degenerate hopf bifurcations in discontinuous planar systems abstract we study the stability of a singular point for planar discontinuous differential equations with a line of discontinuities this is done for the most generic cases by computing some kind of lyapunov constants our computations are based on the so called r p q generalized polar coordinates introduced by lyapunov and they are essentially different from the ones used in the smooth case these lyapunov constants are also used to generate limit cycles for some concrete examples

the number of small amplitude limit cycles in arbitrary polynomial systems abstract in this paper we study the number of small amplitude limit cycles in arbitrary polynomial systems it is found that almost all the results for the number of small amplitude limit cycles are obtained by calculating lyapunov constants and determining the order of the corresponding hopf bifurcation it is well known that the difficulty in calculating the lyapunov constants increases with the increasing of the degree of polynomial systems so it is necessary and valuable for us to achieve some general results about the number of small amplitude limit cycles in arbitrary polynomial systems with degree m which is denoted by m m in this paper by applying the method developed by c christopher and n lloyd in 1995 and m han and j li in 2012 we first obtain the lower bounds for m 6 m 14 and then prove that m m m 2 if m 23 finally we obtain that m m grows as least as rapidly as 18 25 1 2 ln 2 m 2 2 ln m 2 for all large m it is proved by m han j li lower bounds for the hilbert number of polynomial systems j differential equations 252 2012 3278 3304 that the number of all limit cycles in arbitrary polynomial systems with degree m grows as least as rapidly as 1 2 ln 2 m 2 2 ln m 2

cauchy problems for fractional differential equations with riemann liouville fractional derivatives abstract in this paper we are concerned with cauchy problems of fractional differential equations with riemann liouville fractional derivatives in infinite dimensional banach spaces we introduce the notion of fractional resolvent obtain some its properties and present a generation theorem for exponentially bounded fractional resolvents moreover we prove that a homogeneous order cauchy problem is well posed if and only if its coefficient operator is the generator of an order fractional resolvent and we give sufficient conditions to guarantee the existence and uniqueness of weak solutions and strong solutions of an inhomogeneous order cauchy problem

dynamics of a predator prey model we describe the bifurcation diagram of limit cycles that appear in the first realistic quadrant of the predator prey model proposed by r m may stability and complexity in model ecosystems princeton university press princeton nj 1974 in particular we give a qualitative description of the bifurcation curve when two limit cycles collapse on a semistable limit cycle and disappear moreover we show that locally asymptotic stability of a positive equilibrium point does not imply global stability for this class of predator prey models

spectral element discretization of the maxwell equations on considere un probleme variationnel equivalent aux equations de l electromagnetisme avec conditions aux limites absorbantes sur une partie de la frontiere qu on prouve etre bien pose puis on propose une discretisation de ce probleme par schema aux differences finies en temps et elements spectraux en espace et on etablit des estimations d erreur entre solutions exacte et approchee

implementation of a new algorithm of computation of the poincare liapunov constants in the last years many papers giving different methods to compute the poincare liapunov constants have been published in appl math warsaw 28 2002 17 a new method to compute recursively all the poincare liapunov constants as a function of the coefficients of the system for an arbitrary analytic system which has a perturbed linear center at the origin was given and thus a theoretical answer to the classical center problem was given the method also computes the coefficients of the poincare series as a function of the same coefficients we describe its implementation in two different ways by means of a computer algebra system and an algorithm in any computer language if this second alternative is used later it is necessary to translate the results so that they can be manipulated with a computer algebra system we describe also how the availability of symbolic manipulation procedures has recently led to a significant progress in the resolution of the different problems related with the poincare liapunov constants as they are the central problems like the small amplitude limit cycles

the number of limit cycles of certain polynomial differential equations two dimensional differential systems are considered where p and q are polynomials the question of interest is the maximum possible numberof limit cycles of such systems in terms of the degree of p and q an algorithm is described for determining a so called focal basis this can be implemented on a computer estimates can then be obtained for the number of small amplitude limit cycles the technique is applied to certain cubic systems a class of examples with exactly five small amplitude limit cycles is constructed quadratic systems are also considered

