Versal deformations of a singular point of a vector field on the plane in the case of zero eigenvalues

Богданов, Р. И.

doi:10.1007/bf01075453

Cited by 239 publications

(209 citation statements)

References 0 publications

Supporting

Mentioning

201

Contrasting

Unclassified

Order By: Relevance

“…Thus one sees that the RG equation gives the normal form of the reduced evolution equation on the invariant manifold. We remark that when a 1 = b 1 = 0, the RG equation is nothing but the Bogdanov equation [51].…”

Section: Normal Formmentioning

confidence: 99%

“…In §4, we show how to deal with generic systems which involve a linear operator A having zero-eigenvalue where A may or may not be diagonalizable; when A is not diagonalizable, the eigenvalues are degenerate and A is equivalent with a matrix having a Jordan cell 0 1 0 0 . As examples with a Jordan cell, we shall show that the RG equation gives the normal forms [50] of the two-dimensional equations including the Takens and the Bogdanov equation [51], and deal with an extended version of Takens equation with three-degrees of freedom. In §5, we apply the method to some problems such as the unstable motion in the Lotka-Volterra system and the Hopf bifurcation in the Brusselator [52].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Renormalization-Group Method for Reduction of Evolution Equations; Invariant Manifolds and Envelopes

Ei¹,

Fujii²,

Kunihiro³

2000

Annals of Physics

145

View full text Add to dashboard Cite

The renormalization group (RG) method as a powerful tool for reduction of evolution equations is formulated in terms of the notion of invariant manifolds. We start with derivation of an exact RG equation which is analogous to the Wilsonian RG equations in statistical physics and quantum field theory. It is clarified that the perturbative RG method constructs invariant manifolds successively as the initial value of evolution equations, thereby the meaning to set t 0 = t is naturally understood where t 0 is the arbitrary initial time. We show that the integral constants in the unperturbative solution constitutes natural coordinates of the invariant manifold when the linear operator A in the evolution equation is semi-simple, i.e., diagonalizable; when A is not semi-simple and has a Jordan cell, a slight modification is necessary because the dimension of the invariant manifold is increased by the perturbation. The RG equation determines the slow motion of the would-be integral constants in the unperturbative solution on the invariant manifold. We present the mechanical procedure to construct the perturbative solutions hence the initial values with which the RG equation gives meaningful results. The underlying structure of the reduction by the RG method as formulated in the present work turns out to completely fit to the universal one elucidated by Kuramoto some years ago. We indicate that the reduction procedure of evolution equations has a good correspondence with the renormalization procedure in quantum field theory; the counter part of the universal structure of reduction elucidated by Kuramoto may be the Polchinski's theorem for renormalizable field theories. We apply the method to interface dynamics such as kink-anti-kink and soliton-soliton interactions in the latter of which a linear operator having a Jordan-cell structure appears.

show abstract

Section: Normal Formmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Renormalization-Group Method for Reduction of Evolution Equations; Invariant Manifolds and Envelopes

Ei¹,

Fujii²,

Kunihiro³

2000

Annals of Physics

145

View full text Add to dashboard Cite

show abstract

“…Then we have to calculate the normal form up to third order. For the BT singularity, this is (see [2] or [14, Section 7.3])…”

Section: Normal Form Reductionmentioning

confidence: 99%

“…Here P is an m-dimensional (m = 2 or 3) subspace spanned by the solutions of (2) corresponding to the m zero real part eigenvalues (sometimes called the centre eigenspace), Q is the complementary space, and P and Q are invariant under the flow associated with (2). Further, for the nonlinear equation (8), there exists a centre manifold in C which is a finite (m = 2 or 3) dimensional, invariant manifold.…”

Section: Centre Manifold Analysismentioning

confidence: 99%

Zero singularities of codimension two and three in delay differential equations

Campbell

Yuan

2008

Nonlinearity

View full text Add to dashboard Cite

We give conditions under which a general class of delay differential equations has a point of Bogdanov-Takens or a triple zero bifurcation. We show how a centre manifold projection of the delay equations reduces the dynamics to a two or three dimensional systems of ordinary differential equations. We put these equations in normal form and determine how the coefficients of the normal forms depend on the original parameters in the model. Finally we apply our results to two neural models and compare the predictions of the theory with numerical bifurcation analysis of the full equations. One model involves a transcritical bifurcation, hence we derive and analyze the appropriate unfoldings for this case.Mathematics Subject Classification: 92B20, 34K20, 34K15

show abstract

“…a stationary point with a double zero eigenvalue of the Jacobian (see Takens, 1974;Bogdanov, 1975;Arnold, 1988;Guckenheimer & Holmes, 1983);…”

Section: Introductionmentioning

confidence: 99%

Numerical analysis of homoclinic orbits emanating from a Takens-Bogdanov point

Beyn

1994

IMA J Numer Anal

View full text Add to dashboard Cite

It is known that branches of homoclinic orbits emanate from a singular point of a dynamical system with a double zero eigenvalue (Takens-Bogdanov point). We develop a robust numerical method for starting the computation of homoclinic branches near such a point. It is shown that this starting procedure relates to branch switching. In particular, for a certain transformed problem the homoclinic predictor is guaranteed to converge to the true orbit under a Newton iteration.

show abstract

Versal deformations of a singular point of a vector field on the plane in the case of zero eigenvalues

Cited by 239 publications

References 0 publications

Renormalization-Group Method for Reduction of Evolution Equations; Invariant Manifolds and Envelopes

Renormalization-Group Method for Reduction of Evolution Equations; Invariant Manifolds and Envelopes

Zero singularities of codimension two and three in delay differential equations

Numerical analysis of homoclinic orbits emanating from a Takens-Bogdanov point

Contact Info

Product

Resources

About