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Extension and unification of singular perturbation methods for ODEs based on the renormalization group method. (English) Zbl 1179.34035

This paper deals with a system of ordinary differential equations (ODEs) on a manifold $M$ of the form

$\frac{dx}{dt}=\epsilon g\left(t,x,\epsilon \right),\phantom{\rule{1.em}{0ex}}x\in M\phantom{\rule{2.em}{0ex}}\left(1\right)$

which is almost periodic in $t$, and where $\epsilon \in ℝ$ or $ℂ$ is a small parameter.

The main tool in the investigation of the above system is the so called renormalization group (RG) method. The RG method is a relatively new method proposed by L. Y. Chen, N. Goldenfeld, and Y. Oono [Phys. Rev. E, 54 (1996); Phys. Rev. Lett. 73, No.10, 1311-1315 (1994; Zbl 1020.81729)] which reduces a problem to a simpler equation called the RG equation, based on an idea of the renormalization group in quantum field theory.

One of the purposes of this paper is to provide basic theorems on the RG method extending the author’s previous works, in which the RG method is discussed for more restricted problems than (1). At first, definitions of higher order RG equations for (1) are given and properties of them are investigated. It is proved that the RG method provides approximate vector fields and approximate solutions along with error estimates. Further, it is shown that if the RG equation has a normally hyperbolic invariant manifold N, the original equation (1) also has an invariant manifold ${N}_{\epsilon }$ which is diffeomorphic to N. The RG equations are shown to have the same symmetries (action of Lie groups) as those for the original equation. In addition, if the original equation is an autonomous system, then the RG equation has an additional symmetry. These facts imply that the RG equation is easier to analyze then the original equation. An illustrative example to verify these theorems is also given.

The other purpose of this paper is to show that the RG method extends and unifies other traditional singular perturbation methods, such as the averaging method, the multiple time scale method, the (hyper)normal forms theory, the center manifold reduction, the geometric singular perturbation methods, the phase reduction, and Kunihiro’s method based on envelopes. Some properties of the infinite-order RG equation are also investigated. It is proved that the infinite-order RG equation converges if and only if the original equation is invariant under an appropriate torus action. The infinite RG equation for a time-dependent linear system proves to be convergent and related to monodromy matrices in Floquet theory.

##### MSC:
 34C20 Transformation and reduction of ODE and systems, normal forms 34C29 Averaging method 34E15 Asymptotic singular perturbations, general theory (ODE) 34C14 Symmetries, invariants (ODE) 34C45 Invariant manifolds (ODE)