Overview

A large number of important stability problems are modeled by multiparameter linear differential equations with periodic coefficients. As direct applications we may mention mechanical systems with periodically varying stiffness, mass, and load (parametric excitation). Other problems are from frequency modulation, warble tone room testing in acoustics, plasma physics etc. Finally, we mention the applications, which originated the study of periodic differential equations, including mean motion of the lunar perigee and wave propagation in stratified media.

The stability analysis for solutions to differential equations with periodic coefficients has been a challenge for more than hundred years. From a historical point of view, the important early studies are [Mathieu (1868); Floquet (1883); Hill (1886); Rayleigh (1887); Liapunov (1892); Poincar (1899)]. For further development we refer to the books [Malkin (1966); Schmidt (1975); Yakubovich and Starzhinskii (1975); Yakubovich and Starzhinskii (1987); Nayfeh and Balachandran (1995)].

Different methods for analysis of stability are available: the classical Floquet method [Floquet (1883); Cesari (1971)], the method of infinite determinants [Bolotin (1964)], the perturbation method [Hsu (1963); Nayfeh and Mook (1979)], and the Galerkin method [Pedersen (1985)]. Few of these methods can from a practical point of view be extended to multiple degrees of freedom systems. For such extension we refer to [Lindh and Likins (1970); Fu and Nemat-Nasser (1972); Hansen (1985); Wu et al. (1995); Turhan (1998)]. It is concluded that the Floquet method is a general and practical method for systems with multiple degrees of freedom. Even with increasing computer power the...