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Chaotic dynamics of the fractional Willis aneurysm system and its control

Gao Fei Li Teng Tong Heng-Qing Ou Zhuo-Ling

Chaotic dynamics of the fractional Willis aneurysm system and its control

Gao Fei, Li Teng, Tong Heng-Qing, Ou Zhuo-Ling
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  • The Willis aneurysm system has some limitations in the description of the complex hemodynamic mechanism of blood with viscoelasticity. The fractional calculus has been used to depict some complex and disordered processes in organisms. Thus, we propose a fractional Willis aneurysm system (FWAS) byusing the Caputo fractional differential and its theory in the present article. Firstly, the existence and uniqueness of solution for FWAS are investigated theoretically. Then, we prove that the FWAS has a chaotic characteristic by analyzing the phase portraits and Poincar section, and it is a rational extension of its integer order form. We investigate the influences of pulse pressure and fractional order on the FWAS by means of bifurcation diagram and period doubling bifurcation. The results show that small changes of pulse pressure and fractional order canlead to a remarkable effect on the motion state of the FWAS. As the chaotic FWAS indicates that the brain blood flow is unstable, and the cerebral aneurysms are more likely to rupture in a very chaotic velocity field. Therefore we use two methods to control the chaotic FWAS. One is to design a suitable controller based on the stability theorem of fractional nonlinear non-autonomous system, and the other is to use a pulse control by taking the inspirit function of drug as impulse function. The numerical simulations show that the proposed two methods can control the blood flow velocity and speed up the periodic fluctuation within a small range, which shows that the cerebral aneurysm is not easy to rupture. The results obtained in this paper display that the fractional differential is a feasible method to characterize the Willis aneurysm system. The theoretical results in our article can provide some theoretical guidance for controlling and utilizing the actual FWAS system.
      Corresponding author: Gao Fei, hgaofei@gmail.com
    • Funds: Project supported by the Major Research plan of the National Natural Science Foundation of China (Grant No. 91324201) and the Natural Science Foundation of Hubei Province, China (Grant No. 2014CFB865).
    [1]

    Austin G 1971 Math. Biosci. 11 163

    [2]

    Cao J D, Liu T Y 1993 J. Biomath. 8 9 (in Chinese)[曹进德, 刘天一1993生物数学学报8 9]

    [3]

    Nieto J J, Torres A 2000 Nonlinear Anal. 40 513

    [4]

    Yang C H, Zhu S M 2003 Acta Sci. Nat. Univ. Sunyatseni 42 1 (in Chinese)[杨翠红, 朱思铭2003中山大学学报42 1]

    [5]

    Gu Y F, Xiao J 2014 Acta Phys. Sin. 63 160506 (in Chinese)[古元凤, 肖剑2014物理学报63 160506]

    [6]

    Li Y M, Yu S 2008 J. Biomath. 23 235 (in Chinese)[李医民, 于霜2008生物数学学报23 235]

    [7]

    Sun M H, Xiao J, Dong H L 2016 Highlights of Sciencepaper Online 9 640 (in Chinese)[孙梦晗, 肖剑, 董海亮2016中国科技论文在线精品论文9 640]

    [8]

    Lu K Q, Liu J X 2009 Physics 38 453 (in Chinese)[陆坤权, 刘寄星2009物理38 453]

    [9]

    Zhu K Q 2009 Mech. Pract. 31 104 (in Chinese)[朱克勤2009力学与实践31 104]

    [10]

    Ahmed E, El-Sayed A M A, El-Saka H A A 2007 J. Math. Anal. Appl. 325 542

    [11]

    Chen M, Jia L B, Yin X Z 2011 Chin. Phys. Lett. 28 88703

    [12]

    Dokoumetzidis A, Macheras P 2009 J. Pharmaceut. Biomed. 36 165

    [13]

    Verotta D 2010 J. Pharmaceut. Biomed. 37 257

    [14]

    Podlubny I 1999 Fractional Differential Equations (New York:Academic Press) pp41-120

    [15]

    Daftardar-Gejji V, Jafari H 2007 J. Math. Anal. Appl. 328 1026

    [16]

    Hu J B, Zhao L D 2013 Acta Phys. Sin. 62 060504 (in Chinese)[胡建兵, 赵灵冬2013物理学报62 060504]

    [17]

    Kai D, Ford N J 2004 Appl. Math. Comput. 154 621

    [18]

    Diethelm K, Ford N J, Freed A D 2005 Comput. Method Appl. M. 194 743

  • [1]

    Austin G 1971 Math. Biosci. 11 163

    [2]

    Cao J D, Liu T Y 1993 J. Biomath. 8 9 (in Chinese)[曹进德, 刘天一1993生物数学学报8 9]

