A Note on the Triple-Zero Linear Degeneracy: Normal Forms, Dynamical and Bifurcation Behaviors of an Unfolding

Freire, Emilio; Gamero, E.; Rodríguez-Luis, Alejandro J.; Algaba, Antonio

doi:10.1142/s0218127402006175

Cited by 57 publications

(48 citation statements)

References 19 publications

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“…In this context it will occur in one-parameter families, and will then have codimension one. The unfoldings of this singularity have been studied by several authors [Tak73a], [Tak74], [Tak73b], [Guc81], [BV84], [AMF + 03], [FGRLA02], [DI98], [GH83] looking at the different type of bifurcations that a two (or one) parameter family of vector fields unfolding this singularity can present.…”

Section: Analytic Unfoldings Of the Central Singularitymentioning

confidence: 99%

Breakdown of Heteroclinic Orbits for Some Analytic Unfoldings of the Hopf-Zero Singularity

Baldomá

Seara

2006

J Nonlinear Sci

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Summary. In this paper we study the exponentially small splitting of a heteroclinic connection in a one-parameter family of analytic vector fields in R 3 . This family arises from the conservative analytic unfoldings of the so-called Hopf zero singularity (central singularity). The family under consideration can be seen as a small perturbation of an integrable vector field having a heteroclinic orbit between two critical points along the z axis. We prove that, generically, when the whole family is considered, this heteroclinic connection is destroyed. Moreover, we give an asymptotic formula of the distance between the stable and unstable manifolds when they meet the plane z = 0. This distance is exponentially small with respect to the unfolding parameter, and the main term is a suitable version of the Melnikov integral given in terms of the Borel transform of some function depending on the higher-order terms of the family. The results are obtained in a perturbative setting that does not cover the generic unfoldings of the Hopf singularity, which can be obtained as a singular limit of the considered family. To deal with this singular case, other techniques are needed. The reason to study the breakdown of the heteroclinic orbit is that it can lead to the birth of some homoclinic connection to one of the critical points in the unfoldings of the Hopf-zero singularity, producing what is known as a Shilnikov bifurcation.

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Section: Analytic Unfoldings Of the Central Singularitymentioning

confidence: 99%

Breakdown of Heteroclinic Orbits for Some Analytic Unfoldings of the Hopf-Zero Singularity

Baldomá

Seara

2006

J Nonlinear Sci

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“…As we will see in this section, the origin of system (1.1) can be found in the study of the analytic unfoldings of the so called Hopf-zero singularity. More concretely, let us consider a vector field in R 3 which has the origin as a critical point and, for some positive α * , the eigenvalues of the linear part at the origin are 0, ±α The unfoldings of this singularity in the conservative case and all the different behaviour these families can present have been broadly studied [17,19,18,14,5,1,10,7,11,6,15]. The standard way to proceed in the study of these unfoldings, is to use the normal form theory to write the vector field as simple as possible up to some order and then to study the effects of the non symmetric terms in the dynamics.…”

Section: Baldomá and T M Searamentioning

confidence: 99%

The inner equation for generic analytic unfoldings of the Hopf-zero singularity

Baldomá¹,

Seara²

2008

DCDS-B

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Abstract. A classical problem in the study of the (conservative) unfoldings of the so called Hopf-zero bifurcation, is the computation of the splitting of a heteroclinic connection which exists in the symmetric normal form along the z-axis. In this paper we derive the inner system associated to this singular problem, which is independent on the unfolding parameter. We prove the existence of two solutions of this system related with the stable and unstable manifolds of the unfolding, and we give an asymptotic formula for their difference. We check that the results in this work agree with the ones obtained in the regular case by the authors.

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“…Based on the Lie transforms, Gamero et al [9,10] proposed a recursive procedure to compute the normal form and the associated coefficients of a dynamical system in 3 R , and the results were applied in the analysis of bifurcation behaviors in the autonomous electronic oscillator. In this paper, we are concerned with the unique normal form of the nonlinear dynamical systems in …”

Section: Introductionmentioning

confidence: 99%

Computation of the Unique Normal Form for Nonlinear Dynamical Systems in the Mechanical Applications

Kou¹,

Li²,

Feng³

2017

dtetr

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Abstract. In the mechanical applications, research for degenerate bifurcations is usually connected with the normal form theory of nonlinear dynamical systems. In this paper, we mainly concern with the unique normal form for a class of three dimensional vector fields by the method of transformation with parameters. With the aid of the Maple software, a recursive formula for computing the coefficients of the normal form is given. Our results can be easily applied to investigate the direct bifurcation and stability of any model which can be written in this.

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A Note on the Triple-Zero Linear Degeneracy: Normal Forms, Dynamical and Bifurcation Behaviors of an Unfolding

Cited by 57 publications

References 19 publications

Breakdown of Heteroclinic Orbits for Some Analytic Unfoldings of the Hopf-Zero Singularity

Breakdown of Heteroclinic Orbits for Some Analytic Unfoldings of the Hopf-Zero Singularity

The inner equation for generic analytic unfoldings of the Hopf-zero singularity

Computation of the Unique Normal Form for Nonlinear Dynamical Systems in the Mechanical Applications

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