unique normal forms the nilpotent hamiltonian case the problem we consider in this paper namely the characterization of unique normal forms derives from bifurcation theory in a typical situation one has an unperturbed vector field near a degenerate equilibrium in order to study the effect of generic perturbations suitable coordinates are introduced and the simplified system is said to be in normal form a well known case is the hopf bifurcation where a pair of conjugate eigenvalues cross the imaginary axis using averaging one can in this case compute the periodic orbit that splits off from the equilib um a second familiar example is the takens bogdanov bifurcation where two eigenvalues are zero in all these situations it is necessary to make certain choices both in the definition of what constitutes a normal form and in the computation of the transformation the idea here is to use this second kind of choice to sharpen the very definition of normal form while in standard treatments only the linear part of the field plays a role we show that one can effectively use its non linear part to remove additional terms and thus simplify the problem further the underlying philosophy in this type of game is that

normal forms for hopf zero singularities with nonconservative nonlinear part abstract in this paper we are concerned with the simplest normal form computation of the systems 0 1 x 2 x f x y 2 z 2 y z y f x y 2 z 2 z y z f x y 2 z 2 where f is a formal function with real coefficients and without any constant term these are the classical normal forms of a larger family of systems with hopf zero singularity indeed these are defined such that this family would be a lie subalgebra for the space of all classical normal form vector fields with hopf zero singularity the simplest normal forms and simplest orbital normal forms of this family with nonzero quadratic part are computed we also obtain the simplest parametric normal form of any non degenerate perturbation of this family within the lie subalgebra the symmetry group of the simplest normal forms is also discussed this is a part of our results in decomposing the normal forms of hopf zero singular systems into systems with a first integral and nonconservative systems

an improved theory of asymptotic unfoldings abstract an asymptotic unfolding of a dynamical system near a rest point is a system with additional parameters such that every one parameter deformation of the original system can be embedded in the unfolding preserving all properties that can be detected by asymptotic methods asymptotic unfoldings are computed using normal and hypernormal form methods we present a simplified and improved method of computing such unfoldings that can be used in any normal form style

freedom in small parameter expansion for nonlinear perturbations the freedom of choice of the zero order term in the perturbative analysis of harmonic oscillators that are perturbed by a nonlinear perturbation is investigated in detail within the framework of the method of normal forms in the case of the unforced duffing oscillator it is demonstrated that the choice leading to minimal normal forms mnf is by far the best indicating that mnf may be a way to significantly improve the convergence properties of the perturbation series relative to the traditional expansions

unique normal forms for planar vector fields soit x un germe d un champ vectoriel analytique ou formel en equilibre a o r 2 avec des valeurs propres purement imaginaires a 0 i a 0 0 alors x est formellement equivalent a precisement un des champs vectoriels polynomiaux suivant ou x 1 2 cos y 1 2 sin pour des coordonnees appropriees x y 7b a 0 a 0 7b a k a 0 k 7b b j a 0 a 1 a j j j 1 b j j 0

construction of darboux coordinates and poincare birkhoff normal forms in noncanonical hamiltonian systems abstract we demonstrate a general method to construct darboux coordinates via normal form expansions in noncanonical hamiltonian system obtained from e g a variational approach to quantum systems the procedure serves as a tool to naturally extract canonical coordinates out of the variational parameters and at the same time to transform the energy functional into its poincare birkhoff normal form the method is general in the sense that it is applicable for arbitrary degrees of freedom in arbitrary orders of the local expansion and it is independent of the precise form of the hamilton operator the method presented allows for the general and systematic investigation of quantum systems in the vicinity of fixed points which e g correspond to ground excited or transition states moreover it directly allows to calculate classical and quantum reaction rates by applying transition state theory

unipotent normal forms for symplectic maps abstract a normal form theory for symplectic maps with non diagonalizable linear part is presented the theory is obtained by working directly with the group of symplectic diffeomorphisms using the fact that non diagonalizable linear symplectic maps have a unique semisimple unipotent decomposition the normalization of a symplectic map with respect to its linear part proceeds in two steps first a semisimple normalization for which the theory is well established and second a unipotent normalization which is presented here the role of normal form symmetry and reduction of the normalized map is also considered the theory is applied to two examples of symplectic maps on r 4 first a non semisimple multiple 1 eigenvalue and second the collision of floquet multipliers of opposite sign

idempotent connected abundant semigroups a general theory for a class of abundant semigroups is developed for a semigroup s in this class let e be its set of idempotents and the subsemigroup of s generated by e when is regular there is a homomorphism with a number of desirable properties from s onto a full subsemigroup of the hall semigroup t from this fact analogues of results in the regular case are obtained for simple and simple abundant semigroups

