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Chaotic analysis of fractional Willis delayed aneurysm system

## Chaotic analysis of fractional Willis delayed aneurysm system

Gao Fei, Hu Dao-Nan, Tong Heng-Qing, Wang Chuan-Mei
• #### Abstract

The dynamic system of Willis aneurysm (WAS) has played an important role in theoretical and clinical research of cerebral aneurysms. Fractional differential is an effective mathematical tool that can describe the cerebral aneurysm system accurately and profoundly. However, the existing fractional Willis aneurysm system (FWAS) cannot describe the delayed aneurysm rupture of unknown cause in reality. Therefore, by introducing the time-delay factors into the existing fractional Willis aneurysm system as a rational extension, a new fractional Willis aneurysm system with time-delay (FWASTD) is proposed in this paper.First, FWASTD is introduced in the context, and the comparison of time sequences map between FWAS and FWASTD proves that FWASTD is feasible in the depiction of time-delay situations. The bifurcation diagram and the largest Lyapunov exponent diagram as well as the phase diagram of fractional order also confirm the chaotic characteristics of the FWASTD.Then, the classical analysis methods in chaotic dynamics, such as time series diagram, phase diagram and Poincar section are used to analyze FWASTD in detail. When studying the diagrams of time-delay factors for the important physiological parameters of the system, we find that blood flow resistance coefficient can exert a remarkable effect on the system stability under time-delay. Besides, the experimental results show that the FWASTD becomes stable with the increase of blood flow resistance under a certain condition. Usually, promoting thrombosis is a kind of adjunctive therapy in clinic for cerebral aneurysm. The results of this part can accord with the treatment in clinic and has great significance in clinical diagnosis.Finally, as the chaotic state of the time-delay system indicates that cerebral aneurysm is in a dangerous situation, the primary task of the control for this new system is to achieve stability rather than synchronization. The stability theory of fractional time-delayed system is adopted in a strict proof of the uniqueness of solution for the FWASTD. To make FWASTD stable, a corresponding linear controller is designed based on the stability theory of fractional order delay system. The numerical simulation indicates that the linear controller can control the blood flow velocity and speed up the periodic fluctuation within a small range, which illustrates that it is not easy to rupture the cerebral aneurysm. We also make self-synchronization control between FWASTD and FWAS just in case that the coefficients of the system are not clear.The research results in this paper, to some extent, can serve as theoretical guidance for the clinical diagnosis and the treatment of aneurysm.

#### Authors and contacts

###### 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), the Fundamental Research Funds for the Central Universities, China (Grant No. 2018IB017), the Natural Science Foundation of Hubei Province, China (Grant No. 2014CFB865), and the Humanity and Social Science Youth foundation of Ministry of Education of China (Grant No. 14YJCZH143).

