О бифуркациях хаотических аттракторов в широтно-импульсной системе управления
DOI:
https://doi.org/10.21638/11701/spbu10.2024.106Аннотация
Численно и аналитически исследуются нелинейные явления, индуцированные локальными и глобальными бифуркациями в системе управления с широтно-импульсной модуляцией первого рода. Показано, что переход от регулярных колебаний к хаотическим при вариации параметров происходит через последовательность классической суперкритической бифуркации удвоения периода и бифуркаций «граничного столкновения» (“border collision”). В области хаотической динамики наблюдается бифуркация слияния (“merging bifurcation”) циклов хаотических интервалов (cycles of chaotic intervals), которая связана с гомоклинической бифуркацией неустойчивых периодических орбит. Такая бифуркация относится к кризисам хаотических аттракторов. В момент бифуркации неустойчивая периодическая орбита сталкивается с некоторыми из границ хаотического аттрактора, становясь гомоклинической. Найдены уравнения бифуркационных границ в форме явной зависимости от параметров. На плоскости управляющих параметров построены области устойчивости периодических режимов и области существования четырех-, двух- и однополосных хаотических аттракторов.
Ключевые слова:
кусочно-гладкое бимодальное отображение, бифуркация граничного столкновения, гомоклинические бифуркации неустойчивых периодических орбит, бифуркации хаотических аттракторов
Скачивания
Библиографические ссылки
Кипнис М. М. Хаотические явления в детерминированной одномерной широтно-импульсной системе управления // Техническая кибернетика. 1992. № 1. С. 108–112.
Gelig A. Kh., Churilov A. N. Stability and oscillations of nonlinear pulse-modulated systems. Boston: Birkhäuser, 1998. XVI + 362 p.
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. N 2. P. 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. (еds.) 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. N 023122.
Zhusubaliyev Zh. T., Avrutin V., Sushko I., Gardini L. Border collision bifurcation of a resonant closed invariant curve // Chaos. 2022. Vol. 32. Art. N 043101.
Radi D., Gardini L. A piecewise smooth model of evolutionary game for residential mobility and segregation // Chaos. 2018. Vol. 28. Art. N 055912.
Avrutin V., Zhusubaliyev Zh. T. Nested closed invariant curves in piecewise smooth maps // International Journal of Bifurcation and Chaos. 2019. Vol. 29. Art. N 1930017.
Patra M., Banerjee S. Hyperchaos in 3D piecewise smooth maps // Chaos, Solitons and Fractals. 2020. Vol. 133. Art. N 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. N 2. P. 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. N 2. P. 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. P. 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. N 3. Art. N 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. N 2. Art. N 022211.
Jeffrey M. R., Glendinning P. Hidden dynamics for piecewise smooth maps // Nonlinearity. 2021. Vol. 34. N 5. P. 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. N 1–2. P. 39–57.
Feigin M. I. Doubling of the oscillation period with C-bifurcations in piecewise continuous systems // Journal of Appl. Math. 1970. Vol. 34. N 5–6. P. 822–830.
Feigin M. I. On the structure of C-bifurcation boundaries of piecewise continuous systems // Journal of Appl. Math. Mech. 1978. Vol. 42. N 5. P. 820–829.
Di Bernardo M., Feigin M. I., Hogan S. J., Homer M. E. Local analysis of C-bifurcations in n-dimensional piecewise-smooth dynamical systems // Chaos, Solitons and Fractals. 1999. Vol. 19. N 11. P. 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. P. 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. P. 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. P. 126–136.
Avrutin V., Eckstein E., Schanz M. On detection of multi-band chaotic attractors // Proceedings of Royal Society A. 2007. Vol. 463. P. 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.
References
Kipnis M. M. Kchaoticeskie yavlenia v determinirovannoy odnomernoy shirotno-impyl'snoy sisteme upravlenia [Chaotic phenomenas in a determination a single width modulated control system]. Technical kibernetics, 1992, no. 1, pp. 108–112. (In Russian)
Gelig A. Kh., Churilov A. N. Stability and oscillations of nonlinear pulse-modulated systems. Boston, Birkhäuser, 1998, XVI + 362 p.
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.
Загрузки
Опубликован
Как цитировать
Выпуск
Раздел
Лицензия
Статьи журнала «Вестник Санкт-Петербургского университета. Прикладная математика. Информатика. Процессы управления» находятся в открытом доступе и распространяются в соответствии с условиями Лицензионного Договора с Санкт-Петербургским государственным университетом, который бесплатно предоставляет авторам неограниченное распространение и самостоятельное архивирование.