    [3]

    Nieto J J, Torres A 2000 Nonlinear Anal. 40 513

    [4]

    Yang C H, Zhu S M 2003 Acta Sci. Nat. Univ. Sunyatseni 42 1 (in Chinese)[杨翠红, 朱思铭2003中山大学学报42 1]

    [5]

    Gu Y F, Xiao J 2014 Acta Phys. Sin. 63 160506 (in Chinese)[古元凤, 肖剑2014物理学报63 160506]

    [6]

    Li Y M, Yu S 2008 J. Biomath. 23 235 (in Chinese)[李医民, 于霜2008生物数学学报23 235]

    [7]

    Sun M H, Xiao J, Dong H L 2016 Highlights of Sciencepaper Online 9 640 (in Chinese)[孙梦晗, 肖剑, 董海亮2016中国科技论文在线精品论文9 640]

    [8]

    Lu K Q, Liu J X 2009 Physics 38 453 (in Chinese)[陆坤权, 刘寄星2009物理38 453]

    [9]

    Zhu K Q 2009 Mech. Pract. 31 104 (in Chinese)[朱克勤2009力学与实践31 104]

    [10]

    Ahmed E, El-Sayed A M A, El-Saka H A A 2007 J. Math. Anal. Appl. 325 542

    [11]

    Chen M, Jia L B, Yin X Z 2011 Chin. Phys. Lett. 28 88703

    [12]

    Dokoumetzidis A, Macheras P 2009 J. Pharmaceut. Biomed. 36 165

    [13]

    Verotta D 2010 J. Pharmaceut. Biomed. 37 257

    [14]

    Podlubny I 1999 Fractional Differential Equations (New York:Academic Press) pp41-120

    [15]

    Daftardar-Gejji V, Jafari H 2007 J. Math. Anal. Appl. 328 1026

    [16]

    Hu J B, Zhao L D 2013 Acta Phys. Sin. 62 060504 (in Chinese)[胡建兵, 赵灵冬2013物理学报62 060504]

    [17]

    Kai D, Ford N J 2004 Appl. Math. Comput. 154 621

    [18]

    Diethelm K, Ford N J, Freed A D 2005 Comput. Method Appl. M. 194 743

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    [8] Gao Xin, Liu Xing-Wen. Delayed fuzzy control of a unified chaotic system. Acta Physica Sinica, 2007, 56(1): 84-90. doi: 10.7498/aps.56.84
    [9] Qian Fu-Cai, Liu Ding, Zhu Shao-Ping. Optimal control for uncertainty dynamic chaotic systems. Acta Physica Sinica, 2010, 59(4): 2250-2255. doi: 10.7498/aps.59.2250
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  • Received Date:  05 June 2016
  • Accepted Date:  06 September 2016
  • Published Online:  05 December 2016

Chaotic dynamics of the fractional Willis aneurysm system and its control

    Corresponding author: Gao Fei, hgaofei@gmail.com
  • 1. School of Science, Wuhan University of Technology, Wuhan 430070, China
Fund Project:  Project supported by the Major Research plan of the National Natural Science Foundation of China (Grant No. 91324201) and the Natural Science Foundation of Hubei Province, China (Grant No. 2014CFB865).

Abstract: The Willis aneurysm system has some limitations in the description of the complex hemodynamic mechanism of blood with viscoelasticity. The fractional calculus has been used to depict some complex and disordered processes in organisms. Thus, we propose a fractional Willis aneurysm system (FWAS) byusing the Caputo fractional differential and its theory in the present article. Firstly, the existence and uniqueness of solution for FWAS are investigated theoretically. Then, we prove that the FWAS has a chaotic characteristic by analyzing the phase portraits and Poincar section, and it is a rational extension of its integer order form. We investigate the influences of pulse pressure and fractional order on the FWAS by means of bifurcation diagram and period doubling bifurcation. The results show that small changes of pulse pressure and fractional order canlead to a remarkable effect on the motion state of the FWAS. As the chaotic FWAS indicates that the brain blood flow is unstable, and the cerebral aneurysms are more likely to rupture in a very chaotic velocity field. Therefore we use two methods to control the chaotic FWAS. One is to design a suitable controller based on the stability theorem of fractional nonlinear non-autonomous system, and the other is to use a pulse control by taking the inspirit function of drug as impulse function. The numerical simulations show that the proposed two methods can control the blood flow velocity and speed up the periodic fluctuation within a small range, which shows that the cerebral aneurysm is not easy to rupture. The results obtained in this paper display that the fractional differential is a feasible method to characterize the Willis aneurysm system. The theoretical results in our article can provide some theoretical guidance for controlling and utilizing the actual FWAS system.

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