×

zbMATH — the first resource for mathematics

Criteria of resonance origin in a single-circuit control system with saturation. (English. Russian original) Zbl 1156.93021
Autom. Remote Control 69, No. 8, 1297-1310 (2008); translation from Avtom. Telemekh. 2008, No. 8, 33-47 (2008).
Summary: Studies are made of forced periodic oscillations in a single-circuit control system with a parameter, the dynamics of which is described by resonance equations in the linear approximation. Criteria are suggested of the origin of resonance that is understood as an increase to infinity of the amplitude of forced oscillations in the approximation of the parameter to certain critical values. The principal results relate to the case when answers are defined by bounded nonlinearities of the order of a constant; the summands that decrease to zero are of no importance.
MSC:
93C15 Control/observation systems governed by ordinary differential equations
94C05 Analytic circuit theory
34C25 Periodic solutions to ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Lazer, A.C. and Leach, D.E., Bounded Perturbations of Forced Harmonic Oscillators at Resonance, Ann. Mat. Pura Appl., 1969, vol. 82, pp. 49–68. · Zbl 0194.12003 · doi:10.1007/BF02410787
[2] Alonso, J.M. and Ortega, R., Unbounded Solutions of Semilinear Equations at Resonance, Nonlinear, 1996, vol. 9, pp. 1099–1111. · Zbl 0896.34026 · doi:10.1088/0951-7715/9/5/003
[3] Ward, J.R. and Ruiz, D., Some Notes on Periodic System with Linear Parts at Resonance, Discret. Continuous Dynam. Syst., 2004, vol. 11, pp. 337–350. · Zbl 1063.34034 · doi:10.3934/dcds.2004.11.337
[4] Bonheure, D., Fabry, C., and Smets, D., Periodic Solutions of Forced Isochronous Oscillators at Resonance, Discret. Continuous Dynam. Syst., 2002, vol. 8, pp. 907–930. · Zbl 1021.34032 · doi:10.3934/dcds.2002.8.907
[5] Krasnosel’skii, A.M., Asimptotika nelineinostei i operatornye uravneniya (Asymptotics of Nonlinearities and Operator Equations), Moscow: Nauka, 1992.
[6] Fabry, C. and Fonda, A., Bifurcations from Infinity in Asymmetric Nonlinear Oscillators, Nonlinear. Diff. Equat. Appl., 2000, vol. 7, pp. 23–42. · Zbl 0979.34033 · doi:10.1007/PL00001421
[7] Ma, R., Bifurcation from Infinity and Multiple Solutions for Periodic Boundary Value Problems, Nonlinear Anal., 2000, vol. 42, pp. 27–39. · Zbl 0966.34015 · doi:10.1016/S0362-546X(98)00327-7
[8] Krasnosel’skii, M.A., Topologicheskie metody v teorii nelineinykh integral’nykh uravnenii (Topological Methods in the Theory of Nonlinear Integral Equations) Moscow: Tekhn.-Teoret. Lit., 1956.
[9] Krasnosel’skii, M.A. and Zabreiko, P.P., Geometricheskie metody nelineinogo analiza (Geometrical Methods of Nonlinear Analysis), Moscow: Nauka, 1975.
[10] Krasnosel’skii, A.M. and Krasnosel’skii, M.A., Vector Fields in a Product of Spaces and Applications to Differential Equations, Diff. Uravn., 1997, vol. 33, pp. 60–67.
[11] Bogolyubov, N.N. and Krulov, N.M., Vvedenie v nelineinuyu mekhaniku (Introduction to Nonlinear Mechanics), Izhevsk: R&C Dynamics, 2004.
[12] Vidyasagar, M., Nonlinear Systems Analysis, Englewood Cliffs: Prentice Hall, 1993. · Zbl 0900.93132
[13] Krasnosel’skii, A.M. and Mawhin, J., Periodic Solutions of Equations with Oscillating Nonlinearities, Math. Comput. Modell., 2000, vol. 32, pp. 1445–1455. · Zbl 0974.34042 · doi:10.1016/S0895-7177(00)00216-8
[14] Krasnosel’skii, M.A. and Pokrovskii, A.V., Sistemy s gisterezisom (Systems with Hysteresis), Moscow: Nauka, 1983.
[15] The Science of Hysteresis, Mayergoyz, I.D. and Bertotti, G., Eds., Amsterdam: Elsevier, 2005.
[16] Krasnosel’skii, A.M., Asymptotic Homogeneity of Hysteresis Operators, ZAMM Z. Angew. Math. Mech., 1996, vol. 76, no. 2, pp. 313–316.
[17] Krasnosel’skii, A.M. and Rachinskii, D.I., On a Number of Unbounded Branches of Solutions in the Neighborhood of an Asymptotic Bifurcation Point, Funkts. Analis. Prilozhen., 2005, vol. 39, no. 3, pp. 37–53. · doi:10.4213/faa73
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.