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)
Chaos in fractional-order Genesio-Tesi system and its synchronization. (English) Zbl 1239.93020
Summary: In this paper, we study the chaotic dynamics of fractional-order Genesio-Tesi system. Theoretically, a necessary condition for occurrence of chaos is obtained. Numerical investigations on the dynamics of this system have been carried out and properties of the system have been analyzed by means of Lyapunov exponents. It is shown that in case of a commensurate system the lowest order of fractional-order Genesio-Tesi system to yield chaos is 2.79. Further, chaos synchronization of fractional-order Genesio-Tesi system is investigated via two different control strategies. Active control and sliding mode control are proposed and the stability of the controllers are studied. Numerical simulations have been carried out to verify the effectiveness of controllers.
MSC:
93B12Variable structure systems
93D09Robust stability of control systems
34A08Fractional differential equations
34H10Chaos control (ODE)
References:
[1]Grigorenko, I.; Grigorenko, E.: Chaotic dynamics of the fractional Lorenz system, Phys rev lett 91, 034101 (2003)
[2]Hartley, T. T.; Lorenzo, C. F.; Qammer, H. K.: Chaos in a fractional order Chua’s system, IEEE trans circ syst I 42, 485-490 (1995)
[3]Matouk, A. E.: Chaos, feedback control and synchronization of a fractional-order modified autonomous van der Pol – Duffing circuit, Commun nonlinear sci numer simul 16, 975-986 (2011) · Zbl 1221.93227 · doi:10.1016/j.cnsns.2010.04.027
[4]Ge, Z. M.; Ou, C. Y.: Chaos in a fractional order modified Duffing system, Chaos solitons fract 34, 262-291 (2007) · Zbl 1132.37324 · doi:10.1016/j.chaos.2005.11.059
[5]Ahmad, W. M.; Sprott, J. C.: Chaos in fractional order autonomous nonlinear systems, Chaos solitons fract 16, 339-351 (2003) · Zbl 1033.37019 · doi:10.1016/S0960-0779(02)00438-1
[6]Yu, Y.; Li, H. X.; Wang, S.; Yu, J.: Dynamic analysis of a fractional-order Lorenz chaotic system, Chaos solitons fract 42, 1181-1189 (2009) · Zbl 1198.37063 · doi:10.1016/j.chaos.2009.03.016
[7]Lu, J. G.; Chen, G.: A note on the fractional order Chen system, Chaos solitons fract 27, 685-688 (2006) · Zbl 1101.37307 · doi:10.1016/j.chaos.2005.04.037
[8]Lu, J. G.: Chaotic dynamics of the fractional order Lü system and its synchronization, Phys lett A 354, 305-311 (2006)
[9]Li, C.; Chen, G.: Chaos and hyperchaos in the fractional order Rössler equations, Physica A: statis mech appl 341, 55-61 (2004)
[10]Lu, J. G.: Chaotic dynamics and synchronization of fractional order arneodo’s systems, Chaos solitons fract 26, 1125-1133 (2005) · Zbl 1074.65146 · doi:10.1016/j.chaos.2005.02.023
[11]Sheu, L. J.; Chen, H. K.; Chen, J. H.; Tam, L. M.; Chen, W. C.; Lin, K. T.: Chaos in the Newton – leipnik system with fractional order, Chaos solitons fract 36, 98-103 (2008) · Zbl 1152.37319 · doi:10.1016/j.chaos.2006.06.013
[12]Gejji, V. D.; Bhalekar, S.: Chaos in fractional ordered Liu system, Comput math appl 59, 1117-1127 (2010) · Zbl 1189.34081 · doi:10.1016/j.camwa.2009.07.003
[13]Shahiri, M.; Ghaderi, R.; N., A. Ranjbar; Hosseinnia, S. H.; Momani, S.: Chaotic fractional-order coullet system: synchronization and control approach, Commun nonlinear sci numer simul 15, 665-674 (2010) · Zbl 1221.37222 · doi:10.1016/j.cnsns.2009.05.054
[14]Pecora, L. M.; Carroll, T. L.: Synchronization in chaotic systems, Phys rev lett 64, 821-824 (1990)
[15]Chen, G.; Dong, X.: From chaos to order: methodologies, Perspectives and applications (1998)
[16]Bocaletti, S.; Kurths, J.; Osipov, G.; Valladares, D.; Zhou, C.: The synchronization of chaotic systems, Phys rep 366, 1-101 (2002) · Zbl 0995.37022 · doi:10.1016/S0370-1573(02)00137-0
