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

Li, Changpin; Peng, Guojun

Chaos Solitons Fractals 22, No. 2, 443-450 (2004).

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.

MSC:

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

Keywords:

numerical simulation; fractional calculus; chaos; Chen’s system; phase diagrams

Software:

FracPECE

References:

[1]	Butzer, P.L.; Westphal, U., An introduction to fractional calculus, (), 1-85 · Zbl 0987.26005
[2]	Podlubny, I., Fractional differential equations, (1999), 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 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, (), 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)