Spectral properties of the Preisach hysteresis model with random input. I. General results

We derive exact results for the spectral density $S\left(\omega \right)$ of the output of the Preisach model, a standard model for complex, nonlocal hysteresis. We obtain general results for uncorrelated input signals with arbitrary input and Preisach densities. It is shown analytically that uncorrelated input signals are transformed into output exhibiting long-time correlations. For the simplest example of uniform input and Preisach distributions we prove that correlations decay asymptotically with a $t^{-3}$ power law corresponding to a logarithmic low frequency divergence of the second derivative of the spectrum $S\left(\omega \right)$. A simpler expression for symmetric Preisach models is also obtained, which is discussed in detail in Part II, showing that long-time tails or even $1/f$ noise are general features of this class of models.

Figure 1

A rectangular elementary loop characterized by thresholds $\alpha$ and $\beta$ is shown. The output of the Preisach model is given by a weighted superposition of outputs of such elements.

Figure 2

The spectrum $S\left(\omega \right)$ as given by Eq. (7) with Eq. (25) is shown (full line). Its first derivative $S^{\left(1\right)}\left(\omega \right)$ (dot-dashed) crosses the origin with infinite slope as given by the logarithmic divergence of the second derivative $S^{\left(2\right)}\left(\omega \right)$ (dashed).

Figure 3

The output correlation function $C\left(t\right)$ obtained from a numerical Fourier inversion of $S\left(\omega \right)$ is shown in a double logarithmic plot (full line). For comparison the asymptotic long-time tail according to Eq. (31) is plotted (dashed line).