Resonant Vector Fields Research Articles

Time-reversibility and integrability of $ p:-q $ resonant vector fields

<abstract><p>We study the local analytical integrability in a neighborhood of $ p:-q $ resonant singular point of a two-dimensional vector field and its connection to time-reversibility with respect to the non-smooth involution $ \varphi(x, y) = (y^{p/q}, x^{q/p}). $ Some generalizations of the theory developed by Sibirsky for the $ 1:-1 $ resonant case to the $ p:-q $ resonant case are presented.</p></abstract>

Gevrey series in compensators linearizing a planar resonant vector field and its unfolding

We consider a planar vector field $X$ near a saddle type $p:-q$ resonant singular point. Assuming that it has a normal form with a Gevrey-$d$ expansion (like $d=p+q$ which is in particular the case when starting from an analytic vector field) we show that $X$ can be linearized working with a change of coordinates that is of Gevrey order $d$ in certain $\log$-like variables, called compensators or also tags, multiplied by the first integral $u=x^qy^p$ of the linear part. Next we consider the unfolding of such a resonance, and provide (weaker) Gevrey-type linearization using compensators.

From dynamical systems to renormalization

In this paper we study logarithmic derivatives associated to derivations on completed graded Lie algebra, as well as the existence of inverses. These logarithmic derivatives, when invertible, generalize the exp-log correspondence between a Lie algebra and its Lie group. Such correspondences occur naturally in the study of dynamical systems when dealing with the linearization of vector fields and the non linearizability of a resonant vector fields corresponds to the non invertibility of a logarithmic derivative and to the existence of normal forms. These concepts, stemming from the theory of dynamical systems, can be rephrased in the abstract setting of Lie algebra and the same difficulties as in perturbative quantum field theory (pQFT) arise here. Surprisingly, one can adopt the same ideas as in pQFT with fruitful results such as new constructions of normal forms with the help of the Birkhoff decomposition. The analogy goes even further (locality of counter terms, choice of a renormalization scheme) and shall lead to more interactions between dynamical systems and quantum field theory.

Linearization of resonant vector fields

A method allowing the formal linearization of a large class of vector fields with resonant eigenvalues is presented, the admissible nonlinearities being characterized by conditions that are easy to check. This method also gives information on the terms that are actually present in a nonlinear normal form of a given resonant vector field.

A Real Normal Form Approach to the Study of Resonant Power Systems

A novel formulation for computing real normal forms of resonant vector fields is presented. Making use of a special real-valued nonlinear transformation of coordinates in physical space, a general technique for the computation of real normal forms is proposed. Symbolic and numerical calculations are presented to illustrate the effectiveness and applicability of the proposed analytic technique.

Normal forms and linearization of resonant vector fields with multiple eigenvalues

We discuss the linearization and normal forms of resonant vector fields X ( x ) = A x + a ( x ) , where A has one double or triple eigenvalue or a pair of double eigenvalues: we present a simple way of identifying the resonant monomials that have to appear in its normal form, and also effective conditions on the nonlinearity a ( x ) for the resonant vector field to be linearizable.

Linearization of germs of hyperbolic vector fields

We develop a normal form to express asymptotically a conjugacy between a germ of resonant vector field and its linear part. We show that such an asymptotic expression can be written in terms of functions of the Logarithmic Mourtada type. To cite this article: P. Bonckaert et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).

Analytic linearizability of some resonant vector fields

A method allowing the linearization of vector fields with resonant eigenvalues is presented, the admissible nonlinearities being characterized by conditions that are easy to check.

Correction and linearization of resonant vector fields and diffeomorphisms

We extend the classical Siegel-Brjuno-Russmann linearization theorem to the resonant case by showing that under A. D. Brjuno's diophantine condition, any resonant local analytic vector field (resp. diffeomorphism) possesses a well-defined correction which (1) depends on the chart but, in any given chart, is unique (2) consists solely of resonant terms and (3) has the property that, when substracted from the vector field (resp. when factored out of the diffeomorphism), the vector field or diffeomorphism thus “corrected” becomes analytically linearizable (with a privileged or “canonical” linearizing map). Moreover, in spite of the small denominators and contrary to a hitherto prevalent opinion, the correction's analyticity can be established by pure combinatorics, without any analysis.

Bifurcation from a Periodic Orbit for a Strongly Resonant Reversible Autonomous Vector Field

The author considers a reversible system admitting a symmetric periodic orbit such that the Jordan block belonging to the Floquet exponent zero is four-dimensional, nonsemisimple. Using the normal form theory around closed orbits, it is shown that, generically, such a solution is part of a one-parameter family of symmetric periodic orbits. The existence in such a system of two one-parameter families of symmetric solutions homoclinic to some periodic orbits is also proven. Finally, the author shows how this problem is related to the perturbed reversible 1-1 resonance vector fields, and allows its study to be completed.

The nilpotent part and distinguished form of resonant vector fields or diffeomorphisms

The nilpotent part and distinguished form of resonant vector fields or diffeomorphisms