#### References

 [1] Lan Q 2015 Chin. J. Neurosurg. 31 541 (in Chinese) [兰青 2015 中华神经外科杂志 31 541] [2] Liang S K, Jiang C H 2016 Chin. J. Neurosurg. 32 1071 (in Chinese) [梁士凯, 姜除寒 2016 中华神经外科杂志 32 1071] [3] Liu A H 2017 Chin. J. Stroke 12 850 (in Chinese) [刘爱华 2017 中国卒中杂志 12 850] [4] Fiorella D, Woo H H, Albuquerque F C, Nelson P K 2008 Neurosurgery 62 1115 [5] Liu J, Jing L K, Wang C, Paliwal N, Wang S Z, Zhang Y, Xiang J P, Siddiqui A H, Meng H, Yang X J 2016 J. Neurointerv. Surg. 8 1140 [6] Zhang Y, Yang X J 2016 Chin. J. Cerebrovasc. Dis. 7 372 (in Chinese) [张莹, 杨新健 2016 中国脑血管病杂志 7 372] [7] Radaelli A G, Augsburger L, Cebral J R, Ohta M, Rufenacht D A, Balossino R, Benndorf G, Hose D R, Marzo A, Metcalfe R, Mortier P, Mut F, Reymond P, Socci L, Verhegghe B, Frangi A F 2008 J. Bio. 41 2069 [8] Connolly J E S, Rabinstein A A, Carhuapoma J R, Derdeyn C P, Dion J, Higashida R T, Hoh B L, Kirkness C J, Naidech A M, Ogilvy C S, Patel A B, Thompson B G, Vespa P, Council A H A S, Int C C R, Nursing C C, Anesthes C C S, Cardiology C C 2012 Stroke 43 1711 [9] Gonzalez C F, Cho Y I, Ortega H V, Moret J 1992 Am. J. Neuroradiol. 13 181 [10] Dai X, Qiao A K 2016 J. Med. Biomech. 31 461 (in Chinese) [戴璇, 乔爱科 2016 医用生物力学 31 461] [11] Austin G 1971 Math. Biosci. 11 163 [12] 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] [13] Yang C H, Zhu S M 2003 Acta Sci. Nat. Univ. Sunyatseni 43 1 (in Chinese) [杨翠红, 朱思铭 2003 中山大学学报(自然科学版) 43 1] [14] Gu Y F, Xiao J 2014 Acta Phys. Sin. 63 160501 (in Chinese) [古元凤, 肖剑 2014 物理学报 63 160501] [15] 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] [16] Sun M H 2016 M. S. Thesis (Chongqing: University of Chongqing) (in Chinese) [孙梦晗 2016 硕士学位论文 (重庆: 重庆大学)] [17] Gao F, Li T, Tong H Q, Ou Z L 2016 Acta Phys. Sin. 65 230502 (in Chinese) [高飞, 李腾, 童恒庆, 欧卓玲 2016 物理学报 65 230502] [18] Lu K Q, Liu J X 2009 Physics 38 453 (in Chinese) [陆坤权, 刘寄星 2009 物理 38 453] [19] Zhu K Q 2009 Mech. Pract. 31 104 (in Chinese) [朱克勤 2009 力学与实践 31 104] [20] Ahmed E, El-Sayed A M A, El-Saka H A A 2007 J. Math. Anal. Appl. 325 542 [21] Dokoumetzidis A, Macheras P 2009 J. Pharmacokinet. Pharmacodyn. 36 165 [22] Liang Y, Wang X Y 2013 Acta Phys. Sin. 62 018901 (in Chinese) [梁义, 王兴元 2013 物理学报 62 018901] [23] Ouyang C, Lin W T, Cheng R J, Mo J Q 2013 Acta Phys. Sin. 62 060201 (in Chinese) [欧阳成, 林万涛, 程荣军, 莫嘉琪 2013 物理学报 62 060201] [24] Podlubny I 1999 Fractional Differential Equations (New York: Academic Press) p41 [25] Huo R, Wang X L, Wu G R 2014 J. Inner Mongolia Agric. Univ. (Nat. Sci. Edn.) 35 167 (in Chinese) [霍冉, 王晓丽, 吴国荣 2014 内蒙古农业大学学报 35 167] [26] Hu J, Lu G, Zhang S, Zhao L 2015 Commun. Nonlinear Sci. 20 905 [27] Bhalekar S, Daftardar-Gejji V 2010 Commun. Nonlinear Sci. 15 2178 [28] Diethelm K, Neville F 2002 Nonlinear Dynam. 29 3 [29] Guan M, Shi H, Zhang G 2017 Chin. J. Cerebrovasc Dis. 14 46 (in Chinese) [关明浩, 史怀璋, 张广 2017 中国脑血管病杂志 14 46]

#### Cited By

•  [1] Lan Q 2015 Chin. J. Neurosurg. 31 541 (in Chinese) [兰青 2015 中华神经外科杂志 31 541] [2] Liang S K, Jiang C H 2016 Chin. J. Neurosurg. 32 1071 (in Chinese) [梁士凯, 姜除寒 2016 中华神经外科杂志 32 1071] [3] Liu A H 2017 Chin. J. Stroke 12 850 (in Chinese) [刘爱华 2017 中国卒中杂志 12 850] [4] Fiorella D, Woo H H, Albuquerque F C, Nelson P K 2008 Neurosurgery 62 1115 [5] Liu J, Jing L K, Wang C, Paliwal N, Wang S Z, Zhang Y, Xiang J P, Siddiqui A H, Meng H, Yang X J 2016 J. Neurointerv. Surg. 8 1140 [6] Zhang Y, Yang X J 2016 Chin. J. Cerebrovasc. Dis. 7 372 (in Chinese) [张莹, 杨新健 2016 中国脑血管病杂志 7 372] [7] Radaelli A G, Augsburger L, Cebral J R, Ohta M, Rufenacht D A, Balossino R, Benndorf G, Hose D R, Marzo A, Metcalfe R, Mortier P, Mut F, Reymond P, Socci L, Verhegghe B, Frangi A F 2008 J. Bio. 41 2069 [8] Connolly J E S, Rabinstein A A, Carhuapoma J R, Derdeyn C P, Dion J, Higashida R T, Hoh B L, Kirkness C J, Naidech A M, Ogilvy C S, Patel A B, Thompson B G, Vespa P, Council A H A S, Int C C R, Nursing C C, Anesthes C C S, Cardiology C C 2012 Stroke 43 1711 [9] Gonzalez C F, Cho Y I, Ortega H V, Moret J 1992 Am. J. Neuroradiol. 13 181 [10] Dai X, Qiao A K 2016 J. Med. Biomech. 31 461 (in Chinese) [戴璇, 乔爱科 2016 医用生物力学 31 461] [11] Austin G 1971 Math. Biosci. 11 163 [12] 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] [13] Yang C H, Zhu S M 2003 Acta Sci. Nat. Univ. Sunyatseni 43 1 (in Chinese) [杨翠红, 朱思铭 2003 中山大学学报(自然科学版) 43 1] [14] Gu Y F, Xiao J 2014 Acta Phys. Sin. 63 160501 (in Chinese) [古元凤, 肖剑 2014 物理学报 63 160501] [15] 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] [16] Sun M H 2016 M. S. Thesis (Chongqing: University of Chongqing) (in Chinese) [孙梦晗 2016 硕士学位论文 (重庆: 重庆大学)] [17] Gao F, Li T, Tong H Q, Ou Z L 2016 Acta Phys. Sin. 65 230502 (in Chinese) [高飞, 李腾, 童恒庆, 欧卓玲 2016 物理学报 65 230502] [18] Lu K Q, Liu J X 2009 Physics 38 453 (in Chinese) [陆坤权, 刘寄星 2009 物理 38 453] [19] Zhu K Q 2009 Mech. Pract. 31 104 (in Chinese) [朱克勤 2009 力学与实践 31 104] [20] Ahmed E, El-Sayed A M A, El-Saka H A A 2007 J. Math. Anal. Appl. 325 542 [21] Dokoumetzidis A, Macheras P 2009 J. Pharmacokinet. Pharmacodyn. 36 165 [22] Liang Y, Wang X Y 2013 Acta Phys. Sin. 62 018901 (in Chinese) [梁义, 王兴元 2013 物理学报 62 018901] [23] Ouyang C, Lin W T, Cheng R J, Mo J Q 2013 Acta Phys. Sin. 62 060201 (in Chinese) [欧阳成, 林万涛, 程荣军, 莫嘉琪 2013 物理学报 62 060201] [24] Podlubny I 1999 Fractional Differential Equations (New York: Academic Press) p41 [25] Huo R, Wang X L, Wu G R 2014 J. Inner Mongolia Agric. Univ. (Nat. Sci. Edn.) 35 167 (in Chinese) [霍冉, 王晓丽, 吴国荣 2014 内蒙古农业大学学报 35 167] [26] Hu J, Lu G, Zhang S, Zhao L 2015 Commun. Nonlinear Sci. 20 905 [27] Bhalekar S, Daftardar-Gejji V 2010 Commun. Nonlinear Sci. 15 2178 [28] Diethelm K, Neville F 2002 Nonlinear Dynam. 29 3 [29] Guan M, Shi H, Zhang G 2017 Chin. J. Cerebrovasc Dis. 14 46 (in Chinese) [关明浩, 史怀璋, 张广 2017 中国脑血管病杂志 14 46]
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•  Citation:
##### Metrics
• Abstract views:  183
• Cited By: 0
##### Publishing process
• Received Date:  02 February 2018
• Accepted Date:  16 April 2018
• Published Online:  05 August 2018

