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Chaos generation via a switching fractional multi-model system. (English) Zbl 1186.34012
Summary: This paper introduces a system with switching multi-model structure which can generate chaos. Sub-models in this structure are fractional-order linear systems with any desired commensurate order less than 1. It shows that this system is capable of demonstrating chaotic behavior if its parameters and switching rule are suitably chosen. The structure of the proposed system is defined in a general form; consequently various chaotic attractors can be created by this system with different choices of order, parameters and switching rule. Numerical simulations illustrate behavior of the introduced system in some different situations.

34A08 Fractional ordinary differential equations
34C28 Complex behavior and chaotic systems of ordinary differential equations
34A36 Discontinuous ordinary differential equations
Full Text: DOI
[1] Khadra, A.; Liu, X.Z.; Shen, X., Impulsively synchronizing chaotic systems with delay and applications to secure communication, Automatica, 41, 1491-1502, (2005) · Zbl 1086.93051
[2] Yang, T.; Wu, C.W.; Chua, L.O., Cryptography based on chaotic system, IEEE transactions on circuits and systems I: fundamental theory and applications, 44, 469-472, (1997) · Zbl 0884.94021
[3] Tavazoei, M.S.; Haeri, M., An optimization algorithm based on chaotic behavior and fractal nature, Journal of computational and applied mathematics, 206, 2, 1070-1081, (2007) · Zbl 1151.90555
[4] D.C. Hamill, J.H.B. Deane, P.J. Aston, Some applications of chaos in power converters, IEE Colloquium on Update on New Power Electronic Techniques (Digest No: 1997/091), 23 May 1997, pp. 5/1-5/5
[5] Kyriazis, M., Practical applications of chaos theory to the modulation of human ageing: nature prefers chaos to regularity, Biogerontology, 4, 2, 75-90, (2003)
[6] W.L. Ditto, Applications of chaos in biology and medicine, in Chaos and the Changing Nature of Science and Medicine: An Introduction, AIP Conference Proceedings, 376, June 20, 1996, pp. 175-202
[7] H. Aref, Mixing well using chaos, American Physical Society, Annual March Meeting, March 17-21, 1997
[8] Chen, G.; Dong, X., From chaos to order: methodologies, perspectives and applications, (1998), World Scientific Singapore
[9] Tang, K.S.; Man, K.F.; Zhong, G.Q.; Chen, G., Generating chaos via \(x | x |\), IEEE transactions on circuits and systems I: fundamental theory and applications, 48, 5, 636-641, (2001) · Zbl 1010.34033
[10] Wang, X.F.; Chen, G., Generating topologically conjugate chaotic systems via feedback control, IEEE transactions on circuits and systems I: fundamental theory and applications, 50, 6, 812-817, (2003) · Zbl 1368.93108
[11] Zhong, Q.; Man, K.F.; Chen, G., Generating chaos via a dynamical controller, International journal of bifurcation and chaos, 11, 3, 865-869, (2001)
[12] Lü, J.; Chen, G., Generating multiscroll chaotic attractors: theories, methods and applications, International journal of bifurcation and chaos, 16, 775-858, (2006) · Zbl 1097.94038
[13] ()
[14] L. Sommacal, P. Melchior, J.M. Cabelguen, A. Oustaloup, A. Ijspeert, Fractional multimodels of the gastrocnemius frog muscle, in: 2nd IFAC Workshop on Fractional Differentiation and its Applications, Porto, Portugal, July 19-21, 2006 · Zbl 1125.92021
[15] L. Sommacal, P. Melchior, A. Dossat, J. Petit, J.M Cabelguen, A. Oustaloup, A.J. Ijspeert, A comparison between two fractional multimodels structures for rat muscles modeling, in: 6th IFAC Symposium on Modeling and Control in Biomedical Systems (Including Biological System), Reims, France, September 20-22, 2006
[16] L. Sommacal, A. Dossat, P. Melchior, J. Petit, J.M. Cabelguen, A. Oustaloup, N-step predictive algorithm based on fractional multimodel for rat muscle, in: 32nd Annual Conference of the IEEE Industrial Electronics Society, Paris, France, November 7-10, 2006
[17] L. Sommacal, P. Melchior, M. Aoun, J.M. Cabelguen, J. Petit, A. Oustaloup, A.J. Ijspeert, Modeling of a rat muscle using fractional multimodel, in: 2nd International Symposium on Communications, Control and Signal Processing, ISCCSP’06, Marrakech, Morocco, March 13-15, 2006
[18] L. Sommacal, P. Melchior, J.M. Cabelguen, A. Oustaloup, A.J. Ijspeert, Fractional model of a gastrocnemius muscle for tetanus pattern, in: 20th ASME International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, ASME IDETC/CIE’05, Long Beach, California, USA, September 26-28, 2005 · Zbl 1125.92021
[19] Hartley, T.T.; Lorenzo, C.F.; Qammer, H.K., Chaos in a fractional-order chua’s system, IEEE transactions on circuits and systems I, 42, 485-490, (1995)
[20] Petras, I., A note on the fractional-order chua’s system, Chaos, solitons and fractals, 38, 1, 140-147, (2008)
[21] P. Arena, R. Caponetto, L. Fortuna, D. Porto, Chaos in a fractional-order Duffing system, in: Proceedings ECCTD, Budapest, Hungry, 1997, pp. 1259-1262
