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In this paper, the general mathematical principle is overall explained and a new general technique is presented inorder to calculate uniformly asymptotic expansions of solutions of the perturbed bifurcation problem (1.6) in the vicinity of y=0, A=0. ~:0 , by means of singular perturbation method. Simultaneously, Newton's polygon [4] is generalized. Finally, the calculating results of two examples are given. Null(A*) :span{C*}, [Ir = I~ J then under certain conditions, the solutions of [l] solution at (0,60) The meaning of the nonlinear operator eq. (I.I) is very generalized. For ex ~ ar~ple, in elasticity, ~ represents a load, and the bifurcation is called the buck ling; in fluid dynamics, ( stands5 for a fluid parameter, such as Mach number, and the bifurcation is called the critical point. When a small effect of an imperfection or a heterogeneity in practical problems is considered, the mathematical description (i.i) of problems will be

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