non existence and uniqueness of limit cycles for planar polynomial differential systems with homogeneous nonlinearities abstract in this paper we study the limit cycles of the planar polynomial differential systems x a x y p n x y y x a y q n x y where p n and q n are homogeneous polynomials of degree n 2 and a r consider the functions p n cos sin cos q n cos sin sin q n cos sin cos p n cos sin sin 1 a 2 n 1 2 a first we prove that these differential systems have at most 1 limit cycle if there exists a linear combination of 1 and 2 with definite sign this result improves previous known results furthermore if 1 1 a 2 0 for some 1 2 0 we provide necessary and sufficient conditions for the non existence and the existence and uniqueness of the limit cycles of these differential systems when one of these mentioned limit cycles exists it is hyperbolic and surrounds the origin

bifurcation of limit cycles and integrability of planar dynamical systems in complex form we describe a new algorithm for the calculation of liapunov quantities for two dimensional systems in terms of the complex form of the system it is more transparent conceptually and more efficient computationally we demonstrate the advantages of this approach by deriving the integrability conditions and information about the bifurcation of limit cycles for some cubic systems

rotationally symmetric p harmonic maps from d2 to s2 we consider rotationally symmetric p harmonic maps from the unit disk d2 r2 to the unit sphere s2 r3 subject to dirichlet boundary conditions and with 1 p we show that the associated energy functional admits a unique minimizer which is of class c in the interior and c1 up to the boundary we also show that there exist infinitely many global solutions to the associated euler lagrange equation and we completely characterize them

multiparametric quantum gl 2 lie bialgebras quantum r matrices and non relativistic limits multiparametric quantum deformations of gl 2 are studied through a complete classification of gl 2 lie bialgebra structures from these the non relativistic limit leading to harmonic oscillator lie bialgebras is implemented by means of a contraction procedure new quantum deformations of gl 2 together with their associated quantum r matrices are obtained and other known quantizations are recovered and classified several connections with integrable models are outlined

a moderate deviation principle for 2 d stochastic navier stokes equations in this paper we prove a central limit theorem and establish a moderate deviation principle for twodimensional stochastic navier stokes equations with multiplicative noise the weak convergence method plays an important role

the tight constant in the dvoretzky kiefer wolfowitz inequality let hat f n denote the empirical distribution function for a sample of n i i d random variables with distribution function f in 1956 dvoretzky kiefer and wolfowitz proved that p big sqrt n sup x hat f n x f x lambda big leq c exp 2 lambda 2 where c is some unspecified constant we show that c can be taken as 1 as conjectured by birnbaum and mccarty in 1958 provided that exp 2 lambda 2 leq frac 1 2 in particular the two sided inequality p big sqrt n sup x hat f n x f x lambda big leq 2 exp 2 lambda 2 holds without any restriction on lambda in the one sided as well as in the two sided case the constants cannot be further improved

on the centers of the weight homogeneous polynomial vector fields on the plane we classify all centers of a planar weight homogeneous polynomial vector field of weight degree 1 2 3 and 4

growth at infinity and index of polynomial maps abstract let f k n k n be a polynomial map such that f 1 0 is compact where k r or c then we give a condition implying that there is a uniform bound for the ojasiewicz exponent at infinity in certain deformations of f this fact gives a result about the invariance of the global index of f

rational integrability of two dimensional quasi homogeneous polynomial differential systems abstract we characterize in terms of the conservative dissipative decomposition of a vector field the two dimensional quasi homogeneous polynomial differential systems which have a rational first integral we obtain the kowalevskaya exponents of these vector fields and relate the rational integrability of these fields to their kowalevskaya exponents

structural stability of planar homogeneous polynomial vector fields applications to critical points and to infinity lethmbe the space of planar homogeneous polynomial vector fields of degreemendowed with the coefficient topology we characterize the set mof the vector fields ofhmthat are structurally stable with respect to perturbations inhmand we determine the exact number of the topological equivalence classes in m the study of structurally stable homogeneous polynomial vector fields is very useful for understanding some interesting features of inhomogeneous vector fields thus by using this characterization we can do first an extension of the hartman grobman theorem which allows us to study the critical points of planar analytical vector fields whosek jets are zero for allk munder generic assumptions and second the study of the flows of the planar polynomial vector fields in a neighborhood of the infinity also under generic assumptions

