A Necessary Condition for the Existence of Chaos in Fractional Order Delay Differential Equations
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A Necessary Condition for the Existence of Chaos in Fractional Order Delay Differential Equations

Authors: Sachin Bhalekar

Abstract:

In this paper we propose a necessary condition for the existence of chaos in delay differential equations of fractional order. To explain the proposed theory, we discuss fractional order Liu system and financial system involving delay.

Keywords: Caputo derivative, delay, stability, chaos.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1087916

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[1] Podlubny I. Fractional Differential Equations. Academic Press, San Diego, 1999.
[2] Sabatier J, Poullain S, Latteux P, Thomas J, Oustaloup A. Robust speed control of a low damped electromechanical system based on CRONE control: Application to a four mass experimental test bench. Nonlinear Dyn. 2004; 38: 383–400.
[3] Caputo M, Mainardi F. A new dissipation model based on memory mechanism. Pure Appl. Geophys. 1971; 91: 134–147.
[4] Mainardi F, Luchko Y, Pagnini G. The fundamental solution of the spacetime fractional diffusion equation. Frac. Calc. Appl. Anal. 2001; 4(2): 153–192.
[5] Agrawal OP. Formulation of Euler–Lagrange equations for fractional variational problems. J. Math. Anal. Appl. 2002; 272: 368–379.
[6] Anastasio TJ. The fractional-order dynamics of Brainstem Vestibulo– Oculomotor neurons. Biol. Cybernet. 1994; 72: 69–79.
[7] Magin RL. Fractional Calculus in Bioengineering. Begll House Publishers, USA, 2006.
[8] Mainardi F, Raberto M, Gorenflo R, Scalas E. Fractional calculus and continuous-time finance, II: The waiting-time distribution. Physica A 2000; 287: 468–481.
[9] Machado JAT. And I say to myself: ”What a fractional world!”. Frac. Calc. Appl. Anal. 2011; 14: 635–654.
[10] Si-Ammour A, Djennoune S, Bettayeb M. A sliding mode control for linear fractional systems with input and state delays. Commun. Nonlinear Sci. Numer. Simul. 2009; 14: 2310–2318.
[11] Feliu V, Rivas R, Castillo F. Fractional order controller robust to time delay for water distribution in an irrigation main canal pool. Computers and Electronics in Agriculture 2009; 69(2): 185-197.
[12] Wang D, Yu J. Chaos in the fractional order logistic delay system. J. Electronic Sci. Tech. of China 2008; 6(3): 225–229.
[13] Bhalekar S, Daftardar-Gejji V, Baleanu D, Magin R. Fractional bloch equation with delay, Comput. Math. Appl. 2011; 61: 1355–1365.
[14] Smith H. An Introduction to Delay Differential Equations with Applications to the Life Sciences, Texts in Applied Mathematics 57. Springer, New York, 2010.
[15] Matignon D. Stability results for fractional differential equations with applications to control processing. In: Computational Engineering in Systems and Application multi conference, IMACS, IEEE-SMC proceedings, Lille, France, July, vol. 2; 1996. p. 963–8.
[16] Hwang C, Cheng YC. A numerical algorithm for stability testing of fractional delay systems. Automatica 2006; 42: 825-831.
[17] S.E. Hamamci, An algorithm for stabilization of fractional order time delay systems using fractional-order PID Controllers, IEEE Trans. Automatic Control 52 (2007) 1964-1969.
[18] Kamran AM, Mohammad H. On robust stability of LTI fractional-order delay systems of retarded and neutral type, Automatica 2010; 46: 362- 368.
[19] Lazarevic M, Spasic AM. Finite time stability analysis of fractional order time delay systems: Gronwall’s approach. Math. Comp. Model. 49 (2009) 475–481.
[20] Zhang X. Some results of linear fractional order time-delay system. Appl. Math. Comput. 2008; 197: 407-411.
[21] Luchko Y, Gorenflo R. An operational method for solving fractional differential equations with the Caputo derivatives, Acta Math. Vietnamica 1999; 24: 207–233.
[22] Diethelm K, Ford NJ, Freed AD. A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 2002; 29: 3–22.
[23] Diethelm K. An algorithm for the numerical solution of differential equations of fractional order. Elec. Trans. Numer. Anal. 5 (1997) 1–6.
[24] Diethelm K, Ford NJ. Analysis of fractional differential equations, J. Math. Anal. Appl. 2002; 265: 229–48.
[25] Bhalekar S, Daftardar-Gejji V. A predictor-corrector scheme for solving nonlinear delay differential equations of fractional order. J. Fractional Calculus and its Applications 1(5) (2011) 1-8. http://www.fcaj.webs.com/Download/JFCAA-Vol.1,%20No.5.pdf
[26] Deng W, Li C, Lu J. Stability analysis of linear fractional differential system with multiple time delays Nonlinear Dyn. 2007; 48: 409–416.
[27] Bhalekar S, Daftardar-Gejji V., Fractional Ordered Liu system with delay. Comm. Nonlin. Sci. Num. Simul. 2010; 15: 2178–2191.
[28] Zhen W, Xia H, Guodong S. Analysis of nonlinear dynamics and chaos in a fractional order financial system with time delay. Comput. Math. Appl. 2011; In press, doi:10.1016/j.camwa.2011.04.057