×

Fractional-order attractors synthesis via parameter switchings. (English) Zbl 1222.37107

Summary: We provide numerical evidence, via graphics generated with the help of computer simulations, that switching the control parameter of a dynamical system belonging to a class of fractional-order systems in a deterministic way, one obtains an attractor which belongs to the class of all admissible attractors of the considered system. For this purpose, while a multistep numerical method for fractional-order differential equations approximates the solution to the mathematical model, the control parameter is switched periodically every few integration steps. The switch is made inside of a considered set of admissible parameter values. Moreover, the synthesized attractor matches the attractor obtained with the control parameter replaced with the averaged switched parameter values. The results are verified in this paper on a representative system, the fractional-order Lü system. In this way we were able to extend the applicability of the algorithm presented in earlier papers using a numerical method for fractional differential equations.

MSC:

37N35 Dynamical systems in control
34A08 Fractional ordinary differential equations
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
26A33 Fractional derivatives and integrals
37C70 Attractors and repellers of smooth dynamical systems and their topological structure
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Danca, M.-F.; Tang, W. K.S.; Chen, G., A switching scheme for synthesizing attractors of dissipative chaotic systems, Appl Math Comput, 201, 650-667 (2008) · Zbl 1147.65104
[2] Danca, M.-F., Random parameter-switching synthesis of a class of hyperbolic attractors, Chaos, 18, 033111 (2008) · Zbl 1309.93062
[3] Lu, J. G., Chaotic dynamics of the fractional-order Lü system and its synchronization, Phys Lett A, 354, 305-311 (2006)
[4] Hirsch, M.; Pugh, C., Stable manifolds and hyperbolic sets, Proc Sympos Pure Math, 14, 133-164 (1970) · Zbl 0215.53001
[5] Hirsch, W. M.; Smale, S.; Devaney, L. R., Differential equations dynamical systems and an introduction to chaos (2004), Elsevier: Elsevier Amsterdam · Zbl 1135.37002
[6] Kapitanski, L.; Rodnianski, I., Shape and Morse theory of attractors, Commun Pure Appl Math, 53, 218-242 (2000) · Zbl 1026.37007
[7] Temam, R., Infinite dimensional dynamical systems in mechanics and physics (1988), Springer: Springer Berlin · Zbl 0662.35001
[8] Foias, C.; Jolly, M. S., On the numerical algebraic approximation of global attractors, Nonlinearity, 8, 95-319 (1995) · Zbl 0837.34052
[9] Milnor, J., On the concept of attractor, Commun Math Phys, 99, 177-195 (1985) · Zbl 0595.58028
[10] Podlubny, I., Fractional differential equations (1999), Academic Press: Academic Press San Diego · Zbl 0918.34010
[11] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional integrals and derivatives (1999), Gordon and Breach: Gordon and Breach Yverdon
[12] Ahmed, E.; Elgazzar, A. S., On fractional order differential equations model for nonlocal epidemics, Physica A, 379, 607-614 (2007)
[13] Chen, W., Nonlinear dynamics and chaos in a fractional-order financial system, Chaos, Solitons & Fractals, 36, 1305-1314 (2008)
[14] El-Sayed, A. M.A.; El-Mesiry, A. E.M.; El-Saka, H. A.A., On the fractional-order logistic equation, Appl Math Lett, 20, 817-823 (2007) · Zbl 1140.34302
[15] (Hilfer, R., Applications of fractional calculus in physics (2001), World Scientific: World Scientific New Jersey) · Zbl 0998.26002
[16] Kiani, B. A.; Fallahi, K.; Pariz, N.; Leung, H., A chaotic secure communication scheme using fractional chaotic systems based on an extended fractional Kalman filter, Commun Nonlinear Sci Numer Simul, 14, 863-879 (2009) · Zbl 1221.94049
[17] Laskin, N., Fractional market dynamics, Physica A, 287, 482-492 (2000)
[18] Oustaloup A. La Dérivation Non Entière: Théorie, Synthèse et Applications (Hermes, Paris); 1995.; Oustaloup A. La Dérivation Non Entière: Théorie, Synthèse et Applications (Hermes, Paris); 1995.
[19] Podlubny, I.; Petráš, I.; Vinagre, B. M.; O’Leary, P.; Dorcák, L., Analogue realization of fractional-order controllers, Nonlinear Dynam, 29, 281-296 (2002) · Zbl 1041.93022
[20] Sun, H. H.; Abdelwahab, A. A.; Onaral, B., Linear approximation of transfer function with a pole of fractional order, IEEE Trans Automat Contr, 29, 441-444 (1984) · Zbl 0532.93025
[21] Diethelm, K.; Ford, N. J.; Freed, A. D., A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynam, 29, 3-22 (2002) · Zbl 1009.65049
[22] Diethelm, K.; Ford, N. J.; Freed, A. D., Detailed error analysis for a fractional Adams method, Numer Algorithms, 36, 31-52 (2004) · Zbl 1055.65098
[23] Tavazoei, M. S., Comments on stability analysis of a class of nonlinear fractional-order systems, IEEE Trans Circ Syst II, 56, 519-520 (2009)
[24] Tavazoei, M. S.; Haeri, M.; Bolouki, S.; Siami, M., Stability preservation analysis for frequency-based methods in numerical simulation of fractional-order systems, SIAM J Numer Anal, 47, 321-338 (2008) · Zbl 1203.26012
[25] Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P., Numerical recipes in C: the art of scientific computing (1992), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0845.65001
[26] Petráš, I., Chaos in the fractional-order Volta’s system: modeling and simulation, Nonlinear Dynam, 57, 157-170 (2009) · Zbl 1176.34050
[27] Hartley, T. T.; Lorenzo, C. F.; Killory Qammar, H., Chaos in a fractional order Chua’s system, IEEE Trans Circ Syst I, 42, 485-490 (1995)
[28] Lü, J.; Chen, G.; Cheng, D.; Celikovsky, S., Bridge the gap between the Lorenz system and the Chen system, Int J Bifurcat Chaos, 12, 2917-2926 (2002) · Zbl 1043.37026
[29] Falconer, K., Fractal geometry: mathematical foundations and applications (1990), John Wiley & Sons: John Wiley & Sons Chichester · Zbl 0689.28003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.