Existence of Duck Solutions of A Kind of Differential Equations with Small Parameter
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摘要: 本文应用微分方程定性理论、 渐进分析方法、 隐函数定理以及不动点理论的方法研究一类单参数二维奇异摄动系统.给出了当系统的奇点在破坏点的小邻域时鸭解和鸭极限环存在的充分条件.证明了存在参数值a=a\-c(ε),使得对a\-c(ε)小邻域中的所有参数a, 系统存在鸭极限环.并给出了鸭解和鸭极限环的渐近估计式以及鸭极限环随参数变化的规律.本文推广了文献[1]和文献[2]的结果.Abstract: A kind of one parameter planar singular perturbation equation is studied by the qualitative theory of ordinary differential equations,asymptotic analysis methods,implicit function theorem and fixed point methods.Some sufficient conditions are given to support the existence of duck solutions and duck cycles when the singular points of the system are in the small neighbourhood of turnning points.It is proved that there exists a value of parameter a=ac(ε) such that for a in a small neighbourhood of ac(ε),the systems have duck cycles.Moreover,the asymptotic estimation of corresponding duck solutions and duck cycles and the rule of changing of the duck cycles with parameter are obtained.The article 1 and 2 are the special cases of this paper.
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Key words:
- asymptotic methods /
- qualitative theory /
- differential equations /
- singular perturbation /
- bifurcation /
- duck solution /
- duck cycle
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[1]李翠萍.奇异摄动方程与鸭解[J].北京航空航天大学学报,1993,19(4):84~89. [2]Li Cuiping.Duck solutions.a new kind of bifurcation phenomenon in relaxation oscillations[J].Acta Math Sinica(New Series),1996,12(1):89~104. [3]Mischenko E F,Rozov N.Differential equations with small parameters and relaxation oscillation[M].Moscow:Academic Press,1975. [4]张芷芬,丁同仁,黄文灶,等.微分方程定性理论[M].北京:科学出版社,1985. -
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