A fuzzy crisis in a Duffing－van der Pol system

doi:10.1088/1674-1056/19/3/030513

Abstract A crisis in a Duffing--van del Pol system with fuzzy uncertainties is studied by means of the fuzzy generalised cell mapping (FGCM) method. A crisis happens when two fuzzy attractors collide simultaneously with a fuzzy saddle on the basin boundary as the intensity of fuzzy noise reaches a critical point. The two fuzzy attractors merge discontinuously to form one large fuzzy attractor after a crisis. A fuzzy attractor is characterized by its global topology and membership function. A fuzzy saddle with a complicated pattern of several disjoint segments is observed in phase space. It leads to a discontinuous merging crisis of fuzzy attractors. We illustrate this crisis event by considering a fixed point under additive and multiplicative fuzzy noise. Such a crisis is fuzzy noise-induced effects which cannot be seen in deterministic systems.

Keywords: fuzzy dynamical systems fuzzy noise fuzzy bifurcation cell mapping methods

Received: 17 June 2009
Revised: 22 August 2009
Accepted manuscript online:

PACS:	05.40.Ca	(Noise)
	05.45.-a	(Nonlinear dynamics and chaos)
	02.10.Ab	(Logic and set theory)

Fund: Project supported by the National Natural Science Foundation of China (Grant Nos.~10772140 and 10872155).

Cite this article:

Hong Ling(洪灵) A fuzzy crisis in a Duffing－van der Pol system 2010 Chin. Phys. B 19 030513

[1]	Moss F and McClintock P V E 1989 Noise in Nonlinear DynamicalSystems (Cambridge: Cambridge University Press)
[2]	Klir G J and Folger T A 1988 Fuzzy Sets, Uncertainty, andInformation (New Jersey: Prentice-Hall, Englewood Cliffs)
[3]	Bucolo M, Fazzino S, Rosa M L and Fortuna L 2003 Chaos,Solitons and Fractals 17 557
[4]	Sandler U and Tsitolovsky L 2001 Fuzzy Sets and Systems 121 237
[5]	Grebogi C, Ott E and York J A 1982 Phys. Rev. Lett.48 1507
[6]	Ott E 2002 Chaos in Dynamical Systems (Cambridge: Cambridge UniversityPress)
[7]	Xu P M and Wen B C 2004 Chin. Phys. 13 618
[8]	Zou H L, Xu J X and Jiang J 2008 Chin. Phys. B 17117
[9]	Meunier C and Verga A D 1988 J. Stat. Phys.50 345
[10]	Doi S, Inoue J and Kumagai S 1988 J. Stat.Phys. 90 1107
[11]	Xu W, He Q, Rong H W and Fang T 2003 Acta Phys. Sin.52 1365 (in Chinese)
[12]	He Q, Xu W, Li S A and Xiao Y Z 2008 Acta Phys. Sin.57 4021 (in Chinese)
[13]	He Q, Xu W, Li S A and Xiao Y Z 2008 Acta Phys. Sin.57 743 (in Chinese)
[14]	Tomonaga Y and Takatsuka K 1998 Physica D 111 51
[15]	Cuesta F, Ponce E and Aracil J 2001 IEEE Trans. on FuzzySystems 9 355
[16]	Satpathy P K, Das D and Gupta P B D 2004 Int. J.Electrical Power and Energy System 26 531
[17]	Hong L and Sun J Q 2006 Communications in Nonlinear Scienceand Numerical Simulation 11 1
[18]	Hong L and Sun J Q 2006 Physica D 213 181
[19]	Hsu C S 1987 Cell-to-Cell Mapping: A Method of Global Analysisfor Non-linear Systems (New York: Springer-Verlag)
[20]	Yoshida Y 2000 Fuzzy Sets and Systems 113 453
[21]	Friedman Y and Sandler U 1996 Fuzzy Sets and Systems 84 61
[22]	Friedman Y and Sandler U 1999 Fuzzy Sets and Systems 106 61
[23]	Risken H 1996 The Fokker-Planck Equation (New York:Springer-Verlag)
[24]	Hsu C S 1995 Int. J. Bifurc. and Chaos 5 1085
[25]	Guckenheimer J and Holmes P J 1983 Nonlinear Oscillations,Dynamical Systems and Bifurcations of Vector Fields (New York:Springer-Verlag)
[26]	Holmes P and Rand D 1980 Int. J. Non-LinearMechanics 15 449-458
[27]	Namachchivaya N S 1990 J. Appl. Math. andComput. 38 101
[28]	Schenk-Hoppe K R 1996 Nonlinear Dynamics 11 255
[29]	Namachchivaya N S 1991 J. Appl. Mech. 58 259