unique normal forms for vector fields and hamiltonians on construit des formes canoniques pour tous les champs vectoriels resp hamiltoniens en equilibre a 0 r n relativement au groupe des diffeomorphismes formels resp des transformations canoniques tangentiels a l identite

a perturbation method for computing the simplest normal forms of dynamical systems abstract a previously developed perturbation method is generalized for computing the simplest normal form at each level of computation the minimum number of terms are retained of general n dimensional differential equations this direct approach combining the normal form theory with center manifold theory in one unified procedure can be used to systematically compute the simplest or unique normal form two particular singularities of the jacobian of the system are considered in this paper the first one is associated with one pair of purely imaginary eigenvalues hopf type singularity and the other corresponds to a simple zero and a pair of purely imaginary eigenvalues hopf zero type singularity the approach can be easily formulated and implemented using a computer algebra system maple programs have been developed in this paper which can be automatically executed by a user without the knowledge of computer algebra a physical oscillator model is studied in detail to show the computational efficiency of the direct method and the advantage of using the simplest normal form which greatly simplifies the analysis on dynamical systems in particular for bifurcations and stability

unique normal form of a class of 3 dimensional vector fields with symmetries abstract we present the unique normal form of a class of 3 dimensional vector fields bt zero singularity with symmetries the main technique applied to the computation is the combination of a linear grading function and the method of multiple lie brackets we introduce new notations for block matrices to simplify the expression of block matrices the new notations help to prove the non degeneracy of huge size matrices by an induction technique

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

rees matrix covers for locally inverse semigroups byd b mcalisterabstract a regular semigroup s is locally inverse if each local submonoid ese ean idempotent is an inverse semigroup it is shown that every locally inversesemigroup is an image of a regular rees matrix semigroup over an inversesemigroup by a homomorphism 0 which is one to one on each local submonoid such a homomorphism is called a local isomorphism regular semigroups which arelocally isomorphic images of regular rees matrix semigroups over semilattices arealso characterized

orbital formal normal forms for general bogdanov takens singularity abstract we solve completely the problem of classification of germs of complex planar vector fields with nilpotent singularity with respect to formal orbital equivalence

reduction of secular error in approximations to harmonic oscillator systems with nonlinear perturbations abstract the freedon inherent in perturbative expansions is exploited within the framework of the method of normal forms to demonstrate that the secular errors evolving in approximations to solutions for harmonic oscillator systems with small nonlinear perturbations can be reduced significantly by the minimal normal form mnf choice of the free resonant terms in the expansion the mnf choice previously developed for systems with one degree of freedom is extended to higher dimensional ones by applying it to the phase component of the normal form equations the results are shown for a sequence of examples of growing complexity starting with single oscillators and ending with three coupled oscillators

spectral sequences and parametric normal forms abstract this paper extends recent developments on spectral sequence approach on the simplest normal form theory to parametric cases in this paper the method is first applied to two simple examples parametric single zero and parametric resonant saddle singularities and then to parametric generalized hopf singularity we provide the simplest parametric normal form for these cases

further reductions of normal forms for dynamical systems abstract we propose in this paper a method for obtaining a significant refinement of normal forms for dynamical systems or vector fields with concrete and interesting applications we use lower order nonlinear terms in the normal form for the simplifications of higher order terms our approach is applicable for both the non nilpotent and the nilpotent cases for dynamical systems of dimensions 2 and 3 we give an algorithm that leads to interesting finite order normal forms which are optimal or unique with respect to equivalence by formal near identity transformations we can compute at the same time a formal diffeormorphism that realizes the normalization comparisons with other methods are given for several examples

linear grading function and further reduction of normal forms abstract in this note an idea of quasi homogeneous normal form theory using new grading functions is introduced the definition ofnth order normal form is given and some sufficient conditions for the uniqueness of normal forms are derived a special case of the unsolved problem in a paper of baider and sanders for the unique normal form of bogdanov takens singularities is solved

radius renormalization in limit cycles oscillatory problems that exhibit limit cycles due to small nonlinearities are studied within the framework of the method of normal forms it is shown that the zero order term in the perturbative expansion of the solution can be chosen so that the radius of the asymptotic circle approximation to the limit cycle is unaffected by higher order corrections

hypernormal form theory foundations and algorithms abstract hypernormal forms unique normal forms simplest normal forms are investigated both from the standpoint of foundational theory and algorithms suitable for use with computer algebra the baider theory of the campbell hausdorff group is refined by a study of its subgroups to determine the smallest substages into which the hypernormalization process can be divided this leads to a linear algebra algorithm to compute the generators needed for each substage with the least amount of work a concrete interpretation of jan sanders spectral sequence for hypernormal forms is presented examples are given and a proof is given for a little known theorem of belitskii expressing the hypernormal form space in the inner product style as the kernel of a higher order differential operator

