On hysteresis and retardation

Hysteresis and retardation are two kinds of popularphenomena 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 ofphenomena, presents their general and individual characteristics, as well astheir relations. The illustrative examples in the paper show that thehysteresis implies the phase delay of two processes varying periodicallywith a physical parameter, while the retardation reflects the time delay oftwo dynamic processes varying arbitrarily in the time domain. In the case ofboth linear hysteresis and harmonically time-varying input, they lookidentical. The nonlinear hysteresis, however, will reduce their relevanceeven the harmonically time-varying input remains unchanged. In general, theyare two kinds of quite different phenomena by nature. In the aspect ofmemory, for example, the hysteresis and retardation characterize localmemory and global memory, respectively. As for their transfer property, ahysteretic system corresponds to the rational fractional and a delayedsystem corresponds to that with one or more exponential functions.Even though there is a closed hysteretic loop for the linear hystereticsystem, the output of a nonlinear system under harmonic input may not behaveperiodically. The nature of hysteresis comes from the multiple branches of ahysteretic loop, instead of the closed loop.The nature of a delayed system defines a continuous mapping between twocontinuous functions in their corresponding closed intervals. Such a delayedsystem, hence, is infinitely dimensional, no matter how short the time delayis and how many degrees of freedom the system has. As a matter of fact, alinear dynamic system involving any time delays has to be modeled as a delaydifferential equation, which has infinite dimensions and infinite number ofeigenvalues. Furthermore, nonlinear dynamic systems with time delays exhibiteven more complicated dynamics. Time delays are usually very short inmechanical systems. However, the neglect of time delays in the dynamicanalysis of a delayed system may result in essential mistakes.