Extension and Unification of Singular Perturbation Methods for ODEs Based on the Renormalization Group Method

Chiba, Hayato

doi:10.1137/090745957

Cited by 47 publications

(66 citation statements)

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“…In his paper [21], Chiba proved that the multiple scale analysis and the RG method are equivalent or produce the identical results for the ordinary differential equations. A simple proof for the case of autonomous system of differential equations will be presented in the remark (3.6).…”

Section: The Rg Methodsmentioning

confidence: 99%

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Renormalization Group Method

Muta¹

2009

World Scientific Lecture Notes in Physics

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Renormalization Group (RG) method is a general method whose aim is to globally approximate solutions to differential equations involving a small parameter. In this thesis, we will give an algorithm for the RG method to generate the RG equation needed in the process of finding an approximate solution for ODEs. In chapter 1, we have some introduction to perturbation theory and introducing some traditional methods in perturbation theory. In chapter 2 we compare the results of RG and other conventional methods using numerical or explicit methods. Thereafter, in chapter 3, we rigorously compare the approximate solution obtained using the RG method and the true solution using two classes of system of ordinary differential equations. In chapter 4, we present a simplified RG method and apply it to the second order RG. In chapter 5 we briefly explain the first order Normal Form (NF) theory and then its relation to the RG method. Also a similar geometric interpretation for the RG equation and NF's outcome has been provided. In the Appendix, we have added definitions and proofs used in this thesis. The RG method is much more straightforward than other traditional methods and does not require prior information about the solutions. One begins with a naive perturbative expansion which already contains all the necessary information that we need to construct a solution. Using RG, there is no need to asymptotically match the solutions in the overlapping regions, which is a key point in some other methods. In addition, the RG method is applicable to most of perturbed differential equations and will produce a closed form solution which is, most of the times, as accurate as or even more accurate than the solutions obtained by other conventional methods.ii

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Section: The Rg Methodsmentioning

confidence: 99%

“…analysis and RG method are equivalent (see [21] for more information about the equivalence of these two methods for ODEs). In this remark, we want to show that, in the case of autonomous system of differential equations, the RG method and the multiple scale analysis produce the identical results.…”

Section: Reproducing the Rg Equationmentioning

confidence: 99%

Renormalization Group Method

Muta¹

2009

World Scientific Lecture Notes in Physics

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“…A comparison of averaging and multiple timing by a number of important examples can be found in [10]. The relation between multiple timing and the renormalization method was discussed in [2], [3], and [14]. In [17], Perko established the equivalence of the averaging method and multiple timing for standard equations likė…”

Section: The Basic Idea Of Two (Or Multiple) Timescalesmentioning

confidence: 99%

“…In the same spirit, perturbation methods for ODEs can also be seen as normal form methods, for the idea is always to transform the ODEs to equations that are simpler and can be solved or at least partly solved. For ODEs, normalization has, for instance, been formulated in [2], [3], [14], and [20]. A number of examples will demonstrate this.…”

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confidence: 99%

Profits and Pitfalls of Timescales in Asymptotics

Verhulst¹

2015

SIAM Rev.

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Abstract. The method of multiple timescales is widely used in engineering and mathematical physics.In this article we draw attention to the literature on the comparison of various perturbation methods. We indicate where we can obtain an advantage from the concept of timescales, and we present examples where the anticipation of timescales makes sense and cases where, because of resonances or bifurcations, the analysis is less straightforward. In a number of problems second order approximations are essential to understand the phenomena. We will conclude from these examples that the anticipation of timescales as in the multiple timescales method may give misleading results; methods that do not anticipate timescales a priori such as averaging and renormalization should be used in research problems.

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“…These different formats result in the same normal form Hamiltonian but these formats can result in different normal form transformations. The convergence or divergence of the normal form and normal form transformation of order infinity has been investigated in the previous studies [12][13][14][15][16][17][18][19][20][21][22][23][24] under various conditions. Under some of these conditions, the normal form and normal form transformation converges globally, but, since generic Hamiltonians are non-integrable [25], in most cases, there is no hope to seek for (nontrivial) global integral of motions without any symmetries.…”

Section: Introductionmentioning

confidence: 99%

A new method to improve validity range of Lie canonical perturbation theory: with a central focus on a concept of non-blow-up region

2014

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Validity ranges of Lie Canonical Perturbation Theory (LCPT) are investigated in terms of non blow-up regions. We investigate how the validity ranges depend on the perturbation order in two systems, one of which is a simple Hamiltonian system with one degree of freedom and the other is a HCN molecule. Our analysis of the former system indicates that non blow-up regions become reduced in size as the perturbation order increases. In case of LCPT by Dragt and Finn and that by Deprit, the non blow-up regions enclose the region inside the separatrix of the Hamiltonian but it may not be the case for LCPT by Hori. We also analyze how well the actions constructed by these LCPTs approximate the true action of the Hamiltonian in the non blow-up regions and find that the conventional truncated LCPT does not work over the whole region inside the separatrix whereas LCPT by Dragt and Finn without truncation does. Our analysis of the latter system indicates that non blow-up regions do not necessarily cover the whole regions inside the HCN well.We propose a new perturbation method to improve non blow-up regions and validity ranges inside them. Our method is free from blowing up and retains the same normal form as the conventional LCPT. We demonstrate our method in the two systems and show that the actions constructed by our method have larger validity ranges than those by the conventional and our previous methods proposed in [1,2].

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Extension and Unification of Singular Perturbation Methods for ODEs Based on the Renormalization Group Method

Cited by 47 publications

References 50 publications

Renormalization Group Method

Renormalization Group Method

Profits and Pitfalls of Timescales in Asymptotics

A new method to improve validity range of Lie canonical perturbation theory: with a central focus on a concept of non-blow-up region

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