quasi adequate semigroups the investigation of general quasi adequate semigroups is initiated these are semigroups which are abundant and in which the idempotents form a subsemigroup for such a semigroup s we study the minimum good congruence such that s is adequate results on together with results from a previous paper of the authors are used to obtain a structure theorem for a class of quasi adequate semigroups

vector potential normal form classification for completely integrable solenoidal nilpotent singularities abstract we introduce an sl 2 invariant family of polynomial vector fields with an irreducible nilpotent singularity in this paper we are concerned with characterization and normal form classification of these vector fields we show that the family is a lie subalgebra and each vector field from this family is volume preserving completely integrable and rotational all such vector fields share a common quadratic invariant we provide a poisson structure for the lie subalgebra 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 which can be found in applications 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

simplified renormalization group method for ordinary differential equations abstract the renormalization group rg method for differential equations is one of the perturbation methods which allows one to obtain invariant manifolds of a given ordinary differential equation together with approximate solutions to it this article investigates higher order rg equations which serve to refine an error estimate of approximate solutions obtained by the first order rg equations it is shown that the higher order rg equation maintains the similar theorems to those provided by the first order rg equation which are theorems on well definedness of approximate vector fields and on inheritance of invariant manifolds from those for the rg equation to those for the original equation for example since the higher order rg equation is defined by using indefinite integrals and is not unique for the reason of the undetermined integral constants the simplest form of rg equation is available by choosing suitable integral constants it is shown that this simplified rg equation is sufficient to determine whether the trivial solution to time dependent linear equations is hyperbolically stable or not and thereby a synchronous solution of a coupled oscillators is shown to be stable

computation of the simplest normal forms with perturbation parameters based on lie transform and rescaling normal form theory is one of the most power tools for the study of nonlinear differential equations in particular for stability and bifurcation analysis recently many researchers have paid attention to further reduction of conventional normal forms cnf to so called the simplest normal form snf however the computation of normal forms has been restricted to systems which do not contain perturbation parameters unfolding the computation of the snf is more involved than that of cnfs and the computation of the snf with unfolding is even more complicated than the snf without unfolding although some author mentioned further reduction of the snf no results have been reported on the exact computation of the snf of systems with perturbation parameters this paper presents an efficient method for computing the snf of differential equations with perturbation parameters unlike cnf theory which uses an independent nonlinear transformation at each order this approach uses a consistent nonlinear transformation through all order computations the particular advantage of the method is able to provide an efficient recursive formula which can be used to obtain the nth order equations containing the nth order terms only this greatly saves computational time and computer memory the recursive formulations have been implemented on computer systems using maple as an illustrative example the snf for single zero singularity is considered using the new approach

constant mean curvature surfaces in a half space of mathbf r 3 with boundary in the boundary of the half space the structure of the set of compact constant mean curvature surfaces whose boundary is a given jordan curve in r seems far from being understood we shall consider a simple but interesting situation concerning this problem assume that m be an embedded compact h surface in r x3 0 with m p x3 0 then there is little known about the geometry and topology of m in terms of for example if is convex is m of genus zero when is a circle it follows from alexandrov a that m is neccesarily a spherical cap or the planar disk bounded by we first show that if n p is a sequence of embedded perhaps nonconnected curves converging to a point p and mn r is a sequence of 1 surfaces h 1 with mn n then some subsequence of mn converges either to p or to the unit sphere tangent to p at p the convergence being smooth in r p the same kind of result was obtained by wente w when n is an arbitrary jordan curve in r converging to a point p and mn is an immersed topological disc bounded by n which minimizes area among disks bounding a fixed algebraic volume

quasi homogeneous normal forms in this paper we extend the concepts of the normal form theory for vector fields that are expanded in quasi homogeneous components of a fixed type these expansions have been used by some authors in the analysis of the determinacy of a given singularity also the use of reparametrizations in the time are considered namely beyond the use of c conjugation to determine normal forms we present a method useful to determine how much a vector field can be simplified by using c equivalence the results obtained are applied in the case of the bogdanov takens singularity firstly using c conjugation and later showing the improvements provided by the c equivalence

bifurcation of critical periods for plane vector fields on etudie un probleme de bifurcation dans des familles de champs de vecteurs analytiques plans qui ont un centre non degenere a l origine pour toutes les valeurs d un parametre r n on definit la fonction periode p on determine les points critiques de cette fonction pres du centre avec comme parametre de bifurcation

