Spectral properties of the Preisach hysteresis model with random input. II. Universality classes for symmetric elementary loops

The Preisach model with symmetric elementary hysteresis loops and uncorrelated input is treated analytically in detail. It is shown that the appearance of long-time tails in the output correlations is a quite general feature of this model. The exponent $\eta$ of the algebraic decay $t^{-\eta }$, which may take any positive value, is determined by the tails of the input and the Preisach density. We identify the system classes leading to identical algebraic tails. These results imply the occurrence of $1/f$ noise for a large class of hysteretic systems.

Figure 1

The exact form of the output spectral density according to Eq. (11) (for $\nu =1$) and Eq. (17) (for $\nu \ne 1$) is shown for several input distributions with charcteristic exponents $\nu$ (left: linear; right: double-logarithmic scale). For $\nu =3$ (dotted) the spectrum $S\left(\omega \right)$ diverges like ${\left|\omega \right|}^{-1/3}$ and for $\nu =2$ (dot-dashed) logarithmically. For $\nu =1$ (dot-dot-dashed) the first derivative $S^{\left(1\right)}\left(\omega \right)$ diverges logarithmically, for $\nu =2/3$ (dashed) $S^{\left(2\right)}\left(\omega \right)$, and for $\nu =2/5$ (full) $S^{\left(3\right)}\left(\omega \right)$ diverges like ${\left|\omega \right|}^{-1/2}$.

Figure 2

The output correlation functions as obtained by numerical Fourier transformation of the spectra in Fig. 1 (same plot style) are shown on a double-logarithmic scale. For comparison their asymptotic behavior according to Eqs. (14, 23) is also shown (full lines). From top to bottom they decay as $t^{-2/3}$, $t^{-1}$, $t^{-2}\text{ln}t$, $t^{-5/2}$, and $t^{-7/2}$.