Box products in nilpotent normal form theory: The factoring method

Murdock, James

doi:10.1016/j.jde.2015.09.018

Cited by 5 publications

(3 citation statements)

References 34 publications

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“…This is a systematic procedure, but it leads to too many terms, which then have to be recombined to compactify the result, a problem that is familiar from the vector field case, cf. [9,17,16].…”

Section: The (2 3)-reducible Casementioning

confidence: 99%

Normal form for maps with nilpotent linear part

Mokhtari¹,

Roell²,

Sanders³

2020

Preprint

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The normal form for an n-dimensional map with irreducible nilpotent linear part is determined using sl 2 -representation theory. We sketch by example how the reducible case can also be treated in an algorithmic manner. The construction (and proof) of the sl 2 -triple from the nilpotent linear part is more complicated than one would hope for, but once the abstract sl 2 theory is in place, both the description of the normal form and the computational splitting to compute the generator of the coordinate transformation can be handled explicitly in terms of the nilpotent linear part without the explicit knowledge of the triple. If one wishes one can compute the normal form such that it is guaranteed to lie in the kernel of an operator and one can be sure that this is really a normal form with respect to the nilpotent linear part; one can state that the normal form is in sl 2 -style.Although at first sight the normal form theory for maps is more complicated than for vector fields in the nilpotent case, it turns out that the final result is much better. Where in the vector field case one runs into invariant theoretical problems when the dimension gets larger if one wants to describe the general form of the normal form, for maps we obtain results without any restrictions on the dimension.In the literature only the 2-dimensional nilpotent case has been described sofar, as far as we know.

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Section: The (2 3)-reducible Casementioning

confidence: 99%

Normal form for maps with nilpotent linear part

Mokhtari¹,

Roell²,

Sanders³

2020

Preprint

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“…Even in the twodimensional case, there have been numerous important contributions in various types and approaches; e.g., see [1-3, 5, 10, 13, 18, 20, 22, 36-39, 41, 43, 45]. There have only been a few contributions in three dimensional state space cases; see [11,44] where hypernormalization is performed up to degree three; also see [15][16][17] and [27][28][29][30][31]. In this paper we provide a complete normal form classification for all vector fields v in equations (1.1)-(1.2), that is, the set of all completely integrable solenoidal nilpotent singularities where ∆ is one of their invariants and a multiple scalar of N is their linear part.…”

Section: Introductionmentioning

confidence: 99%

Vector potential normal form classification for completely integrable solenoidal nilpotent singularities

Gazor

Mokhtari

Sanders

2019

Journal of Differential Equations

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We introduce a sl 2 -invariant family of nonlinear vector fields with a non-semisimple triple zero singularity. In this paper we are concerned with characterization and normal form classification of these vector fields. We show that the family constitutes a Lie algebra structure and each vector field from this family is solenoidal, completely integrable and rotational. All such vector fields share a common quadratic invariant. We provide a Poisson structure for the Lie algebra from which the second invariant for each vector field can be readily derived. We show that each vector field from this family can be uniquely characterized by two alternative representations; one uses a vector potential while the other uses two functionally independent Clebsch potentials. Our normal form results are designed to preserve these structures and representations. The results are implemented in Maple in order to compute vector potential and the Clebsch potential normal forms of a given vector field from this family. Some practical normal form coefficient formulas for degrees of up to four are presented.

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“…Thus, normal form style determines what terms are simplified and what terms shall remain in the normal form system. The inner product, semisimple, sl(2), simplified and formal basis styles are among the main examples; see [36, Page ix] and [21,38]. A formal basis style basically determines the priority of elimination between different alternative omittable terms.…”

Section: Introductionmentioning

confidence: 99%

Bifurcation Control and Universal Unfolding for Hopf-Zero Singularities with Leading Solenoidal Terms

Gazor

Sadri²

2016

SIAM J. Appl. Dyn. Syst.

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In this paper we introduce universal asymptotic unfolding normal forms for nonlinear singular systems. Next, we propose an approach to find the parameters of a parametric singular system that they play the role of the universal unfolding parameters. These parameters effectively influence the local dynamics of the system. We propose a systematic approach to locate local bifurcations in terms of these parameters. Here, we apply the proposed approach on Hopf-zero singularities whose the first few low degree terms are incompressible. In this direction, we obtain novel orbital and parametric normal form results for such families by assuming a nonzero quadratic condition. Moreover, we give a truncated universal asymptotic unfolding normal form and prove the finite determinacy of the steady-state bifurcations for two most generic subfamilies of the associated amplitude systems. We analyze the local primary bifurcations of equilibria and limit cycles, and the secondary Hopf bifurcation of invariant tori. The results are successfully implemented and verified using Maple. By employing the proposed approach, we design an effective multiple-parametric quadratic state feedback controller for a singular system on a three dimensional central manifold with two imaginary uncontrollable modes. We illustrate how our program systematically identifies the distinguished (universal unfolding) parameters, derives the estimated transition varieties in terms of these parameters, and locates the local primary and secondary bifurcations of equilibria, limit cycles and invariant tori. This approach is useful in designing efficient nonlinear feedback controllers (single or multiple inputs) for local bifurcation control in engineering problems.

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Box products in nilpotent normal form theory: The factoring method

Cited by 5 publications

References 34 publications

Normal form for maps with nilpotent linear part

Normal form for maps with nilpotent linear part

Vector potential normal form classification for completely integrable solenoidal nilpotent singularities

Bifurcation Control and Universal Unfolding for Hopf-Zero Singularities with Leading Solenoidal Terms

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