Dept. of Appl.Math @ ucc.ie
Systems with Hysteresis
Welcome

Contents

What is `Hysteresis'

Preisach Model

Closed loop systems

Complex behavior
   Equations
   ``Poincare mapping''
   Visualization
   Chaotic behavior
   Try me!
   One relay equation
   Weiss model equation
   All in one

Other Hysterons

Advanced modelling

Bibliography

``Poincare mapping''

In general, the phase space of the differential equation with Preisach nonlinearity is infinite dimensional since it involves the ``memory'' of the nonlinearity, and it is not clear how to reduce the dynamics of the system to a discrete mapping. But if the assumption that for all $t_0$,
\begin{displaymath} \max_{t\ge t_0} x(t)>\gamma \end{displaymath} (5)

holds, we can correctly define the two-dimensional cross-section. More formally, if $t$ satisfies
\begin{displaymath} x(t)=\gamma , x'(t)> 0 \end{displaymath} (6)

then the state of the system (3) can be described by the pair $\left(t {\rm mod} \sqrt{2}\pi,x'(t)\right)$. We introduce a map $F_{b,\gamma }$ that maps the section $S$, described by (6), into itself as follows. Begin with a point of $S$ and follow a trajectory of (3) until it has its next intersection with $S$. Of course, this mapping may be only partially defined.