Periodic Solutions of a System of Differential Equations with Hysteresis Nonlinearity in the Presence of Eigenvalue Zero
- 20 Downloads
We study a system of ordinary differential equations of order n containing a nonlinearity of imperfectrelay type with hysteresis and external periodic perturbation. We consider the problem of existence of solutions with periods equal to (or multiple of) the period of external perturbation with two points of switching within the period. The problem is solved in the case where the collection of simple real eigenvalues of the matrix of the system contains an eigenvalue equal to zero. By a nonsingular transformation, the system is reduced to a canonical system of special form, which enables us to perform its analysis by the analytic methods. We propose an approach to finding the points of switching for the representative point of periodic solution and to the choice of the parameters of nonlinearity and the feedback vector. A theorem on necessary conditions for the existence of periodic solutions of the system is proved. Sufficient conditions for the existence of the required solutions are established. We also perform the analysis of stability of solutions by using the point mapping and the fixed-point method. We present an example that confirms the established results.
Unable to display preview. Download preview PDF.
- 2.V. I. Zubov, Oscillations in Nonlinear and Controllable Systems [in Russian], Sudpromgiz, Leningrad (1962).Google Scholar
- 3.R. A. Nelepin, Exact Analytic Methods in the Theory of Nonlinear Automatic Systems [in Russian], Sudpromgiz, Leningrad (1967).Google Scholar
- 4.A.V. Pokrovskii, “Existence and calculation of stable modes in on-off systems,” Avtomat. Telemekh., No. 4, 16–23 (1986).Google Scholar
- 5.D. K. Potapov, “Optimal control over distributed systems of elliptic type of high order with spectral parameter and discontinuous nonlinearity,” Izv. Ros. Akad. Nauk. Teor. Sist. Uprav., No. 2, 19–24 (2013).Google Scholar
- 6.I. L. Nyzhnyk and A. O. Krasneeva, “Periodic solutions of second-order differential equations with discontinuous nonlinearity,” Nelin. Kolyvannya, 15, No. 3, 381–389 (2012); English translation : J. Math. Sci., 191, No. 3, 421–430 (2013).Google Scholar
- 10.A. M. Kamachkin, D. K. Potapov, and V. V. Yevstafyeva, “Solution to second-order differential equations with discontinuous righthand side,” Electron. J. Different. Equat., No. 221, 1–6 (2014).Google Scholar
- 12.A. M. Samoilenko and I. L. Nizhnik, “Differential equations with bistable nonlinearity,” Ukr. Mat. Zh., 67, No. 4, 517–554 (2015); English translation : Ukr. Math. J., 67, No. 4, 584–624 (2015).Google Scholar
- 13.G. Bonanno, G. D’Agui, and P.Winkert, “Sturm–Liouville equations involving discontinuous nonlinearities,” Minimax Theory Appl., 1, No. 1, 125–143 (2016).Google Scholar
- 15.A. M. Kamachkin, D. K. Potapov, and V. V. Yevstafyeva, “Existence of solutions for second-order differential equations with discontinuous right-hand side,” Electron. J. Different. Equat., No. 124, 1–9 (2016).Google Scholar
- 17.V. V. Yevstafyeva, “Existence of a unique kT-periodic solution for one class of nonlinear systems,” J. Sib. Fed. Univ. Math. Phys., 6, No. 1, 136–142 (2013).Google Scholar
- 19.A. M. Kamachkin, D. K. Potapov, and V. V. Yevstafyeva, “Existence of subharmonic solutions to a hysteresis system with sinusoidal external influence,” Electron. J. Different. Equat., No. 140, 1–10 (2017).Google Scholar
- 23.I. D. Mayergoyz, Mathematical Models of Hysteresis and Their Applications, Elsevier, Amsterdam (2003).Google Scholar