Normal Forms Of Vector Fields Research Articles

Realizability of the normal forms for the non-semisimple 1:1 resonant Hopf bifurcation in a vector field

In this paper, a new and efficient mechanism to compute the normal forms for 1:1 resonant Hopf bifurcation is developed. For a vector field given by ordinary differential equations, by assuming that eigenvalues at an equilibrium point are purely imaginary, double, and non-semisimple, the mechanism provides a direct method to calculate the coefficients for the normal forms. In particular, we present the following results: (1) a simple direct method to determine a basis of the complementary spaces for the Lie transform; (2) a simple direct method to determine the projection of any vector in H43 to the complementary spaces; and (3) the normal forms for vector fields with double purely imaginary eigenvalues with geometric multiplicity one. In addition, explicit formulas for coefficients with three unfolding parameters are obtained and the normal forms for the non-semisimple 1:1 resonant Hopf bifurcation are obtained.

Smooth Gevrey normal forms of vector fields near a fixed point

We study germs of smooth vector fields in a neighborhood of a fixed point having an hyperbolic linear part at this point. It is well known that the ''small divisors'' are invisible either for the smooth linearization or normal form problem. We prove that this is completely different in the smooth Gevrey category. We prove that a germ of smooth $\al$-Gevrey vector field with an hyperbolic linear part admits a smooth $\be$-Gevrey transformation to a smooth $\be$-Gevrey normal form. The Gevrey order $\be$ depends on the rate of accumulation to $0$ of the small divisors. We show that a formally linearizable Gevrey smooth germ with the linear part satisfies Brjuno's small divisors condition can be linearized in the same Gevrey class.

Analysis of the Interaction between Modes by Normal Forms and Singular Perturbation Theory

Due to the nonlinear characteristics of modern power systems, strong interaction between system angle stability and voltage stability will present. However, because of extreme complexity, these two kinds of stability are often investigated separately and the interactions between them are normally neglected in traditional studies. In this paper, a new method based on the theory of normal forms of vector fields and singular perturbation theory is introduced to deal with the power system structure preserving model for system stability modes interaction analysis. The relationship between angle and voltage stability is discovered. The method is validated with a sample power system.

Parametric normal forms of vector fields and their further simplification

Given a family of vector fields parametrized by ξ in its linear part, one usually obtains its versal unfolding by the so-called two-step approach, i.e. find a Poincaré normal form for the system with fixed ξ0 and then apply the obtained near-identity transformation to the system with general ξ. It is also a common practice to treat ξ as a component together with those state components and calculate normal forms of the extended system. In this paper we reformulate normal forms on modules of homogeneous polynomials over the ring of all continuous functions of ξ and give a direct computation of versal unfolding. Our procedure enables us to determine coefficients of all terms of a certain degree in the normal form before we give a near-identity transformation of this degree. We can give all available near-identity transformations and choose an appropriate one to eliminate more terms of higher degree for a simpler normal form. We prove that the normal form reduced in our procedure is the simplest and unique. We illustrate our method with systems of linear centre and nilpotent linear parts separately.

Reversibility and quasi-homogeneous normal forms of vector fields

This paper uses tools in quasi-homogeneous normal form theory to discuss certain aspects of reversible vector fields around an equilibrium point. Our main result provides an algorithm, via Lie Triangle, that detects the non-reversibility of vector fields. That is, it is possible to decide whether a planar center is not reversible. Some of the theory developed is also applied to get further results on nilpotent and degenerate polynomial vector fields. We find several families of nilpotent centers which are non-reversible.

Calcul Moulien

This paper is an introduction to mould calculus, as introduced by Jean Écalle. We give a precise definition of moulds and describe their main properties. We translate mould symmetries (alterna(e)l and symetra(e)l) using non commutative formal power series in two given bialgebras 𝔸 and 𝔼, corresponding to two coproducts structure given by Δ(a)=a⊗1+1⊗a and Δ * (a i )=∑ l+k=i a l ⊗a k . We apply this formalism to the problem of normal forms for vector fields and diffeomorphisms.

Gevrey normal forms of vector fields with one zero eigenvalue

We study normal forms of isolated singularities of vector fields in R n or C n . When all eigenvalues of the linear part of the vector field are nonzero, one can eliminate all so-called nonresonant terms from the equation provided some spectral condition (like Siegel) is satisfied. In this paper, we discuss the case where there is one zero eigenvalue (in that case Siegel's condition is not satisfied), and show that the formal normalizing transformations are either convergent or divergent of at most Gevrey type. In some cases, we show the summability of the normalizing transformations, which leads to the existence of analytic normal forms in complex sectors around the singularity.