classification of the centers their cyclicity and isochronicity for a class of polynomial differential systems generalizing the linear systems with cubic homogeneous nonlinearities abstract in this paper we classify the centers the cyclicity of its hopf bifurcation and their isochronicity for the polynomial differential systems in r 2 of arbitrary degree d 3 odd that in complex notation z x i y can be written as z i z z z d 3 2 a z 3 b z 2 z c z z 2 d z 3 where r and a b c d c if d 3 we obtain the well known class of all polynomial differential systems of the form a linear system with cubic homogeneous nonlinearities

abel like differential equations with no periodic solutions we present various criteria for the non existence of positive periodic solutions of generalized abel differential equations with periodic coefficients that can change sign as an application we obtain some families of planar vector fields without limit cycles

periodic solutions for equation x a t x m b t x n c t x l with a t and b t changing signs in this paper we investigate the differential equation x s x t a t xm b t xn c t xl where a b c c 0 1 m n l and m n l z a solution x t with x 1 x 0 is called a periodic solution under some hypotheses which admit a t and b t without fixed sign we obtain the upper bound sometimes sharp for the number of isolated periodic solutions of the equation applying these results for the abel equation i e m 3 n 2 l 1 we get that if there exists 0 such that s t c t 0 resp s t a t b t 0 then the equation has at most 2 resp 4 non zero isolated periodic solutions furthermore suppose that a t t is a smooth curve which lies in r 0 0 1 with a 0 a 1 we obtain that if vector fields s x t 1 resp and c t x 1 are transverse to resp s x t 1 on in opposite directions then the number of non zero isolated periodic solutions of this abel equation is still no more than 2 resp 4 these conclusions generalize the known criteria about the abel equation which only refer to the cases with either a t or b t keeping sign finally as an application we study a kind of trigonometrical abel equation

the index of singularities of vector fields and finite jets abstract we describe when the index of a singularity of a smooth vector field is determined by a finite jet at the singularity we also give some criteria to determine some terms from the formal series expansion of the vector field at the singularity which determine the index

moderate deviations of minimum contrast estimators under contamination since statistical models are simplifications of reality it is important in estimation theory to study the behavior of estimators also under distributions slightly different from the proposed model in testing theory when dealing with test statistics where nuisance parameters are estimated knowledge of the behavior of the estimators of the nuisance parameters is needed under alternatives to evaluate the power in this paper the moderate deviation behavior of minimum contrast estimators is investigated not only under the supposed model but also under distributions close to the model a particular example is the multivariate maximum likelihood estimator determined within the proposed model the set up is quite general including for instance also discrete distributions the rate of convergence under alternatives is determined both when comparing the minimum contrast estimator with a natural parameter in the parameter space and when comparing it with the proposed true value in the parameter space it turns out that under the model the asymptotic optimality of the maximum likelihood estimator in the local sense continues to hold in the moderate deviation area

classical r matrices for the osp 2 2 lie superalgebra the co lie structures compatible with the osp 2 2 lie superalgebra structure are investigated and found to be all of the coboundary type the corresponding classical r matrices are classified into several disjoint families the osp 1 2 u 1 lie superbialgebras are also classified

on probabilities of excessive deviations for kolmogorov smirnov cramer von mises and chi square statistics let n be the classical empirical process and t d 0 1 r assume t satisfies the lipschitz condition using the komlos major tusnady inequality bounds for p t n x n n are obtained for every n and x n 0 hence expansions for large deviations as well as some moderate and cramer type large deviations results for t n are derived

an upper bound of the index of an equilibrium point in the plane abstract we give an upper bound of the index of an isolated equilibrium point of a c 1 vector field in the plane the vector field is decomposed in gradient and hamiltonian components this decomposition is related with the loewner vector field associated to this decomposition we consider the set where the gradient and hamiltonian components are linearly dependent the number of branches of starting at the equilibrium point determines the upper bound of the index

central limit theorems for the wasserstein distance between the empirical and the true distributions if x is integrable f is its cdf and f n is the empirical cdf based on an i i d sample from f then the wasserstein distance between f n and f which coincides with the l 1 norm f n t f t dt of the centered empirical process tends to zero a s the object of this article is to obtain rates of convergence and distributional limit theorems for this law of large numbers or equivalently stochastic boundedness and distributional limit theorems for the l 1 norm of the empirical process some limit theorems for the ornstein uhlenbeck process are also derived as a by product

