×

zbMATH — the first resource for mathematics

Chaotic dynamics and synchronization of fractional-order Arneodo’s systems. (English) Zbl 1074.65146
Summary: We numerically investigate the chaotic behaviors of the fractional-order Arneodo’s system [cf. A. Arneodo, P. H. Coullet, E. A. Spiegel and C. Tresser [Physica D 14, 327–347 (1985; Zbl 0595.58030)]. We find that chaos exists in the fractional-order Arneodo’s system with order less than 3. The lowest order we find to have chaos is 2.1 in this fractional-order Arneodo’s system. Our results are validated by the existence of a positive Lyapunov exponent. The linear and nonlinear drive-response synchronization methods are also presented for synchronizing the fractional-order chaotic Arneodo’s systems only using a scalar drive signal. The two approaches, based on stability theory of fractional-order systems, are simple and theoretically rigorous. They do not require the computation of the conditional Lyapunov exponents. Simulation results are used to visualize and illustrate the effectiveness of the proposed synchronization methods.

MSC:
65P20 Numerical chaos
37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37M25 Computational methods for ergodic theory (approximation of invariant measures, computation of Lyapunov exponents, entropy, etc.)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Podlubny, I., Fractional differential equations, (1999), Academic Press New York · Zbl 0918.34010
[2] Bagley, R.L.; Calico, R.A., Fractional order state equations for the control of viscoelastically damped structures, J guid control dynam, 14, 304-311, (1991)
[3] Sun, H.H.; Abdelwahad, 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
[4] 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)
[5] Heaviside, O., Electromagnetic theory, (1971), Chelsea New York · JFM 30.0801.03
[6] Laskin, N., Fractional market dynamics, Physica A, 287, 482-492, (2000)
[7] Kusnezov, D.; Bulgac, A.; Dang, G.D., Quantum levy processes and fractional kinetics, Phys rev lett, 82, 1136-1139, (1999)
[8] Oustaloup, A.; Levron, F.; Nanot, F.; Mathieu, B., Frequency band complex non integer differentiator: characterization and synthesis, IEEE trans CAS-I, 47, 25-40, (2000)
[9] Chen, Y.Q.; Moore, K., Discretization schemes for fractional-order differentiators and integrators, IEEE trans CAS-I, 49, 363-367, (2002) · Zbl 1368.65035
[10] Hartley, T.T.; Lorenzo, C.F., Dynamics and control of initialized fractional-order systems, Nonlinear dynam, 29, 201-233, (2002) · Zbl 1021.93019
[11] Hwang, C.; Leu, J.-F.; Tsay, S.-Y., A note on time-domain simulation of feedback fractional-order systems, IEEE trans auto contr, 47, 625-631, (2002) · Zbl 1364.93772
[12] Podlubny, I.; Petras, I.; Vinagre, B.M.; O’Leary, P.; Dorcak, L., Analogue realizations of fractional-order controllers, Nonlinear dynam, 29, 281-296, (2002) · Zbl 1041.93022
[13] Hartley, T.T.; Lorenzo, C.F.; Qammer, H.K., Chaos in a fractional order chua’s system, IEEE trans CAS-I, 42, 485-490, (1995)
[14] Arena P, Caponetto R, Fortuna L, Porto D. Chaos in a fractional order Duffing system. In: Proc. ECCTD. Budapest; 1997. p. 1259-62
[15] Ahmad, W.M.; Sprott, J.C., Chaos in fractional-order autonomous nonlinear systems, Chaos, solitons & fractals, 16, 339-351, (2003) · Zbl 1033.37019
[16] Ahmad, W.M.; Harb, W.M., On nonlinear control design for autonomous chaotic systems of integer and fractional orders, Chaos, solitons & fractals, 18, 693-701, (2003) · Zbl 1073.93027
[17] Ahmad, W.; El-Khazali, R.; El-Wakil, A., Fractional-order wien-bridge oscillator, Electr lett, 37, 1110-1112, (2001)
[18] Grigorenko, I.; Grigorenko, E., Chaotic dynamics of the fractional Lorenz system, Phys rev lett, 91, 034101, (2003)
[19] Arena, P.; Caponetto, R.; Fortuna, L.; Porto, D., Bifurcation and chaos in noninteger order cellular neural networks, Int J bifuract chaos, 7, 1527-1539, (1998) · Zbl 0936.92006
[20] Arena, P.; Fortuna, L.; Porto, D., Chaotic behavior in noninteger-order cellular neural networks, Phys rev E, 61, 776-781, (2000)
[21] Li, C.G.; Chen, G., Chaos and hyperchaos in fractional order Rössler equations, Physica A, 341, 55-61, (2004)
[22] Li, C.G.; Chen, G., Chaos in the fractional order Chen system and its control, Chaos, solitons & fractals, 22, 549-554, (2004) · Zbl 1069.37025
[23] Li, C.P.; Peng, G.J., Chaos in chen’s system with a fractional order, Chaos, solitons & fractals, 22, 443-450, (2004) · Zbl 1060.37026
[24] Zaslavsky, G.M., Chaos, fractional kinetics, and anomalous transport, Phys rep, 371, 461-580, (2002) · Zbl 0999.82053
[25] Chen G, Fradkov AL. Chaos control and synchronization bibliographies (1987-2001). Available: http://www.ee.cityu.edu.hk/ gchen/chaos-papers.html
[26] Pecora, L.M.; Carroll, T.L., Synchronization in chaotic systems, Phys rev lett, 64, 821-824, (1990) · Zbl 0938.37019
[27] Li, C.; Liao, X.; Yu, J., Synchronization of fractional order chaotic systems, Phys rev E, 68, 067203, (2003)
[28] Arneodo, A.; Coullet, P.; Spiegel, E.; Tresser, C., Asymptotic chaos, Physica D, 14, 3, 327-347, (1985) · Zbl 0595.58030
[29] Podlubny, I., Geometric and physical interpretation of fractional integration and fractional differentiation, Fract calc appl anal, 5, 4, 367-386, (2002) · Zbl 1042.26003
[30] Charef, A.; Sun, H.H.; Tsao, Y.Y.; Onaral, B., Fractal system as represented by singularity function, IEEE trans auto contr, 37, 1465-1470, (1992) · Zbl 0825.58027
[31] Chen, C.T., Linear system theory and design, (1984), Holt Rinehart, Winston, Inc. New York
[32] Matignon D. Stability results of fractional differential equations with applications to control processing. In: IMACS, IEEE-SMC, Lille, France, p. 963-8
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.