Linear Grading Function and Further Reduction of Normal Forms

Kokubu, Hiroshi; Oka, Hiroe; Wang, Duo

doi:10.1006/jdeq.1996.0181

Cited by 76 publications

(56 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…(4.28) is an element of P d 1 (R n ) ⊕ P d 2 (R n ) ⊕ · · · ⊕ P d k (R n ), we can choose B k−1 so that R k may take a value in C d 1 ⊕ C d 2 ⊕ · · · ⊕ C d k (see Examples 4.6, 4.7). Even if R 1 is a polynomial vector field, we can simplify R i (A) systematically by using the grading function (see Kokubu, Oka and Wang [9]) under appropriate assumptions, although we do not give care to this method in this paper. Example 4.6.…”

Section: If Eq (326) Is Autonomous the Rg Equation (41) Has The Pmentioning

confidence: 99%

Simplified renormalization group method for ordinary differential equations

Chiba

2009

Journal of Differential Equations

View full text Add to dashboard Cite

The renormalization group (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. This article investigates higher order RG equations which serve to refine an error estimate of approximate solutions obtained by the first order RG equations. It is shown that the higher order RG equation maintains the similar theorems to those provided by the first order 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 the higher order RG equation is defined by using indefinite integrals and is not unique for the reason of the undetermined integral constants, the simplest form of RG equation is available by choosing suitable integral constants. It is shown that this simplified 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.

show abstract

Section: If Eq (326) Is Autonomous the Rg Equation (41) Has The Pmentioning

confidence: 99%

Simplified renormalization group method for ordinary differential equations

Chiba

2009

Journal of Differential Equations

View full text Add to dashboard Cite

show abstract

“…In the proof of the main theorem, the sixteen 3rd-order algebraic equations used are where A jkl 's are explicitly given in terms of the original system's coefficients a i jlm 's in Equation (22).…”

Section: Appendixmentioning

confidence: 99%

The simplest parametrized normal forms of Hopf and generalized Hopf bifurcations

Chen

2007

Nonlinear Dyn

View full text Add to dashboard Cite

This paper considers the computation of the simplest parameterized normal forms (SPNF) of Hopf and generalized Hopf bifurcations. Although the notion of the simplest normal form has been studied for more than two decades, most of the efforts have been spent on the systems that do not involve perturbation parameters due to the restriction of the computational complexity. Very recently, two singularities -single zero and Hopf bifurcation -have been investigated, and the SPNFs for these two cases have been obtained. This paper extends a recently developed method for Hopf bifurcation to compute the SPNF of generalized Hopf bifurcations. The attention is focused on a codimension-2 generalized Hopf bifurcation. It is shown that the SPNF cannot be obtained by using only a near-identity transformation. Additional transformations such as time and parameter rescaling are further introduced. Moreover, an efficient recursive formula is presented for computing the SPNF. Examples are given to demonstrate the applicability of the new method.

show abstract

“…Un autre type de forme normale a été défini par H. Kokubu, H. Oka et D. Wang dans [3] ainsi que par G. Belitskii [6].…”

Section: Forme Normale D'un Bonne Perturbation D'un Champs Quasi-homounclassified