Normal forms for Hopf-Zero singularities with nonconservative nonlinear part

Gazor, Majid; Mokhtari, Fahimeh; Sanders, Jan A.

doi:10.1016/j.jde.2012.11.004

Cited by 22 publications

(17 citation statements)

References 31 publications

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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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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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“…This paper deals with hypernormalization for the cases when the nonconservative part is zero. For the zero conservative part, the hypernormalization follows from the method introduced in [20]. Finally, for the cases of both nonzero conservative and nonconservative parts, the normal form computation in [19] must be implemented.…”

Section: Introductionmentioning

confidence: 99%

Volume-preserving normal forms of Hopf-zero singularity

Gazor¹,

Mokhtari²

2013

Nonlinearity

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A practical method is described for computing the unique generator of the algebra of first integrals associated with a large class of Hopf-zero singularity. The set of all volume-preserving classical normal forms of this singularity is introduced via a Lie algebra description. This is a maximal vector space of classical normal forms with first integral; this is whence our approach works. Systems with a non-zero condition on their quadratic parts are considered. The algebra of all first integrals for any such system has a unique (modulo scalar multiplication) generator. The infinite level volume-preserving parametric normal forms of any non-degenerate perturbation within the Lie algebra of any such system is computed, where it can have rich dynamics. The associated unique generator of the algebra of first integrals are derived. The symmetry group of the infinite level normal forms are also discussed. Some necessary formulas are derived and applied to appropriately modified Rössler and generalized Kuramoto-Sivashinsky equations to demonstrate the applicability of our theoretical results. An approach (introduced by Iooss and Lombardi) is applied to find an optimal truncation for the first level normal forms of these examples with exponentially small remainders. The numerically suggested radius of convergence (for the first integral) associated with a hypernormalization step is discussed for the truncated first level normal forms of the examples. This is achieved by an efficient implementation of the results using Maple.

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“…We have already used the following lemma in [18][19][20]. Since it plays a central role in efficient use of time rescaling for simplifying Θ-terms, we here state it as a Lemma.…”

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“…The two-jet estimated varieties in terms of original distinguished parameters (µ 1 , µ 4 ) in equation (7.1). The vertical and horizontal axes are µ 1 and µ 4 , respectively.at integrating and enhancing our results[17][18][19][20][21][22] into a user-friendly Maple library for normal form analysis of singularities.Any control system [24, Equations (2.1-2.2)] on a three dimensional central manifold with two imaginary uncontrollable modes can be transformed intȯx := u + f (x, z 1 , z 2 , u),ż 1 := z 2 + g(x, z 1 , z 2 , u),ż 2 := −z 1 + h(x, z 1 , z 2 , u),(7.1) using linear changes in state and feedback variables; see [24, Equation (2.3)]. Here u stands for a quadratic multiple-feedback controller and assume that it is given by u := µ 1 + µ 2 z 2 + µ 3 z 1 + µ 4 x + µ 5 z 2 2 + µ 6 z 1 z 2 + µ 7 xz 2 + µ 8 xz 1 , (7.2)…”

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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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Normal forms for Hopf-Zero singularities with nonconservative nonlinear part

Cited by 22 publications

References 31 publications

Vector potential normal form classification for completely integrable solenoidal nilpotent singularities

Vector potential normal form classification for completely integrable solenoidal nilpotent singularities

Volume-preserving normal forms of Hopf-zero singularity

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

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