zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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 necessary condition for double scroll attractor existence in fractional-order systems. (English) Zbl 1209.37037
Summary: Based on the stability theorem in fractional differential equations, a necessary condition is given to check existence of double scroll attractor in a fractional-order system. Numerical simulations are presented to evaluate accuracy of this condition in fractional-order Chen and Lü systems. Also, we show that using frequency domain approximation in the numerical simulations of fractional systems may result in wrong consequences. For example, this approximation can numerically demonstrate chaos in the non-chaotic fractional-order systems. Unfortunately, this mistake has occurred in the recent literature that found the lowest-order chaotic systems among fractional-order systems.

37D45Strange attractors, chaotic dynamics
34A08Fractional differential equations
Full Text: DOI
[1] Chen, G.; Yu, X.: Chaos control: theory and applications. (2003) · Zbl 1029.00015
[2] Sun, H. H.; Abdelwahad, A. A.; Onaral, B.: IEEE trans. Automat. control. 29, 441 (1984)
[3] Ichise, M.; Nagayanagi, Y.; Kojima, T.: J. electro-anal. Chem.. 33, 253 (1971)
[4] Heaviside, O.: Electromagnetic theory. (1971) · Zbl 30.0801.03
[5] Bagley, R. L.; Calico, R. A.: J. guid. Control dyn.. 14, 304 (1991)
[6] Kusnezov, D.; Bulgac, A.; Dang, G. D.: Phys. rev. Lett.. 82, 1136 (1999)
[7] Laskin, N.: Physica A. 287, 482 (2000)
[8] El-Sayed, A. M. A.: Int. J. Theor. phys.. 35, No. 2, 311 (1996)
[9] Hartley, T. T.; Lorenzo, C. F.; Qammer, H. K.: IEEE trans. CAS-I. 42, 485 (1995)
[10] P. Arena, R. Caponetto, L. Fortuna, D. Porto, Chaos in a fractional order Duffing system, in: Proc. ECCTD, Budapest, 1997, pp. 1259 -- 1262
[11] Ahmad, W. M.; Sprott, J. C.: Chaos solitons fractals. 16, 339 (2003)
[12] Lu, J. G.; Chen, G.: Chaos solitons fractals. 27, No. 3, 685 (2006)
[13] Lu, J. G.: Phys. lett. A. 354, No. 4, 305 (2006)
[14] Li, C.; Chen, G.: Physica A: stat. Mech. appl.. 341, 55 (2004)
[15] Lu, J. G.: Chaos solitons fractals. 26, No. 4, 1125 (2005)
[16] L.J. Sheu, H.K. Chen, J.H. Chen, L.M. Tam, W.C. Chen, K.T. Lin, Y. Kang, Chaos in the Newton -- Leipnik system with fractional order, Chaos Solitons Fractals (2006), in press · Zbl 1152.37319
[17] Podlubny, I.: Fractional differential equations. (1999) · Zbl 0924.34008
[18] Diethelm, K.; Ford, N. J.; Freed, A. D.: Numer. algorithms. 36, 31 (2004)
[19] Li, C.; Peng, G.: Chaos solitons fractals. 22, 443 (2004)
[20] Diethelm, K.; Ford, N. J.; Freed, A. D.: Nonlinear dyn.. 29, 3 (2002)
[21] Diethelm, K.: Electron. trans. Numer. anal.. 5, 1 (1997)
[22] Diethelm, K.; Ford, N. J.: J. math. Anal. appl.. 265, 229 (2002)
[23] Charef, A.; Sun, H. H.; Tsao, Y. Y.; Onaral, B.: IEEE trans. Automat. control. 37, No. 9, 1465 (1992)
[24] Ahmed, E.; El-Sayed, A. M. A.; El-Saka, H. A. A.: J. math. Anal. appl.. 325, No. 1, 542 (2007)
[25] D. Matignon, Stability result on fractional differential equations with applications to control processing, in: IMACS-SMC Proceedings, Lille, France, 1996, pp. 963 -- 968
[26] Chua, L. O.; Komuro, M.; Matsumoto, T.: IEEE trans. Circuits syst.. 33, 1072 (November 1986)
[27] Silva, C. P.: IEEE trans. Circuits syst. I. 40, 675 (October 1993)
[28] Cafagna, D.; Grassi, G.: Int. J. Bifur. chaos. 13, No. 10, 2889 (2003)
[29] Lü, J.; Chen, G.; Yu, X.; Leung, H.: IEEE trans. Circuits syst. I (Reg. Papers). 51, No. 12, 2476 (December 2004)
[30] Chen, G.; Ueta, T.: Int. J. Bifur. chaos. 9, 1465 (1999)
[31] Lü, J.; Chen, G.; Zhang, S.: Int. J. Bifur. chaos. 12, No. 3, 659 (2002)
[32] Oustaloup, A.: La commande CRONE: commande robuste d’ordre non entier. (1991)