Resonances of Linear Differential Equations with Piecewise Constant Coefficients

This paper develops a method to study and control parametric resonance in systems, which are governed by linear differential equations of the second order with periodic coefficients. This method allows finding a monodromy matrix as a result of composition of elementary phase flux transformations. The method is based on piecewise constant approximation of coefficients of the equations. A criterion for parametric resonance is found which is taking into account multiplicity of roots of the characteristic equation of monodromy and action of a dissipative forces as well. Complete study of parametric resonance is presented for Hill's equation and for equation of Mathieu's type in a special case of periodic two-step piecewise constant functions. For a case of a dissipative force the range of parameters for resonance to occur is shown as well.