## Chaotic analysis of fractional Willis delayed aneurysm system

###### 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), the Fundamental Research Funds for the Central Universities, China (Grant No. 2018IB017), the Natural Science Foundation of Hubei Province, China (Grant No. 2014CFB865), and the Humanity and Social Science Youth foundation of Ministry of Education of China (Grant No. 14YJCZH143).

Abstract: The dynamic system of Willis aneurysm (WAS) has played an important role in theoretical and clinical research of cerebral aneurysms. Fractional differential is an effective mathematical tool that can describe the cerebral aneurysm system accurately and profoundly. However, the existing fractional Willis aneurysm system (FWAS) cannot describe the delayed aneurysm rupture of unknown cause in reality. Therefore, by introducing the time-delay factors into the existing fractional Willis aneurysm system as a rational extension, a new fractional Willis aneurysm system with time-delay (FWASTD) is proposed in this paper.First, FWASTD is introduced in the context, and the comparison of time sequences map between FWAS and FWASTD proves that FWASTD is feasible in the depiction of time-delay situations. The bifurcation diagram and the largest Lyapunov exponent diagram as well as the phase diagram of fractional order also confirm the chaotic characteristics of the FWASTD.Then, the classical analysis methods in chaotic dynamics, such as time series diagram, phase diagram and Poincar section are used to analyze FWASTD in detail. When studying the diagrams of time-delay factors for the important physiological parameters of the system, we find that blood flow resistance coefficient can exert a remarkable effect on the system stability under time-delay. Besides, the experimental results show that the FWASTD becomes stable with the increase of blood flow resistance under a certain condition. Usually, promoting thrombosis is a kind of adjunctive therapy in clinic for cerebral aneurysm. The results of this part can accord with the treatment in clinic and has great significance in clinical diagnosis.Finally, as the chaotic state of the time-delay system indicates that cerebral aneurysm is in a dangerous situation, the primary task of the control for this new system is to achieve stability rather than synchronization. The stability theory of fractional time-delayed system is adopted in a strict proof of the uniqueness of solution for the FWASTD. To make FWASTD stable, a corresponding linear controller is designed based on the stability theory of fractional order delay system. The numerical simulation indicates that the linear controller can control the blood flow velocity and speed up the periodic fluctuation within a small range, which illustrates that it is not easy to rupture the cerebral aneurysm. We also make self-synchronization control between FWASTD and FWAS just in case that the coefficients of the system are not clear.The research results in this paper, to some extent, can serve as theoretical guidance for the clinical diagnosis and the treatment of aneurysm.

Reference (29)

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