vector fields with homogeneous nonlinearities and many limit cycles abstract consider planar real polynomial differential equations of the form x l x x n x where x x y r 2 l is a 2 2 matrix and x n is a homogeneous vector field of degree n 1 most known results about these equations valid for infinitely many n deal with the case where the origin is a focus or a node and give either non existence of limit cycles or upper bounds of one or two limit cycles surrounding the origin in this paper we improve some of these results and moreover we show that for n 3 odd there are equations of this form having at least n 1 2 limit cycles surrounding the origin our results include cases where the origin is a focus a node a saddle or a nilpotent singularity we also discuss a mechanism for the bifurcation of limit cycles from infinity

reversibility and quasi homogeneous normal forms of vector fields abstract 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

a new chebyshev family with applications to abel equations we prove that a family of functions defined through some definite integrals forms an extended complete chebyshev system the key point of our proof consists of reducing the study of certain wronskians to the gram determinants of a suitable set of new functions our result is then applied to give upper bounds for the number of isolated periodic solutions of some perturbed abel equations

classification of low dimensional lie super bialgebras a thorough analysis of lie super bialgebra structures on lie super algebras osp 1 2 and super e 2 is presented a combined technique of computer algebraic computations and subsequent identification of equivalent structures is applied in all the cases poisson lie brackets on supergroups are found the possibility of quantizing them in order to obtain quantum groups is discussed it turns out to be straightforward for all but one structure for the super e 2 group

polynomial first integrals for weight homogeneous planar polynomial differential systems of weight degree 3 our main result is the classification of all weight homogeneous planar polynomial differential systems of weight degree 3 having a polynomial first integral

statistical inference for the intensity in a partially observed jump diffusion abstract this article focuses on a partially observed linear diffusion with jumps described by a poisson process precisely we study an inferential problem for the intensity of the poisson process by establishing a moderate deviation principle for the least squares estimator of the intensity

classical and quantum sl 1 2 superalgebras casimir operators and quantum chain hamiltonians we examine in this paper the two parameter deformed superalgebra uqs sl 1 2 and use the results in the construction of quantum chain hamiltonians this study is done both in the framework of the serre presentation and in the r matrix scheme of faddeev reshetikhin and takhtajan frt we show that there exists an infinite number of casimir operators indexed by integers p 2 in the undeformed case and by p z in the deformed case which obey quadratic relations the construction of the dual superalgebra of functions of slqs 1 2 is also given and higher tensor product representations are discussed finally we construct quantum chain hamiltonians based on the casimir operators in the deformed case we find two hamiltonians which describe deformed t j models

classification of the quantum deformations of the superalgebra gl 1 1 we present a classification of the possible quantum deformations of the supergroup and its lie superalgebra in each case the super commutation relations and the hopf structures are explicitly computed for each r matrix one finds two inequivalent co products whether one chooses an unbraided or a braided framework while the corresponding structures are isomorphic as algebras in the braided case one recovers the classical algebra for suitable limits of the deformation parameters but this is no longer true in the unbraided case

limit cycles for planar semi quasi homogeneous polynomial vector fields abstract this paper is concerned with the limit cycles for planar semi quasi homogeneous polynomial systems we give some explicit criteria for the nonexistence and existence of periodic orbits a lower bound is given for the maximum number of limit cycles of such a system the cyclicity and center problems are studied for some subfamilies of semi quasi homogeneous polynomial systems our results generalize those obtained for polynomial semi homogeneous systems

on the existence of polynomial first integrals of quadratic homogeneous systems of ordinary differential equations we consider systems of ordinary differential equations with a quadratic homogeneous right hand side we give a new simple proof of an earlier result which gives the necessary conditions for the existence of polynomial first integrals the necessary conditions for the existence of a polynomial symmetry field are given it is proved that an arbitrary homogeneous first integral of a given degree is a linear combination of a fixed set of polynomials pacs numbers 0210 0230h 0260l

cubic systems and abel equations abstract we investigate cubic systems which can be transformed to an equation of abel form the conditions for the origin to be a centre and in particular an isochronous centre are obtained the maximum number of limit cycles which can bifurcate from a fine focus is determined and some information is obtained about the global phase portrait

