EI、Scopus 收录
中文核心期刊
Chen Yani, Meng Wenjing, Qian Youhua. FIXED POINT CHAOS AND FOLD/FOLD BURSTING OF A CLASS OF DUFFING SYSTEMS AND THE MECHANISM ANALYSIS[J]. Chinese Journal of Theoretical and Applied Mechanics, 2020, 52(5): 1475-1484. DOI: 10.6052/0459-1879-20-098
Citation: Chen Yani, Meng Wenjing, Qian Youhua. FIXED POINT CHAOS AND FOLD/FOLD BURSTING OF A CLASS OF DUFFING SYSTEMS AND THE MECHANISM ANALYSIS[J]. Chinese Journal of Theoretical and Applied Mechanics, 2020, 52(5): 1475-1484. DOI: 10.6052/0459-1879-20-098

FIXED POINT CHAOS AND FOLD/FOLD BURSTING OF A CLASS OF DUFFING SYSTEMS AND THE MECHANISM ANALYSIS

  • Received Date: March 31, 2020
  • In this paper, a class of bistable Duffing type system with two slow variables under new materials is explored. The system is simulated by time history diagram, phase diagram and bifurcation diagram, then the dynamic mechanism of the system under different parameters is analyzed theoretically. Firstly, this manuscript describes that when the amplitude parameter value is greater than 1, the system may exhibit fixed point chaos and explains the reason of fixed point chaos. Secondly, this manuscript introduces the phenomenon of Fold/Fold bursting in parameter space which is caused by the movement of the system from one side of the saddle-node surface to the other side. We also call it saddle-node bursting. In fact, when the system passes through the saddle-node surface, the number of equilibrium points changes. Then this manuscript uses the path of longitudinal parabolic to explain the mechanism of Fold/Fold bursting. And it is found that regardless of the value of constant coefficient term and amplitude, as long as a certain relationship is satisfied, there will always be Fold/Fold bursting. Next this manuscript uses the linear path to discuss the influence of newly added constant coefficient term. It is found that the position where the path intersects the saddle-node surface will affect the symmetry of the bursting, and the span of the path will affect the magnitude of the bursting oscillation. Finally, this manuscript uses the multiple inflection curve path to discuss the phenomenon when two incentive terms have specific relation. When , the change of the constant coefficient term will make the system show Fold/Fold bursting with different times, and the maximum can reach triple bursting. Moreover, it is found that if you can find a path that can be divided into segments, and each segment will have an intersection with the saddle surface, then times Fold/Fold bursting will occur.
  • [1] Popovi? N, Marr C, Swain PS. A geometric analysis of fast-slow models for stochastic gene expression. Journal of Mathematical Biology, 2016,72(1):87-122
    [2] Chtouki A, Lakrad F, Belhaq M. Quasi-periodic bursters and chaotic dynamics in a shallow arch subject to a fast--slow parametric excitation. Nonlinear Dynamics, 2020,99(570):283-298
    [3] Podogova SD, Mishagin KG, Medvedev SY, et al. An algorithm for a group time scale using a moving average over multiple time scales. Measurement Techniques, 2015,58(3):532-538
    [4] Karami H, Karimi S, Bonakdari H, et al. Predicting discharge coefficient of triangular labyrinth weir using extreme learning machine, artificial neural network and genetic programming. Neural Computing and Applications, 2018,29(7):983-989
    [5] Sun J, Wu ZH. Isolating spatiotemporally local mixed Rossby-gravity waves using multi-dimensional ensemble empirical mode decomposition. Climate Dynamics, 2020,54(3):1383-1405
    [6] Zhang Y, Li T, Gao JY, et al. Origins of quasi-biweekly and intraseasonal oscillations over the South China Sea and Bay of Bengal and scale selection of unstable equatorial and off-equatorial modes. Journal of Meteorological Research, 2020,34(2):137-149
    [7] Ngueuteu GSM, Yamapi R, Woafo P. Quasi-static transient and mixed mode oscillations induced by fractional derivatives effect on the slow flow near folded singularity. Nonlinear Dynamics, 2014,78(4):2717-2729
    [8] Han XJ, Bi QS, Zhang C, et al. Study of mixed-mode oscillations in a parametrically excited van der Pol system. Nonlinear Dynamics, 2014,77(4):1285-1296
    [9] Han XJ, Xia FB, Zhang C, et al. Origin of mixed-mode oscillations through speed escape of attractors in a Rayleigh equation with multiple-frequency excitations. Nonlinear Dynamics, 2017,88(4):2693-2703