the criterion of the center focus for a class of planar perturbed polynomial systems with 2n 1 th degree abstract in this paper we investigate the problem of center focus for the system d x d t y k 1 n p 2 k 1 x y d y d t x k 1 n q 2 k 1 x y which is regarded as a perturbed one of a planar linear system d x d t y d y d t x where p 2 k 1 x y q 2 k 1 x y k 1 2 n are 2 k 1 th degree homogeneous polynomials in x y we shall give a simple and convenient method which can immediately distinguish that the singular point o is a center or fine focus and the stability of the singular point can be determined by the matrices consist of the coefficients of perturbed terms

time discretization scheme for quasi static maxwell s equations with a non linear boundary condition we study a time dependent eddy current equation for the magnetic field h accompanied with a non linear degenerate boundary condition bc which is a generalization of the classical silver muller condition for a non perfect conductor more exactly the relation between the normal components of electrical e and magnetic h fields obeys the following power law email protected email protected a email protected for some email protected 0 1 we establish the existence and uniqueness of a weak solution in a suitable function space under the minimal regularity assumptions on the boundary c and the initial data h 0 we design a non linear time discrete approximation scheme based on rothe s method and prove convergence of the approximations to a weak solution we also derive the error estimates for the time discretization

coexistence of limit cycles and invariant algebraic curves for a kukles system abstract we consider a kukles system of the form x y y f x y where f x y is a polynomial with real coefficients of degree d without y as a divisor we study the maximum number of small amplitude limit cycles for these kind of systems which can coexist with invariant algebraic curves we give all the possible distributions of invariant straight lines for a kukles system and we give some bounds for the number of limit cycles we also give some necessary conditions for the existence of an invariant algebraic curve of degree 2 and we study the possible coexistence of this curve and a limit cycle finally we give two examples of cubic kukles systems both with an invariant hyperbola in the first example the hyperbola coexists with a center and in the second one it coexists with two small amplitude limit cycles these two examples contradict a previous result given in ann differential equations 7 3 1991 323

existence theorems of global solutions for nonlinear volterra type integral equations in banach spaces in this paper first a new fixed point theorem is established and then by the use of it the existence theorems of global solutions for nonlinear volterra type integral equations in banach spaces are investigated the results obtained in this paper generalize and improve the results corresponding to those obtained by others

numerical solution to the time dependent maxwell equations in axisymmetric singular domains the singular complement method in this paper we present a method to solve numerically the axisymmetric time dependent maxwell equations in a singular domain in math methods appl sci 25 2002 49 math methods appl sci 26 2003 861 the mathematical tools and an in depth study of the problems posed in the meridian half plane were exposed the numerical method and experiments based on this theory are now described here it is also the generalization to axisymmetric problems of the singular complement method that we developed to solve maxwell equations in 2d singular domains see c r acad sci paris t 330 2000 391 it is based on a splitting of the space of solutions in a regular subspace and a singular one derived from the singular solutions of the laplace problem numerical examples are finally given to illustrate our purpose in particular they show how the singular complement method captures the singular part of the solution

existence results for fractional order semilinear functional differential equations with nondense domain abstract in this paper we establish sufficient conditions for existence and uniqueness of solutions for some nondensely defined semilinear functional differential equations involving the riemann liouville derivative our approach is based on integrated semigroup theory the banach contraction principle and the nonlinear alternative of leray schauder type

approximate controllability of fractional order semilinear systems with bounded delay abstract in this paper sufficient conditions are established for the approximate controllability of a class of semilinear delay control systems of fractional order the existence and uniqueness of mild solution of the system is also proved the results are obtained by using contraction principle and the schauder fixed point theorem some examples are given to illustrate the theory