a large deviation result for parameter estimators and its application to nonlinear regression analysis elaborating on the work of ibragimov and has minskii 1981 we prove a law of large deviations lld for m estimators i e those estimators which maximize a functional continuous in the parameter of the observations this lld is applied using the results of petrov 1975 to the problem of parametrical nonlinear regression in the situation of discrete time independent errors and regression functions which are continuous in the parameter this improves a result of prakasa rao 1984

moderate deviations and nonparametric inference for monotone functions this paper considers self normalized limits and moderate deviations of nonparametric maximum likelihood estimators for monotone functions we obtain their self normalized cramer type moderate deviations and limit distribution theorems for the nonparametric maximum likelihood estimator in the current status model and the grenander type estimator as applications of the results we present a new procedure to construct asymptotical confidence intervals and asymptotical rejection regions of hypothesis testing for monotone functions the theoretical results can guarantee that the new test has the probability of type ii error tending to 0 exponentially simulation studies also show that the new nonparametric test works well for the most commonly used parametric survival functions such as exponential and weibull survival distributions

topological equivalence of a plane vector field with its principal part defined through newton polyhedra abstract we introduce a blowing up of singularities of vector fields associated with newton polyhedra in the space of the exponents by means of which we prove a generalization of the hartman grobman theorem namely a plane vector field possessing characteristic orbits is locally topologically equivalent with its principal part under suitable non degeneracy hypotheses under stronger hypotheses a similar equivalence result is proven between a plane vector field and a single quasihomogeneous component of its principal part the case of second order equations is also studied

a new uniqueness criterion for the number of periodic orbits of abel equations abstract a solution of the abel equation x a t x 3 b t x 2 such that x 0 x 1 is called a periodic orbit of the equation our main result proves that if there exist two real numbers a and b such that the function a a t b b t is not identically zero and does not change sign in 0 1 then the abel differential equation has at most one non zero periodic orbit furthermore when this periodic orbit exists it is hyperbolic this result extends the known criteria about the abel equation that only refer to the cases where either a t 0 or b t 0 does not change sign we apply this new criterion to study the number of periodic solutions of two simple cases of abel equations the one where the functions a t and b t are 1 periodic trigonometric polynomials of degree one and the case where these two functions are polynomials with three monomials finally we give an upper bound for the number of isolated periodic orbits of the general abel equation x a t x 3 b t x 2 c t x when a t b t and c t satisfy adequate conditions

planar quasi homogeneous polynomial differential systems and their integrability abstract in this paper we study the quasi homogeneous polynomial differential systems and provide an algorithm for obtaining all these systems with a given degree using this algorithm we obtain all quasi homogeneous vector fields of degree 2 and 3 the quasi homogeneous polynomial differential systems are liouvillian integrable in particular we characterize all the quasi homogeneous vector fields of degree 2 and 3 having a polynomial rational or global analytical first integral

existence of non trivial limit cycles in abel equations with symmetries abstract we study the periodic solutions of the generalized abel equation x a 1 a 1 t x n 1 a 2 a 2 t x n 2 a 3 a 3 t x n 3 where n 1 n 2 n 3 1 are distinct integers a 1 a 2 a 3 r and a 1 a 2 a 3 are 2 periodic analytic functions such that a 1 t sin t a 2 t cos t a 3 t sin t cos t are periodic positive even functions when n 3 n 1 n 3 n 2 0 we prove that the equation has no non trivial different from zero limit cycle for any value of the parameters a 1 a 2 a 3 when n 3 n 1 n 3 n 2 0 we obtain under additional conditions the existence of non trivial limit cycles in particular we obtain limit cycles not detected by abelian integrals

limit cycles of polynomial differential equations with quintic homogeneous nonlinearities abstract in this paper we mainly study the number of limit cycles which can bifurcate from the periodic orbits of the two centers x y y x x y 1 x 2 y 2 2 y x 1 x 2 y 2 2 when they are perturbed inside the class of all polynomial differential systems with quintic homogeneous nonlinearities we do this study using the averaging theory of first second and third orders

