Asymptotic analysis of linear periodic systems of homogeneous differential equations with a large or small parameter (Q881240)

An for the algorithm investigation of a wide class of systems of linear ordinary differential equations with periodic coefficients, involving large or small parameters is described. The proposed algorithm allows to investigate the mentioned system constructively by means of a new version of the splitting method known in the theory of regular and singular perturbations. The analysis of certain classes of real physical systems gives rise (in the linear approximation) to the investigation of systems of differential equations in the form \[ {\dot{x}}=A(t)x, \] where \(A(t)\) is a \(T\)-periodic matrix. If the averaged value \(A_0\) of the matrix \(A(t)\) exists, then one can investigate systems of the form: \[ {\dot{x}}=(A_0+\delta A_1(t))x \] where \(\delta>0\) is a parameter which characterizes the oscillation amplitude. The paper considers the cases \(\delta=\epsilon>0\) and \(\delta=\epsilon^{-1}\). Several properties, as uniform boundedness of the solution, are proved.