continuous galerkin methods for solving the time dependent maxwell equations in 3d geometries a few years ago costabel and dauge proposed a variational setting which allows one to solve numerically the time harmonic maxwell equations in 3d geometries with the help of a continuous approximation of the electromagnetic field in this paper we investigate how their framework can be adapted to compute the solution to the time dependent maxwell equations in addition we propose some extensions such as the introduction of a mixed variational setting and its discretization to handle the constraint on the divergence of the field 2007 elsevier inc all rights reserved

on a solution to fractional order integrodifferential equations with analytic semigroups abstract in this paper we study a class of fractional order integrodifferential equations considered in an arbitrary banach space using the theory of analytic semigroups we establish the existence uniqueness and regularity of a mild solution to these fractional order integrodifferential equations

on a finite element method for solving the three dimensional maxwell equations abstract the aim of this paper is to present a method for solving the time domain three dimensional maxwell equations which can be coupled with a particle solver for this purpose maxwell s equations are reformulated as a constrained wave equation system with lagrange multipliers associated to the conditions 0 and e e 0 we approximate both the fields and the lagrange multipliers with a finite element method using a taylor hood element

bifurcation of a non smooth planar limit cycle from a vertex abstract we investigate non smooth planar systems of differential equations with discontinuous right hand sides discontinuity sets intersect at a vertex and are of quasilinear nature by means of the b equivalence method which was introduced in m akhmetov asymptotic representation of solutions of regularly perturbed systems of differential equations with a nonclassical right hand side ukrainian math j 43 1991 1209 1214 m akhmetov on the expansion of solutions to differential equations with discontinuous right hand side in a series in initial data and parameters ukrainian math j 45 1993 786 789 m akhmetov n a perestyuk differential properties of solutions and integral surfaces of nonlinear impulse systems differential equations 28 1992 445 453 see also e akalin m u akhmet the principles of b smooth discontinuous flows math comput simul 49 2005 981 995 m u akhmet perturbations and hopf bifurcation of the planar discontinuous dynamical system nonlinear anal 60 2005 163 178 these systems are reduced to impulsive differential equations sufficient conditions are established for the existence of foci and centers both in the noncritical and critical cases hopf bifurcation is considered from a vertex which unites several curves in the critical case an appropriate example is provided to illustrate the results

polynomial first integrals via the poincare series in appl math 28 2001 17 a method to compute the poincare liapunov constants for an arbitrary analytic differential system which has a linear center at the origin in function of the coefficients of the system was given this method also computes the coefficients of the poincare series in function of the same coefficients in this work we use this method to determine polynomial differential systems which have a polynomial first integral

small amplitude limit cycles of symmetric cubic systems hilbert s sixteenth problem concerns the number and relative position of limit cycles in a planar polynomial system of differential equations we show using multiple hopf bifurcation from multiple fine foci that limit cycle configurations of types 3 3 2 2 and 2 2 1 1 1 occur in symmetric cubic systems

nonexistence results for a class of fractional elliptic boundary value problems abstract in this paper we study a class of fractional elliptic problems of the form s u f x u in u 0 in r n where s 0 1 we prove nonexistence of positive solutions when is star shaped and f is supercritical we also derive a nonexistence result for subcritical f in some unbounded domains the argument relies on the method of moving spheres applied to a reformulated problem using the caffarelli silvestre extension caffarelli and silvestre 2007 11 of a solution of the above problem

poincare s reversibility condition we consider a real planar analytic vector field x such that the origin o is a centre for the linearization of x poincare s condition of reversibility with respect to a line passing through o is then a sufficient condition for o to be a centre for the vector field x we provide necessary and sufficient conditions involving the vanishing of certain polynomials in the coefficients in the expansion of x for reversibility we also show that if the linearization l x of the divergence of x is non trivial then the only possible reversibility line is given by l x 0 in such cases this provides the basis for a simple test of reversibility we examine the consequences of our various tests for quadratic and cubic vector fields all non hamiltonian cases are discussed when l x 0 in cubic systems it is possible for the reversibility line if it exists to be unique but it is also possible for there to be two such lines these possibilities are characterized algebraically and a prescription is provided for determining the reversibility line s in each case

