Geodesic
On bifurcations of chaotic attractors in a pulse width modulated control system
Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 20 (2024) no. 1, pp. 62-78

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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, merging bifurcation of a chaotic attractor. 
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     author = {Zh. T. Zhusubaliyev and U. A. Sopuev and D. A. Bushuev and A. S. Kucherov and A. Z. Abdirasulov},
     title = {On bifurcations of chaotic attractors in a pulse width modulated control system},
     journal = {Vestnik Sankt-Peterburgskogo universiteta. Prikladna\^a matematika, informatika, processy upravleni\^a},
     pages = {62--78},
     publisher = {mathdoc},
     volume = {20},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VSPUI_2024_20_1_a5/}
}
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%A A. Z. Abdirasulov
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Zh. T. Zhusubaliyev; U. A. Sopuev; D. A. Bushuev; A. S. Kucherov; A. Z. Abdirasulov. On bifurcations of chaotic attractors in a pulse width modulated control system. Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 20 (2024) no. 1, pp. 62-78. http://geodesic.mathdoc.fr/item/VSPUI_2024_20_1_a5/