Mathematical Problems in Engineering
Volume 2006 (2006), Article ID 94929, 21 pages

Modeling, chaotic behavior, and control of dissipation properties of hysteretic systems

J. Awrejcewicz1 and L. Dzyubak2

1Department of Automatics and Biomechanics, Technical University of Lodz, 1/15 Stefanowskiego Street, Łódź 90924, Poland
2Department of Applied Mathematics, National Technical University “Kharkov Polytechnic Institute,”, 21 Frunze Street, Kharkov 61002, Ukraine

Received 28 August 2004; Revised 17 June 2005; Accepted 27 June 2005

Copyright © 2006 J. Awrejcewicz and L. Dzyubak. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


In the present work, hysteresis is simulated by means of internal variables. Analytical models of different types of hysteresis loops enable reproduction of major and minor loops and provide good agreement with experimental data. Using an effective algorithm based on the analysis of wandering trajectories, an evolution of chaotic behavior regions of oscillators with hysteresis is presented in various parametric planes. A substantial influence of a hysteretic dissipation value on the form and location of these regions and also restraining and generating effects of the hysteretic dissipation on chaos occurrence are shown. It was demonstrated that for fixed parameters, which govern the shape of a hysteresis loop, the extent of pinch of this loop could be controlled by means of the amplitude and frequency of external periodic excitation.