Applied problems of perturbation theory

In this paper, we study solutions to nonlinear singularly perturbed differential equations with initial conditions. Proving the asymptotic closeness of the solutions of the perturbed and unperturbed problems on the real axis is a top-priority task. But it doesn't always work out. For the first time in works in this direction, the concept of bistability of solutions was introduced. The definition of stability to the right and to the left is given. As well as definitions of bistability of solutions. Examples are given. If the solution is bistable, then it is always possible to show the asymptotic closeness of the solutions of the perturbed and unperturbed problems on the real domain.