Simplified renormalization group method for ordinary differential equations
This article investigates higher order renormalization group (RG) equations which serve to refine an error estimate of approximate solutions obtained by the first order RG equations. The RG method for differential equations is one of the perturbation methods which allows one to obtain invariant manifolds of a given ordinary differential equation together with approximate solutions to it. It is shown that a higher order RG equation maintains similar theorems to those provided by the first RG equation, which are theorems on well-definedness of approximate vector fields, and on inheritance of invariant manifolds from those for the RG equation to those for the original equation, for example. Since a higher order RG equation is defined by using indefinite integrals and is not unique for the reason of undetermined integral constants, the simplest form of RG equation is available by choosing suitable integral constants. It is shown that a simplified a RG equation is sufficient to determine whether the trivial solution to time-dependent linear equations is hyperbolically stable or not, and thereby a synchronous solution of a coupled oscillators is shown to be stable.
scientific article; zbMATH DE number 5530628