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 ordered Liu system. (English) Zbl 1189.34081
Summary: Present paper deals with fractional version of a dynamical system introduced by C. Liu, L. Liu and T. Liu [A novel three-dimensional autonomous chaos system, Chaos Solitons Fractals 39, No. 4, 1950–1958 (2009)]. Numerical investigations on the dynamics of this system have been carried out. Properties of the system have been analyzed by means of Lyapunov exponents. Furthermore the minimum effective dimensions have been identified for chaos to exist in commensurate and incommensurate orders. It is noteworthy that the results obtained are consistent with the analytical conditions given in the literature.
MSC:
34C28Complex behavior, chaotic systems (ODE)
26A33Fractional derivatives and integrals (real functions)
34A08Fractional differential equations
45J05Integro-ordinary differential equations
References:
[1]Podlubny, I.: Fractional differential equations, (1999)
[2]Samko, S. G.; Kilbas, A. A.; Marichev, O. I.: Fractional integrals and derivatives: theory and applications, (1993) · Zbl 0818.26003
[3]Grigorenko, I.; Grigorenko, E.: Chaotic dynamics of the fractional Lorenz system, Phys. rev. Lett. 91, 034101 (2003)
[4]Hartley, T. T.; Lorenzo, C. F.; Qammer, H. K.: Chaos in a fractional order Chua’s system, IEEE trans circuits syst. I 42, 485-490 (1995)
[5]Li, C.; Chen, G.: Chaos and hyperchaos in the fractional order Rössler equations, Physica A 341, 55-61 (2004)
[6]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 fractals 36, No. 1, 98-103 (2008) · Zbl 1152.37319 · doi:10.1016/j.chaos.2006.06.013
[7]Tavazoei, M. S.; Haeri, M.: Regular oscillations or chaos in a fractional order system with any effective dimension, Nonlinear dynam. 54, No. 3, 213-222 (2008) · Zbl 1187.70043 · doi:10.1007/s11071-007-9323-1
[8]D. Matignon, Stability results for fractional differential equations with applications to control processing. In: Computational Engineering in Systems and Application Multiconference, vol. 2, IMACS, IEEE-SMC Proceedings, Lille, France, July 1996, pp. 963–968
[9]Liu, C.; Liu, L.; Liu, T.: A novel three-dimensional autonomous chaos system, Chaos solitons fractals 39, No. 4, 1950-1958 (2009) · Zbl 1197.37039 · doi:10.1016/j.chaos.2007.06.079
[10]Rosenstein, M. T.; Collins, J. J.; De Luca, C. J.: A practical method for calculating largest Lyapunov exponents from small data sets, Physica D 65, 117-134 (1993) · Zbl 0779.58030 · doi:10.1016/0167-2789(93)90009-P
[11]Diethelm, K.; Ford, N. J.; Freed, A. D.: A predictor–corrector approach for the numerical solution of fractional differential equations, Nonlinear dynam. 29, 3-22 (2002) · Zbl 1009.65049 · doi:10.1023/A:1016592219341
[12]Diethelm, K.: An algorithm for the numerical solution of differential equations of fractional order, Electron trans. Numer. anal. 5, 1-6 (1997) · Zbl 0890.65071 · doi:emis:journals/ETNA/vol.5.1997/pp1-6.dir/pp1-6.html
[13]Diethelm, K.; Ford, N. J.: Analysis of fractional differential equations, J. math. Anal. appl. 265, 229-248 (2002) · Zbl 1014.34003 · doi:10.1006/jmaa.2001.7194
[14]Vukić, Z.; Kuljaća, L.: Nonlinear control systems, (2003)
[15]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
[16]Deng, W.; Li, C.; Lu, 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
[17]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
[18]Chua, L. O.; Komuro, M.; Matsumoto, T.: The double-scroll family, IEEE trans. Circuits syst. 33, 1072-1118 (1986) · Zbl 0634.58015 · doi:10.1109/TCS.1986.1085869
[19]Silva, C. P.: Shil’nikov’s theorem–A tutorial, IEEE trans. Circuits syst. I 40, 675-682 (1993) · Zbl 0850.93352 · doi:10.1109/81.246142
[20]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
[21]Lu, J.; Chen, G.; Yu, X.; Leung, H.: Design and analysis of multiscroll chaotic attractors from saturated function series, IEEE trans. Circuits syst. I 51, No. 12, 2476-2490 (2004)