COMPLEX BURSTING OSCILLATION STRUCTURES IN A TWO-DIMENSIONAL NON-AUTONOMOUS DISCRETE SYSTEM

Chen Zhenyang; Han Xiujing; Bi Qinsheng

doi:10.6052/0459-1879-16-267

Volume 49 Issue 1

Turn off MathJax

Article Contents

Abstract

References

Chinese Journal of Theoretical and Applied Mechanics > 2017 > 49(1): 165-174. > DOI: 10.6052/0459-1879-16-267 CSTR: 32045.14.0459-1879-16-267

Chen Zhenyang, Han Xiujing, Bi Qinsheng. COMPLEX BURSTING OSCILLATION STRUCTURES IN A TWO-DIMENSIONAL NON-AUTONOMOUS DISCRETE SYSTEM[J]. Chinese Journal of Theoretical and Applied Mechanics, 2017, 49(1): 165-174. DOI: 10.6052/0459-1879-16-267

Citation:

PDF (3079 KB)

COMPLEX BURSTING OSCILLATION STRUCTURES IN A TWO-DIMENSIONAL NON-AUTONOMOUS DISCRETE SYSTEM

Faculty of Civil Engineering and Mechanics, Jiangsu University, Zhenjiang 212013, Jiangsu, China

Received Date: September 21, 2016
Revised Date: November 05, 2016
Available Online: November 10, 2016

Graphical Abstract

Abstract

Abstract

Bursting oscillations are the archetypes of complex dynamical behaviors in systems with multiple time scales, and the problem related to dynamical mechanisms and classifications of bursting oscillations is one of the important problems in bursting research.However, up to now, most of the structures of bursting revealed by researchers are relatively simple.In this paper, we take the non-autonomous discrete Duffing system as an example to explore novel bursting patterns with complex bifurcation structures, which are divided into two groups, i.e., symmetric bursting induced by Fold bifurcations and asymmetric bursting induced by delayed Flip bifurcations.Typically, the fast subsystem exhibits an Sshaped fixed point curve with two Fold points, and the stable upper and lower branches evolve into chaos by a cascade of Flip bifurcations.When the non-autonomous term (i.e., the slow variable) passes through Fold points, transitions to various attractors (e.g., periodic orbits and chaos) on the stable branches may take place, which accounts for the appearance of Fold-bifurcation-induced symmetric bursting patterns.If the non-autonomous term is not able to pass through Fold points, but to go through Flip points, delayed Flip bifurcations can be observed.Based on this, delayed-Flip-bifurcation-induced asymmetric bursting patterns are obtained.In particular, the bursting patterns reported here exhibit complex structures containing inverse Flip bifurcations, which has been found to be related to the fact that the nonautonomous term slowly passes through Flip points of the fast subsystem in an inverse way.Our results enrich dynamical mechanisms and classifications of bursting in discrete systems.
- discrete Duffing system,
- complex bursting patterns,
- delayed Flip bifurcations,
- inverse Flip bifurcation structures

FullText(HTML)

References (31)

