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Chaotic behavior of a class of discontinuous dynamical systems of fractional-order. (English) Zbl 1194.93087
Summary: In this paper, the chaos persistence in a class of discontinuous dynamical systems of fractional-order is analyzed. To that end, the initial value problem is first transformed, by using the Filippov regularization [A. F. Filippov, Differential equations with discontinuous right-hand side. Moskva: “Nauka” (1985; Zbl 0571.34001)], into a set-valued problem of fractional-order, then by Cellina’s approximate selection theorem [J. P. Aubin and A. Cellina, Differential inclusions. Set-valued maps and viability theory, Grundlehren der Mathematischen Wissenschaften, 264. Berlin etc.: Springer-Verlag (1984; Zbl 0538.34007); J.-P. Aubin and H. Frankowska, Set-valued analysis, Boston: Birkhäuser (1990; Zbl 0713.49021)]. The problem is approximated into a single-valued fractional-order problem, which is numerically solved by using a numerical scheme proposed by [K. Diethelm, N. J. Ford, A. D. Freed, Nonlinear Dyn. 29, No. 1–4, 3–22 (2002; Zbl 1009.65049)]. Two typical examples of systems belonging to this class are analyzed and simulated.
MSC:
93C15Control systems governed by ODE
37M99Approximation methods and numerical treatment of dynamical systems
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