freedom in minimal normal forms abstract the purpose of this paper is to clarify the amount of freedom that is available in a complete normalization of a symplectic map in particular we prove that the minimal normal form mnf of kahn and zarmi is a symplectic normal form in disguise we show that there is no reason to believe that the minimal normal has an extended region of good behavior despite the fact that it is better in the immediate neighborhood of the origin our results apply also to multi dimensional problems however the minimal normal form of kahn and zarmi cannot be trivially extended to higher dimensionality without loosing analiticity thus our discussion will clear up some of the issues raised recently by mane and weng

filtration preserving mappings and centralizers within graded lie algebras we use spectral sequence techniques to compute centralizers of elements within graded lie algebras and the methods are then applied to the calculation of unique normal forms of elements within one parameter matrix lie algebras a finiteness criterion for unique normal forms is presented

normal form theory and spectral sequences abstract the concept of unique normal form is formulated in terms of a spectral sequence as an illustration of this technique some results of baider and churchill concerning the normal form of the anharmonic oscillator are reproduced the aim of this paper is to show that spectral sequences give us a natural framework in which to formulate normal form theory

computation of normal forms for high dimensional non linear systems and application to non planar non linear oscillations of a cantilever beam a new and efficient computation of the normal forms is developed in this paper for high dimensional non linear systems and the computational method is applied to non planar non linear oscillations of a cantilever beam the method developed here has the advantage of directly calculating the coefficients of the normal forms and the associated near identity non linear transformations for three different cases that is 1 the case of two pairs of pure imaginary eigenvalues 2 the case of one non semisimple double zero and a pair of pure imaginary eigenvalues and 3 the case of two non semisimple double zero eigenvalues the final partial differential equations of various resonant cases appear in a canonical form whose solutions can be conveniently obtained using polynomial equations with the aid of the maple software a symbolic program for computing the normal forms of high dimensional non linear systems is given comparing the method developed here with other methods of computing the normal forms it is understood that we may respectively obtain the normal forms the coefficients of the normal forms and the associated near identity non linear transformations for three resonant cases by using a same main maple symbolic program moreover the method is easy to apply to engineering problems the normal forms of the averaged equations and their coefficients for non planar non linear oscillations of the cantilever beam under combined parametric and forcing excitations are calculated for two resonant cases

on averaging reduction and symmetry in hamiltonian systems abstract the existence of periodic orbits for hamiltonian systems at low positive energies can be deduced from the existence of nondegenerate critical points of an averaged hamiltonian on an associated reduced space alternatively in classical kinetic plus potential energy hamiltonians the existence of such orbits can often be established by elementary geometrical arguments the present paper unifies the two approaches by exploiting discrete symmetries including reversing diffeomorphisms that occur in a given system the symmetries are used to locate the periodic orbits in the averaged hamiltonian and thence in the original hamiltonian when the periodic orbits are continued under perturbations admitting the same symmetries in applications to the henon heiles hamiltonian it is illustrated how higher order averaging can sometimes be used to overcome degeneracies encountered at first order

unique normal form of bogdanov takens singularities abstract in this note the method introduced by kokubu oka and wang is improved and the unsolved problem in a paper of baider and sanders for the unique normal form of bogdanov takens singularities is solved under a very general condition

minimal normal forms in harmonic oscillations with small nonlinear perturbations the freedom of choice of the zeroth order approximation in the perturbative expansion of solutions to oscillatory problems with small nonlinearities is exploited within the framework of the method of normal forms a priori the normal form may contain an infinite number of resonant terms it is shown that in a large class of problems of interest the normal form can be simplified so that the number of such terms is small or to have other desired properties conservative as well as dissipative systems are discussed

on the representations and formula presented equivariant normal form for solenoidal hopf zero singularities abstract in this paper we deal with the solenoidal conservative lie algebra associated to the classical normal form of hopf zero singular system we concentrate on the study of some representations and z 2 equivariant normal form for such singular differential equations first we list some of the representations that this lie algebra admits the vector fields from this lie algebra could be expressed by the set of ordinary differential equations where the first two of them are in the canonical form of a one degree of freedom hamiltonian system and the third one depends upon the first two variables this representation is governed by the associated poisson algebra to one sub family of this lie algebra euler s form vector potential and clebsch representation are other representations of this lie algebra that we list here we also study the non potential property of vector fields with hopf zero singularity from this lie algebra finally we examine the unique normal form with non zero cubic terms of this family in the presence of the symmetry group z 2 the theoretical results of normal form theory are illustrated with the modified chua s oscillator