References

[1]	Izhikevich EM. Neural excitability, spiking and bursting. Int J Bifurcat Chaos, 2000, 10(6):1171-1266 doi: 10.1142/S0218127400000840
[2]	Izhikevich EM. Resonance and selective communication via bursts in neurons having subthreshold oscillations. Bio Systems, 2002, 67(1-3):95-102 doi: 10.1016/S0303-2647(02)00067-9
[3]	古华光,朱洲,贾冰. 一类新的混沌神经放电的动力学特征的实验和数学模型研究. 物理学报, 2011, 60(10):1005051-10050512 http://www.cnki.com.cn/Article/CJFDTOTAL-WLXB201110013.htm Gu Huaguang, Zhu Zhou, Jia Bing. Dynamics of a novel chaotic neural firing pattern discovered in experiment and simulated in mathematical model. Acta Phys Sin, 2011, 60(10):1005051-10050512(in Chinese) http://www.cnki.com.cn/Article/CJFDTOTAL-WLXB201110013.htm
[4]	王帅, 于文浩, 陈巨辉等. 鼓泡流化床中流动特性的多尺度数值模拟. 力学学报,2016, 48(3):585-592 http://lxxb.cstam.org.cn/CN/abstract/abstract145781.shtml Wang Shuai, Yu Wenhao, Chen Juhui, et al. Multi-scale simulation on hydrodynamic characteristics in bubbling fluidized bed. Chinese Journal of Theoretical and Applied Mechanics, 2016, 48(3):585-592(in Chinese) http://lxxb.cstam.org.cn/CN/abstract/abstract145781.shtml
[5]	Li XH, Bi QS. Single-Hopf Bursting in periodic perturbed Belousov——Zhabotinsky reaction with two time scales. Chinese Phys Lett, 2013, 30(1):10503-10506 doi: 10.1088/0256-307X/30/1/010503
[6]	Li XH, Zhang C, Yu Y, et al. Periodic switching oscillation and mechanism in a periodically switched BZ reaction. Sci China E, 2012, 55(10):2820-2828 doi: 10.1007/s11431-012-4988-1
[7]	Lashina EA, Chumakova NA, Chumakov GA, et al. Chaotic dynamics in the three-variable kinetic model of CO oxidation on platinum group metals. Chem Eng J, 2009, 154(1-3):82-87 doi: 10.1016/j.cej.2009.02.017
[8]	贾宏涛, 许春晓, 崔桂香. 槽道湍流近壁区多尺度输运特性研究. 力学学报, 2007, 39(2):181-187 http://lxxb.cstam.org.cn/CN/abstract/abstract141540.shtml Jia Hongtao, Xu Chunxiao, Cui Guixiang. Multi-scale energy transfer in near-wall region of turbulent channel flow. Chinese Journal of Theoretical and Applied Mechanics, 2007, 39(2):181-187(in Chinese) http://lxxb.cstam.org.cn/CN/abstract/abstract141540.shtml
[9]	Clarke RJ, Apell HJ, Kong BY. Allosteric effect of ATP on, Na⁺, K⁺- ATPase conformational kinetics. Biochemistry, 2007, 46(23):7034-7044 doi: 10.1021/bi700619s
[10]	郭建侠, 卞林根, 戴永久. 玉米生育期地表能量平衡的多时间尺度特征分析及不平衡原因的探索. 中国科学D, 2008, 38(9):1103-1111 http://www.cnki.com.cn/Article/CJFDTOTAL-JDXK200809006.htm Guo Jianxia, Bian Lingen, Dai Yongjiu. Analysis of characteristics of multiple time scale in surface energy balance of corn in growth period, and exploration of the reasons for unbalance. Sci China D, 2008, 38(9):1103-1111(in Chinese) http://www.cnki.com.cn/Article/CJFDTOTAL-JDXK200809006.htm
[11]	Courbage M, Nekorkin VI, Vdovin LV. Chaotic oscillations in a map-based model of neural activity. Chaos, 2007, 17(4):155-160 http://cn.bing.com/academic/profile?id=16c39feaee8d1ce7e25b7e1366fd9c88&encoded=0&v=paper_preview&mkt=zh-cn
[12]	Butera Jr RJ, Rinzel J, Smith JC. Models of respiratory rhythm generation in the pre-Bötzinger complex. Ⅱ. populations of coupled pacemaker neurons. J Neurophysiol, 1999, 82(1):398-415 http://d.scholar.cnki.net/detail/SJPD2059_U/SJPD12102800176896
[13]	Rinzel J. Bursting oscillation in an excitable membrane model//Everitt WN, Sleeman BD. Lecture Note In Mathematics. Vol. 1151. New York:Springer, 1985:304-316
[14]	杨卓琴,陆启韶. 神经元Chay模型中不同类型的簇放电模式. 中国科学G, 2007, 37(4):440-450 http://www.cnki.com.cn/Article/CJFDTOTAL-JGXK200704004.htm Yang Zhuoqin Lu Qishao. Different types of bursting spiking mode in Chay model of neurons. Sci China G, 2007, 37(4):440-450(in Chinese) http://www.cnki.com.cn/Article/CJFDTOTAL-JGXK200704004.htm
[15]	Rulkov NF. Regularization of synchronized chaotic bursts. Phys Rev Lett, 2001,86(1):183-186 doi: 10.1103/PhysRevLett.86.183