Power System Nonlinearity Modal Interaction by the Normal Forms of Vector Fields

Because of the robust nonlinear characteristics appearing in today's modern power system, a strong interaction exists between the angle stability and the voltage stability, which were conventionally studied insularly. However, as the power system is a complex unified system, angle instability always happens in conjunction with voltage instability. The authors propose a novel method to analyze this type of stability problem. In the proposed method, the theory of normal forms of vector fields is utilized to treat the auxiliary dynamic system. By use of this method, the interaction between response modes caused by the nonlinearity of the power system can be analyzed. Consequently, the eigenvalue analysis method is extended to cope with performance analysis of the power system with heavy nonlinearity. The effectiveness of the proposed methodology is verified on a 3-bus power system.

FURTHER REDUCTION OF NORMAL FORMS AND UNIQUE NORMAL FORMS OF SMOOTH MAPS

Further reduction for classical normal forms of smooth maps is considered in this paper. Firstly, based on the idea of computation of simplest normal forms for vector fields [Yu & Yuan, 2003], we compute the transformed map of a given smooth map under a near identity formal transformation, and give a recursive formula for the homogeneous terms of the transformed map, which is a powerful tool for further reduction of classical normal forms of smooth maps. Secondly, by using the recursive formula, the idea of [Chen & Della Dora, 1999] for further reduction of normal forms for maps and the method introduced by Kokubu et al. [1996] for further reduction of normal forms of vector fields, we develop the concepts of Nth order normal forms and infinite order normal forms of smooth maps, and give some sufficient conditions for uniqueness of normal forms of smooth maps. As an application, we show the occurrence of the flip–Neimark–Sacker bifurcation in a financial model.

Normal forms of vector fields with perturbation parameters and their application

Abstract This paper is concerned with the normal forms of vector fields with perturbation parameters. Usually, such a normal form is obtained via two steps: first find the normal form of a “reduced” system of the original vector field without perturbation parameters, and then add an unfolding to the normal form. This way, however, it does not yield the relation (transformation) between the original system and the normal form. A study is given in this paper to consider the role of near-identity transformations in the computation of normal forms. It is shown that using only near-identity transformations cannot generate the normal form with unfolding as expected. Such normal forms, which contain many nonlinear terms involving perturbation parameters, are not very useful in bifurcation analysis. Therefore, additional transformations are needed, resulting in a further reduction of normal forms – the simplest normal form. Examples are presented to illustrate the theoretical results.

Normal forms of vector fields on Poisson manifolds

We study formal and analytic normal forms of radial and Hamiltonian vector fields on Poisson manifolds near a singular point.

Borel summability of formal solutions of some first order singular partial differential equations and normal forms of vector fields

Let L = ∑ i = 1 d X i ( z ) ∂ z i be a holomorphic vector field degenerating at z = 0 such that Jacobi matrix ( ( ∂ X i / ∂ z j ) ( 0 ) ) has zero eigenvalues. Consider L u = F ( z , u ) and let u ˜ ( z ) be a formal power series solution. We study the Borel summability of u ˜ ( z ) , which implies the existence of a genuine solution u ( z ) such that u ( z ) ∼ u ˜ ( z ) as z → 0 in some sectorial region. Further we treat singular equations appearing in finding normal forms of singular vector fields and study to simplify L by transformations with Borel summable functions.

Unique Orbital Normal Form for Vector Fields of Hopf-Zero Singularity

Normal forms for vector fields of Hopf-zero singularity in R3 are studied. Multiple Lie bracket method is used to give unique normal forms under both conjugacy and orbital equivalence for such vector fields with a generic quadratic part.

Examples of non-conjugated holomorphic vector fields and foliations

We consider germs of holomorphic vector fields near the origin of C 2 with a saddle-node singularity, and the induced singular foliations. In a previous article we described the invariants addressing the analytical classification of these vector fields. They split into three parts: a formal, an orbital and a tangential component. For a fixed formal class, the orbital invariant (associated to the foliation) was obtained by Martinet and Ramis; we give it an integral representation. We then derive examples of non-orbitally conjugated foliations by the use of a “first-step” normal form, whose first-significative jet is an invariant. The tangential invariant also admits an integral representation, hence we derive explicit examples of vector fields, inducing the same foliation, that are not mutually conjugated. In addition, we provide a family of normal forms for vector fields orbitally equivalent to the model of Poincaré–Dulac.

