Approximate analytical solution in slow-fast system based on modified multi-scale method

doi:10.1007/s10483-020-2598-9

Applied Mathematics and Mechanics (English Edition) ›› 2020, Vol. 41 ›› Issue (4): 605-622.doi: https://doi.org/10.1007/s10483-020-2598-9

Approximate analytical solution in slow-fast system based on modified multi-scale method

Xianghong LI¹, Jianhua TANG¹, Yanli WANG¹, Yongjun SHEN²

1. Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China;
2. Department of Mechanical Engineering, Shijiazhuang Tiedao University, Shijiazhuang 050043, China

收稿日期:2019-11-19 修回日期:2020-01-23 发布日期:2020-03-26
通讯作者: Xianghong LI E-mail:lxhll601@163.com
基金资助:
Project supported by the National Natural Science Foundation of China (Nos. 11672191, 11772206, and U1934201) and the Hundred Excellent Innovative Talents Support Program in Hebei University (No. SLRC2017053)

Approximate analytical solution in slow-fast system based on modified multi-scale method

Xianghong LI¹, Jianhua TANG¹, Yanli WANG¹, Yongjun SHEN²

1. Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China;
2. Department of Mechanical Engineering, Shijiazhuang Tiedao University, Shijiazhuang 050043, China

Received:2019-11-19 Revised:2020-01-23 Published:2020-03-26
Contact: Xianghong LI E-mail:lxhll601@163.com
Supported by:
Project supported by the National Natural Science Foundation of China (Nos. 11672191, 11772206, and U1934201) and the Hundred Excellent Innovative Talents Support Program in Hebei University (No. SLRC2017053)

摘要/Abstract

摘要： A simple, yet accurate modified multi-scale method (MMSM) for an approximately analytical solution in nonlinear oscillators with two time scales under forced harmonic excitation is proposed. This method depends on the classical multi-scale method (MSM) and the method of variation of parameters. Assuming that the forced excitation is a constant, one could easily obtain the approximate analytical solution of the simplified system based on the traditional MSM. Then, this solution for the oscillator under forced harmonic excitation could be established after replacing the harmonic excitation by the constant excitation. To certify the correctness and precision of the proposed analytical method, the van der Pol system with two scales subject to slowly periodic excitation is investigated; this system presents rich dynamical phenomena such as spiking (SP), spiking-quiescence (SP-QS), and quiescence (QS) responses. The approximate analytical expressions of the three types of responses are given by the MMSM, and it can be found that the precision of the new analytical method is higher than that of the classical MSM and better than that of the harmonic balance method (HBM). The results obtained by the present method are considerably better than those obtained by traditional methods, quantitatively and qualitatively, particularly when the excitation frequency is far less than the natural frequency of the system.

关键词: modified multi-scale method (MMSM), slow-fast system, multi-scale method (MSM), van der Pol system

Abstract: A simple, yet accurate modified multi-scale method (MMSM) for an approximately analytical solution in nonlinear oscillators with two time scales under forced harmonic excitation is proposed. This method depends on the classical multi-scale method (MSM) and the method of variation of parameters. Assuming that the forced excitation is a constant, one could easily obtain the approximate analytical solution of the simplified system based on the traditional MSM. Then, this solution for the oscillator under forced harmonic excitation could be established after replacing the harmonic excitation by the constant excitation. To certify the correctness and precision of the proposed analytical method, the van der Pol system with two scales subject to slowly periodic excitation is investigated; this system presents rich dynamical phenomena such as spiking (SP), spiking-quiescence (SP-QS), and quiescence (QS) responses. The approximate analytical expressions of the three types of responses are given by the MMSM, and it can be found that the precision of the new analytical method is higher than that of the classical MSM and better than that of the harmonic balance method (HBM). The results obtained by the present method are considerably better than those obtained by traditional methods, quantitatively and qualitatively, particularly when the excitation frequency is far less than the natural frequency of the system.

Key words: modified multi-scale method (MMSM), slow-fast system, multi-scale method (MSM), van der Pol system

中图分类号:

34E13
74H10

Xianghong LI, Jianhua TANG, Yanli WANG, Yongjun SHEN. Approximate analytical solution in slow-fast system based on modified multi-scale method[J]. Applied Mathematics and Mechanics (English Edition), 2020, 41(4): 605-622.

