bifurcation analysis of a generalized gause model with prey harvesting and a generalized holling response function of type iii abstract in this paper we study a generalized gause model with prey harvesting and a generalized holling response function of type iii p x m x 2 a x 2 b x 1 the goal of our study is to give the bifurcation diagram of the model for this we need to study saddle node bifurcations hopf bifurcation of codimension 1 and 2 heteroclinic bifurcation and nilpotent saddle bifurcation of codimension 2 and 3 the nilpotent saddle of codimension 3 is the organizing center for the bifurcation diagram the hopf bifurcation is studied by means of a generalized lienard system and for b 0 we discuss the potential integrability of the system the nilpotent point of multiplicity 3 occurs with an invariant line and can have a codimension up to 4 but because it occurs with an invariant line the effective highest codimension is 3 we develop normal forms in which the invariant line is preserved for studying of the nilpotent saddle bifurcation for b 0 the reversibility of the nilpotent saddle is discussed we study the type of the heteroclinic loop and its cyclicity the phase portraits of the bifurcations diagram partially conjectured via the results obtained allow us to give a biological interpretation of the behavior of the two species

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

center cyclicity of lorenz chen and lu systems abstract this work provides upper bounds on the cyclicity of the centers on center manifolds in the well known lorenz family and also in the chen and lu families we prove that at most one limit cycle can be made to bifurcate from any center of any element of these families perturbing within the respective family with the exception of one specific lorenz system where the cyclicity increases we also show that this bound is sharp

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

computation of lyapunov values for two planar polynomial differential systems the complex formulas of computing the lyapunov values are investigated for two planar polynomial systems with a pair of pure imaginary eigenvalues the relations between the two systems and their standard forms are discussed and the transformation formulas are given the corresponding complex formulas of computing the lyapunov values are obtained the results obtained here can be applied to judging the type of singular points and the number of limit cycles in planar polynomial systems

bifurcation of limit cycles in a cubic order planar system around a nilpotent critical point abstract in this paper bifurcation of limit cycles is considered for planar cubic order systems with an isolated nilpotent critical point normal form theory is applied to compute the generalized lyapunov constants and to prove the existence of at least 9 small amplitude limit cycles in the neighborhood of the nilpotent critical point in addition the method of double bifurcation of nilpotent focus is used to show that such systems can have 10 small amplitude limit cycles near the nilpotent critical point these are new lower bounds on the number of limit cycles in planar cubic order systems near an isolated nilpotent critical point moreover a set of center conditions is obtained for such cubic systems

counting real zeros of analytic functions satisfying linear ordinary differential equations abstract we suggest an explicit procedure to establish upper bounds for the number of real zeros of analytic functions satisfying linear ordinary differential equations with meromorphic coefficients if the equation formula has no singular points in a small neighborhooduof a real segmentk all the coefficientsaj t have absolute value aonuanda0 t 1 then any solution of this equation may have no more than a zeros onk where u k is a geometric constant depending only onkandu if the principal coefficienta0 t is nonconstant but its modulus is at leasta 0 somewhere onk then the number of real zeros onkof any solution analytic inu does not exceed a a with some u k

generating limit cycles from a nilpotent critical point via normal forms abstract it is well known that the normal form theory can be applied to solve the center focus problem for monodromic planar nilpotent singularities in this paper we see how this theory can also be applied to generate limit cycles from this type of singularities

a d c optimization algorithm for solving the trust region subproblem this paper is devoted to difference of convex functions d c optimization d c duality local and global optimality conditions in d c programming the d c algorithm dca and its application to solving the trust region problem the dca is an iterative method that is quite different from well known related algorithms thanks to the particular structure of the trust region problem the dca is very simple requiring only matrix vector products and in practice converges to the global solution the inexpensive implicitly restarted lanczos method of sorensen is used to check the optimality of solutions provided by the dca when a nonglobal solution is found a simple numerical procedure is introduced both to find a feasible point having a smaller objective value and to restart the dca at this point it is shown that in the nonconvex case the dca converges to the global solution of the trust region problem using only matrix vector products and requiring at most 2m 2 restarts where m is the number of distinct negative eigenvalues of the coefficient matrix that defines the problem numerical simulations establish the robustness and efficiency of the dca compared to standard related methods especially for large scale problems

