In this paper, a kind of singularly perturbed first-order differential equations with integral boundary condition are considered. With the method of boundary layer function and the Banach fixed-point theorem, the uniformly valid asymptotic solution of the original problem is obtained.
A singularly perturbed second-order semilinear differential equation with integral boundary conditions is considered. By the method of boundary functions, the conditions under which there exists an internal transition layer for the original problem are established. The existence of spike-type solution is obtained by smoothly connecting the solutions of left and right associated problems, and the asymptotic expansion of the spike-type solution is also presented.
The canard phenomenon occurring in planar fast-slow systems under non-generic conditions is investigated.When the critical manifold has a non-generic fold point,by using the method of asymptotic analysis combined with the recently developed blow-up technique,the existence of a canard is established and the asymptotic expansion of the parameter for which a canard exists is obtained.
A class of delayed oscillators of El Nifio-southern oscillation （ENSO） models is considered. Using the delayed theory, the perturbed theory and other methods, the asymptotic expansions of the solutions for ENSO models are obtained and the asymptotic behaviour of solution of corresponding problem is studied.
