Abstract
A system of \(n\)th-order ordinary differential equations with relay nonlinearity and periodic perturbation function on the right-hand side is studied. The matrix of the system has real nonzero eigenvalues, among which there is at least one positive and one multiple eigenvalue. A nonsingular transformation that reduces the matrix of the system to Jordan form is used. Continuous periodic solutions with two switching points in the phase space of the system are considered. It is assumed that the period of the perturbation function is a multiple of the periods of these solutions. Necessary conditions for the existence of such solutions are established. An existence theorem for a solution of period equal to the period of the perturbation function is proved. A numerical example confirming the obtained results is presented.
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References
M. A. Krasnosel’skii and A. V. Pokrovskii, “Periodic oscillations in systems with relay nonlinearities,” Dokl. Akad. Nauk SSSR 216 (4), 733–736 (1974).
M. A. Krasnosel’skii and A. V. Pokrovskii, “Equations with discontinuous nonlinearities,” Dokl. Akad. Nauk SSSR 248 (5), 1056–1059 (1979).
M. A. Krasnosel’skii and A. V. Pokrovskii, Systems with Hysteresis (Nauka, Moscow, 1983) [in Russian].
A. V. Pokrovskii, “Existence and calculation of stable modes in switching systems,” Autom. Remote Control 47 (4), 451–458 (1986).
A. M. Krasnosel’skii, “Forced oscillations in systems with hysteresis nonlinearities,” Dokl. Akad. Nauk SSSR 292 (5), 1078–1082 (1987).
M. A. Krasnosel’skii, A. V. Pokrovskii, V. V. Chernorutskii, and G. Tronel’, “On the dynamics of controlled systems described by equations of parabolic type with hysteresis nonlinearities,” Autom. Remote Control 53 (11), 1705–1711 (1992).
J. W. Macki, P. Nistri, and P. Zecca, “Mathematical models for hysteresis,” SIAM Rev. 35 (1), 94–123 (1993).
A. Visintin, Differential Models of Hysteresis (Springer- Verlag, Berlin, 1994).
M. Brokate and J. Sprekels, Hysteresis and Phase Transitions (Springer, New York, 1996).
I. D. Mayergoyz, Mathematical Models of Hysteresis and Their Applications (Elsevier, Amsterdam, 2003).
V. V. Yevstafyeva, “On necessary conditions for existence of periodic solutions in a dynamic system with discontinuous nonlinearity and an external periodic influence,” Ufa Math. J. 3 (2), 19–26 (2011).
V. V. Yevstafyeva, “Existence of the unique kT-periodic solution for one class of nonlinear systems,” J. Sib. Fed. Univ. Math. Phys. 6 (1), 136–142 (2013).
A. Visintin, “Ten issues about hysteresis,” Acta Appl. Math. 132 (1), 635–647 (2014).
L. Fang, J. Wang, and Q. Zhang, “Identification of extended Hammerstein systems with hysteresis-type input nonlinearities described by Preisach model,” Nonlinear Dynam. 79 (2), 1257–1273 (2015).
V. V. Yevstafyeva, “On existence conditions for a two-point oscillating periodic solution in an non- autonomous relay system with a Hurwitz matrix,” Autom. Remote Control 76 (6), 977–988 (2015).
A. M. Kamachkin, D. K. Potapov, and V. V. Yevstafyeva, “Existence of periodic solutions to automatic control system with relay nonlinearity and sinusoidal external influence,” Internat. J. Robust Nonlinear Control 27 (2), 204–211 (2017).
A. M. Kamachkin, D. K. Potapov, and V. V. Yevstafyeva, “Existence of subharmonic solutions to a hysteresis system with sinusoidal external influence,” Electron. J. Differential Equations, No. 140 (2017).
A. M. Kamachkin, D. K. Potapov, and V. V. Yevstafyeva, “On uniqueness and properties of periodic solution of second-order nonautonomous system with discontinuous nonlinearity,” J. Dyn. Control Syst. 23 (4), 825–837 (2017).
V. V. Yevstafyeva, “Periodic solutions of a system of differential equations with hysteresis nonlinearity in the presence of a zero eigenvalue,” Ukrainian Math. J. 70 (8), 1252–1263 (2019).
A. M. Kamachkin, D. K. Potapov, and V. V. Yevstafyeva, “Existence of periodic modes in automatic control system with a three-position relay,” Internat. J. Control 93 (4), 763–770 (2020).
A. M. Kamachkin, G. M. Khitrov, and V. N. Shamberov, “Normal matrix forms in decomposition and control problems for multidimensional systems,” Vestn. St.-Peterbg. Univ. Prikl. Mat. Inform. Protsessy Upr. 13 (4), 417–430 (2017).
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Translated from Matematicheskie Zametki, 2021, Vol. 109, pp. 529-543 https://doi.org/10.4213/mzm12411.
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Yevstafyeva, V.V. Existence of \(T/k\)-Periodic Solutions of a Nonlinear Nonautonomous System Whose Matrix Has a Multiple Eigenvalue. Math Notes 109, 551–562 (2021). https://doi.org/10.1134/S0001434621030238
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DOI: https://doi.org/10.1134/S0001434621030238