parallelization of the lyapunov constants and cyclicity for centers of planar polynomial vector fields christopher in 2006 proved that under some assumptions the linear parts of the lyapunov constants with respect to the parameters give the cyclicity of an elementary center this paper is devote to establish a new approach namely parallelization to compute the linear parts of the lyapunov constants more concretely it is showed that parallelization computes these linear parts in a shorter quantity of time than other traditional mechanisms to show the power of this approach we study the cyclicity of the holomorphic center iz z 2 z 3 z n under general polynomial perturbations of degree n for n 13 we also exhibit that from the point of view of computation among the hamiltonian time reversible and darboux centers the holomorphic center is the best candidate to obtain high cyclicity examples of any degree for n 4 5 13 we prove that the cyclicity of the holomorphic center is at least n 2 n 2 this result give the highest lower bound for m 6 m 7 m 13 among the existing results where m n is the maximum number of limit cycles bifurcating from an elementary monodromic singularity of polynomial systems of degree n as a direct corollary we also obtain the highest lower bound for the hilbert numbers h 6 40 h 8 70 and h 10 108 because until now the best result was h 6 39 h 8 67 and h 10 100

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

the infinitesimal 16th hilbert problem in the quadratic case let h x y be a real cubic polynomial with four distinct critical values in a complex domain and let x h h y frac partial partial x h x frac partial partial y be the corresponding hamiltonian vector field we show that there is a neighborhood of x h in the space of all quadratic plane vector fields such that any x has at most two limit cycles

restricted independence in displacement function for better estimation of cyclicity abstract since the independence of focal values is a sufficient condition to give a number of limit cycles arising from a center focus equilibrium in this paper we consider a restricted independence to a parametric curve which gives a method not only to increase the lower bound for the cyclicity of the center focus equilibrium but also to be available when those focal values are not independent we apply this method to a nondegenerate cubic center focus variety and prove that the cyclicity reaches its an upper bound

bifurcations of certain family of planar vector fields tangent to axes abstract the topological versality of the family v x x 1 x y y y 2 ax by cy 2 of vector fields on the plane is proven the asymptotic formulas of the bifurcational curves are given the proofs rely on an analysis of the limit cycles and the non abelian integrals along irrational curves

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

nine limit cycles around a singular point by perturbing a cubic hamiltonian system with a nilpotent center in this paper we study bifurcation of limit cycles in planar cubic near hamiltonian systems with a nilpotent center we use normal form theory to compute the generalized lyapunov constants and show that there exist at least 9 limit cycles around the nilpotent center this is a new lower bound on the number of limit cycles in planar cubic near hamiltonian systems with a nilpotent center

limit cycle bifurcations from a non degenerate center in this work we discuss the computational problems which appear in the computation of the poincare liapunov constants and the determination of their functionally independent number moreover we calculate the minimum number of bautin ideal generators which give the number of small limit cycles under certain hypothesis about the generators in particular we consider polynomial systems of the form x y pn x y y x qn x y where pn and qn are a homogeneous polynomial of degree n we use center bifurcation rather than multiple hopf bifurcations used a previous work 19 to estimate the cyclicity of a unique singular point of focus center type for n 4 5 6 7 and compare with the results given by the conjecture presented in 18

the transverse dynamics of a holomorphic flow on considere un champ vectoriel polynomial homogene p z dans c n 1 les solutions du systeme d equations differentielles dz i dt p i z t c i 0 n feuillette c n 1 par des courbes holomorphes ayant des singularites le long du lieu commun p i z 0 on definit la notion de dynamique transverse du feuilletage

random periodic solutions of random dynamical systems abstract in this paper we give the definition of the random periodic solutions of random dynamical systems we prove the existence of such periodic solutions for a c 1 perfect cocycle on a cylinder using a random invariant set the lyapunov exponents and the pullback of the cocycle

a study of the limit cycles associated with a generalized codimension 3 lienard oscillator in this paper melnikov and petrov methods are used to study the limit cycles associated with the degenerate bifurcations of a representative generalized codimension 3 lienard oscillator it is shown that the coexistence of multiple limit cycles is possible and that jumping phenomena exist between limit cycles moreover it is found that the stable limit cycles oscillations and the unstable limit cycles oscillations can occur alternatively

bifurcation of 2 periodic orbits from non hyperbolic fixed points we introduce the concept of 2 cyclicity for families of one dimensional maps with a non hyperbolic fixed point by analogy to the cyclicity for families of planar vector fields with a weak focus this new concept is useful in order to study the number of 2 periodic orbits that can bifurcate from the fixed point as an application we study the 2 cyclicity of some natural families of polynomial maps

