Abstract
The 0:1 and 1:1 resonances at Hopf bifurcations in control systems with a parameter are investigated. Conditions for the generation of cycles in the neighborhood of the equilibrium position and at infinity are formulated. Nonlinearities with a principle quadratic part and with a principle homogeneous part of the general (nonpolynomial) type in the neighborhood of the equilibrium position are separately studied. The main case of bounded saturation nonlinearities at infinity is also studied.
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REFERENCES
Marsden, J. and McCracken, M., Hopf Bifurcation and Its Applications, New York: Springer, 1982. Translated under the title Bifurkatsiya rozhdeniya tsikla i ee prilozheniya, Moscow: Mir, 1980.
Hopf, E., Abzweigung einer periodischen Lösung von einer stationaren Lösung eines differential Systems, Ber. Math. Phys. Sachsische Ac. der Wiss. Leipzig, 1942, vol. 94, pp. 1-22.
Andronov, A.A. and Witt, A., Sur la theorie mathematiques des autooscillations, C. R. Acad. Sci. Paris, 1930, vol. 190, pp. 256-258.
Krasnosel'skii, A.M. and Krasnosel'skii, M.A., Large-Amplitude Cycles in Autonomous Hysteresis Systems, Dokl. Akad. Nauk SSSR, 1985, vol. 283, no. 1, pp. 23-26.
Guckenheimer, J. and Holmes, Ph., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, New York: Springer, 1990.
Shilnikov, L.P., Shilnikov, A.L., Turaev, D.V., and Chua, L.O., Methods of Qualitative Theory in Nonlinear dynamics. Part I, River Edge: World Scientific Publ., 1998.
Schneider, K.R., Perturbed Center Manifolds and Applications to Hopf Bifurcation, IX Mezhdunar. Konf. po Nelineinym Kolebaniyam (Int. Conf. on Nonlinear Oscillations), Kiev: Naukova Dumka, 1984, vol. 3, pp. 450-455.
Kozyakin, V.S. and Krasnosel'skii, M.A., A Functionalization Method for the Parameter in the Problem of Bifurcation Points, Dokl. Akad. Nauk SSSR, 1980, vol. 254, no. 5, pp. 1061-1064.
Krasnosel'skii, M.A. and Zabreiko, P.P., Geometricheskie metody nelineinogo analiza, Moscow: Nauka, 1975. Translated under the title Geometrical Methods of Nonlinear Analysis, New York: Springer, 1984.
Arnol'd, V.I., Dopolnitel'nye glavy teorii obyknobennykh differentsial'nykh uravnenii, Moscow: Nauka, 1978. Translated under the title Geometric Methods in the Theory of Ordinary Differential Equations, New York: Springer, 1983.
Knobloch, E. and Proctor, M.R.E., The Double Hopf Bifurcation with 2:1 Resonance, Proc. R. Soc., 1988, vol. 415, pp. 61-90.
Krasnosel'skii, A.M., Rachinskii, D.I., and Schneider, K., Hopf Bifurcations in Resonance 2:1, Inst. Nonlinear Sci., National Univ. Ireland, Univ. College, Cork, Report 00-005, May, 2000.
Besekerskii, V.A. and Popov, E.P., Teoriya sistem avtomaticheskogo regulirovaniya (Theory of Automatic Control Systems), Moscow: Nauka, 1972.
Vidyasagar, M., Nonlinear System Analysis, Englewood Cliffs: Prentice Hall, 1993.
Krasnosel'skii, A.M., Kuznetsov, N.A., and Rachinskii, D.I., Resonance Equations with Unbounded Nonlinearities, Dokl. Ross. Akad. Nauk, 2000, vol. 373, no. 3, pp. 295-299.
Krasnosel'skii, A.M., Kuznetsov, N.A., and Rachinskii, D.I., Nonlinear Hopf Bifurcations, Dokl. Ross. Akad. Nauk, 2000, vol. 373, no 4, pp. 455-458.
Grigor'ev, F.N. and Kuznetsov, N.A., Time-Optimal Control for a Nonlinear Problem, Avtom. Telemekh., 2000, no. 8, pp. 11-24.
Krasnosel'skii, A.M., McInerney, J., and Pokrovskii A.V., Weak Resonances in Hopf Bifurcations for Control Systems with Nonpolynomial Nonlinearities, Dokl. Ross. Akad. Nauk, 2000, vol. 375, no. 5, pp. 600-605.
Alonso, J.M. and Ortega, R., Unbounded Solutions of Semilinear Equations at Resonance, Nonlinearity, 1996, vol. 9, pp. 1099-1111.
Krasnosel'skii, A.M. and Rachinskii, D.I., Hopf Bifurcations from Infinity Generated by Bounded Nonlinear Terms, Funct. Differ. Equat., 1999, vol. 6, nos. 3, 4, pp. 357-374.
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Bliman, PA., Krasnosel'skii, A.M. & Rachinskii, D.I. Strong Resonances at Hopf Bifurcations in Control Systems. Automation and Remote Control 62, 1783–1802 (2001). https://doi.org/10.1023/A:1012786122737
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DOI: https://doi.org/10.1023/A:1012786122737