augmented formulations for solving maxwell equations we consider augmented variational formulations for solving the static or time harmonic maxwell equations for that a term is added to the usual h curl conforming formulations it consists of a weighted l2 scalar product between the divergence of the em and the divergence of test fields in this respect the methods we present are h curl div conforming we also build mixed augmented variational formulations with either one or two lagrange multipliers to dualize the equation on the divergence and when applicable the relation on the tangential or normal trace of the field it is proven that one can derive formulations which are equivalent to the original static or time harmonic maxwell equations in the latter case spurious modes are automatically excluded numerical analysis and experiments will be presented in the forthcoming paper augmented formulations for solving maxwell equations numerical analysis and experiments in preparation

existence of the mild solution for fractional semilinear initial value problems abstract in this paper we will study the existence and uniqueness of mild solution for the semilinear initial value problem of non integer order u t a u t f t u t g u t s u t where 0 1 and f t u t g u t s u t is a given function

the existence of mild solutions for impulsive fractional partial differential equations abstract this paper is concerned with the existence of mild solutions for a class of impulsive fractional partial semilinear differential equations some errors in mophou 2010 2 are corrected and some previous results are generalized

algorithmic derivation of centre conditions the conditions for a critical point of a polynomial differential system to be a centre are of particular significance because of the frequency with which they are required in applications we demonstrate how computer algebra can be effectively employed in the search for necessary and sufficient conditions for critical points of such systems to be centres we survey recent developments and illustrate our approach by means of examples

a singular field method for maxwell s equations numerical aspects for 2d magnetostatics the present paper deals with the solution of maxwell type problems by means of nodal h1 conforming finite elements in a nonconvex piecewise regular domain surrounded by a perfect conductor such a discretization cannot in general approximate the singular behavior of the electromagnetic field near reentrant corners or edges the singular field method consists of adding to the finite element discretization space some particular fields which take into account the singular behavior the latter are deduced from the singular functions associated with the scalar laplace operator the theoretical justification of this approach as well as the analysis of the convergence of the approximation are presented for a very simple model problem arising from magnetostatics in a translation invariant setting but the study can be easily extended to numerous maxwell type problems the numerical implementation of both variants is studied for a domain containing a single reentrant corner

fractional cauchy problems with almost sectorial operators this paper concerns the abstract cauchy problem of fractional evolution equations with almost sectorial operators the suitable mild solutions of evolution equations with riemann liouville derivative and caputo derivative are introduced respectively the existence theorems of mild solutions of cauchy problems are established the results obtained here improve and generalize some known results finally an example is given for demonstration

on recent developments in the theory of abstract differential equations with fractional derivatives abstract this note is motivated from some recent papers treating the problem of the existence of a solution for abstract differential equations with fractional derivatives we show that the existence results in agarwal et al 2009 1 belmekki and benchohra 2010 2 darwish et al 2009 3 hu et al 2009 4 mophou and n guerekata 2009 6 7 mophou 2010 8 9 muslim 2009 10 pandey et al 2009 11 rashid and el qaderi 2009 12 and tai and wang 2009 13 are incorrect since the considered variation of constant formulas is not appropriate in this note we also consider a different approach to treat a general class of abstract fractional differential equations

a finite volume method for the approximation of maxwell s equations in two space dimensions on arbitrary meshes a new finite volume method is presented for discretizing the two dimensional maxwell equations this method may be seen as an extension of the covolume type methods to arbitrary possibly non conforming or even non convex n sided polygonal meshes thanks to an appropriate choice of degrees of freedom an equivalent formulation of the scheme is given in terms of discrete differential operators obeying discrete duality principles the main properties of the scheme are its energy conservation its stability under a cfl like condition and the fact that it preserves gauss law and divergence free magnetic fields second order convergence is demonstrated numerically on non conforming and distorted meshes

