On the Hopf bifurcation in control systems with a bounded nonlinearity asymptotically homogeneous at infinity. (English) Zbl 0984.34029

Summary: The authors study existence, uniqueness, and stability of large-amplitude periodic cycles arising in Hopf bifurcation at infinity of autonomous control systems with bounded nonlinear feedback. They consider systems with functional nonlinearities of Landesman-Lazer type and a class of systems with hysteresis nonlinearities. The method is based on the technique of parameter functionalization and methods of monotone concave and convex operators.


34C23 Bifurcation theory for ordinary differential equations
34C55 Hysteresis for ordinary differential equations
34K18 Bifurcation theory of functional-differential equations
Full Text: DOI


[1] Bliman, P.-A.; Krasnosel’skii, A.M.; Sorine, M.; Vladimirov, A.A., Nonlinear resonance in systems with hysteresis, Nonlinear anal., 27, 561-577, (1996) · Zbl 0864.34027
[2] Brokate, M.; Sprekels, J., Hysteresis and phase transitions, (1996), Springer-Verlag New York · Zbl 0951.74002
[3] P. Diamond, D. I. Rachinskii, and, M. G. Yumagulov, Stability of large cycles in a nonsmooth problem with Hopf bifurcation at infinity, submitted for publication. · Zbl 0963.34034
[4] Fučik, S.; Kufner, A., Nonlinear differential equations, (1980), Elsevier Oxford · Zbl 0647.35001
[5] Fučik, S.; Nečas, J.; Kučera, M., Ranges of nonlinear asymptotically linear operators, J. differential equations, 17, 375-394, (1975) · Zbl 0299.47035
[6] Hassard, B.; Kazarinoff, N.; Wan, Y.-H., Theory and applications of Hopf bifurcations, (1981), Cambridge Univ. Press London · Zbl 0474.34002
[7] Kozyakin, V.S.; Krasnosel’skii, M.A., The method of parameter functionalization in the Hopf bifurcation problem, Nonlinear anal., 11, 149-161, (1987) · Zbl 0661.34036
[8] Krasnosel’skii, A.M., Asymptotics of nonlinearities and operator equations, Operator theory advances and appl., 76, (1995), Birkhäuser Basel · Zbl 0863.47047
[9] Krasnosel’skii, A.M., On nonlinear resonance in systems with hysteresis, Models of hysteresis, Pitman research notes in mathematics, 286, (1993), Longman London, p. 71-76 · Zbl 0808.34042
[10] Krasnosel’skii, M.A., Positive solutions of operator equations, (1964), Noordhoff Groningen · Zbl 0121.10604
[11] Krasnosel’skii, M.A.; Kuznetsov, N.A.; Yumagulov, M.G., Localization and construction of cycles in Hopf’s bifurcation at infinity, Dokl. math., 52, 223-226, (1995) · Zbl 0880.34038
[12] Krasnosel’skii, M.A.; Kuznetsov, N.A.; Yumagulov, M.G., Conditions of cycle stability for the Hopf bifurcation at infinity, Automat. remote control, 58, 43-48, (1997) · Zbl 0920.34040
[13] Krasnosel’skii, M.A.; Lifshits, E.A.; Sobolev, V.A., Linear positive systems: the method of positive operators, (1990), Hilderman Berlin · Zbl 0674.47036
[14] Krasnosel’skii, M.A.; Pokrovskii, A.V., Systems with hysteresis, (1989), Springer-Verlag Berlin · Zbl 1092.47508
[15] Krasnosel’skii, M.A.; Zabreiko, P.P., Geometrical methods of nonlinear analysis, (1984), Springer-Verlag Berlin · Zbl 0546.47030
[16] Krejčı́, P., Hysteresis, convexity and dissipation in hyperbolic equations, (1996), Gakkōtosho Tokyo · Zbl 1187.35003
[17] Landesman, E.N.; Lazer, A.C., Nonlinear perturbations of linear elliptic boundary problems at resonance, J. math. mech., 19, 609-623, (1970) · Zbl 0193.39203
[18] Marsden, J.; McCracken, M., Hopf bifurcation and its applications, (1982), Springer-Verlag New York
[19] Mawhin, J.; Willem, M., Critical point theory and Hamiltonian systems, (1989), Springer-Verlag Berlin · Zbl 0676.58017
[20] Visintin, A., Differential models of hysteresis, (1994), Springer-Verlag Berlin · Zbl 0820.35004
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.