Abstract
We study local bifurcations in a control system with pulse-width modulation of the first kind (PWM-1) whose state is described by a piecewise smooth mapping. It is shown that, in addition to classical bifurcations, the so-called border collision bifurcations, which have no analogs in smooth systems, are possible in pulsed systems. The main bifurcation transitions are described using a piecewise linear continuous mapping as a normal form.
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REFERENCES
Kipnis, M.M., Phase portraits of impulse-modulated systems, Autom. Remote Control, 1990, vol. 51, no. 12, pp. 1693–1701.
Kipnis, M.M., Chaotic phenomena in a deterministic one-dimensional pulse-width control system, Tekh. Kibern., 1992, no. 1, pp. 108–112.
Kipnis, M.M., Symbolic and chaotic dynamics of a pulse-width control system, Dokl. Math., 1992, vol. 37, no. 5, pp. 217–219.
Gelig, A.Kh. and Churilov, A.N., Kolebaniya i ustoichivost’ nelineinykh impul’snykh sistem (Oscillations and Stability of Nonlinear Pulsed Systems), St. Petersburg: Izd. S.-Peterburg. Gos. Univ., 1993.
Filippov, A.F., Differentsial’nye uravneniya s razzryvnoi pravoi chast’yu (Differential Equations with Discontinuous Right-Hand Side), Moscow: Nauka, 1985.
Rozenvasser, E.N., Periodicheski nestatsionarnye sistemy upravleniya (Periodically Time-Varying Control Systems), Moscow: Nauka, 1973.
Avrutin, V., Mosekilde, E., Zhusubaliyev, Zh.T., and Gardini, L., Onset of chaos in a single-phase power electronic inverter, Chaos, 2015, no. 25, pp. 043114-1–043114-14.
Avrutin, V., Zhusubaliyev, Zh.T., and Mosekilde, E., Cascades of alternating pitchfork and flip bifurcations in h-bridge inverters, Physica D, 2017, no. 345, pp. 27–39.
Avrutin, V., Gardini, L., Sushko, I., and Tramontana, F., Continuous and Discontinuous Piecewise-Smooth One-Dimensional Maps: Invariant Sets and Bifurcation Structures, Singapore: World Sci., 2019.
Mira, C., Gardini, L., Barugola, A., and Cathala, J.C., Chaotic Dynamics in Two-Dimensional Noninvertible Maps, Singapore: World Sci., 1996.
Nusse, H.E. and Yorke, J.A., Border-collision bifurcations including “period two to period three” for piecewise smooth systems, Physica D, 1992, vol. 57, no. 1–2, pp. 39–57.
Nusse, H.E. and Yorke, J.A., Border-collision bifurcation: an explanation for observed bifurcation phenomena, Phys. Rev. E, 1994, vol. 49, pp. 1073–1076.
Nusse, H.E. and Yorke, J.A., Border-collision bifurcations for piecewise smooth one dimensional maps, Int. J. Bifurcation Chaos, 1995, vol. 5, no. 1, pp. 189–207.
Feigin, M.I., Doubling of oscillation period in C-bifurcations in piecewise continuous systems, Prikl. Mat. Mekh., 1970, vol. 34, no. 5, pp. 861–869.
Feigin, M.I., On the generation of a family of subharmonic modes in a piecewise continuous system, Prikl. Mat. Mekh., 1974, vol. 38, no. 5, pp. 810–818.
Feigin, M.I., On the structure of C-bifurcation boundaries of piecewise continuous systems, Prikl. Mat. Mekh., 1978, vol. 42, no. 5, pp. 820–819.
Feigin, M.I., Vynuzhdennye kolebaniya sistem s razryvnymi nelineinostyami (Forced Oscillations of Systems with Discontinuous Nonlinearities), Moscow: Nauka, 1994.
Di Bernardo, M., Feigin, M.I., Hogan, S.J., and Homer, M.E., Local analysis of C-bifurcations in \( n \)-dimensional piecewise smooth dynamical systems, Chaos Solitons Fractals, 1999, vol. 19, no. 11, pp. 1881–1908.
Banerjee, S., Ranjan, P., and Grebogi, C., Bifurcations in two-dimensional piecewise smooth maps: theory and applications in switching circuits, IEEE Trans. Circ. Syst. I , 2000, vol. 47, no. 5, pp. 633–643.
Nonlinear Phenomena in Power Electronics, Banerjee, S. and Verghese, C.C., Eds., New York: IEEE Press, 2001.
Zhusubaliyev, Zh.T. and Mosekilde, E., Bifurcations and Chaos in Piecewise-Smooth Dynamical Systems, Singapore: World Sci., 2003.
Di Bernardo, M., Budd, C.J., Champneys, A.R., and Kowalczyk, P., Piecewise-Smooth Dynamical Systems: Theory and Applications, London: Springer-Verlag, 2008.
Di Bernardo, M., Budd, C.J., Champneys, A.R., Kowalczyk, P., Nordmark, A.B, Tost, G.O., and Piiroinen, P.T., Bifurcations in nonsmooth dynamical systems, SIAM Rev., 2008, vol. 50, pp. 629–701.
Simpson, D.J.W., Border-collision bifurcations in \( R^N \), SIAM Rev., 2016, vol. 58, pp. 177–226.
Sushko, I., Avrutin, V., and Gardini, L., Bifurcation structure in the skew tent map and its application as a border collision normal form, J. Difference Equat. Appl., 2016, vol. 22, no. 8, pp. 1040–1087.
Zhusubaliyev, Zh.T. and Mosekilde, E., Equilibrium-torus bifurcation in nonsmooth systems, Phys. D, 2008, vol. 237, no. 7, pp. 930–936.
ACKNOWLEDGMENTS
The work was carried out at the International Scientific Laboratory of the Dynamics of Nonsmooth Systems of the Southwest State University under the guidance of prof. Zh.T. Zhusubaliyev.
Funding
The work was supported by the Ministry of Education and Science of the Russian Federation, grant “Implementation of the strategic academic leadership program Priority–2030” (agreements nos. 075-15-2021-1155 and 075-15-2021-1213).
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Translated by V. Potapchouck
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Yanochkina, O.O., Titov, D.V. Bifurcation Analysis of a Pulse-Width Control System. Autom Remote Control 83, 204–213 (2022). https://doi.org/10.1134/S0005117922020047
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DOI: https://doi.org/10.1134/S0005117922020047