fully discrete finite element scheme for maxwell s equations with non linear boundary condition abstract we study a full maxwell s system accompanied with a non linear degenerate boundary condition which represents a generalization of the classical silver muller condition for a non perfect conductor the relationship between the normal components of electric e and magnetic h field obeys the following power law h e 1 e for some 0 1 we establish the existence and uniqueness of a weak solution in a suitable function spaces under the minimal regularity assumptions on the boundary and the initial data e 0 and h 0 we design a non linear time discrete approximation scheme and prove convergence of the approximations to a weak solution we also derive the error estimates for the time discretization as a next step we study the fully discrete problem using curl conforming edge elements and derive the corresponding error estimates finally we present some numerical experiments

multiplicity of limit cycles and analytic m solutions for planar differential systems abstract this work deals with limit cycles of real planar analytic vector fields it is well known that given any limit cycle of an analytic vector field it always exists a real analytic function f 0 x y defined in a neighborhood of and such that is contained in its zero level set in this work we introduce the notion of f 0 x y being an m solution which is a merely analytic concept our main result is that a limit cycle is of multiplicity m if and only if f 0 x y is an m solution of the vector field we apply it to study in some examples the stability and the bifurcation of periodic orbits from some non hyperbolic limit cycles

existence and uniqueness of mild solutions to impulsive fractional differential equations this paper is concerned with the existence and uniqueness of a mild solution to impulsive fractional semilinear differential equations the results are obtained by means of fixed point methods we also give an example of such problems

a p adaptive method for electromagnetic wave propagation summary the discontinuous galerkin dg finite element method is used for the numerical solution of the three dimensional maxwell equations control of errors in the numerical level for the divergence free constraint of the magnetic field can be obtained through the use of divergence free vector bases in this work the so called perfectly hyperbolic ph formulation of the maxwell equations is employed to retain both divergence free magnetic field and in the presence of charges to satisfy the gauss constraint for the electric field at the numerical level for both approaches it is found that higher order approximations have favorable effect on the preservation of the divergence constraints and that the ph formulations retains these errors to a lower level it is shown that high order accuracy in space and time is achieved in unstructured meshes using implicit time marching for nonuniform meshes local resolution refinement is employed using p type adaptivity to ensure accurate electromagnetic wave propagation thus the potential of the method to reach the required higher resolution in anisotropic meshes and obtain accurate electromagnetic wave propagation with reduced computational effort is demonstrated this article is protected by copyright all rights reserved

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 method of constructing cycles without contact around a weak focus has a weak focus then limit cycles can be made to appear in its neighborhood by varying slightly the right sides 11 frommer 2 bautin 3 chin et al 4 and tung 5 are examples of excellent expositions of poincare s idea it is necessary to go from numerical examples to general theory in the first part of this paper we present a method of constructing cycles without contact around a weak focus which makes clear poincart s idea used in the above work 2 51 in the second part another method to analyze global phase portraits is used quadratic systems with at least four limit cycles are produced by means of both the above methods this contradicts the results of petrovskii and landis 6

generation of stable limit cycles in controllable linear systems in this paper we prove that any controllable linear systems x ax bu x n u m admits a polynomial feedback u u x such that the closed loop system x ax bu x admits an orbitally asymptotically stable limit cycle moreover we prove that for any positive integer n there exists an nth order polynomial autonomous ordinary differential equation with a unique limit cycle

separatrix and limit cycles of quadratic systems and dulac s theorem separatrix cycles for a planar quadratic vector field are studied the results obtained are used to show that in any bounded region of the plane a quadratic vector field has at most a finite number of limit cycles