stability of linear quadratic minimization over euclidean balls stability properties of the problem of minimizing a nonconvex linear quadratic function over a euclidean ball known as the trust region subproblem are studied in this paper we investigate in detail the case where the linear part of the objective function is perturbed and obtain necessary and sufficient conditions for the upper lower semicontinuity of the karush kuhn tucker kkt point set map and the global solution map explicit formulas for computing the directional derivative and the frechet derivative of the optimal value function stability of the karush kuhn tucker point set under the perturbation of the quadratic form is also studied

a class of linear generalized equations solution stability of a class of linear generalized equations in finite dimensional euclidean spaces is investigated by means of generalized differentiation exact formulas for the frechet and the mordukhovich coderivatives of the normal cone mappings of perturbed euclidean balls are obtained necessary and sufficient conditions for the local lipschitz like property of the solution maps of such linear generalized equations are derived from these coderivative formulas since the trust region subproblems in nonlinear programming can be regarded as linear generalized equations these conditions lead to new results on stability of the parametric trust region subproblems

limit cycles in slow fast codimension 3 saddle and elliptic bifurcations abstract this paper deals with local bifurcations occurring near singular points of planar slow fast systems in particular it is concerned with the study of the slow fast variant of the unfolding of a codimension 3 nilpotent singularity the slow fast variant of a codimension 1 hopf bifurcation has been studied extensively before and its study has lead to the notion of canard cycles in the van der pol system similarly codimension 2 slow fast bogdanov takens bifurcations have been characterized here the singularity is of codimension 3 and we distinguish slow fast elliptic and slow fast saddle bifurcations we focus our study on the appearance on small amplitude limit cycles and rely on techniques from geometric singular perturbation theory and blow up

a system with three limit cycles appearing in a hopf bifurcation and dying in a homoclinic bifurcation the cusp of order 4 abstract in this paper we give an example of a family of polynomial vector fields with three limit cycles appearing simultaneously on a hopf bifurcation h of order 3 and vanishing simultaneously in a homoclinic loop bifurcation hl of order 3 the region with three limit cycles is a topological 3 simplex the system is a generalization of bogdanov s system at the same time we give the bifurcation diagram for the universal unfolding of the cusp of order 4 this bifurcation diagram is a cone it contains a cone on the bifurcation diagram with the three limit cycles inside a 3 simplex region plus saddle node and cusp bifurcations of lower order

liouvillian first integrals of differential equations liouvillian functions are functions that are built up from rational functions using exponentiation integration and algebraic functions we show that if a system of differential equations has a generic solution that satisfies a liouvillian relation that is there is a liouvillian function of several variables vanishing on the curve defined by this solution then the system has a liouvillian first integral that is a nonconstant liouvillian function that is constant on solution curves in some nonempty open set we can refine this result in special cases to show that the first integral must be of a very special form

polynomial systems a lower bound for the hilbert numbers let h n be the maximum possible number of limit cycles of systems dot x p x y dot y q x y where p and q are polynomials of degree at most n we are concerned with the rate of growth of h n as n increases it is known that h n geq kn 2 for some constant k in this paper we show that h n grows at least as rapidly as n 2 log n

on local behavior of holomorphic functions along complex submanifolds of ℂ n in this paper we establish some general results on local behavior of holomorphic functions along complex submanifolds of n as a corollary we present multi dimensional generalizations of an important result of coman and poletsky on bernstein type inequalities on transcendental curves in 2

on some properties of quadratic programs with a convex quadratic constraint in this paper we consider the problem of minimizing a possibly nonconvex quadratic function with a quadratic constraint we point out some new properties of the problem in particular in the first part of the paper we show that i given a kkt point that is not a global minimizer it is easy to find a better feasible point ii strict complementarity holds at the local nonglobal minimizer in the second part of this paper we show that the original constrained problem is equivalent to the unconstrained minimization of a piecewise quartic merit function using the unconstrained formulation we give in the nonconvex case a new second order necessary condition for global minimizers in the third part of this paper algorithmic applications of the preceding results are briefly outlined and some preliminary numerical experiments are reported