Examples of non-conjugated holomorphic vector fields and foliations

We consider germs of holomorphic vector fields near the origin of C 2 with a saddle-node singularity, and the induced singular foliations. In a previous article we described the invariants addressing the analytical classification of these vector fields. They split into three parts: a formal, an orbital and a tangential component. For a fixed formal class, the orbital invariant (associated to the foliation) was obtained by Martinet and Ramis; we give it an integral representation. We then derive examples of non-orbitally conjugated foliations by the use of a “first-step” normal form, whose first-significative jet is an invariant. The tangential invariant also admits an integral representation, hence we derive explicit examples of vector fields, inducing the same foliation, that are not mutually conjugated. In addition, we provide a family of normal forms for vector fields orbitally equivalent to the model of Poincaré–Dulac.

An Algorithm for Computing Quasi-Homogeneous Formal Normal Forms under Equivalence

This paper presents a recursive algorithm which is useful for computing normal forms of vector fields using smooth orbital equivalence. The case of vector fields with a singularity corresponding to a triple-zero eigenvalue with geometric multiplicity one is considered in detail. The results obtained are applied to the study of a simple electronic device, with only one nonlinearity.

A matching pursuit technique for computing the simplest normal forms of vector fields

This paper presents a matching pursuit technique for computing the simplest normal forms of vector fields. First a simple, explicit recursive formula is derived for general differential equations, which reduces computation to the minimum. Then a matching pursuit technique is introduced and applied to the Takens–Bogdanov dynamical singularity. It is shown that unlike other methods for computing normal forms, the technique using matching pursuit does not need any algebraic constraints which are required for the existence of the simplest normal form. The efficient method and matching pursuit technique, which have been implemented using Maple, can be “automatically” executed on various computer systems. A number of examples are presented to demonstrate the advantages of the technique.

AN EFFICIENT METHOD FOR COMPUTING THE SIMPLEST NORMAL FORMS OF VECTOR FIELDS

A computationally efficient method is proposed for computing the simplest normal forms of vector fields. A simple, explicit recursive formula is obtained for general differential equations. The most important feature of the approach is to obtain the "simplest" formula which reduces the computation demand to minimum. At each order of the normal form computation, the formula generates a set of algebraic equations for computing the normal form and nonlinear transformation. Moreover, the new recursive method is not required for solving large matrix equations, instead it solves linear algebraic equations one by one. Thus the new method is computationally efficient. In addition, unlike the conventional normal form theory which uses separate nonlinear transformations at each order, this approach uses a consistent nonlinear transformation through all order computations. This enables one to obtain a convenient, one step transformation between the original system and the simplest normal form. The new method can treat general differential equations which are not necessarily assumed in a conventional normal form. The method is applied to consider Hopf and Bogdanov–Takens singularities, with examples to show the computation efficiency. Maple programs have been developed to provide an "automatic" procedure for applications.

Analyzing dynamic performance of power systems over parameter space using normal forms of vector fields. I. Identification of vulnerable regions (Republished)

This is the first part of a two-part paper discussing the dynamic performance of power systems over parameter space using the method of normal forms. The normal form transformation is also derived under the second order resonance condition. By using the resonance condition as a guide to indicate unsatisfactory system dynamic performance, the authors propose a computationally efficient method to study the system under varying parameters. An approach to determine the resonance and near-resonance region in the parameter space is developed. For the resonance case, conclusions are drawn by simply examining the analytic solutions. For the near-resonance case, the machine states showing poor performance can be found by tracing the dominant nonlinear modal interaction and mode-machine interaction. The method is tested on the IEEE 50-generator system. The results reveal many interesting characteristics of the system related to resonance and near-resonance, which validates the effectiveness of the method as a useful analytical tool for system operation and design. Further work on quantifying the effect of the modal interactions on the machine states is presented in part II. [An incorrect version of this paper originally appeared in the IEEE Transactions on Power Systems, vol.16, no.3, p.445-50, 2001. This a republished corrected version.]

Analyzing dynamic performance of power systems over parameter space using normal forms of vector fields, part I: Identification of vulnerable regions

This is the first part of a two-part paper discussing the dynamic performance of power systems over parameter space using the method of normal forms. A new normal form transformation is derived under the second-order resonance condition. By using the resonance condition as a guide to indicate detrimental system dynamic performance, the authors propose a computationally efficient method to study the system under varying parameters. An approach to determine the resonance and near-resonance region in the parameter space is developed. For the resonance case, conclusions are drawn by simply examining the analytic solutions. For the near-resonance case, the machine states showing poor performance can be found by tracing the dominant nonlinear modal interaction and mode-machine interaction. The method is tested on the IEEE 50-generator system. The results reveal many interesting characteristics of the system related to resonance and near resonance, which validates the effectiveness of the method as a useful analytical tool for system operation and design. Further work on quantifying the effect of the modal interactions on the machine states is presented in Part II.