Rocky Mountain Journal of Mathematics

Periodic Oscillations in Systems with Hysteresis

Jack W. Macki, Paolo Nistri, and Peitro Zecca

Full-text: Open access

Article information

Rocky Mountain J. Math., Volume 22, Number 2 (1992), 669-681.

First available in Project Euclid: 5 June 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 34A60: Differential inclusions [See also 49J21, 49K21]
Secondary: 34K10: Boundary value problems

Hysteresis harmonic balance differential inclusions multifunctions


Macki, Jack W.; Nistri, Paolo; Zecca, Peitro. Periodic Oscillations in Systems with Hysteresis. Rocky Mountain J. Math. 22 (1992), no. 2, 669--681. doi:10.1216/rmjm/1181072758.

Export citation


  • D.P. Atherton, Nonlinear control engineering, Van Nostrand, New York, 1975.
  • A.R. Bergen and R.L. Franks, Justification of the describing function method, SIAM J. Control, 9 (1971), 568-589.
  • É.M. Braverman, S.M. Meerkov and E.S. Pyatnitskii, A small parameter in the problem of justifying the harmonic balance method (in the case of the filter hypothesis). I, II. Avtomatika i Telemekhanika, No. 1 & No. 2, Jan. 1975 and Feb. 1975, 5-21 and 5-12, translated as Automation and Remote Control 36 (1975).
  • --------, Conditions for applicability of the method of harmonic balance for systems with hysteresis nonlinearity (in the case of filter hypothesis), Avtomatika i Telemekhanika, No. 11 (1976), 16-27, translated as Automation and Remote Control 37 (1976), 1640-1650.
  • A. Damlamian and A. Visintin, Une généralisation vectorielle du modèle de Preisach pour l'hysterésis, Comptes Rendus Acad. Sci. Paris 297 (1983), 437-440.
  • M. Fabrizio and C. Giorgi, L'isteresi per un Sistema elettromagnetico, Boll. U.M.I., 1981, Supplemento 1, 33-45.
  • A. Gelb and W.E. Van der Velde, Multiple input describing functions and nonlinear system design, McGraw-Hill, New York, 1968.
  • U. Hornung, The mathematics of hysteresis, Bull. Australian Math. Soc. 30 (1984), 271-287.
  • A.M. Krasnosel'skii, Forced oscillations in systems with hysteresis nonlinearities, Dokl. Akad. Nauk, SSSR 292 (1987), 1078-1082.
  • A.M. Krasnosel'skii and M.A. Krasnosel'skii, Cycles of large amplitude in autonomous systems with hysteresis, Dokl. Akad. Nauk. SSSR 283 (1985), 23-26.
  • A.M. Krasnosel'skii, Equations with hysteresis nonlinearities, Abhandl. Akad. Wiss. DDR, Abteil. Math.-Naturw. Tech., Band I, Teil 2, 437-458 (1977).
  • A.M. Krasnosel'skii and A.V. Pokrovskii, System with hysteresis (in Russian), Nauka, Moscow, 1983; English transl., Springer, New York, 1989.
  • P. Krejči, Periodic solutions of partial differential equations with hysteresis, Equadiff 6, Springer, New York, 1986.
  • T.W. Ma, Topological degree for set-valued compact vector fields in locally convex spaces, Dissertationes Math. 92 (1972), 1-43.
  • J.W. Macki, P. Nistri and P. Zecca, A theoretical justification of the method of harmonic balance for systems with discontinuities, Rocky Mountain J. Math. 20 (1990), 1079-1098.
  • A.I. Mees, Describing functions: ten years on, IMA J. Appl. Math. 32 (1984), 221-233.
  • A.I. Mees and A.R. Bergen, Describing functions revisited, IEEE Trans. Autom. Control 20 (1975), 473-478.
  • R.K. Miller and A.N. Michel, Sinusoidal input-periodic response in nonlinear differential equations containing discontinuous elements, Proc. Conf. in Integral and Functional Differential Equations, Lecture Notes Pure Appl. Math. 67, Springer, New York, 1980.
  • R.K. Miller, A.N. Michel and G.S. Krenz, On limit cycles of feedback systems which contain a hysteresis nonlinearity, SIAM J. Control Optim. 24 (1986), 276-305.
  • A.V. Pokrovskii, On the theory of hysteresis nonlinearities, Soviet Math. Doklady 14 (1973), 896-900.
  • D.D. Šiljak, Nonlinear systems, Wiley, New York, 1963.
  • S.J. Skar and R.K. Miller, On the existence of self-excited oscillations in systems with discontinuous elements, Circuits, Systems, Signal Processes 2 (1983), 57-76.
  • YA.Z. Tsypkin, Automatic Relay Systems (in Russian), Nauka, Moscow, 1974; English translation publ. by Cambridge Univ. Press, 1984.
  • A. Visintin, On the Preisach model for hysteresis, Nonlinear Anal. T.M.A. 8 (1984), 977-996.
  • --------, Partial differential equations with hysteresis, in Nonlinear parabolic equations: Qualitative properties of solutions, Pitman Research Notes #149, Longman, Harlow, 1987.
  • --------, Evolution problems with hysteresis in the source term, SIAM J. Math. Anal. 17 (1986), 1113-1138.
  • --------, A collection of references on hysteresis, Istituto di Analisi Numerica del C.N.R. 630, Pavia, 1988.