[17]Kolumbán, G.; Kennedy, M. P.; Chua, L. O.: The role of synchronization in digital communication using chaos, IEEE trans circ syst I 44, 927-936 (1997)
[18]Bai, E. W.; Lonngren, K. E.; Ucar, A.: Secure communication via multiple parameter modulation in a delayed chaotic system, Chaos solitons fract 23, 1071-1076 (2005) · Zbl 1068.94500 · doi:10.1016/j.chaos.2004.06.072
[19]Čelikovský, S.; Chen, G.: Secure synchronization of a class of chaotic systems from a nonlinear observer approach, IEEE trans automat control 50, 76-82 (2005)
[20]Wu, X.; Li, J.: Chaos control and synchronization of a three species food chain model via Holling functional response, Int J comput math 87, 199-214 (2010) · Zbl 1179.92068 · doi:10.1080/00207160801993232
[21]Yang, T.; Chua, L. O.: Secure communication via chaotic parameter modulation, IEEE trans circ syst I: Fund theor appl 43, No. 9 (1996)
[22]Liao, T. L.; Tsai, S. H.: Adaptive synchronization of chaotic systems and its application to secure communications, Chaos solitons fract 11, 1387-1396 (2000) · Zbl 0967.93059 · doi:10.1016/S0960-0779(99)00051-X
[23]Feki, M.: An adaptive chaos synchronization scheme applied to secure communication, Chaos solitons fract 18, 141-148 (2003) · Zbl 1048.93508 · doi:10.1016/S0960-0779(02)00585-4
[24]Chen, M.; Zhou, D.; Shang, Y.: A new observer-based synchronization scheme for private communication, Chaos solitons fract 24, 1025-1030 (2005) · Zbl 1069.94508 · doi:10.1016/j.chaos.2004.09.096
[25]Puebla, H.; Alvarez-Ramirez, J.: More secure communication using chained chaotic oscillators, Phys lett A 283, 96-108 (2001) · Zbl 0974.94002 · doi:10.1016/S0375-9601(01)00226-2
[26]Wang, Y. W.; Wen, C.; Yang, M.; Xiao, J. W.: Adaptive control and synchronization for chaotic systems with parametric uncertainties, Phys lett A 372, 2409-2414 (2008) · Zbl 1220.37030 · doi:10.1016/j.physleta.2007.11.066
[27]Lina, J. S.; Yan, J. J.: Adaptive synchronization for two identical generalized Lorenz chaotic systems via a single controller, Nonlinear anal: real world appl 10, 1151-1159 (2009) · Zbl 1167.37329 · doi:10.1016/j.nonrwa.2007.12.005
[28]Park, J. H.; Lee, S. M.; Kwon, O. M.: Adaptive synchronization of Genesio – Tesi chaotic system via a novel feedback control, Phys lett A 371, 263-270 (2007) · Zbl 1209.93122 · doi:10.1016/j.physleta.2007.06.020
[29]Falahpoor, M.; Ataei, M.; Kiyoumarsi, A.: A chattering-free sliding mode control design for uncertain chaotic systems, Chaos solitons fract 42, 1755-1765 (2009) · Zbl 1198.93014 · doi:10.1016/j.chaos.2009.03.082
[30]Vincent, U. E.: Synchronization of identical and non-identical 4-D chaotic systems using active control, Chaos solitons fract 37, 1065-1075 (2008) · Zbl 1153.37359 · doi:10.1016/j.chaos.2006.10.005
[31]Vincent, U. E.: Synchronization of identical and non-identical 4-D chaotic systems using active control, Chaos solitons fract 37, 1065-1075 (2008) · Zbl 1153.37359 · doi:10.1016/j.chaos.2006.10.005
[32]Bai, E. W.; Lonngren, K. E.: Synchronization of two Lorenz systems using active control, Chaos solitons fract 8, 51-58 (1997) · Zbl 1079.37515 · doi:10.1016/S0960-0779(96)00060-4
[33]Bai, E. W.; Lonngren, K. E.: Sequential synchronization of two Lorenz systems using active control, Chaos solitons fract 11, 1041-1044 (2000) · Zbl 0985.37106 · doi:10.1016/S0960-0779(98)00328-2
[34]Agiza, H. N.; Yassen, M. T.: Synchronization of Rössler and chaotic dynamical systems using active control, Phys lett A 278, 191-197 (2001) · Zbl 0972.37019 · doi:10.1016/S0375-9601(00)00777-5
[35]Ho, M. C.; Hung, Y. C.: Synchronization of two different systems by using generalized active control, Phys lett A 301, 424 (2002) · Zbl 0997.93081 · doi:10.1016/S0375-9601(02)00987-8
[36]Hung, M. L.; Lin, J. S.; Yan, J. J.; Liao, T. L.: Optimal PID control design for synchronization of delayed discrete chaotic systems, Chaos solitons fract 35, 781-785 (2008)
