Effects of varying nonlinearity and their singular perturbation flavour

Y .Autonomous differential equations y q f y, p s 0 whose nonlinearity varies Y .with a parameter p are studied. As a prototype, one may think of y q f y q0 < < py1 .y g y s 0. We discuss periodic solutions with initial values taken from various domains and their different types of convergence as p “ ‘. Equations yY y . .f y, p s 0, yf y, p) 0 are similarly discussed, with the ``period' ' of a solution replaced by its ``maximal interval of existence.' ' The study shows a natural link to singularly perturbed problems. It turns out that the family of ODEs under consideration are essentially a family of singularly perturbed problems. Solutions may develop ``kinks' ' and higher order derivatives of solutions possess ``boundary layers,' ' namely sets of non-uniform convergence. Similarities and differences between this family and the more common singularly perturbed problems which abound in the literature emerge. Q 2000 Academic Press Key Words: varying nonlinearity; convergence of solutions; delta method; bound-ary layer; interval of existence; singular perturbations