algebraic and topological classification of the homogeneous cubic vector fields in the plane tsutomu date and masao iri in di gave an algebraic classification of systems i p x y i q x y where p and q are homogeneous polynomials of degree 2 for this they used the classification of the binary cubic forms and also the simultaneous classification of a linear binary form and a cubic binary form given by the algebraic invariant theory we begin by doing a similar study for systems p x y q x y where p and q are homogeneous polynomials of degree three i e cubic systems the classification s theorem of such systems is based on the classification of fourth order binary forms gurevich in gu did the classification of fourth order binary forms on the field of complex numbers since we did not find the classification on the real domain we adapt gurevich s proof to obtain it in section 1 we give some definitions and preliminary results while in section 2 we give the theorem of classification of fourth order binary forms on the real domain the method used in the proof caley s method let us obtain canonical forms of the fourth order binary forms and the algebraic characteristics section 3 is devoted to obtaining the algebraic classification of systems p x y j q x y where p and q are homogeneous polynomials of degree 3 given an arbitrary system x p q with p and q homogeneous polynomials of degree 3 we can know the equivalence class at which it belongs through the algebraic characteristics in section 4 we study the phase portraits of systems 1 p x y j q x y where p and q are homogeneous polynomials of degree n and p and q have no common factor such systems have been studied by j argemi in a here we give a shorter new proof of his results by using

the projective vector field of a kind of three dimensional quasi homogeneous system on s2 abstract in this paper we study the projective vector field q t of a three dimensional quasi homogeneous system with weight 1 1 3 and degree 2 3 2 projective vector fields q t of this kind are classified into two types for one type q t has no closed orbit and at most eight singularities which lead to a global topology of the three dimensional system for the other type q t has at most ten singularities in addition we show a relationship between q t and a lienard system of this type for both of them we obtain some conditions for the existence of limit cycles

periodic orbits in complex abel equations abstract this paper is devoted to prove two unexpected properties of the abel equation d z d t z 3 b t z 2 c t z where b and c are smooth 2 periodic complex valuated functions t r and z c the first one is that there is no upper bound for its number of isolated 2 periodic solutions in contrast recall that if the functions b and c are real valuated then the number of complex 2 periodic solutions is at most three the second property is that there are examples of the above equation with b and c being low degree trigonometric polynomials such that the center variety is formed by infinitely many connected components in the space of coefficients of b and c this result is also in contrast with the characterization of the center variety for the examples of abel equations d z d t a t z 3 b t z 2 studied in the literature where the center variety is located in a finite number of connected components

integrability partial integrability and nonintegrability for systems of ordinary differential equations the integrability of systems of ordinary differential equations with polynomial vector fields is investigated by using the singularity analysis methods three types of results are obtained first a general relationship between the degrees of first integrals and the so called kowalevskaya exponents is derived second it is shown that all solutions of algebraically integrable systems can be expanded in puiseux series third a new method to study partially integrable systems is studied these different aspects allow us to study algorithmically the integrability partial integrability and nonintegrability of differential systems

monodromy center focus and integrability problems for quasi homogeneous polynomial systems abstract this paper deals with planar quasi homogeneous polynomial vector fields and addresses three major questions the monodromy the center focus and the integrability problems we characterize the monodromic planar quasi homogeneous polynomial vector fields and we give a condition to distinguish between a center and a focus in this case also we provide conditions which characterize the integrability of quasi homogeneous polynomial systems under non resonance conditions the results obtained allow us to analyse two monodromic planar systems with degenerate linear part one of them with nilpotent linearization and another one with null linear part

limit cycles bifurcating from planar polynomial quasi homogeneous centers in this paper we find an upper bound for the maximum number of limit cycles bifurcating from the periodic orbits of any planar polynomial quasi homogeneous center which can be obtained using first order averaging method this result improves the upper bounds given in 7

delta method in large deviations and moderate deviations for estimators the delta method is a popular and elementary tool for deriving limiting distributions of transformed statistics while applications of asymptotic distributions do not allow one to obtain desirable accuracy of approximation for tail probabilities the large and moderate deviation theory can achieve this goal motivated by the delta method in weak convergence a general delta method in large deviations is proposed the new method can be widely applied to driving the moderate deviations of estimators and is illustrated by examples including the wilcoxon statistic the kaplan meier estimator the empirical quantile processes and the empirical copula function we also improve the existing moderate deviations results for m estimators and l statistics by the new method some applications of moderate deviations to statistical hypothesis testing are provided

centers of quasi homogeneous polynomial differential equations of degree three we characterize the centers of the quasi homogeneous planar polynomial differential systems of degree three such systems do not admit isochronous centers at most one limit cycle can bifurcate from the periodic orbits of a center of a cubic homogeneous polynomial system using the averaging theory of first order

