Existence of Duck Solutions of A Kind of Differential Equations with Small Parameter

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=a_c(ε) such that for a in a small neighbourhood of a_c(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.