zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
A note on phase synchronization in coupled chaotic fractional order systems. (English) Zbl 1238.34121
Summary: The dynamic behaviors of fractional order systems have received increasing attention in recent years. This paper addresses the reliable phase synchronization problem between two coupled chaotic fractional order systems. An active nonlinear feedback control scheme is constructed to achieve phase synchronization between two coupled chaotic fractional order systems. We investigated the necessary conditions for fractional order Lorenz, Lü and Rössler systems to exhibit chaotic attractor similar to their integer order counterpart. Then, based on the stability results of fractional order systems, sufficient conditions for phase synchronization of the fractional models of Lorenz, Lü and Rössler systems are derived. The synchronization scheme that is simple and global enables synchronization of fractional order chaotic systems to be achieved without the computation of the conditional Lyapunov exponents. Numerical simulations are performed to assess the performance of the presented analysis.
MSC:
34H10Chaos control (ODE)
34A08Fractional differential equations
34D06Synchronization
93B52Feedback control
References:
[1]Chen, G.; Yu, X.: Chaos control: theory and applications, (2003)
[2]Yamada, T.; Fujisaka, H.: Stability theory of synchronized motion in coupled-oscillator systems, Progr. theoret. Phys. 70, 1240-1248 (1983) · Zbl 1171.70307 · doi:10.1143/PTP.70.1240
[3]Pecora, L. M.; Carrol, T. L.: Synchronization in chaotic systems, Phys. rev. Lett. 64, 821-824 (1990)
[4]Ott, E.; Grebogi, C.; Yorke, J. A.: Controlling chaos, Phys. rev. Lett. 64, 1196-1199 (1990) · Zbl 0964.37501 · doi:10.1103/PhysRevLett.64.1196
[5]Aziz-Alaoui, M. A.: Synchronization of chaos, Encyclopedia math. Phys., 213-226 (2006)
[6]Carroll, T. L.; Heagy, J. F.; Pecora, L. M.: Transforming signals with chaotic synchronization, Phys. rev. E 54, No. 5, 4676-4680 (1996)
[7]Kocarev, L.; Parlitz, U.: Generalized synchronization, predictability, and equivalence of unidirectionally coupled dynamical systems, Phys. rev. E 76, No. 11, 1816-1819 (1996)
[8]Rosenblum, M. G.; Pikovsky, A. S.; Kurths, J.: From phase to lag synchronization in coupled chaotic oscillators, Phys. rev. Lett. 78, No. 22, 4193-4196 (1996)
[9]Bai, E.; Lonngren, K.: Synchronization and control of chaotic systems, Chaos solitons fractals 10, No. 9, 1571-1575 (1999) · Zbl 0958.93513 · doi:10.1016/S0960-0779(98)00204-5
[10]Liu, J.; Ye, C.; Zhang, S.; Song, W.: Anti-phase synchronization in coupled map lattices, Phys. lett. A 274, No. 1–2, 27-29 (2000) · Zbl 1050.37518 · doi:10.1016/S0375-9601(00)00522-3
[11]Ho, M.; Hung, Y.; Chou, C.: Phase and anti-phase synchronization of two chaotic systems by using active control, Phys. lett. A 296, No. 1, 43-48 (2002) · Zbl 1098.37529 · doi:10.1016/S0375-9601(02)00074-9
[12]Petráš, I.: A note on the fractional-order Chua’s system, Chaos solitons fractals 38, No. 1, 140-147 (2008)
[13]Ge, Z. M.; Ou, C. Y.: Chaos in a fractional order modified Duffing system, Chaos solitons fractals 34, No. 2, 262-291 (2007) · Zbl 1132.37324 · doi:10.1016/j.chaos.2005.11.059
[14]Hartley, T.; Lorenzo, C.; Qammer, H.: Chaos in a fractional order Chua’s system, IEEE trans. Circuits systems 42, 485-490 (1995)
[15]Li, C.; Chen, G.: Chaos and hyperchaos in fractional order Rössler equations, Physica A 341, 55-61 (2004)
[16]Li, C.; Chen, G.: Chaos in the fractional order Chen system and its control, Chaos solitons fractals 22, No. 3, 549-554 (2004) · Zbl 1069.37025 · doi:10.1016/j.chaos.2004.02.035
[17]Lü, J. G.; Chen, G.: A note on the fractional-order Chen system, Chaos solitons fractals 27, No. 3, 685-688 (2006) · Zbl 1101.37307 · doi:10.1016/j.chaos.2005.04.037
[18]Grigorenko, I.; Grigorenko, E.: Chaotic dynamics of the fractional Lorenz system, Phys. rev. Lett. 91, No. 3, 034101 (2003)
[19]Arneodo, A.; Coullet, P.; Spiegel, E.; Tresser, C.: Asymptotic chaos, Physica D 14, No. 3, 327-347 (1985) · Zbl 0595.58030 · doi:10.1016/0167-2789(85)90093-4
[20]Deng, W. H.; Li, C. P.: Chaos synchronization of the fractional Lü system, Physica A 353, 61-72 (2005)
[21]Li, C.; Zhou, T.: Synchronization in fractional-order differential systems, Physica D 212, No. 1–2, 111-125 (2005) · Zbl 1094.34034 · doi:10.1016/j.physd.2005.09.012