phase portraits for quadratic systems with a higher order singularity with two zero eigenvalues abstract this paper presents a classification of all possible phase portraits of quadratic systems of differential equations with a singular point with two zero eigenvalues the singular points include the fourth order saddle node the third order point having an elliptic and a hyperbolic sector the third order saddle point and the second order cusp point

abstract fractional cauchy problems with almost sectorial operators abstract of concern are the cauchy problems for linear and semilinear time fractional evolution equations involving in the linear part a linear operator a whose resolvent satisfies the estimate of growth 1 0 in a sector of the complex plane which occurs when one considers for instance the partial differential operators in the limit domain of dumb bell with a thin handle or in the space of holder continuous functions by constructing a pair of families of operators in terms of the generalized mittag leffler type functions and the resolvent operators associated with a for the first time and a deep analysis on the properties for these families we obtain the existence and uniqueness of mild solutions and classical solutions to the cauchy problems moreover we present three examples to illustrate the feasibility of our results

time dependent maxwell s equations with charges in singular geometries this paper is devoted to the solution of the instationary maxwell equations with charges the geometry of the domain can be singular in the sense that its boundary can include reentrant corners or edges the difficulties arise from the fact that those geometrical singularities generate in their neighborhood strong electromagnetic fields the time dependency of the divergence of the electric field is addressed to tackle this problem some new theoretical and practical results are presented on curl free singular fields and on singular fields with l2 non vanishing divergence the method which allows to compute the instationary electromagnetic field is based on a splitting of the spaces of solutions into a two term direct sum first the subspace of regular fields it coincides with the whole space of solutions provided that the domain is either convex or with a smooth boundary second a singular subspace defined and characterized via the singularities of the laplace operator several numerical examples are presented to illustrate the mathematical framework this paper is the generalization of the singular complement method

three dimensional perfectly matched layer for the absorption of electromagnetic waves the perfectly matched layer is a technique of free space simulation developed for solving unbounded electromagnetic problems with the finite difference time domain method referred to as pml this technique has been described in a previous paper for two dimensional problems the present paper is devoted to the three dimensional case the theory of the perfectly matched layer is generalized to three dimensions and some numerical experiments are shown to illustrate the efficiency of this new method of free space simulation

three limit cycles in a leslie gower predator prey model with additive allee effect in this work a bidimensional continuous time differential equations system is analyzed which is derived of leslie type predator prey schemes by considering a nonmonotonic functional response and allee effect on population prey for the system obtained we describe the bifurcation diagram of limit cycles that appears in the first quadrant the only quadrant of interest for the sake of realism we show that under certain conditions over the parameters the system allows the existence of three limit cycles the first two cycles are infinitesimal ones generated by hopf bifurcation the third one arises from a homoclinic bifurcation furthermore we give conditions over the parameters such that the model allows long term extinction or survival of both populations in particular the presence of a weak allee effect does not imply extinction of populations necessarily for our model

energy conserved splitting spectral methods for two dimensional maxwell s equations in this paper the energy conserved splitting legendre galerkin method and energy conserved splitting legendre collocation method for maxwell s equations in two dimensions are proposed the schemes are energy conserved unconditionally stable and can be implemented efficiently the both methods are of second order convergence in time the high order error estimates in space for the two methods are also obtained for the ec s lc method an identity of the discrete divergence of electric fields is also obtained numerical examples are provided to confirm the theoretical analysis

on the number of algebraically independent poincare liapunov constants abstract in this paper an upper bound for the number of algebraically independent poincare liapunov constants in a certain basis for planar polynomial differential systems is given finally it is conjectured that an upper bound for the number of functionally independent poincare liapunov quantities would be m 2 3 m 7 where m is the degree of the polynomial differential system moreover the computational problems which appear in the computation of the poincare liapunov constants and in the determination of the center cases are also discussed

semilinear fractional integro differential equations with compact semigroup abstract in this paper we study the local and global existence of mild solutions to a class of fractional integro differential equations in an arbitrary banach space associated with operators generating compact semigroup on the banach space

