Nilpotent Linear Part Research Articles

An Organizing Center of Codimension Four in a Predator-Prey Model with Generalist Predator: From Tristability and Quadristability to Transients in a Nonlinear Environmental Change

In this paper, we take the Rosenzweig–MacArthur (RM) model with generalist predator as an example in a constant or changing environment. When the environment is fixed, we provide a more easily verifiable classification, in terms of the coefficients of the system with nilpotent linear part and general higher terms, to determine the types and codimension of nilpotent singularities in a general planar system. Second, by using the existing classification and some algebraic methods, we show that the highest codimension of a nilpotent focus is 4 and the sample RM model with generalist predator can exhibit nilpotent focus bifurcation of codimension 4. Our results indicate that generalist predation can cause not only richer bifurcations and dynamics (such as multitype tristability and quadristability, a figure-eight loop) but also the possible extirpation of prey. When the environment is changing, we study the impact of the rate and intensity of a nonlinear environmental change on dynamics. The key observations on the asymptotic and transient dynamics include (i) transient tracking on unstable steady states or oscillations, and transient-related regime shifts; (ii) slow and fast regime shifts; (iii) regulation of transient dynamics by the environmental change parameters and ; (iv) slow negative or fast positive environmental change can delay or even avoid population extirpation.

Normal form for maps with nilpotent linear part

The normal form for an n -dimensional map with irreducible nilpotent linear part is determined using s l 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 s l 2 -triple from the nilpotent linear part is more complicated than one would hope for, but once the abstract s l 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 s l 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 two-dimensional nilpotent case has been described so far, as far as we know.

Equivariant decomposition of polynomial vector fields

To compute the unique formal normal form of families of vector fields with nilpotent linear part, we choose a basis of the Lie algebra consisting of orbits under the action of the nilpotent linear part. This creates a new problem: to find explicit formulas for the structure constants in this new basis. These are well known in the 2D case, and recently expressions were found for the 3D case by ad hoc methods. The goal of the this paper is to formulate a systematic approach to this calculation. We propose to do this using a rational method for the inversion of the Clebsch–Gordan coefficients. We illustrate the method on a family of 3D vector fields and compute the unique formal normal form for the Euler family both in the 2D and 3D cases, followed by an application to the computation of the unique normal form of the Rössler equation.

Abelian Integrals from an Unfolding of Codimension-3 Singularities with Nilpotent Linear Part

In this paper, we study the maximum number of limit cycles for the unfolding of codimension-3 planar singularities with nilpotent linear parts. In [J. Math. Anal. Appl. 499 (2017)], the authors proved that when parameter [Formula: see text] is rational, the corresponding problem could be transformed to solving semi-algebraic systems. At the same time, it is pointed out that when [Formula: see text], the logarithmic function will appear according to the method, which makes it impossible to solve the problem. In this paper, we use some techniques to avoid the occurrence of logarithmic function, and get the corresponding system to produce at most two limit cycles.

Holomorphic normal form of nonlinear perturbations of nilpotent vector fields

We consider germs of holomorphic vector fields at a fixed point having a nilpotent linear part at that point, in dimension $n \geq 3$. Based on Belitskii's work, we know that such a vector field is formally conjugate to a (formal) normal form. We give a condition on that normal form which ensure that the normalizing transformation is holomorphic at the fixed point. We shall show that this sufficient condition is a nilpotent version of Bruno's condition (A). In dimension 2, no condition is required since, according to Str{\'o}zyna- Zoladek, each such germ is holomorphically conjugate to a Takens normal form. Our proof is based on Newton's method and $\frak{sl}\_2(\Bbb C)$-representations.

Center conditions for nilpotent cubic systems using the Cherkas method

In this study, we consider the center problem of a cubic polynomial differential system with a nilpotent linear part. The analysis is based on the application of the Cherkas method to the Takens normal form. The analysis requires many computations, which are verified by employing one algebraic manipulator and extensive use of the computer algebra system called Singular.

Reduced Forms of a Singular Vector Field in <b><i>C</i></b><sup>3</sup> with Nilpotent Linear Part

We study the reduction problem for a holomorphic singular vector field with nilpotent linear part at the origin, P = (y + a(X)) ∂x + (z + b(X)) ∂y + c(X)∂z, where X = (x, y, z) ∈ C3. By introducing a notion of quasi-valuation, which was given in the previous paper [M-S] with A. Shirai, we characterize a class which can be reduced into simpler ones by formal change of coordinates which admit various variations in Theorems A and B.

EXPLICIT DERIVATION OF RETURN MAPS FOR PLANAR VECTOR FIELDS WITH NILPOTENT LINEAR PART WITH A CENTER OR A FOCUS

For planar vector fields with nilpotent linear part with a center or a focus it is shown that, remarkably, there exist substitutions after which the equations become regular perturbations of Hamiltonian systems. The return map is then obtained as a convergent series by iterating a contractive map, yielding formulas suitable for numerical calculations and estimates for the errors.

Normal forms with exponentially small remainder and Gevrey normalization for vector fields with a nilpotent linear part

Bonckaert, P (reprint author), [Bonckaert, Patrick; Verstringe, Freek] Univ Hasselt, B-3590 Diepenbeek, Belgium. Bonckaert, P (reprint author)

Stability analysis of planar systems with nilpotent (non-zero) linear part

A simple necessary and sufficient condition for asymptotic stability of the origin is derived for a large class of planar nonlinear systems that cannot be studied by means of their linearization, which is nilpotent and not zero. The condition is based on the use of Darboux polynomials, to derive a Lyapunov function for the first approximation of the system with respect to a vector function representing a dilation. Results by Andreev (1958), on the qualitative behaviour of equilibrium points, and the Belitskii normal form are used in the proofs. The proposed condition is applied to the asymptotic stabilization of the origin, for a class of planar systems whose linearization cannot be stabilized.