center conditions at infinity for abel differential equations an abel differential equation y p x y 2 q x y 3 is said to have a center at a set a a 1 a r of complex numbers if y a 1 y a 2 y a r for any solution y x with the initial value y a 1 small enough the polynomials p q are said to satisfy the polynomial composition condition on a if there exist polynomials p q and w such that p p and q q are representable as p x p w x q x q w x and w a 1 w a 2 w a r we show that for wide ranges of degrees of p and q restricted only by certain assumptions on the common divisors of these degrees the composition condition provides a very accurate approximation of the center one up to a finite number of configurations not accounted for to our best knowledge this is the first general i e not restricted to small degrees of p and q or to a very special form of these polynomials result in the center problem for abel equations as an important intermediate result we show that at infinity according to an appropriate projectivization of the parameter space the center conditions are given by a system of the moment equations of the form as a1 p k q 0 s 2 r k 0 1

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

quadratic systems with an integrable saddle a complete classification in the coefficient space r 12 a quadratic polynomial differential system can be identified with a single point of r12 through the coefficients using the algebraic invariant theory we classify all the quadratic polynomial differential systems of r12 having an integrable saddle we show that there are only 47 topologically different phase portraits in the poincar e disc associated to this family of quadratic systems up to a reversal of the sense of their orbits moreover each one of these 47 representatives is determined by a set of affine invariant conditions

local minimizers of quadratic functions on euclidean balls and spheres in this paper a characterization of the local nonglobal minimizes of a quadratic function defined on a euclidean ball or a sphere is given it is proven that there exists at most one local nonglobal minimizes and that the lagrange multiplier that corresponds to this minimizes is the largest solution of a nonlinear scalar equation an algorithm is proposed for computing the local nonglobal minimizes

local bifurcations and a survey of bounded quadratic systems abstract this paper presents a survey of the known results for bounded quadratic systems as well as a study of the local bifurcations that occur at critical points of such systems it is shown that the only finite codimension bifurcations that occur at a critical point of a bounded quadratic system are the saddle node and the hopf takens bifurcations of codimensions 1 and 2 and the bogdanov takens bifurcations of codimensions 2 and 3 furthermore it is shown that whenever a bounded quadratic system has one of these critical points then a full generic unfolding of the critical point exists in the class of bounded quadratic systems finally we give a complete list of those limit periodic sets whose finite cyclicity still needs to be established in order to obtain the existence of a finite upper bound for the number of limit cycles that can occur in a bounded quadratic system

compact kahler manifolds homotopic to negatively curved riemannian manifolds in this paper we show that any compact kahler manifold homotopic to a compact riemannian manifold with negative sectional curvature admits a kahler einstein metric of general type moreover we prove that on a compact symplectic manifold x homotopic to a compact riemannian manifold with negative sectional curvature for any almost complex structure j compatible with the symplectic form there is no non constant j holomorphic entire curve f mathbb c rightarrow x

coderivatives of a karush kuhn tucker point set map and applications abstract the problem of minimizing a linear quadratic function over the euclidean ball is encountered frequently in the theory of trust region methods in nonlinear programming by some tools from variational analysis we investigate the stability of the karush kuhn tucker point set map of that problem with respect to total perturbations of its data verifiable sufficient conditions for the local lipschitz like property of the map are obtained and the connection of our results with the existing criteria for the lower semicontinuity of this karush kuhn tucker point set map is shown

minimum number of ideal generators for a linear center perturbed by homogeneous polynomials abstract using the algorithm presented in j gine x santallusia on the poincare liapunov constants and the poincare series appl math warsaw 28 1 2001 17 30 the poincare liapunov constants are calculated for polynomial systems of the form x y p n x y y x q n x y where p n and q n are homogeneous polynomials of degree n the objective of this work is to calculate the minimum number of ideal generators i e the number of functionally independent poincare liapunov constants through the study of the highest fine focus order for n 4 and n 5 and compare it with the results that give the conjecture presented in j gine on the number of algebraically independent poincare liapunov constants appl math comput 188 2 2007 1870 1877 moreover the computational problems which appear in the computation of the poincare liapunov constants and the determination of the number of functionally independent ones are also discussed