[37]Bowong, S.; Kakmeni, F. M.: Synchronization of uncertain chaotic systems via backstepping approach, Chaos solitons fract 21, 999-1011 (2004) · Zbl 1045.37011 · doi:10.1016/j.chaos.2003.12.084
[38]Zhang, J.; Li, C.; Zhang, H.; Yu, J.: Chaos synchronization using single variable feedback based on backstepping method, Chaos solitons fract 21, 1183-1193 (2004) · Zbl 1129.93518 · doi:10.1016/j.chaos.2003.12.079
[39]Delavari, H.; Ghaderi, R.; Ranjbar, A.; Momani, S.: Synchronization of chaotic nonlinear gyros using fractional order controller, New trends nanotechnol fractional calculus appl 5, 479-485 (2010) · Zbl 1222.93129 · doi:10.1007/978-90-481-3293-5_42
[40]Hosseinnia, S. H.; Ghaderi, R.; N., A. Ranjbar; Mahmoudian, M.; Momani, S.: Sliding mode synchronization of an uncertain fractional order chaotic system, Comput math appl 59, 1637-1643 (2010) · Zbl 1189.34011 · doi:10.1016/j.camwa.2009.08.021
[41]Bhalekar, S.; Daftardar-Gejji, V.: Synchronization of different fractional order chaotic systems using active control, Commun nonlinear sci numer simul 15, 3536-3546 (2010) · Zbl 1222.94031 · doi:10.1016/j.cnsns.2009.12.016
[42]Tavazoei, M. S.; Haeri, M.: Synchronization of chaotic fractional-order systems via active sliding mode controller, Physica A 387, 57-70 (2008)
[43]Shahiri, M.; Ghaderi, R.; N., A. Ranjbar; Hosseinnia, S. H.; Momani, S.: Chaotic fractional-order coullet system: synchronization and control approach, Commun nonlinear sci numer simul 15, 665-674 (2010) · Zbl 1221.37222 · doi:10.1016/j.cnsns.2009.05.054
[44]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 · doi:10.1016/j.cnsns.2007.11.011
[45]Genesio, R.; Tesi, A.: A harmonic balance methods for the analysis of chaotic dynamics in nonlinear systems, Automatica 28, 531-548 (1992) · Zbl 0765.93030 · doi:10.1016/0005-1098(92)90177-H
[46]Podlubny, I.: Fractional differential equations, (1999)
[47]Diethelm, K.; Ford, N. J.; Freed, A. D.: A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear dyn 29, 3-22 (2002) · Zbl 1009.65049 · doi:10.1023/A:1016592219341
[48]Diethelm, K.; Ford, N. J.; Freed, A. D.: Detailed error analysis for a fractional Adams method, Numer algor 36, 31-52 (2004) · Zbl 1055.65098 · doi:10.1023/B:NUMA.0000027736.85078.be
[49]Li, C.; Peng, G.: Chaos in Chen’s system with a fractional order, Chaos solitons fract 22, 443-450 (2004) · Zbl 1060.37026 · doi:10.1016/j.chaos.2004.02.013
[50]He, J. H.: A new approach to nonlinear partial differential equations, Comm nonlinear sci numer simul 2, 230-235 (1997) · Zbl 0923.35046 · doi:10.1016/S1007-5704(97)90029-0
[51]Tavazoei, M. S.; Haeri, M.: A necessary condition for double scroll attractor existence in fractional order systems, Phys lett A 367, 102-113 (2007) · Zbl 1209.37037 · doi:10.1016/j.physleta.2007.05.081
[52]Chua, L. O.; Komuro, M.; Matsumoto, T.: The double-scroll family, IEEE trans circ syst 33, 10721118 (1986) · Zbl 0634.58015 · doi:10.1109/TCS.1986.1085869
[53]Silva, C. P.: Shil’nikov’s theorem A tutorial, IEEE trans circ syst I 40, 675682 (1993) · Zbl 0850.93352 · doi:10.1109/81.246142
[54]Cafagna, D.; Grassi, G.: New 3-D-scroll attractors in hyperchaotic Chua’s circuit forming a ring, Int J bifur chaos 13, No. 10, 2889-2903 (2003) · Zbl 1057.37026 · doi:10.1142/S0218127403008284
[55]Lu, J.; Chen, G.; Yu, X.; Leung, H.: Design and analysis of multiscroll chaotic attractors from saturated function series, IEEE trans circ syst I 51, No. 12, 2476-2490 (2004)
[56]Matignon D, Stability results for fractional differential equations with applications to control processing, Computational Engineering in Systems and Application multi-conference, vol. 2, IMACS, In: IEEE-SMC Proceedings, Lille, France, July 1996, pp. 963 – 968.
[57]Slotine, J. J. E.: Applied nonlinear control, (1991) · Zbl 0753.93036