The Return Map for a Planar Vector Field with Nilpotent Linear Part: A Direct and Explicit Derivation

Using a direct approach the return map near a focus of a planar vector field with nilpotent linear part is found as a convergent power series which is a perturbation of the identity and whose terms can be calculated iteratively. The first nontrivial coefficient is the value of an Abelian integral, and the following ones are explicitly given as iterated integrals.

A new transvectant algorithm for nilpotent normal forms

We offer an algorithm to determine the form of the normal form for a vector field with a nilpotent linear part, when the form of the normal form is known for each Jordan block of the linear part taken separately. The algorithm is based on the notion of transvectant, from classical invariant theory.

Singularités nilpotentes et intégrales premières

This paper presents a classification of plane dicritical nilpotent singularities, i.e. singularities which have nilpotent linear part and infinitely many separatrices. In particular the existence of meromorphic first integrals is discussed. The same ideas are applied to other kind of dicritical singularities.

Three dimensional nilpotent singularity and Sil’nikov bifurcation

Abstract In this paper, by using the normal form, blow-up theory and the technique of global bifurcations, we study the singularity at the origin with threefold zero eigenvalue for nonsymmetric vector fields with nilpotent linear part and 4-jet C ∞ -equivalent to y ∂ ∂ x + z ∂ ∂ y + ax 3 y ∂ ∂ z , with a ≠ 0, and analytically prove the existence of Sil’nikov bifurcation, and then of the strange attractor for certain subfamilies of the nonsymmetric versal unfoldings of this singularity under some conditions.

Homoclinic Bifurcations in Symmetric Unfoldings of a Singularity with Three–fold Zero Eigenvalue

In this paper we study the singularity at the origin with three–fold zero eigenvalue for symmetric vector fields with nilpotent linear part and 3–jet C∞–equivalent to \( y\frac{\partial } {{\partial x}} + z\frac{\partial } {{\partial y}} + ax^{2} y\frac{\partial } {{\partial z}} \) with a ≠ 0. We first obtain several subfamilies of the symmetric versal unfoldings of this singularity by using the normal form and blow–up methods under some conditions, and derive the local and global bifurcation behavior, then prove analytically the existence of the Sil'nikov homoclinic bifurcation for some subfamilies of the symmetric versal unfoldings of this singularity, by using the generalized Mel'nikov methods of a homoclinic orbit to a hyperbolic or non–hyperbolic equilibrium in a highdimensional space.

Formal Normal Forms for the Perturbations of a Quasi-Homogeneous Hamiltonian Vector Field

We classify up to a formal change of variable the vector fields of two complex variables together with the foliations that they de.ne which are a perturbation of a quasi-homogeneous Hamiltonian vector field X0 by terms of higher degree of quasi-homogeneity. We do not require anything about the degree δ0 of the initial vector field X0, but we assume that the perturbed vector field still keeps invariant the separatrices of X0. We obtain formal normal forms which extend those obtained in the case of an initial vector field with a semisimple or nilpotent linear part. We give an interpretation of the dual version of these normal forms through the relative cohomology with respect to the dual initial di.erential form ω0.

Divergence and summability of normal forms of systems of differential equations with nilpotent linear part

On considere des formes prenormales associees a des perturbations generiques du systeme x = 2y, y = 3x 2 . Il est connu qu'elles admettent une forme normale formelle x = 2y+2xΔ*, y = 3x 2 +3yΔ*, ou Δ* = A 0 (h)+xA 1 (h) avec h = y 2 - x 3 ([Loray, 1999]). Nous demontrons que A 0 , A 1 et les transformations normalisantes sont divergentes, mais k-sommables. L'entier k depend des premiers termes non nuls de A 0 et A 1 .

On the normal form of a system of differential equations with nilpotent linear part

We consider prenormal forms associated to generic perturbations of the system x ̇ =2y, y ̇ =3x 2 . It is known that they have a formal normal form x ̇ =2y+2x Δ ∗, y ̇ =3x 2+3y Δ ∗ , where Δ ∗=x+A 0(y 2−x 3) [Differential Equations 158 (1) (1999) 152–173]. We show that the series A 0 and the normalizing transformations are divergent, but 1-summable. To cite this article: M. Canalis-Durand, R. Schäfke, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

Further Reduction of Normal Forms for Vector Fields

We study Lie's method in the way of Ushiki for further reduction of normal forms for vector fields with singularity at the origin. We give further reduction of normal forms in two typical cases for vector fields in dimension 2: one with a rotation as its linear part and the other with a nilpotent linear part.

An algebraic approach for obtaining nilpotent normal forms in dimension 4

In this paper, an algebraic approach is presented for determining the normal forms of four-dimensional systems with a nilpotent linear part. Certain transformations introduced in this case result in a simplified procedure for the calculation of normal forms. Thus, one does not need to solve a series of partial differential equations as usually required by the normal-form theory; indeed, algebraic calculations are sufficient. The approach can be applied to higher-order systems with a nilpotent linear part as well. To illustrate the new approach, five examples are presented. Normal forms and the associated coefficients of two physical systems, an electric network and a mechanical system, are fully analyzed.