large deviations of estimators the performance of a sequence of estimators t n of g theta can be measured by its inaccuracy rate lim inf n rightarrow infty n 1 log mathbb p theta t n g theta varepsilon for fixed varepsilon 0 optimality of consistent estimators operatorname wrt the inaccuracy rate is investigated it is shown that for exponential families in standard representation with a convex parameter space the maximum likelihood estimator is optimal if the parameter space is not convex which occurs for instance in curved exponential families in general no optimal estimator exists for the location problem the inaccuracy rate of m estimators is established if the underlying density is sufficiently smooth an optimal m estimator is obtained within the class of translation equivariant estimators tail behaviour of location estimators is studied a connection is made between gross error and inaccuracy rate optimality

phase portraits of planar semi homogeneous vector fields ii we determine all the phase portraits of the semi homogeneous polynomial vector fields of the form ax by ax4 bx3y cx2y2 dxy3 ey4

non stabilisable jets of diffeomorphisms in r2 and of vector fields in r3 we essentially prove two results firstly that for any x germ of a coo vector field on r2 with a certain specific 9 jet there is a germ of a c diffeomorphism f with j f o j x o x denoting the flow of x such that f and x are not c0 conjugate theorem 1 and secondly that whenever x is a germ of a coo vector field on r3 whose 9 jet belongs to a certain class of 9 jets there can be found a germ of a coo vector field y on r3 with jo x 0 jo y 0 such that x and y are not c0 equivalent theorem 2 the second result implies the existence of jets of vector fields in r3 which are not stabilisable for the notion of c0 equivalence the question of the existence of such jets was raised by thom in 1970 and very soon takens found examples in rn with n 2 4 ta ta3 the first result which we use in proving the second also implies the existence of jets of diffeomorphisms in r2 which are not stabilisable for the notion of c conjugacy the diffeomorphisms having such a jet contain hyperbolic or saddle sectors with a different topological structure than the hyperbolic sectors occurring for vector fields in finite codimension in these sectors the singularity scheme of a nice desingularisation after blowing up contains certain successions of semi hyperbolic singularities these cases had not been treated in d r r and our example now shows that the problems we encountered in d r r in trying to add this case were not technical but essential our examples provide germs of coo diffeomorphisms in r2 which are infinitely tangent to the time l mapping of a vector field of finite codimension and not c embeddable in the flow of a vector field of finite codimension this is definitely not possible in r germs of orientable coo diffeomorphisms of finite codimension are coo embeddable in a flow ta4 even the germs of diffeomorphisms infinitely tangent to the identity and with isolated fixed point show up in a c flow but they do not necessarily show up in a c 2 flow se possibly as

limit cycles of abel equations of the first kind abstract consider the scalar differential equation x i 0 m a i t x n i where a i t are t periodic analytic functions and 1 n i n for any polynomial q x x n 0 i 1 m i x n i the equation can be written as x a 0 q x r t x let w be the wronskian of q and r with respect to x and q w the previous polynomials after removing multiplicity of roots and solutions of the differential equation we prove that if the vector field defined by the differential equation is transversal at every point of q x 0 or w t x 0 then the number of limit cycles isolated periodic solutions in the set of periodic solutions of the differential equation is at most 3 n 1

a moderate deviation principle for 2 d stochastic navier stokes equations driven by multiplicative levy noises abstract in this paper we establish a moderate deviation principle for two dimensional stochastic navier stokes equations driven by multiplicative levy noises the weak convergence method introduced by budhiraja dupuis and ganguly in 3 plays a key role

singularity analysis in planar vector fields the aim of singularity analysis is to build local solutions around their singularities in the complex time integrability properties or the existence of particular solutions follow from the detailed analysis of the series expansions of the local solutions once they are constructed in this work we made a survey of the most important results of the singularity analysis which we apply to planar vector fields

structural stability of planar quasi homogeneous vector fields abstract in this work we study the structural stability of planar quasi homogeneous vector fields under quasi homogeneous perturbations and provide a complete classification this study which has been the subject of previous works is only complete in the homogeneous case the main tool in our analysis is a splitting of planar quasi homogeneous vector fields into conservative dissipative parts moreover we describe the topological equivalence classes in the set of the structurally stable planar quasi homogeneous vector fields finally we include some examples where the equivalence classes are determined