Asymptotic Solution to a Class of Nonlinear Singular Perturbation Autonomous Differential Systems
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摘要: 研究了一类广义Lienard奇异摄动系统.首先, 求出了系统的退化解;其次, 利用奇异摄动方法得到了系统的外部解,并用伸长变量方法, 求得了系统的初始层校正项;最后, 得到了系统解的任意次渐近解析展开式,并证明了解的一致有效性.该文所用的方法和理论, 具有广泛的实际应用价值.Abstract: A class of generalized Lienard singular perturbation systems were considered. Firstly, the reduced solution to the system was obtained. Next, the outer solution was constructed by means of the singular perturbation method. Then, a stretch variable was introduced and the initial layer corrective term was found. Finally, the arbitrary-order asymptotic analytic expansion of the system solution was given and the uniform validity of the solution was proved. The proposed method with the basic theory has wide application values.
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Key words:
- stretch variable /
- singular perturbation /
- autonomous system
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[1] CALLOT J L, DIENER F, DIENER M. Le probleme deia “chase au canard”[J]. C R Acad Sci Paris, Ser 1,1978,286(22): 1059-1061. [2] ZVOKIN A K, SHUBI M A. Nonstandard analysis and singular perturbation of ordinary differential equations[J].Usp Mat Nauk,1984,39(2): 77-127. [3] ECKHAUS W. Relaxation Oscillations Including a Standard Chase on French Ducks [M]. Lect Notes in Math,Vol985. Berlin: Springer-Verlag, 1983: 449-494. [4] BENOIT E, LOBRY C. Les canards de R3[J]. C R Acad Sci Paris, Ser 1,1982,294(14): 483-488. [5] XU Yun, ZHANG Jianxia, ZU Xia, et al. Nonchaotic random behaviour in the second order autonomous system[J]. Chin Phys,2007,16(8): 2285-2290. [6] MISCHENKO E F, ROZOV N. Differential Equations With Small Parameters and Relaxation Oscillations [M]. Moscow: Academic Press, 1975. [7] LUTZ R, GOZE M. Nonstandard Analysis, a Practical Guide With Applications [M]. Lect Notes in Math,Vol881. Berlin : Springer-Verlag, 1981. [8] EI S I, MIMURA M, NAGAYAMA M. Interacting spots in reaction diffusion systems[J]. Discrete Contin Dyn Syst,2006,14(1): 31-62. [9] 李翠萍. 奇异摄动与鸭解[J]. 北京航空航天大学学报, 1993,4(1): 84-89.(LI Cuiping. Singular perturbation and duck solutions[J]. Journal of Beijing University of Aeronautics and Astronautics, 1993,4(1): 84-89.(in Chinese)) [10] LI Cuiping. Duck solutions: a new kind of bifurcation phenomenon in relaxation oscillations[J]. Acta Math Sinica (New Ser),1996,12(1): 89-104. [11] 李翠萍. 奇异摄动中的鸭解问题[J]. 中国科学(A辑), 1999,29(12): 1084-1093.(LI Cuiping. The duck solution problems in singular perturbation[J]. Science in China(Series A), 1999,29(12): 1084-1093.(in Chinese)) [12] XIE Feng, HAN Maoan, ZHANG Weijiang. The persistence of canards in 3-D singularly perturbed systems with two fast variables[J].Asymptotic Anal,2006,47(1/2): 95-106. [13] 徐云, 张建峡, 徐霞, 等. Canard轨迹原理[J]. 物理学报, 2008,57(7): 4029-4033.(XU Yun, ZHANG Jianxia, XU Xia, et al. The principle of the phase track of Canard[J].Acta Phys Sin,2008,57(7): 4029-4033.(in Chinese)) [14] 欧阳成, 姚静荪, 温朝晖, 等. 一类广义鸭轨迹系统轨线的构造[J]. 物理学报, 2012,61(3): 030202.(OUYANG Cheng, YAO Jingsun, WEN Zhaohui, et al. Constructing path curve for a class of generalized phase tracks of canard system[J]. Acta Phys Sin, 2012,61(3): 030202.(in Chinese)) [15] MO Jiaqi. A singularly perturbed nonlinear boundary value problem[J]. J Math Anal Appl,1993,178(1): 289-293. [16] MO Jiaqi. Singular perturbation for a class of nonlinear reaction diffusion systems[J]. Science in China(Series A),1989,32(11): 1306-1315. [17] MO Jiaqi, LIN Wantao. A nonlinear singular perturbed problem for reaction diffusion equations with boundary perturbation[J]. Acta Math Appl Sin,2005,21(1): 101-104. [18] MO Jiaqi, LIN Wantao, ZHU Jiang. A variational iteration method for studying the ENSO mechanism[J]. Progress in Natural Science,2004,14(12): 1126-1128. [19] MO Jiaqi, LIN Wantao. Asymptotic solution for a class of sea-air oscillator model for El-Nino-southern oscillation[J]. Chinese Physics B, 2008,17(2): 370-372. [20] MO Jiaqi, LIN Wantao. Asymptotic solution of activator inhibitor systems for nonlinear reaction diffusion equations[J]. J Sys Sci & Complexity,2008,20(1): 119-128. [21] MO Jiaqi. Variational iteration solving method for a class of generalized Boussinesq equation[J]. Chin Phys Lett,2009,26(6): 060202. [22] MO Jiaqi. Homotopiv mapping solving method for gain fluency of a laser pulse amplifier[J]. Science in China, Ser G,2009,52(7): 1007-1010. [23] FENG Yihu, MO Jiaqi. The shock asymptotic solution for nonlinear elliptic equatiom with two parameters[J]. Math Appl, 2015,27(3): 579-585. [24] 冯依虎, 石兰芳, 汪维刚, 等. 一类广义非线性强阻尼扰动发展方程的行波解[J]. 应用数学和力学, 2015,36(3): 315-324.(FENG Yihu, SHI Lanfang, WANG Weigang, et al. The traveling wave solution for a class of generalized nonlinear strong damping disturbed evolution equations[J]. Applied Mathematics and Mechanics,2015,36(3): 315-324.(in Chinese)) [25] 冯依虎, 莫嘉琪. 一类非线性非局部扰动LGH方程的孤立子行波解[J]. 应用数学和力学, 2016,37(4): 426-433.(FENG Yihu, MO Jiaqi. A class of soliton travelling wave solution for the nonlinear nonlocal disturbed LGH equation[J]. Applied Mathematics and Mechanics, 2016,37(4): 426-433.(in Chinese)) [26] FENG Yihu, MO Jiaqi. Asymptotic solution for singularly perturbed fractional order differential equation[J]. J Math,2016,36(2): 239-245. [27] FENG Yihu, CHEN Xianfeng, MO Jiaqi. The generalized interior shock layer solution of a class of nonlinear singularly perturbed reaction diffusion problem[J]. Math Appl,2016,29(1): 161-165.
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