[16]	Wagenaar DA, Pine J, Potter SM. An extremely rich repertoire of bursting patterns during the development of cortical cultures. BMC Neurosci, 2006, 7(1):1-18 doi: 10.1186/1471-2202-7-1
[17]	古华光,惠磊,贾冰. 一类位于加周期分岔中的貌似混沌的随机神经放电规律的识别. 物理学报, 2012, 61(8):0805041-08050410 http://www.cnki.com.cn/Article/CJFDTOTAL-WLXB201208012.htm Gu Huaguang, Hui Lei, Jia Bing. Identification of a stochastic neural firing rhythm lying in period-adding bifurcation and resembling chaos. Acta Phys Sin, 2012, 61(8):0805041-08050410(in Chinese) http://www.cnki.com.cn/Article/CJFDTOTAL-WLXB201208012.htm
[18]	Channell P, Cymbalynk G, Shilnikov A. Applications of the Poincaré mapping technique to analysis of neuronal dynamics. Neurocomputing, 2007, 70(10-12):2107-2111 doi: 10.1016/j.neucom.2006.10.091
[19]	Izhikevich EM, Hoppensteadt F. Classfication of bursting mapping. Int J Bifurcat Chaos, 2004, 14(11):3847-3854 doi: 10.1142/S0218127404011739
[20]	Courbage M, Maslennikov OV, Nekorkin VI. Synchronization in time-discrete model of two electrically coupled spike-bursting neurons. Chaos Soliton Fract, 2012, 45(5):645-659 doi: 10.1016/j.chaos.2011.12.018
[21]	Maslennikov OV, Nekorkin VI. Discrete model of the olivo-cerebellar system:structure and dynamics. Radiophys Quant El, 2012, 55(55):198-214 http://cn.bing.com/academic/profile?id=df75f9d09592b4449f3380115ad985cd&encoded=0&v=paper_preview&mkt=zh-cn
[22]	Nekorkin VI, Maslennikov OV. Spike-burst synchronization in an ensemble of electrically coupled discrete model neurons. Radiophys Quant El, 2011, 54(1):56-73 doi: 10.1007/s11141-011-9271-y
[23]	Stoyanov B, Kordov K. Novel image encryption scheme based on Chebyshev polynomial and Duffing map. Scientific World J, 2014, 2014:283639-283639 http://cn.bing.com/academic/profile?id=a4a7b5f68efd50e48dc56d98a544b204&encoded=0&v=paper_preview&mkt=zh-cn
[24]	Kuehn C. Multiple Time Scale Dynamics. New York:Springer, 2015:8-14
[25]	Han XJ, Bi QS. Bursting oscillations in Duffing's equation with slowly changing external forcing. Commun Nonlinear Sci Numer Simul, 2011, 16(10):4146-4152 doi: 10.1016/j.cnsns.2011.02.021
[26]	张晓芳,陈小可,毕勤胜. 快慢耦合振子的张驰簇发及其非光滑分岔机制. 力学学报,2012,44(3):576-583 http://lxxb.cstam.org.cn/CN/abstract/abstract143327.shtml Zhang Xiaofang, Chen Xiaoke, Bi Qinsheng. Relaxation bursting of a fast-slow coupled oscillation as well as the mechanism of non-smooth bifurcation. Chinese Journal of Theoretical and Applied Mechanics, 2012,44(3):576-583(in Chinese) http://lxxb.cstam.org.cn/CN/abstract/abstract143327.shtml
[27]	Han XJ, Bi QS. Slow passage through canard explosion and mixed mode oscillations in the forced Van der Pol's equation. Nonlinear Dyn, 2012, 68(1-2):275-283 doi: 10.1007/s11071-011-0226-9
[28]	Bear SM, Erneux T, Rinzel J. The slow passage through a Hopf bifurcation:delay, memory effects, and resonance. SIAM J Appl Math, 1989, 49(1):55-71 doi: 10.1137/0149003
[29]	Baesens C. Slow sweep through a period-doubling cascade:Delayed bifurcations and renormalisation. Physica D, 1991, 53(2-4):319-375 doi: 10.1016/0167-2789(91)90068-K
[30]	Han XJ, Bi QS, Zhang C, et al. Study of mixed-mode oscillations in a parametrically excited van der Pol system. Nonlinear Dyn, 2014, 77(4):1285-1296 doi: 10.1007/s11071-014-1377-2
[31]	Tzou JC, Ward MJ, Kolokolnikov T. Slowly varying control parameters, delayed bifurcations, and the stability of spikes in reaction——diffusion systems. Physica D, 2015, 290:24-43 doi: 10.1016/j.physd.2014.09.008

Cited By

Get Citation

PDF

XML

Article Metrics

Article views PDF downloads

COMPLEX BURSTING OSCILLATION STRUCTURES IN A TWO-DIMENSIONAL NON-AUTONOMOUS DISCRETE SYSTEM

Abstract

References

Catalog

Article Metrics

Related

Download Center

Author Center

COMPLEX BURSTING OSCILLATION STRUCTURES IN A TWO-DIMENSIONAL NON-AUTONOMOUS DISCRETE SYSTEM

Abstract

References

Catalog

Article Metrics

Related

Download Center

Author Center

Export File

Citation

Format

Content