    [10] Zhang XF, Zheng JK, Wu GQ, et al. Mixed mode oscillations as well as the bifurcation mechanism in a Duffing's oscillator with two external periodic excitations. Science China Technological Sciences, 2019,62(10):1816-1824
    [11] Rinzel J. Bursting oscillations in an excitable membrane model. Ordinary and Partial Differential Equations, 1985,1151:304-316
    [12] Izhikevich EM. Neural excitablity, spiking and bursting. International Journal of Bifurcation and Chaos, 2000,10(6):1171-1266
    [13] Meng WJ, Qian YH. Mixed-mode oscillation in a class of delayed feedback system and multistability dynamic response. Complexity, 2020,2020:1-18
    [14] Qian YH, Yan DM. Fast slow dynamics analysis of a coupled duffing system with periodic excitation. International Journal of Bifurcation and Chaos, 2016,28(8):1850148
    [15] Zhang W, Liu YZ, Wu MQ. Theory and experiment of nonlinear vibrations and dynamic snap-through phenomena for bi-stable asymmetric laminated composite square panels under foundation excitation. Composite Structures, 2019,225:111140
    [16] Chen XK, Li SL, Zhang ZD, et al. Relaxation oscillations induced by an order gap between exciting frequency and natural frequency. Science China Technological Sciences, 2017,60(2):289-298
    [17] Li XH, Shen YJ, Sun JQ, et al. New periodic-chaotic attractors in slow-fast Duffing system with periodic parametric excitation. Science Reports, 2019,9(1):422-434
    [18] Lin Y, Liu WB, Bao H, et al. Bifurcation mechanism of periodic bursting in a simple three-element-based memristive circuit with fast-slow effect. Chaos, Solitons and Fractals, 2020,131(3):109524
    [19] Wang N, Zhang GS, Bao H. Bursting oscillations and coexisting attractors in a simple memristor-capacitor-based chaotic circuit. Nonlinear Dynamics, 2019,97(2):1477-1494
    [20] 苟向锋, 韩林勃, 朱凌云 等. 单自由度齿轮传动系统安全盆侵蚀与分岔. 振动与冲击, 2020,39(2):123-131
    [20] ( Gou Xiangfeng, Han Linbo, Zhu Lingyun, et al. Safety basin erosion and bifurcation of single-degree-of-freedom gear transmission system. Chinese Journal of Vibration and Shock, 2020,39(2):123-131 (in Chinese))
    [21] 魏梦可, 韩修静, 张晓芳 等. 正负双向脉冲式爆炸及其诱导的簇发振荡. 力学学报, 2019,51(3):904-911
    [21] ( Wei Mengke, Han Xiujing, Zhang Xiaofang, et al. Positive and negative two-way pulse explosion and its induced cluster oscillation. Chinese Journal of Theoretical and Applied Mechanics, 2019,51(3):904-911 (in Chinese))
    [22] 郑健康, 张晓芳, 毕勤胜. 一类混沌系统中的簇发振荡及其延迟叉形分岔行为. 力学学报, 2019,51(2):540-549
    [22] ( Zheng Jiankang, Zhang Xiaofang, Bi Qinsheng. Clusters oscillation and delayed fork bifurcation in a class of chaotic systems. Chinese Journal of Theoretical and Applied Mechanics, 2019,51(2):540-549 (in Chinese))
    [23] 张正娣, 李静, 刘亚楠 等. 非光滑系统不同簇发模式之间的演化及其机理. 中国科学: 技术科学, 2019,49(9):1031-1039
    [23] ( Zhang Zhengdi, Li Jing, Liu Yanan, et al. Evolution and mechanism of different cluster patterns in non-smooth systems. Chinese Journal of Science in China, 2019,49(9):1031-1039 (in Chinese))
    [24] 曲子芳, 张正娣, 彭淼 等. 双频激励下 Filippov 系统的非光滑簇发振荡机理. 力学学报, 2018,50(5):1145-1155
    [24] ( Qu Zifang, Zhang Zhengdi, Peng Miao, et al. Oscillation mechanism of non-smooth clusters in Filippov system under dual-frequency excitation. Chinese Journal of Theoretical and Applied Mechanics, 2018,50(5):1145-1155 (in Chinese))
    [25] Han XJ, Yu Y, Zhang C, et al. Turnover of hysteresis determines novel bursting in Duffing system with multiple-frequency external forcings. International Journal of Non-Linear Mechanics, 2017,89:69-74
    [26] Han XJ, Zhang Y, Bi QS, et al. Two novel bursting patterns in the Duffing system with multiple-frequency slow parametric excitations. Chaos, 2018,28:043111
    [27] Rinzel J, Lee YS. Dissection of a model for neuronal parabolic bursting. Journal of Mathematical Biology, 1987,25(4):653-675
    [28] Bertram R, Butte MJ, Kiemel T, et al. Topological and phenomenological classifification of bursting oscillations. Bulletin of Mathematical Biology, 1995,57(3):413-439
    [29] Stern JV, Osinga HM, LeBeau A, et al. Resetting behavior in a model of bursting in secretory pituitary cells: distinguishing plateaus from pseudo-plateaus. Bulletin of Mathematical Biology, 2008,70(1):68-88
    [30] Krauskopf B, Osinga HM. A codimension-four singularity with potential for action. Mathematical Sciences with Multidisciplinary Applications, 2016,157:253-268
    [31] Saggio ML, Spiegler A, Bernard C, et al. Fast--slow bursters in the unfolding of a high codimension singularity and the ultra-slow transitions of classes. The Journal of Mathematical Neuroscience, 2017,7:7
  • Related Articles

