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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.

93C15 Control/observation systems governed by ordinary differential equations
37M99 Approximation methods and numerical treatment of dynamical systems
Full Text: DOI
[1] Filippov, A.F.: Differential Equations with Discontinuous Right-Hand Sides. Kluwer Academic, Dordrecht (1988) · Zbl 0664.34001
[2] Aubin, J.-P., Cellina, A.: Differential Inclusions Set-valued Maps and Viability Theory. Springer, Berlin (1984) · Zbl 0538.34007
[3] Aubin, J.-P., Frankowska, H.: Set-valued Analysis. Birkhäuser, Boston (1990) · Zbl 0713.49021
[4] Diethelm, K., Ford, N.J., Freed, A.D.: Predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 29, 3–22 (2002) · Zbl 1009.65049
[5] Andronov, A.A., Vitt, A.A., Khaikin, S.E.: Theory of Oscillators. Pergamon, Oxford (1966) · Zbl 0188.56304
[6] Buhite, J.L., Owen, D.R.: An ordinary differential equation from the theory of plasticity. Arch. Ration. Mech. Anal. 71, 357–383 (1979) · Zbl 0424.73031
[7] Clarke, F.H.: Optimization and Non-smooth Analysis. Wiley, New York (1983) · Zbl 0582.49001
[8] Deimling, K.: Multivalued differential equations and dry friction problems. In: Fink, A.M., Miller, R.K., Kliemann, W. (eds.) Proc. Conf. Delay and Differential Equations, pp. 99–106. World Scientific, Singapore (1992) · Zbl 0820.34009
[9] Schilling, K.: An algorithm to solve boundary value problem for differential equations and applications in optimal control. Numer. Funct. Anal. Optim. 10, 733–764 (1989) · Zbl 0666.49001
[10] Wiercigroch, M., de Kraker, B.: Applied Nonlinear Dynamics and Chaos of Mechanical Systems with Discontinuities. World Scientific, Singapore (2000) · Zbl 0953.70001
[11] Bagley, R.L., Calico, R.A.: Fractional order state equations for the control of viscoelastically damped structures. J. Guid. Control Dyn. 14, 304–311 (1991)
[12] Nakagava, M., Sorimachi, K.: Basic characteristics of a fractance device. IEICE Trans. Fundam. Electron. E75-A(12), 1814–1818 (1992)
[13] Oustaloup, A.: La Derivation Non Entiere: Theorie, Synthese et Applications. Hermes, Paris (1995)
[14] Sun, H.H., Abdelwahad, A.A., Onaral, B.: Linear approximation of transfer function with a pole of fractional order. IEEE Trans. Autom. Control 29, 441–444 (1984) · Zbl 0532.93025
[15] Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999) · Zbl 0924.34008
[16] Ichise, M., Nagayanagi, Y., Kojima, T.: An analog simulation of noninteger order transfer functions for analysis of electrode process. J. Electroanal. Chem. 33, 253–265 (1971)
[17] Podlubny, I., Petráš, I., Vinagre, B.M., O’Leary, P., Dorcák, L.: Analogue realization of fractional-order controllers. Nonlinear Dyn. 29(1–4), 281–296 (2002) · Zbl 1041.93022
[18] Laskin, N.: Fractional market dynamics. Physica A 287, 482–492 (2000)
[19] Kusnezov, D., Bulgac, A., Dang, G.D.: Quantum Levy processes and fractional kinetics. Phys. Rev. Lett. 82, 1136–1139 (1999)
[20] Taubert, K.: Converging multistep methods for initial value problems involving multivalued maps. Computing 27, 123–136 (1981) · Zbl 0465.65038
[21] Dontchev, A., Lempio, F.: Difference methods for differential inclusions. SIAM Rev. 34, 263–294 (1992) · Zbl 0757.34018
[22] Kastner-Maresch, A., Lempio, F.: Difference methods with selection strategies for differential inclusions. Numer. Funct. Anal. Optim. 14, 555–572 (1993) · Zbl 0807.65085
[23] Danca, M.-F., Codreanu, S.: On a possible approximation of discontinuous dynamical systems. Chaos Solitons Fractals 13, 681–691 (2002) · Zbl 1046.34015
[24] Diethelm, K., Ford, N.J.: Analysis of fractional differential equations. J. Math. Anal. Appl. 265, 229–248 (2002) · Zbl 1014.34003
[25] Diethelm, K.: An algorithm for the numerical solution of differential equations of fractional order. Electron. Trans. Numer. Anal. 5, 1–6 (1997) · Zbl 0890.65071
[26] Lubich, C.: Discretized fractional calculus. SIAM J. Math. Anal. 17, 704–719 (1986) · Zbl 0624.65015
[27] Shokooh, A., Suarez, L.E.: A comparison of numerical methods applied to a fractional model of damping materials. J. Vib. Control 5, 331–354 (1999)
[28] Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon & Breach, Yverdon (1993) · Zbl 0818.26003
[29] Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes in C, The Art of Scientific Computing, 2nd edn. Cambridge University Press, Cambridge (1992) · Zbl 0845.65001
[30] Charef, A., Sun, H.H., Tsao, Y.Y., Onaral, B.: Fractal system as represented by singularity function. IEEE Trans. Automat. Control 37, 1465–1470 (1992) · Zbl 0825.58027
[31] Petráš, I.: Chaos in the fractional-order Volta’s system: modeling and simulation. Nonlinear Dyn. 57(1–2), 157–170 (2009) · Zbl 1176.34050
[32] Ahmad, W.M., Sprott, J.C.: Chaos in fractional-order autonomous nonlinear systems. Chaos Solitons Fractals 16, 339–351 (2003) · Zbl 1033.37019
[33] Hartley, T.T., Lorenzo, C.F., Qammer, H.K.: Chaos in a fractional order Chua’s system. IEEE T. Circuits-I 42(8), 485–490 (1995)
[34] Arena, P., Caponetto, R., Fortuna, L., Porto, D.: Chaos in a fractional order Duffing system. In: Proceedings ECCTD, Budapest, September, pp. 1259–1262 (1997)
[35] Wu, X.-J., Shen, S.-L.: Chaos in the fractional-order Lorenz system. Int. J. Comput. Math. 86(7), 1274–1282 (2009) · Zbl 1169.65115
[36] Lu, J.G.: Chaotic dynamics of the fractional-order Lü system and its synchronization. Phys. Lett. A 354, 305–311 (2006)
[37] Aziz-Alaoui, M.A., Chen, G.: Asymptotic analysis of a new piecewise-linear chaotic system. Int. J. Bifurc. Chaos 12(1), 147–157 (2002) · Zbl 1047.34055
[38] Sprott, J.C.: A new class of chaotic circuit. Phys. Lett. A 266, 19–23 (2000)
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