A Necessary Condition for the Existence of Chaos in Fractional Order Delay Differential Equations
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.
<p>[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</p>
<p>[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</p>
@article{"International Journal of Engineering, Mathematical and Physical Sciences:65135", author = "Sachin Bhalekar", title = "A Necessary Condition for the Existence of Chaos in Fractional Order Delay Differential Equations", 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.", volume = "7", number = "3", pages = "421-5", }