参考文献

[1] BERTRAM, R., BUTTE, M. J., KIEMELl, T., and SHERMAN, A. Topological and phenomenological classification of bursting oscillations. Bulletin of Mathematical Biology, 57(3), 413-439(1995)
[2] LV, M., WANG, C. N., REN, G. D., MA, J., and SONG, X. L. Model of electrical activity in a neuron under magnetic flow effect. Nonlinear Dynamics, 85(3), 1479-1490(2016)
[3] IZHIKEVICH, E. M. Neural excitability, spiking and bursting. International Journal of Bifurcation and Chaos, 10(6), 1171-1266(2000)
[4] WU, H. G., BAO, B. C., LIU, Z., XU, Q., and JIANG, P. Chaotic and periodic bursting phenomena in a memristive Wien-bridge oscillator. Nonlinear Dynamics, 83(1/2), 893-903(2016)
[5] RULKOV, N. F. Regularization of synchronized chaotic bursts. Physical Review Letters, 86(1), 183(2001)
[6] NAYFEH, A. H. and BALACHANDRAN, B. Applied Nonlinear Dynamics, John Wiley & Sons, New York (1995)
[7] ZHANG, H., CHEN, D. Y., XU, B. B., WU, C. Z., and WANG, X. Y. The slow-fast dynamical behaviors of a hydro-turbine governing system under periodic excitations. Nonlinear Dynamics, 87(4), 2519-2528(2017)
[8] YANG, S. P., CHEN, L. Q., and LI, S. H. Dynamics of Vehicle-Road Coupled System, Science Press, Beijing (2015)
[9] LI, X. H. and HOU, J. Y. Bursting phenomenon in a piecewise mechanical system with parameter perturbation in stiffness. International Journal of Non-Linear Mechanics, 81, 165-176(2016)
[10] BI, Q. S. The mechanism of bursting phenomena in Belousov-Zhabotinsky (BZ) chemical reaction with multiple time scales. SCIENCE CHINA Technological Sciences, 53(3), 748-760(2010)
[11] BERTRAM, R., SMOLEN, P., SHERMAN, A., MEARS, D., and ATWATER, I. A role for calcium release-activated current (CRAC) in cholinergic modulation of electrical activity in pancreatic beta-cells. Biophysical Journal, 68(6), 2323-2332(1995)
[12] BUTERA, R. J., JR, RINZEL, J., and SMITH, J. C. Models of respiratory rhythm generation in the pre-Bötzinger complex, I:bursting pacemaker neurons. Journal of Neurophysiology, 82(1), 382-397(1999)
[13] KEPECS, A. and WANG, X. J. Analysis of complex bursting in cortical pyramidal neuron models. Neurocomputing, 32, 181-187(2000)
[14] LAJOIE, G. and SHEA-BROWN, E. Shared inputs, entrainment, and desynchrony in elliptic bursters:from slow passage to discontinuous circle maps. SIAM Journal on Applied Dynamical Systems, 10(4), 1232-1271(2011)
[15] DESTEXHE, A., MCCORMICK, D. A., and SEJNOWSKI, T. J. A model for 8-10 Hz spindling in interconnected thalamic relay and reticularis neurons. Biophysical Journal, 65(6), 2473-2477(1993)
[16] RINZEL, J. and LEE, Y. S. Dissection of a model for neuronal parabolic bursting. Journal of Mathematical Biology, 25(6), 653-675(1987)
[17] THEODORE, V., MARK, K. A., and TASSO, K. J. Amplitude-modulated bursting:a novel class of bursting rhythms. Physical Review Letters, 117(26), 268101(2016)
[18] HAN, X., BI, Q., and KURTHS, J. Route to bursting via pulse-shaped explosion. Physical Review E, 98(1), 010201(2018)
[19] WANG, J., LU, B., LIU, S. Q., and JIANG, X. F. Bursting types and bifurcation analysis in the pre-Bötzinger complex respiratory rhythm neuron. International Journal of Bifurcation and Chaos, 27(1), 1750010(2017)
[20] BARRIO, R., RODRIGUEZ, M., SERRANO, S., and SHILNIKOV, A. Mechanism of quasiperiodic lag jitter in bursting rhythms by a neuronal network. Europhysics Letters, 112(3), 38002(2015)
[21] DESROCHES, M., GUILLAMON, A., PONCE, E., PROHENS, R., RODRIGUES, S., and TERUEL, A. E. Canards, folded nodes, and mixed-mode oscillations in piecewise-linear slowfast systems. SIAM Review, 58(4), 653-691(2016)
[22] AMBROSIO, B., AZIZ-ALAOUI, M. A., and YAFIA, R. Canard phenomenon in a slow-fast modified Leslie-Gower model. Mathematical Biosciences, 295, 48-54(2018)
[23] VO, T. Generic torus canards. Physica D:Nonlinear Phenomena, 356, 37-64(2017)
[24] MASLENNIKOV, O. V., NEKORKIN, V. I., and KURTHS, J. Basin stability for burst synchronization in small-world networks of chaotic slow-fast oscillators. Physical Review E, 92(4), 042803(2015)
[25] YU, Y., GAO, Y. B., HAN, X. J., and BI, Q. S. Modified function projective bursting synchronization for fast-slow systems with uncertainties and external disturbances. Nonlinear Dynamics, 79(4), 2359-2369(2015)