[22] Ahmad, W.M.; Sprott, J.C., Chaos in fractional-order autonomous nonlinear systems, Chaos, solitons and fractals, 16, 339-351, (2003) · Zbl 1033.37019
[23] Grigorenko, I.; Grigorenko, E., Chaotic dynamics of the fractional Lorenz system, Physical review letters, 91, 034101, (2003)
[24] Li, C.; Chen, G., Chaos in the fractional order Chen system and its control, Chaos, solitons and fractals, 22, 3, 549-554, (2004) · Zbl 1069.37025
[25] Tavazoei, M.S.; Haeri, M., A necessary condition for double scroll attractor existence in fractional-order systems, Physics letters A, 367, 1-2, 102-113, (2007) · Zbl 1209.37037
[26] Li, C.; Chen, G., Chaos and hyperchaos in the fractional-order Rössler equations, Physica A: statistical mechanics and its applications, 341, 55-61, (2004)
[27] Lu, J.G., Chaotic dynamics and synchronization of fractional-order arneodo’s systems, Chaos, solitons and fractals, 26, 4, 1125-1133, (2005) · Zbl 1074.65146
[28] Sheu, L.J.; Chen, H.K.; Chen, J.H.; Tam, L.M.; Chen, W.C.; Lin, K.T.; Kang, Y., Chaos in the newton – leipnik system with fractional-order, Chaos, solitons and fractals, 36, 1, 98-103, (2008) · Zbl 1152.37319
[29] Lu, J.G., Chaotic dynamics and synchronization of fractional-order genesio – tesi systems, Chinese physics, 14, 1517-1521, (2005)
[30] Lu, J.G., Chaotic dynamics of the fractional-order ikeda delay system and its synchronization, Chinese physics, 15, 301-305, (2006)
[31] Arena, P.; Caponetto, R.; Fortuna, L.; Porto, D., Bifurcation and chaos in noninteger order cellular neural networks, Int. J. bifurcation chaos, 8, 7, 1527-1539, (1998) · Zbl 0936.92006
[32] Deng, W.; Lü, J., Design of multidirectional multiscroll chaotic attractors based on fractional differential systems via switching control, Chaos, 16, 043120, (2006) · Zbl 1146.37316
[33] Deng, W.; Lü, J., Generating multi-directional multi-scroll chaotic attractors via a fractional differential hysteresis system, Physics letters A, 369, 5-6, 438-443, (2007) · Zbl 1209.37032
[34] Podlubny, I., Fractional differential equations, (1999), Academic Press San Diego · Zbl 0918.34010
[35] D. Matignon, Stability results for fractional differential equations with applications to control processing, Computational Engineering in Systems and Application Multi-conference, vol. 2, pp. 963-968, IMACS, in: IEEE-SMC Proceedings, Lille, France, July 1996
[36] Lorenz, E.N., Deterministic non-periods flows, Journal of atmospheric science, 20, 130-141, (1963)
[37] Gleick, J., Chaos: making a new science, (1988), Penguin · Zbl 0706.58002
[38] Argyris, J.; Faust, G.; Haase, M., An exploration of chaos, (1996), Springer New York
[39] Linz, S.J., No-chaos criteria for certain classes of driven nonlinear oscillators, Acta physica polonica B, 34, 7, 3741-3749, (2003)
[40] Lü, J.; Zhou, T.; Chen, G.; Yang, X., Generating chaos with a switching piecewise-linear controller, Chaos, 12, 2, 344-349, (2002)
[41] Zheng, Z.; Lü, J.; Chen, G.; Zhou, T.; Zhang, S., Generating two simultaneously chaotic attractors with a switching piecewise-linear controller, Chaos, solitons and fractals, 20, 277-288, (2004) · Zbl 1045.37018
[42] Lü, J.; Yu, X.; Chen, G., Generating chaotic attractors with multiple merged basins of attraction: a switching piecewise-linear control approach, IEEE transactions on circuits and systems I, 50, 198-207, (2003) · Zbl 1368.37041
[43] Aziz-Alaoui, M.; Chen, G., Asymptotic analysis of a new piecewise-linear chaotic system, Int. J. bifurcation chaos, 12, 147-157, (2002) · Zbl 1047.34055
[44] Morel, C.; Bourcerie, M.; Chapeau-Blondeau, F., Generating independent chaotic attractors by chaos anticontrol in nonlinear circuits, Chaos, solitons and fractals, 26, 541-549, (2005) · Zbl 1153.94471
[45] Liu, X.; Teo, K.L.; Zhang, H.; Chen, G., Switching control of linear systems for generating chaos, Chaos, solitons and fractals, 30, 725-733, (2006) · Zbl 1143.93322
[46] Erramilli, A.; Forys, L.J., Oscillations and chaos in a flow model of a switching system, IEEE journal on selected areas in communications, 9, 2, 171-178, (1991)
[47] Tse, C., Complex behavior of switching power converters, (2003), CRC Press New York · Zbl 1017.94535
[48] Edwards, R., Analysis of continuous-time switching networks, Physica D, 146, 165-199, (2000) · Zbl 0986.94051
[49] Tavazoei, M.S.; Haeri, M., Regular oscillations or chaos in a fractional order system with any effective dimension, Nonlinear dynamics, 54, 213-222, (2008) · Zbl 1187.70043
[50] Diethelm, K.; Ford, N.J.; Freed, A.D., A predictor – corrector approach for the numerical solution of fractional differential equations, Nonlinear dynamics, 29, 3-22, (2002) · Zbl 1009.65049
[51] Tavazoei, M.S.; Haeri, M., Unreliability of frequency-domain approximation in recognizing chaos in fractional-order systems, IET signal processing, 1, 4, 171-181, (2007)
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