Chaos in Chen’s system with a fractional order. (English) Zbl 1060.37026

Summary: By utilizing the fractional calculus techniques, we found that chaos does exist in Chen’s system with a fractional order, and some phase diagrams are constructed.


37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
28A80 Fractals


Full Text: DOI


[1] Butzer, P. L.; Westphal, U., An introduction to fractional calculus, (Hilfer, R., Applications of fractional calculus in physics (2000), World Scientific: World Scientific Singapore), 1-85 · Zbl 0987.26005
[2] Podlubny, I., Fractional differential equations (1999), Academic Press: Academic Press San Diego · Zbl 0918.34010
[3] Samko, S. G.; Kilbas, A. A.; Marichev, O. I., Fractional integrals and derivatives: theory and applications (1993), Gordan and Breach: Gordan and Breach Amsterdam · Zbl 0818.26003
[4] Caputo, M., Linear models of dissipation whose \(Q\) is almost frequency independent, II, Geophys. J. R. Astron. Soc., 13, 529-539 (1967)
[5] Chen, G.; Ueta, T., Yet another chaotic attractor, Int. J. Bifurc. Chaos, 9, 1465-1466 (1999) · Zbl 0962.37013
[6] Diethelm, K., An algorithm for the numerical solution of differential equations of fractional order, Elec. Trans. Numer. Anal., 5, 1-6 (1997) · Zbl 0890.65071
[7] Diethelm, K.; Ford, N. J., Analysis of fractional differential equations, J. Math. Anal. Appl., 265, 229-248 (2002) · Zbl 1014.34003
[8] 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
[9] Diethelm, K.; Freed, A. D., The FracPECE subroutine for the numerical solution of differential equations of fractional order, (Heinzel, S.; Plesser, T., Forschung und wissenschaftliches Rechnen (1999), Gesellschaft für wissenschaftliche Datenverarbeitung: Gesellschaft für wissenschaftliche Datenverarbeitung Göttingen), 57-71
[10] Wolf, A.; Swinney, J. B.; Swinney, H. L.; Vastano, J. A., Determining Lyapunov exponents from a time series, Phys. D, 16, 285-317 (1985) · Zbl 0585.58037
[11] Sato, S.; Sano, M.; Sawada, Y., Practical methods of measuring the generalized dimension and the largest Lyapunov exponent in high dimensional chaotic system, Prog. Theor. Phys., 77, 1-5 (1987)
[12] Rosenstein, M. T.; Collins, J. J.; De Luca, C. J., A practical method for calculating largest Lyapunov exponents from small data sets, Phys. D, 65, 117-134 (1993) · Zbl 0779.58030
[13] Rosenstein, M. T.; Collins, J. J.; De Luca, C. J., Reconstruction expansion as a geometry-based framework for choosing delay times, Phys. D, 73, 82-98 (1994)
[14] Charef, A.; Sun, H. H.; Tsao, Y. Y.; Onaral, B., Fractional systems as represented by singularity function, IEEE Trans. Auto Control, 37, 9, 1465-1470 (1993) · Zbl 0825.58027
[15] Hartley, T. T.; Lorenzo; Qammer, H. K., Chaos in a fractional order Chua’s system, IEEE Trans. CAS I, 42, 8, 485-490 (1995)
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.