Lowering of the order of systems of differential equations in mechanics (Q1086417)

Recently there has been considerable activity in the research of so- called slow-fast systems or singularly perturbed differential systems. The simplest of such systems are of the form (1) \(\epsilon\dot x=f(x,y)\), \(\dot y=F(x,y)\), where \(\epsilon\) is ''small'' that are ''close'' to (2) \(f(x,y)=0\), \(\dot y=F(x,y)\), that were studied by Tikhonov in the early fifties. The asymptotic properties of slow-fast systems, the turning point problems, and equivalent studies of equations with a small parameter in the leading term have an extensive and rich literature. A recent development in this study consists of applications of nonstandard analysis (''the canard theories''). It is a fast growing field of research. The author continues the investigation of Tikhonov, Vasil'eva, Butuzov concerning the ''closeness'' of systems (1) and (2). Specifically, he looks for the range of values of the small parameter \(\epsilon\) for which these systems may be considered close. Estimates are derived through the use of a Lyapunov function.