ABSTRACT
One of the important problems of the theory of nonlinear multifrequency oscilla tions is to establish the conditions for preserving invariant tori of the dynamical systems under perturbations. This problem is closely connected with problems of the existence and integral representation of solutions, bounded on the whole R axis, to systems of linear differential equations
with a matrix A(t) which is continuous and bounded on R , f ( t ) E C°(R). These problems are investigated in this chapter. It is known that the existence of a boun ded, for example, on the semiaxis = [0, oo) solution of the system (1.0.1) is equivalent for every fixed vector function f( t) bounded on R+ to the exponential dichotomy (e-dichotomy) on R+ of the corresponding homogeneous system
On the other hand the property of e-dichotomy on J2+ is equivalent to the existence of the quadratic form (the Lyapunov function)
with a continuously differentiable and bounded on i?+ matrix of the coefficients S(t) which possesses a sign-definite derivative due to the system (1.0.1)
Also, no restrictions are required concerning the matrix S(£), except that it is continuously differentiable and bounded on R + . We are especially interested in the property of e-dichotomy of the system (1.0.2) on the whole R axis, because this very property solely ensures the existence and uniqueness of the solution of the system (1.0.1) bounded on R. Additionally, the condition of non-degeneracy was to be imposed on the quadratic form (1.0.3)
The abandoning of the uniqueness condition for the solution of the system (1.0.1) bounded on R yields, by the same token, the abandonment of conditions (1.0.4) and (1.0.5), requiring instead that the derivative of the quadratic form (1.0.3) due to the system which is conjugated with the system (1.0.2)
In addition, the determinant of the matrix S(t) may be zero for some t = £*, 0 < i < n.
