Orbital formal normal forms for general Bogdanov–Takens singularity

Stróżyna, Ewa; Żołądek, Henryk

doi:10.1016/s0022-0396(03)00137-2

Cited by 17 publications

(13 citation statements)

References 11 publications

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“…Further simplifications can be achieved if time reparametrization is allowed, so that a normal form under the smooth orbital equivalence is considered, see [Bogdanov, 1979;Sadovskiȋ, 1985;Algaba et al, 2002;Stróżyna &Żo ladek, 2002;Stróżyna, 2002;Stróżyna &Żo ladek, 2003]. In particular, this gives (4) and (6).…”

Section: Hypernormalization and Time Reparametrizationmentioning

confidence: 99%

Practical Computation of Normal Forms on Center Manifolds at Degenerate Bogdanov–takens Bifurcations

Kuznetsov

2005

Int. J. Bifurcation Chaos

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Simple computational formulas are derived for the two-, three-, and four-order coefficients of the smooth normal form on the center manifold at the Bogdanov-Takens (nonsemisimple double-zero) bifurcation for n-dimensional systems with arbitrary n ≥ 2. These formulas are equally suitable for both symbolic and numerical evaluation and allow one to classify all codim 3 Bogdanov-Takens bifurcations in generic multidimensional ODEs. They are also applicable to systems with symmetries. We perform no preliminary linear transformations but use only critical (generalized) eigenvectors of the linearization matrix and its transpose. The derivation combines the approximation of the center manifold with the normalization on it. Three known models are used as test examples to demonstrate advantages of the method.

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Section: Hypernormalization and Time Reparametrizationmentioning

confidence: 99%

Practical Computation of Normal Forms on Center Manifolds at Degenerate Bogdanov–takens Bifurcations

Kuznetsov

2005

Int. J. Bifurcation Chaos

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“…A conjugated normal form of a nilpotent singularity in the generalized saddle case is studied in [8] and an orbital normal form in [28]. In this work we show, in Theorem 3.24, another orbital normal form in the generalized saddle more convenient for the analysis of the integrability problem.…”

Section: Normal Formmentioning

confidence: 82%

Analytic integrability around a nilpotent singularity

Algaba

García

Giné

2019

Journal of Differential Equations

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“…Some recent developments of the theory of normal form can be found in [5][6][7][8][9][10][11][12][13][14][15]. Chen et al [5] presented the renormalization group theory, which was used for the search of normal forms for large classes of finite-dimensional vector fields [6][7][8].…”

Section: Introductionmentioning

confidence: 99%

Normal Form for High-Dimensional Nonlinear System and Its Application to a Viscoelastic Moving Belt

Chen

Qian

2014

Abstract and Applied Analysis

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This paper is concerned with the computation of the normal form and its application to a viscoelastic moving belt. First, a new computation method is proposed for significantly refining the normal forms for high-dimensional nonlinear systems. The improved method is described in detail by analyzing the four-dimensional nonlinear dynamical systems whose Jacobian matrices evaluated at an equilibrium point contain three different cases, that are, (i) two pairs of pure imaginary eigenvalues, (ii) one nonsemisimple double zero and a pair of pure imaginary eigenvalues, and (iii) two nonsemisimple double zero eigenvalues. Then, three explicit formulae are derived, herein, which can be used to compute the coefficients of the normal form and the associated nonlinear transformation. Finally, employing the present method, we study the nonlinear oscillation of the viscoelastic moving belt under parametric excitations. The stability and bifurcation of the nonlinear vibration system are studied. Through the illustrative example, the feasibility and merit of this novel method are also demonstrated and discussed.

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Orbital formal normal forms for general Bogdanov–Takens singularity

Cited by 17 publications

References 11 publications

Practical Computation of Normal Forms on Center Manifolds at Degenerate Bogdanov–takens Bifurcations

Practical Computation of Normal Forms on Center Manifolds at Degenerate Bogdanov–takens Bifurcations

Analytic integrability around a nilpotent singularity

Normal Form for High-Dimensional Nonlinear System and Its Application to a Viscoelastic Moving Belt

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