algebraic particular integrals integrability and the problem of the center in this work we clarify the global geometrical phenomena corresponding to the notion of center for plane quadratic vector fields we first show the key role played by the algebraic particular integrals of degrees less than or equal to three in the theory of the center these curves control the changes in the systems as parameters vary the bifurcation diagram used to prove this result is realized in the natural topological space for the situation considered namely the real four dimensional projective space next we consider the known four algebraic conditions for the center for quadratic vector fields one of them says that the system is hamiltonian a condition which has a clear geometric meaning we determine the geometric meaning of the remaining other three algebraic conditions i ii iii

the number and stability of limit cycles for planar piecewise linear systems of node saddle type abstract the objective of this paper is to study the number and stability of limit cycles for planar piecewise linear pwl systems of node saddle type with two linear regions firstly we give a thorough analysis of limit cycles for lienard pwl systems of this type proving one is the maximum number of limit cycles and obtaining necessary and sufficient conditions for the existence and stability of a unique limit cycle these conditions can be easily verified directly according to the parameters in the systems and play an important role in giving birth to two limit cycles for general pwl systems in this step the tool of a bendixon like theorem is successfully employed to derive the existence of a limit cycle secondly making use of the results gained in the first step we obtain parameter regions where the general pwl systems have at least one at least two and no limit cycles respectively in addition for the general pwl systems some sufficient conditions are presented for the existence and stability of a unique one and exactly two limit cycles respectively finally some numerical examples are given to illustrate the results and especially to show the existence and stability of two nested limit cycles

entropie geometrique des feuilletages on definit et on etudie l entropie geometrique d un feuilletage generalisant directement l entropie topologique d un flot

a note on euler number of locally conformally kahler manifolds let m 2n be a compact riemannian manifold of non positive resp negative sectional curvature we call m j theta a d bounded locally conformally kahler manifold if the lifted lee form tilde theta on the universal covering space of m is d bounded we show that if m 2n is homeomorphic to a d bounded lck manifold then its euler number satisfies the inequality 1 n chi m 2n ge resp 0

uniqueness of limit cycles for sewing planar piecewise linear systems abstract this paper proves the uniqueness of limit cycles for sewing planar piecewise linear systems with two zones separated by a straight line and only one singularity of monodromic type the proofs are based in an extension of rolle s theorem for dynamical systems on the plane

new lower bounds for the hilbert number of polynomial systems of lienard type abstract in this paper we study the maximal number of limit cycles of some kinds of polynomial lienard systems with arbitrary degree and obtain some new lower bounds for the hilbert number of the systems which improve truly the certain existing results

on euler characteristic and fundamental groups of compact manifolds let m be a compact riemannian manifold pi widetilde m rightarrow m be the universal covering and omega be a smooth 2 form on m with pi omega cohomologous to zero suppose the fundamental group pi 1 m satisfies certain radial quadratic resp linear isoperimetric inequality we show that there exists a smooth 1 form eta on widetilde m of linear resp bounded growth such that pi omega d eta as applications we prove that on a compact kahler manifold m omega with pi omega cohomologous to zero if pi 1 m is mathrm cat 0 or automatic resp hyperbolic then m is kahler non elliptic resp kahler hyperbolic and the euler characteristic 1 frac dim mathbb r m 2 chi m ge 0 resp 0

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

a criticality result for polycycles in a family of quadratic reversible centers abstract we consider the family of dehomogenized loud s centers x y x 1 x x d x 2 f y 2 y where d f r 2 and we study the number of critical periodic orbits that emerge or disappear from the polycycle at the boundary of the period annulus this number is defined exactly the same way as the well known notion of cyclicity of a limit periodic set and we call it criticality the previous results on the issue for the family x r 2 distinguish between parameters with criticality equal to zero regular parameters and those with criticality greater than zero bifurcation parameters a challenging problem not tackled so far is the computation of the criticality of the bifurcation parameters which form a set b of codimension 1 in r 2 in the present paper we succeed in proving that a subset of b has criticality equal to one

on the structure of poincare lyapunov constants for the weak focus of polynomial vector fields in this paper generalized poincare lyapunov constants vi i 1 2 are defined and an explicit formula is given for the lyapunov functions it is shown that vi is determined modulo v1 vi 1 this is used for the investigation and creation of limit cycles