    [1]Zhao Huan, Huang Yujun, Xing Haonan. ADAPTIVE SPARSE POLYNOMIAL CHAOS-BASED FLOW FIELD/SONIC BOOM UNCERTAINTY QUANTIFICATION UNDER MULTI-PARAMETER UNCERTAINTIES[J]. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(9): 2027-2042. DOI: 10.6052/0459-1879-23-122
    [2]Gong Bingqing, Zheng Zechang, Chen Yanmao, Liu Jike. A FAST CALCULATION FOR THE SYMMETRY BREAKING POINT OF QUASI-PERIODIC RESPONSES[J]. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(11): 3181-3188. DOI: 10.6052/0459-1879-22-324
    [3]Wan Honglin, Li Xianghong, Shen Yongjun, Wang Yanli. STUDY ON VIBRATION REDUCTION OF DYNAMIC VIBRATION ABSORBER FOR TWO-SCALE DUFFING SYSTEM[J]. Chinese Journal of Theoretical and Applied Mechanics, 2022, 54(11): 3136-3146. DOI: 10.6052/0459-1879-22-286
    [4]Yue Yuan. LOCAL DYNAMICAL BEHAVIOR OF TWO-PARAMETER FAMILY NEAR THE NEIMARK-SACKER-PITCHFORK BIFURCATION POINT IN A VIBRO-IMPACT SYSTEM[J]. Chinese Journal of Theoretical and Applied Mechanics, 2016, 48(1): 163-172. DOI: 10.6052/0459-1879-15-144
    [5]Time-space analysis of vortex flow field in compressor at near-stall point[J]. Chinese Journal of Theoretical and Applied Mechanics, 2010, 42(4): 623-628. DOI: 10.6052/0459-1879-2010-4-lxxb2009-056
    [6]THE QUASI-FIXED-POINT TRACING METHOD FOR MULIT-PERIODIC-SOLUTIONS OF A NONLINEAR DYNAMIC SYSTEM[J]. Chinese Journal of Theoretical and Applied Mechanics, 1999, 31(2): 222-229. DOI: 10.6052/0459-1879-1999-2-1995-022
    [7]PRINCIPAL RESPONSE OF DUFFING OSCILLATOR TO COMBINED DETERMINISTIC AND NARROW BAND RANDOM PARAMETERIC EXCITATION 1)[J]. Chinese Journal of Theoretical and Applied Mechanics, 1998, 30(2): 178-185. DOI: 10.6052/0459-1879-1998-2-1995-114
    [8]CML MODELS FOR SPATIOTEMPORAL CHAOS[J]. Chinese Journal of Theoretical and Applied Mechanics, 1996, 28(6): 741-744. DOI: 10.6052/0459-1879-1996-6-1995-395
    [9]EVALUATION OF GREEN'S FUNCTION FOR A POINT DYNAMIC LOAD IN THE INTERIOR OF AN ELASTIC HALF-SPACE[J]. Chinese Journal of Theoretical and Applied Mechanics, 1994, 26(5): 583-592. DOI: 10.6052/0459-1879-1994-5-1995-584
    [10]CENTROSYMMETRIC CHAOS[J]. Chinese Journal of Theoretical and Applied Mechanics, 1992, 24(1): 55-58. DOI: 10.6052/0459-1879-1992-1-1995-711
  • Cited by

    Periodical cited type(4)

    1. 刘玲玉,李向红,王宏斌. 一类非线性金融系统的多尺度效应研究. 统计与决策. 2024(09): 67-72 .
    2. 韩修静,黄启旭,丁牧川,毕勤胜. 谐波齿轮系统的快慢振荡机制研究. 力学学报. 2022(04): 1085-1091 . 本站查看
    3. 龚冰清,郑泽昌,陈衍茂,刘济科. 准周期响应对称破缺分岔点的一种快速计算方法. 力学学报. 2022(11): 3181-3188 . 本站查看
    4. 黄建亮,王腾,陈树辉. 含外激励van der Pol-Mathieu方程的非线性动力学特性分析. 力学学报. 2021(02): 496-510 . 本站查看

    Other cited types(1)

Catalog

    Article Metrics

    Article views (1345) PDF downloads (158) Cited by(5)
    Related

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return