Abstract
Theorems on the existence of continua of periodic solutions for systems with hysteretic nonlinearities of the type of Preisah models are suggested. It is found that in the general situation, the standard criteria for existence of at least one periodic solution, which rest on the use of prior upper-bound estimates of the norm of periodic solutions and the Leray–Schauder method, ensure the existence of a one-parameter continuum of periodic solutions to these systems. The proofs are based on the use of new classes of operators that describe periodic responses of hysteretic nonlinearities to periodic inputs.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
Krasnosel'skii, M.A. and Pokrovskii, A.V., Sistemy s gisterezisom (Systems with Hysteresis), Moscow: Nauka, 1983.
Brokata, M. and Sprekels, J., Hysteresis and Phase Transitions, New York: Springer, 1996.
Visintin, A., Differential Models of Hysteresis, Berlin: Springer, 1994.
Krejčí, P., Hysteresis, Convexity and Dissipation in Hyperbolic Equations, Tokyo: Gakkotosho, 1996.
Krasnosel'skii, A.M. and Rachinskii, D.I., On Continua of Cycles in Systems with Hysteresis, Dokl. Ross. Akad. Nauk, 2001, vol. 378, no.3, pp. 314-319.
Vladimirov, A.A., Krasnosel'skii, M.A., and Chernorutskii, V.V., The Cauchy Problem for Systems with Hysteresis, Dokl. Ross. Akad. Nauk, 1993, vol. 333, no.3, pp. 285-287.
Krasnosel'skii, M.A. and Zabreiko, P.P., Geometcheskie metody nelinenogo analiza (Geometric Methods of Nonlinear Analysis), Moscow: Nauka, 1975.
Bobylev, N.A. and Korovin, S.K., Teoremy rodstvennosti v teorii nelineinykh kolebanii. Metody analiza nelineinykh sistem (Contiguity Theorems in the Theory of Nonlinear Oscillations. Methods of Analysis of Nonlinear Systems), Moscow: Dialog-MGU, 1997.
Krasnosel'skii, M.A., Operator sdviga po traektoriyam differentsial'nykh uravnenii (The Operator of Shift Along Trajectories of Differential Equations), Moscow: Nauka, 1966.
Krasnosel'skii, A.M., Krasosel'skii, M.A., and Pokrovskii, A.V., O prilozheniykh metoda napravlyayushchikh potentsialov k sistemam s gisterezisom. Dinamika neodnorodnykh sistem (On Applications of the Method of Guiding Potentials to Systems with Hysteresis. Dynamics of Inhomogeneous Systems), Moscow: VNIISI, 1984.
Pervozvanskii, A.A., Kurs teorii avtomaticheskogo upravleniya (The Course in Automatic Control Theory), Moscow: Nauka, 1986.
Leonov, G.A., Burkin, I.M., and Shepelyavyi, A.I., Chastotnye metody v teorii kolebanii, I, II (Frequency Methods in the Theory of Oscillations, I, II), St. Petersburg: S.-Peterburg. Gos. Univ., 1992.
Krasnosl'skii, M.A. and Pokrovskii, A.V., Operators of the Problem on Forced Oscillations in Systems with Hysteresis, Dokl. Ross. Akad. Nauk, 1992, vol. 334, no.3, pp. 540-543.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Rachinskii, D.I. On Natural Continua of Periodic Solutions of the Systems with Hysteresis. Automation and Remote Control 64, 420–438 (2003). https://doi.org/10.1023/A:1023213624958
Issue Date:
DOI: https://doi.org/10.1023/A:1023213624958