On bifurcations of chaotic attractors in a pulse width modulated control system
DOI:
https://doi.org/10.21638/11701/spbu10.2024.106Abstract
This paper discusses bifurcational phenomena in a control system with pulse-width modulation of the first kind. We show that the transition from a regular dynamics to chaos occurs in a sequence of classical supercritical period doubling and border collision bifurcations. As a parameter is varied, one can observe a cascade of doubling of the cyclic chaotic intervals, which are associated with homoclinic bifurcations of unstable periodic orbits. Such transition are also refereed as merging bifurcation (known also as merging crisis). At the bifurcation point, the unstable periodic orbit collides with some of the boundaries of a chaotic attractor and as a result, the periodic orbit becomes a homoclinic. This condition we use for obtain equations for bifurcation boundaries in the form of an explicit dependence on the parameters. This allow us to determine the regions of stability for periodic orbits and domains of the existence of four-, two- and one-band chaotic attractors in the parameter plane.
Keywords:
piecewise smooth bimodal map, border collision bifurcations, homoclinic bifurcations of periodic orbits, bifurcations of chaotic attractors
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References
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Yakubovich V. A., Leonov G. A., Gelig A. Kh. Stability of stationary sets in control systems with discontinuous nonlinearities. Singapore, World Scientific, 2004, 334 p.
Yanochkina O. O., Titov D. V. Bifurcation analysis of a pulse-width control system. Autom. Remote Control, 2022, vol. 83, no. 2, pp. 204–213.
Zhusubaliyev Zh. T., Mosekilde E. Bifurcations and chaos in piecewise-smooth dynamical systems. Singapore, World Scientific, 2003, 363 p.
Banerjee S., Verghese C. C. (eds) Nonlinear phenomena in power electronis. New York, IEEE Press, 2001, 441 p.
Di Bernardo M., Budd C. J., Champneys A. R., Kowalczyk P. Piecewise-smooth dynamical systems: Theory and applications. London, Springer-Verlag, 2008, 483 p.
Avrutin V., Gardini L., Sushko I., Tramontana F. Continuous and discontinuous piecewise-smooth one-dimensional maps: Invariant sets and bifurcation structures. Singapore, World Scientific, 2019, 637 p.
Zhusubaliyev Zh. T., Mosekilde E., Maity S. M., Mohanan S., Banerjee S. Border collision route to quasiperiodicity: Numerical investigation and experimental сonfirmation. Chaos, 2006, vol. 16, art. no. 023122.
Zhusubaliyev Zh. T., Avrutin V., Sushko I., Gardini L. Border collision bifurcation of a resonant closed invariant curve. Chaos, 2022, vol. 32, art. no. 043101.
Radi D., Gardini L. A piecewise smooth model of evolutionary game for residential mobility and segregation. Chaos, 2018, vol. 28, art. no. 055912.
Avrutin V., Zhusubaliyev Zh. T. Nested closed invariant curves in piecewise smooth maps. International Journal of Bifurcation and Chaos, 2019, vol. 29, art. no. 1930017.
Patra M., Banerjee S. Hyperchaos in 3D piecewise smooth maps. Chaos, Solitons and Fractals, 2020, vol. 133, art. no. 109681.
Sushko I., Commendatore P., Kubin I. Codimension-two border collision bifurcation in a two-class growth model with optimal saving and switch in behavior. Nonlinear Dynamics, 2020, vol. 102, no. 2, pp. 1071–1095.
Gallegati M., Gardini L., Sushko I. Dynamics of a business cycle model with two types of governmental expenditures: The role of border collision bifurcations. Decisions in Economics and Finance, 2021, vol. 44, no. 2, pp. 613–639.
Anufriev M., Gardini L., Radi D. Chaos, border-collisions and stylized empirical facts in an asset pricing model with heterogeneous agents. Nonlinear Dynamics, 2020, vol. 102, pp. 993–1017.
Zhusubaliyev Zh. T., Avrutin V., Bastian F. Transformations of closed invariant curves and closed-invariant-curve-like chaotic attractors in piecewise smooth systems. International Journal of Bifurcation and Chaos, 2021, vol. 31, no. 3, art. no. 2130009.
Simpson D. J. W., Avrutin V., Banerjee S. Nordmark map and the problem of large-amplitude chaos in impact oscillators. Physical Review E, 2021, vol. 102, no. 2, art. no. 022211.
Jeffrey M. R., Glendinning P. Hidden dynamics for piecewise smooth maps. Nonlinearity, 2021, vol. 34, no. 5, pp. 3184–3198.
Nusse H. E., 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.
Feigin M. I. Doubling of the oscillation period with C-bifurcations in piecewise continuous systems. Journal of Appl. Math., 1970, vol. 34, no. 5–6, pp. 822–830.
Feigin M. I. On the structure of C-bifurcation boundaries of piecewise continuous systems. Journal of Appl. Math. Mech., 1978, vol. 42, no. 5, pp. 820–829.
Di Bernardo M., Feigin M. I., Hogan S. J., Homer M. E. Local analysis of C-bifurcations in ndimensional piecewise-smooth dynamical systems. Chaos, Solitons and Fractals, 1999, vol. 19, no. 11, pp. 1881–1908.
Kuznetsov Yu. A. Elements of applied bifurcation theory. New York, Springer-Verlag, 2004, 633 p.
Grebogi C., Ott E., Yorke J. A. Chaotic attractors in crisis. Phys. Rev. Lett., 1982, vol. 48, pp. 1507–1510.
Maistrenko Yu. L., Maistrenko V. L., Chua L. O. Cycles of chaotic intervals in a time-delayed Chua's circuit. International Journal of Bifurcation and Chaos, 1993, vol. 3, pp. 1557–1572.
Mira C., Gardini L., Barugola A., Cathala J. C. Chaotic dynamics in two-dimensional noninvertible maps. Singapore, World Scientific, 1996, 632 p.
Devaney R. L. An introduction to chaotic dynamical systems. New York, CRC Press, Taylor & Francis Group, 2022, 419 p.
Avrutin V., Sushko I., Gardini L. Cyclicity of chaotic attractors in one-dimensional discontinuous maps. Math. Comp. Sim., 2013, vol. 95, pp. 126–136.
Avrutin V., Eckstein E., Schanz M. On detection of multi-band chaotic attractors. Proceedings of Royal Society A., 2007, vol. 463, pp. 1339–1358.
Power electronics handbook. 4th ed. Ed. by M. H. Rashid. Oxford, Butterworth-Heinemann, 2018, XI + 1496 p.
Control of power electronic converters and systems. Ed. by F. Blaabjerg. London, Academic Press, 2021, vol. 3, XVIII + 700 p.
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