weighted regularization of maxwell equations in polyhedral domains we present a new method of regularizing time harmonic maxwell equations by a bf grad div term adapted to the geometry of the domain this method applies to polygonal domains in two dimensions as well as to polyhedral domains in three dimensions in the presence of reentrant corners or edges the usual regularization is known to produce wrong solutions due the non density of smooth fields in the variational space we get rid of this undesirable effect by the introduction of special weights inside the divergence integral standard finite elements can then be used for the approximation of the solution this method proves to be numerically efficient

numerical approximation of the maxwell equations in inhomogeneous media by ap1conforming finite element method the aim of this paper is to present an extension of thep1conforming finite element method for the time dependent maxwell equations that has been previously exposed we shall consider inhomogeneous media with the piecewise constant dielectric and magnetic parameters and precisely the interface of two such media by analogy with the idea developed we propose a method based on the dualization and then the approximation of the interface conditions in a way consistent with the one derived for the fields and the lagrange multipliers of the divergence constraints inside the domain

limit cycles of polynomial differential systems with homogeneous nonlinearities of degree 4 via the averaging method by using the averaging method we study the limit cycles for a class of quartic polynomial differential systems as well as their global shape in the plane more specifically we analyze the global shape of limit cycles bifurcating from a hopf bifurcation and also from periodic orbits with linear center x y y x the perturbation of these systems is made inside the class of quartic polynomial differential systems without quadratic and cubic terms

the basins of attraction in a modified may holling tanner predator prey model with allee affect abstract i analyse a modified may holling tanner predator prey model considering an allee effect in the prey and alternative food sources for predator additionally the predation functional response or predation consumption rate is linear the extended model exhibits rich dynamics and we prove the existence of separatrices in the phase plane separating basins of attraction related to oscillation co existence and extinction of the predator prey population we also show the existence of a homoclinic curve that degenerates to form a limit cycle and discuss numerous potential bifurcations such as saddle node hopf and bogdanov takens bifurcations we use simulations to illustrate the behaviour of the model

higher order limit cycle bifurcations from non degenerate centers abstract the computational problems which appear in the computation of the poincare liapunov constants and the determination of their functionally independent number has led in recent works to consider only the lowest terms of these constants in this work we improve the results obtained in this direction for polynomials systems of the form x y p n x y y x q n x y where p n and q n are a homogeneous polynomial of degree n we use center bifurcation to estimate the cyclicity of a unique singular point of focus center type for different values of n and compare with the results given by the conjecture presented in 15

linearizability and local bifurcation of critical periods in a cubic kolmogorov system since chicone and jacobs investigated local bifurcation of critical periods for quadratic systems and newtonian systems in 1989 great attention has been paid to some particular forms of cubic systems having special practical significance but less difficulties in computation this paper is devoted to the linearizability and local bifurcation of critical periods for a cubic kolmogorov system we use the darboux method to give explicit linearizing transformations for isochronous centers investigating the finite generation for the ideal of all period constants which are of the polynomial form in six parameters we prove that at most two critical periods can be bifurcated from the interior equilibrium if it is an isochronous center moreover we prove that the maximum number of critical periods is reachable

numerical solution to the time dependent maxwell equations in two dimensional singular domains in this paper we present a method to solve numerically the time dependent maxwell equations in nonsmooth and nonconvex domains indeed the solution is not of regularity h1 in space in general moreover the space of h1 regular fields is not dense in the space of solutions thus an h1 conforming finite element method can fail even with mesh refinement the situation is different than in the case of the laplace problem or of the lame system for which mesh refinement or the addition of conforming singular functions work to cope with this difficulty the singular complement method is introduced this method consists of adding some well chosen test functions these functions are derived from the singular solutions of the laplace problem also the scm preserves the interesting features of the original method easiness of implementation low memory requirements small cost in terms of the cpu time to ascertain its validity some concrete problems are solved numerically