The asymptotic formulae in the problem on constructing hyperbolicity and stability regions of dynamical systems

The paper proposes a new general method allowing one to study the problem on constructing hyperbolicity and stability regions for nonlinear dynamical systems. The method is based on a modification of M. Rozo method for studying the stability of linear systems with periodic coefficients depending on a small parameter and on the asymptotic formulae in the perturbation theory of linear operators. We obtain approximate formulae describing the boundary of hyperbolicity and stability regions. As an example, we provide the scheme for constructing the stability regions for Mathieu equation.