[1]	Sparse identification method of extracting hybrid energy harvesting system from observed data Ya-Hui Sun(孙亚辉), Yuan-Hui Zeng(曾远辉), and Yong-Ge Yang(杨勇歌). Chin. Phys. B, 2022, 31(12): 120203.
[2]	Hyperparameter on-line learning of stochastic resonance based threshold networks Weijin Li(李伟进), Yuhao Ren(任昱昊), and Fabing Duan(段法兵). Chin. Phys. B, 2022, 31(8): 080503.
[3]	A new simplified ordered upwind method for calculating quasi-potential Qing Yu(虞晴) and Xianbin Liu(刘先斌). Chin. Phys. B, 2022, 31(1): 010502.
[4]	Dense coding capacity in correlated noisy channels with weak measurement Jin-Kai Li(李进开), Kai Xu(徐凯), and Guo-Feng Zhang(张国锋). Chin. Phys. B, 2021, 30(11): 110302.
[5]	Chaotic signal denoising algorithm based on sparse decomposition Jin-Wang Huang(黄锦旺), Shan-Xiang Lv(吕善翔), Zu-Sheng Zhang(张足生), Hua-Qiang Yuan(袁华强). Chin. Phys. B, 2020, 29(6): 060505.
[6]	Influence of homodyne-based feedback control on the entropic uncertainty in open quantum system Juju Hu(胡菊菊), Qin Xue(薛琴). Chin. Phys. B, 2019, 28(7): 070303.
[7]	Stationary response of stochastic viscoelastic system with the right unilateral nonzero offset barrier impacts Deli Wang(王德莉), Wei Xu(徐伟), Xudong Gu(谷旭东). Chin. Phys. B, 2019, 28(1): 010203.
[8]	Effect of stochastic electromagnetic disturbances on autapse neuronal systems Liang-Hui Qu(曲良辉), Lin Du(都琳), Zi-Chen Deng(邓子辰), Zi-Lu Cao(曹子露), Hai-Wei Hu(胡海威). Chin. Phys. B, 2018, 27(11): 118707.
[9]	Controlling of entropic uncertainty in open quantum system via proper placement of quantum register Ying-Hua Ji(嵇英华), Qiang Ke(柯强), Ju-Ju Hu(胡菊菊). Chin. Phys. B, 2018, 27(10): 100302.
[10]	Signal-to-noise ratio comparison of angular signal radiography and phase stepping method Wali Faiz, Peiping Zhu(朱佩平), Renfang Hu(胡仁芳), Kun Gao(高昆), Zhao Wu(吴朝), Yuan Bao(鲍园), Yangchao Tian(田扬超). Chin. Phys. B, 2017, 26(12): 120601.
[11]	Stochastic bifurcations of generalized Duffing-van der Pol system with fractional derivative under colored noise Wei Li(李伟), Mei-Ting Zhang(张美婷), Jun-Feng Zhao(赵俊锋). Chin. Phys. B, 2017, 26(9): 090501.
[12]	Using wavelet multi-resolution nature to accelerate the identification of fractional order system Yuan-Lu Li(李远禄), Xiao Meng(孟霄), Ya-Qing Ding(丁亚庆). Chin. Phys. B, 2017, 26(5): 050201.
[13]	Pattern dynamics of network-organized system with cross-diffusion Qianqian Zheng(郑前前), Zhijie Wang(王直杰), Jianwei Shen(申建伟). Chin. Phys. B, 2017, 26(2): 020501.
[14]	Modeling random telegraph signal noise in CMOS image sensor under low light based on binomial distribution Yu Zhang(张钰), Xinmiao Lu(逯鑫淼), Guangyi Wang(王光义), Yongcai Hu(胡永才), Jiangtao Xu(徐江涛). Chin. Phys. B, 2016, 25(7): 070503.
[15]	Multifractal modeling of the production of concentrated sugar syrup crystal Sheng Bi(闭胜), Jianbo Gao(高剑波). Chin. Phys. B, 2016, 25(7): 070502.

No Suggested Reading articles found!

Viewed

Full text

Abstract

Cited

Metrics
Related Articles

A fuzzy crisis in a Duffing－van der Pol system

Cite this article:

Online attention