Periodic Solutions of Linear Systems
In this chapter the existence and stability problems of periodic solutions of linear systems will be treated. For centuries people used linear systems as models of phenomena in Nature, in Mechanics, in Physics, etc. So that in the theories of elasticity, heat propagation, the propagation of waves, electromagnetic phenomena, etc. basic differential equations are linear ones. Besides that, as this can be seen from (1.1.5) and (1.3.7), the variational system with respect to a periodic solution of a periodic or an autonomous system is a linear system with periodic coefficients. As a consequence, results obtained in this chapter will be used extensively in the study of periodic solutions of non-linear systems. In the first Section we treat linear systems with constant coefficients though they are special cases of linear systems with periodic coefficients. In Section 2 we are treating homogeneous linear systems with periodic coefficients and show that they are “reducible” to systems with constant coefficients. In Section 3 “forced oscillations” will be dealt with, i.e. inhomogeneous linear systems with periodic “forcing term”. Stability problems will be treated in the fourth Section, and in the last one we shall study second order linear differential equations with periodic or harmonic coefficients: Hill’s and Mathieu’s equations.
KeywordsPeriodic Solution Fundamental Matrix Minimal Polynomial Characteristic Exponent Adjoint System
Unable to display preview. Download preview PDF.