[26] MBÉ, J. H. T., TALLA, A. F., CHENGUI, G. R. G., COILLET, A., LARGER, L., WOAFO, P., and CHEMBO, Y. K. Mixed-mode oscillations in slow-fast delayed optoelectronic systems. Physical Review E, 91(1), 012902(2015)
[27] HAN, X. J., BI, Q. S., ZHANG, C., and YU, Y. Delayed bifurcations to repetitive spiking and classification of delay-induced bursting. International Journal of Bifurcation and Chaos, 24(7), 1450098(2014)
[28] STRIZHAK, P. E. and KAWCZYNSKI, A. L. Regularities in complex transient oscillations in the Belousov-Zhabotinsky reaction in a batch reactor. Journal of Physical Chemistry, 99(27), 10830-10833(1995)
[29] PEDREÑO, S., PISCO, J. P., LARROUY-MAUMUS, G., KELLY, G., and DE CARVALHO, L. P. S. Mechanism of feedback allosteric inhibition of ATP phospho ribosyl transferase. Biochemistry, 51(40), 8027-8038(2012)
[30] PASE, L., LAYTON, J. E., WITTMANN, C., ELLETT, F., NOWELL, C. J., ROGERS, K. L., HALL, C. J., and KEIGHTLEY, M. C. Neutrophil-delivered myelo peroxidase dampens the hydrogen peroxide burst after tissue wounding in zebra fish. Current Biology, 22(19), 1818-1824(2012)
[31] HOFBAUER, S., BELLEI, M., SÜDERMANN, A., PIRKER, K. F., DAIMS, H., FURTMÜLLER, P. G., and DJINOVIĆ-CARUGO, K. Redox thermodynamics of high-spin and low-spin forms of chlorite dismutases with diverse subunit and oligomeric structures. Biochemistry, 51(47), 9501-9512(2012)
[32] LASHINA, E. A., CHUMAKOVA, N. A., CHUMAKOV, G. A., and BORONIN, A. I. Chaotic dynamics in the three-variable kinetic model of CO oxidation on platinum group metals. Chemical Engineering Journal, 154(1-3), 82-87(2009)
[33] CADENA, A., BARRAGÁN, D., and ÁGREDA, J. Bursting in the Belousov-Zhabotinsky reaction added with phenol in a batch reactor. Journal of the Brazilian Chemical Society, 24(12), 2028-2032(2013)
[34] XU, J., MIAO, Y., and LIU, J. C. Strong averaging principle for slow-fast SPDEs with Poisson random measures. Discrete and Continuous Dynamical Systems-Series B, 20, 2233-2256(2015)
[35] CERRAI, S. and LUNARDI, A. Averaging principle for nonautonomous slow-fast systems of stochastic reaction-diffusion equations:the almost periodic case. SIAM Journal on Mathematical Analysis, 49(4), 2843-2884(2017)
[36] KOKOTOVIC, P., KHALI, H. K., and O'REILLY, J. Singular Perturbation Methods in Control:Analysis and Design, Academic Press, Orlando (1999)
[37] GUCKENHEIMER, J., HOFFMAN, K., and WECKESSER, W. The forced van der Pol equation I:the slow flow and its bifurcations. SIAM Journal on Applied Dynamical Systems, 2(1), 1-35(2003)
[38] GLIZER, V. Y., FEIGIN, Y., FRIDMAN, E., and MARGALIOT, M. A new approach to exact slow-fast decomposition of singularly perturbed linear systems with small delays. 53rd IEEE Conference on Decision and Control, IEEE, Los Angeles, 451-456(2014)
[39] FARAZMAND, M. and SAPSIS, T. P. Dynamical indicators for the prediction of bursting phenomena in high-dimensional systems. Physical Review E, 94(3), 032212(2016)
[40] TZOU, J. C., WARD, M. J., and KOLOKOLNIKOV, T. Slowly varying control parameters, delayed bifurcations, and the stability of spikes in reaction-diffusion systems. Physica D:Nonlinear Phenomena, 290, 24-43(2015)
[41] WIGGINS, S. and SHAW, S. W. Chaos and three-dimensional horseshoes in slowly varying oscillators. Journal of Applied Mechanics, 55(4), 959-968(1988)
[42] NAYFEH, A. H. and MOOK, D. T. Nonlinear Oscillations, John Wiley & Sons, New York (1979)
[43] PAPANGELO, A. and CIAVARELLA, M. On the limits of quasi-static analysis for a simple Coulomb frictional oscillator in response to harmonic loads. Journal of Sound and Vibration, 339, 280-289(2015)
[44] DING, H., HUANG, L. L., MAO, X. Y., and CHEN, L. Q. Primary resonance of a traveling viscoelastic beam under internal resonance. Applied Mathematics and Mechanics (English Edition), 38(1), 1-14(2017) https://doi.org/10.1007/s10483-016-2152-6
[45] DING, H., ZHU, M. H., and CHEN, L. Q. Dynamic stiffness method for free vibration of an axially moving beam with generalized boundary conditions. Applied Mathematics and Mechanics (English Edition), 40(7), 911-924(2019) https://doi.org/10.1007/s10483-019-2493-8

Approximate analytical solution in slow-fast system based on modified multi-scale method

Approximate analytical solution in slow-fast system based on modified multi-scale method

RichHTML

PDF

可视化

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 1

编辑推荐

Metrics

本文评价