saddle quantities and applications abstract in this paper we make the connection between the theoretical study of the generalized homoclinic loop bifurcation ghb and the practical computational aspects for this purpose we first compare the dulac normal form with the joyal normal form these forms were both used to prove the ghb theorem but the second one is far more practical from the algorithmic point of view we then show that the information carried by these normal forms can be computed in a much simpler way using what we shall call dual lyapunov constants the coefficients of a normal form or the dual lyapunov quantities are particular cases of what we shall call saddle quantities we calculate the saddle quantities for quadratic systems and we show that no more than three limit cycles can appear in a homoclinic loop bifurcation we also study the homoclinic loop bifurcation of order 5 appearing in a 6 parameter family close to a hamiltonian system to our knowledge this is the first time that one can find a complete description of a ghb of such high order finally we calculate the saddle quantities for a symmetric cubic vector field and we deduce a bound for the number of limit cycles that appear in a ghb

computer assisted techniques for the verification of the chebyshev property of abelian integrals abstract we develop techniques for the verification of the chebyshev property of abelian integrals these techniques are a combination of theoretical results analysis of asymptotic behavior of wronskians and rigorous computations based on interval arithmetic we apply this approach to tackle a conjecture formulated by dumortier and roussarie in f dumortier r roussarie birth of canard cycles discrete contin dyn syst 2 2009 723 781 which we are able to prove for q 2

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

lower bounds for the hilbert number of polynomial systems let h m denote the maximal number of limit cycles of polynomial systems of degree m it is called the hilbert number the main part of hilbert s 16th problem posed in 1900 is to find its value the problem is still open even for m 2 however there have been many interesting results on the lower bound of it for m 2 in this paper we give some new lower bounds of this number the results obtained in this paper improve all existing results for all m 7 based on some known results for m 3 4 5 6 in particular we obtain that h m grows at least as rapidly as 12ln2 m 2 2ln m 2 for all large m

genericity conditions for finite cyclicity of elementary graphics abstract in this paper we explore the khovanskii method for proving the finite cyclicity of elementary graphics and how it can be applied in practice the genericity conditions needed in that case form a proper subset of the usual methods moreover some of the conditions are non intrinsic and can be artificially created by action of the automorphism groups preserving the normal forms near the singularities and their action on regular transitions hence we introduce an extension of the method which treats the usual functional pfaffian systems together with the admissible changes of coordinates in the functional equations

centennial history of hilbert s 16th problem the second part of hilbert s 16th problem deals with polynomial differential equations in the plane it remains unsolved even for quadratic polynomials there were several attempts to solve it that failed yet the problem inspired significant progress in the geometric theory of planar differential equations as well as bifurcation theory normal forms foliations and some topics in algebraic geometry the dramatic history of the problem as well as related developments are presented below 1 the problem and its counterparts what may be said about the number and location of limit cycles of a planar polynomial vector field of degree n the limit cycle is an isolated closed orbit of a vector field this second part of hilbert s 16th problem appears to be one of the most persistent in the famous hilbert list h second only to the riemann function conjecture traditionally hilbert s question is split into three each one requiring a stronger answer problem 1 is it true that a planar polynomial vector field has but a finite number of limit cycles problem 2 is it true that the number of limit cycles of a planar polynomial vector field is bounded by a constant depending on the degree of the polynomials only the bound on the number of limit cycles in problem 2 is denoted by h n and known as the hilbert number linear vector fields have no limit cycles hence h 1 0 it is still unknown whether or not h 2 exists problem 3 give an upper bound for h n a solution to any of these problems implies a solution for the previous ones only the first problem is solved now the positive answer was established in e92 i91 there are analytic counterparts of problems 1 and 2 received by the editors december 2001 2000 mathematics subject classification primary 34cxx 34mxx 37f75

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

characterization of the finite weak singularities of quadratic systems via invariant theory abstract this article is about weak singularities of quadratic differential systems that is non degenerate singular points with traces of the corresponding linearized systems at such points equal to zero these could be foci centers or saddles necessary and sufficient conditions for a real quadratic system to possess a fixed number of weak singularities of a specific order are given the conditions are stated in terms of affine invariant polynomials in the 12 dimensional space of the coefficients

hilbert s 16th problem for quadratic vector fields abstract we discuss here a systematic approach towards a positive answer to hilbert s 16th problem for quadratic systems namely the existence of a uniform bound for the number of limit cycles of a quadratic system the method is the following describe the limit periodic sets surrounding the origin in a family of quadratic vector fields and prove that they have finite cyclicity in this paper we give the list of all graphics and degenerate graphics that should be considered and describe their general features we also indicate how to find or where to find concrete examples of these limit periodic sets