponds to the rational fractional and a delayed system corresponds to that with one or more exponential functions. Even though there is a closed hysteretic loop for the linear hysteretic system, the output of a nonlinear system under harmonic input may not behave periodically. The nature of hysteresis comes from the multiple branches of a hysteretic loop, instead of the closed loop. The nature of a delayed system defines a continuous mapping between two continuous functions in their corresponding closed intervals. Such a delayed system, hence, is infinitely dimensional, no matter how short the time delay is and how many degrees of freedom the system has. As a matter of fact, a linear dynamic system involving any time delays has to be modeled as a delay differential equation, which has infinite dimensions and infinite number of eigenvalues. Furthermore, nonlinear dynamic systems with time delays exhibit even more complicated dynamics. Time delays are usually very short in mechanical systems. However, the neglect of time delays in the dynamic analysis of a delayed system may result in essential mistakes.