[22]Zhou, S.; Li, H.; Zhu, Z.: Chaos control and synchronization in a fractional neuron network system, Chaos solitons fractals 36, No. 4, 973-984 (2008) · Zbl 1139.93320 · doi:10.1016/j.chaos.2006.07.033
[23]Peng, G.: Synchronization of fractional order chaotic systems, Phys. lett. A 363, No. 5–6, 426-432 (2007) · Zbl 1197.37040 · doi:10.1016/j.physleta.2006.11.053
[24]Sheu, L. J.; Chen, H. K.; Chen, J. H.; Tam, L. M.: Chaos in a new system with fractional order, Chaos solitons fractals 31, No. 5, 1203-1212 (2007)
[25]Yan, J.; Li, C.: On chaos synchronization of fractional differential equations, Chaos solitons fractals 32, No. 2, 725-735 (2007) · Zbl 1132.37308 · doi:10.1016/j.chaos.2005.11.062
[26]Li, C.; Yan, J.: The synchronization of three fractional differential systems, Chaos solitons fractals 32, No. 2, 751-757 (2007)
[27]Wang, J.; Xiong, X.; Zhang, Y.: Extending synchronization scheme to chaotic fractional-order Chen systems, Physica A 370, No. 2, 279-285 (2006)
[28]Li, C. P.; Deng, W. H.; Xu, D.: Chaos synchronization of the Chua system with a fractional order, Physica A 360, No. 2, 171-185 (2006)
[29]Zhu, H.; Zhou, S.; Zhang, J.: Chaos and synchronization of the fractional-order Chua’s system, Chaos solitons fractals 39, No. 4, 1595-1603 (2009) · Zbl 1197.94233 · doi:10.1016/j.chaos.2007.06.082
[30]Wu, X.; Wang, H.: A new chaotic system with fractional order and its projective synchronization, Nonlinear dynam. 61, No. 3, 407-417 (2010) · Zbl 1204.37035 · doi:10.1007/s11071-010-9658-x
[31]Odibat, Z.: Adaptive feedback control and synchronization of non-identical chaotic fractional order systems, Nonlinear dynam. 60, No. 4, 479-487 (2010) · Zbl 1194.93105 · doi:10.1007/s11071-009-9609-6
[32]Odibat, Z.; Corson, N.; Aziz-Alaoui, M. A.; Bertelle, C.: Synchronization of chaotic fractional-order systems via linear control, Internat. J. Bifur. chaos 20, No. 1, 81-97 (2010) · Zbl 1183.34095 · doi:10.1142/S0218127410025429
[33]Li, C.: Phase and lag synchronization in coupled fractional order chaotic oscillators, Internat. J. Modern phys. B 21, No. 30, 5159-5166 (2007)
[34]Erjaee, G. H.; Momani, S.: Phase synchronization in fractional differential chaotic systems, Phys. lett. A 372, No. 14, 2350-2354 (2008) · Zbl 1220.34004 · doi:10.1016/j.physleta.2007.11.065
[35]Oldham, K. B.; Spanier, J.: The fractional calculus, (1974)
[36]Podlubny, I.: Fractional differential equations, (1999)
[37]Hilfer, R.: Applications of fractional calculus in physics, (2000)
[38]R. Gorenflo, F. Mainardi, Fractional calculus: integral and differential equations of fractional order, in: Carpinteri and Mainardi (Ed.), Fractals and Fractional Calculus, New York, 1997.
[39]Caputo, M.: Linear models of dissipation whose Q is almost frequency independent, part II, J. R. Astron. soc. 13, 529-539 (1967)
[40]D. Matignon, Stability results of fractional differential equations with applications to control processing, in: Proceeding of IMACS, IEEE-SMC, Lille, France, 1996, pp. 963–968.
[41]Deng, W.; Li, C.; Lü, J.: Stability analysis of linear fractional differential system with multiple time delays, Nonlinear dynam. 48, 409-416 (2007) · Zbl 1185.34115 · doi:10.1007/s11071-006-9094-0
[42]Ahmed, E.; El-Sayed, A. M.; El-Saka, H.: Equilibrium points, stability and numerical solutions of fractional-order predator–prey and rabies models, J. math. Anal. appl. 325, No. 1, 542-553 (2007) · Zbl 1105.65122 · doi:10.1016/j.jmaa.2006.01.087
[43]Tavazoei, M. S.; Haeri, M.: A note on the stability of fractional order systems, Math. comput. Simulation 79, No. 5, 1566-1576 (2009) · Zbl 1168.34036 · doi:10.1016/j.matcom.2008.07.003
[44]Lorenz, E.: Deterministic nonperiodic flow, J. atmos. Sci. 20, 130-141 (1963)
[45]Tavazoei, M. S.; Haeri, M.: Chaotic attractors in incommensurate fractional order systems, Physica D 237, 2628-2637 (2008) · Zbl 1157.26310 · doi:10.1016/j.physd.2008.03.037
[46]Lü, J.; Chen, G.: A new chaotic attractor coined, Internat. J. Bifur. chaos 12, No. 3, 659-661 (2002) · Zbl 1063.34510 · doi:10.1142/S0218127402004620
[47]Yongguang, Y.; Suochun, Z.: Controlling uncertain Lü system using backstepping design, Chaos solitons fractals 15, No. 5, 897-902 (2003)
[48]Rossler, O. E.: An equation for continuous chaos, Phys. lett. A 57, No. 5, 397-398 (1976)