Generic bifurcation structures in piecewise-smooth maps with extremely high number of borders in theory and applications for power converter systems
Final Report Abstract
The reported project’s overall goal was to explain unusual bifurcation phenomena occurring in piecewisesmooth maps, which serve as mathematical models for AC/DC and DC/AC power converters commonly used in various engineering applications, such as renewable energy sources and electric vehicle power supplies. Previous studies have shown that discrete-time models of these systems belong to a class of piecewise-smooth maps insufficiently investigated before and characterized by a extremely high number of switching manifolds. This property leads to several unusual bifurcation phenomena that need to be explained to predict and control the dynamics of the underlying applications. The specific phenomena investigated within the project included the onset of bubbling, border collision bifurcations, the formation and bifurcations of closed invariant curves, and transformations of chaotic attractors. The project’s goals were met and partially exceeded. Notably, we developed a novel theory explaining the mechanism behind the bubbling phenomenon (high-frequency low-amplitude oscillations disturbing the output signal waveform in a restricted phase interval) differently than previous explanations. We demonstrated that this phenomenon is caused by the cumulative action of expanding functions eventually compensated by contractive ones. This led to the description of a novel phenomenon, namely the noiseinduced bubbling, which can occur alone or in combination with border-collision-induced bubbling. Our theory is expected to be effectively applicable in industrial environments. Additionally, we reported a novel type of border collision bifurcations involving a repelling resonant closed invariant curve that collides with the switching manifold at a point of a repelling cycle. This bifurcation results in the emergence of a different closed invariant curve and a global restructuring of the phase space (which is by contrast to usual local border collision bifurcations in continuous piecewise-smooth systems). We also contributed to the understanding of the doubling of closed invariant curves. Moreover, we described a partially bistability-affected period-adding structure, explaining the mechanism leading to bistability and identifying which cycles in the period-adding structure are affected. Furthermore, we discovered a new group of bifurcations, namely border collision bifurcations of chaotic attractors, which are specific to discontinuous piecewise-smooth maps with multiple switching manifolds. Two novel bifurcations from this group were described in detail. Finally, we demonstrated how the recently developed concept of hidden orbits can unify the bifurcation theory for continuous and discontinuous maps, significantly simplifying the analysis of the bifurcation structures formed by border collision bifurcations in the latter.
Publications
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Bifurcations of hidden orbits in discontinuous maps. Nonlinearity, 34(9), 6140-6172.
Avrutin, Viktor & Jeffrey, Mike R.
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Border collision bifurcations of chaotic attractors in one-dimensional maps with multiple discontinuities. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 477(2253), 20210432.
Avrutin, Viktor; Panchuk, Anastasiia & Sushko, Iryna
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Complex dynamics of a vibration machine caused by a relay feedback control. Physica D: Nonlinear Phenomena, 420, 132870.
Zhusubaliyev, Zhanybai T.; Avrutin, Viktor; Rubanov, Vasily G. & Bushuev, Dmitry A.
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Border collision bifurcation of a resonant closed invariant curve. Chaos: An Interdisciplinary Journal of Nonlinear Science, 32(4).
Zhusubaliyev, Zh. T.; Avrutin, V.; Sushko, I. & Gardini, L.
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Noise-induced and border-collision-induced bubbling. Physica D: Nonlinear Phenomena, 435, 133277.
Avrutin, Viktor; von Schwerin-Blume, Lasse; Zhusubaliyev, Zhanybai T.; Haroun, Reham & El, Aroudi Abdelali
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Period adding with symmetry breaking/recovering in a power inverter with hysteresis control. Physica D: Nonlinear Phenomena, 444, 133600.
Zhusubaliyev, Zhanybai. T.; Avrutin, Viktor; Kucherov, Andrey S.; Haroun, Reham & El, Aroudi Abdelali
