on hopf cyclicity of planar systems we investigate the maximal number of limit cycles which appear under perturbations in hopf bifurcations by using the first order melnikov function with multiple parameters

quadratic systems with center and their perturbations abstract we study the quadratic systems on a plane z i z az2 b z 2 cz2 z x iy we give a simple algebraic proof of the center conditions 0 and b 0 or a 1 2 b 1 or a b 1 c c or a 2 b 1 c 1 and of the theorem of bautin that the number of small amplitude limit cycles bifurcating from a center or a focus is not greater than 3 we present the bifurcational diagrams and phase portraits of the systems with center and describe the cyclicity precisely in each case here one mistake in bautin s work is revealed and some questions concerning focus numbers are answered we study also global not small limit cycles for a small perturbation of the center case with two invariant lines the lotka volterra system using abelian integrals we show that the number of zeroes of the corresponding abelian integral is 0 1 or 2 some attention is devoted to the problem of limit cycles for a small perturbation of a system with a two fold center when two center conditions hold the problem turns out to be equivalent to the problem of zeroes of some integral with polynomial coefficients

perturbations of symmetric elliptic hamiltonians of degree four in this paper four parameter unfoldings x of symmetric elliptic hamiltonians of degree four are studied we prove that in a compact region of the period annulus of x0 the displacement function of x is sign equivalent to its principal part which is given by a family induced by a chebychev system and we describe the bifurcation diagram of x in a full neighborhood of the origin in the parameter space where at most two limit cycles can exist for the corresponding systems

alien limit cycles in lienard equations this paper aims at providing an example of a family of polynomial lienard equations exhibiting an alien limit cycle this limit cycle is perturbed from a 2 saddle cycle in the boundary of an annulus of periodic orbits given by a hamiltonian vector field the hamiltonian represents a truncated pendulum of degree 4 in comparison to a former polynomial example not only the equations are simpler but a lot of tedious calculations can be avoided making the example also interesting with respect to simplicity in treatment

the period function of reversible quadratic centers abstract in this paper we investigate the bifurcation diagram of the period function associated to a family of reversible quadratic centers namely the dehomogenized loud s systems the local bifurcation diagram of the period function at the center is fully understood using the results of chicone and jacobs c chicone m jacobs bifurcation of critical periods for plane vector fields trans amer math soc 312 1989 433 486 most of the present paper deals with the local bifurcation diagram at the polycycle that bounds the period annulus of the center the techniques that we use here are different from the ones in c chicone m jacobs bifurcation of critical periods for plane vector fields trans amer math soc 312 1989 433 486 because while the period function extends analytically at the center it has no smooth extension to the polycycle at best one can hope that it has some asymptotic expansion another major difficulty is that the asymptotic development has to be uniform with respect to the parameters in order to prove that a parameter is not a bifurcation value we study also the bifurcations in the interior of the period annulus and we show that there exist three germs of curves in the parameter space that correspond to this type of bifurcation moreover we determine some regions in the parameter space for which the corresponding period function has at least one or two critical periods finally we propose a complete conjectural bifurcation diagram of the period function of the dehomogenized loud s systems our results can also be viewed as a contribution to the proof of chicone s conjecture c chicone review in mathscinet ref 94h 58072

finite cyclicity of graphics with a nilpotent singularity of saddle or elliptic type abstract in this paper we prove finite cyclicity of several of the most generic graphics through a nilpotent point of saddle or elliptic type of codimension 3 inside c families of planar vector fields in some cases our results are independent of the exact codimension of the point and depend only on the fact that the nilpotent point has multiplicity 3 by blowing up the family of vector fields we obtain all the limit periodic sets we calculate two different types of dulac maps in the blown up family and develop a general method to prove that some regular transition maps have a nonzero higher derivative at a point the finite cyclicity theorems are derived by a generalized derivation division method introduced by roussarie

limit cycles bifurcated from a class of quadratic reversible center of genus one abstract we investigate the bifurcation of limit cycles in a class of planar quadratic reversible non hamiltonian systems this systems has a center of genus one the exact upper bound of the number of limit cycles is given

limit cycle bifurcations by perturbing a cuspidal loop in a hamiltonian system abstract in this paper we first study the analytical property of the first melnikov function for general hamiltonian systems exhibiting a cuspidal loop and obtain its expansion at the hamiltonian value corresponding to the loop then by using the first coefficients of the expansion we give some conditions for the perturbed system to have 4 5 or 6 limit cycles in a neighborhood of loop as an application of our main results we consider some polynomial lienard systems and find 4 5 or 6 limit cycles

a chebyshev criterion for abelian integrals we present a criterion that provides an easy sufficient condition in order for a collection of abelian integrals to have the chebyshev property this condition involves the functions in the integrand of the abelian integrals and can be checked in many cases in a purely algebraic way by using this criterion several known results are obtained in a shorter way and some new results which could not be tackled by the known standard methods can also be deduced

abelian integrals and limit cycles abstract the paper deals with generic perturbations from a hamiltonian planar vector field and more precisely with the number and bifurcation pattern of the limit cycles in this paper we show that near a 2 saddle cycle the number of limit cycles produced in unfoldings with one unbroken connection can exceed the number of zeros of the related abelian integral even if the latter represents a stable elementary catastrophe we however also show that in general finite codimension of the abelian integral leads to a finite upper bound on the local cyclicity in the treatment we introduce the notion of simple asymptotic scale deformation

the loop quantities and bifurcations of homoclinic loops abstract the stability and bifurcations of a homoclinic loop for planar vector fields are closely related to the limit cycles for a homoclinic loop of a given planar vector field a sequence of quantities the homoclinic loop quantities were defined to study the stability and bifurcations of the loop among the sequence of the loop quantities the first nonzero one determines the stability of the homoclinic loop there are formulas for the first three and the fifth loop quantities in this paper we will establish the formula for the fourth loop quantity for both the single and double homoclinic loops as applications we present examples of planar polynomial vector fields which can have five or twelve limit cycles respectively in the case of a single or double homoclinic loop by using the method of stability switching

limit cycle bifurcations near a 2 polycycle or double 2 polycycle of planar systems abstract in this paper we study the bifurcation problem of limit cycles for a near hamiltonian system near a 2 polycycle or double 2 polycycle we obtain new results on the number of limit cycles by using the method of stability changing of a homoclinic loop this method also provides a new way to find alien limit cycles especially we found some alien limit cycles of new type

a square root bound on the minimum weight in quasi cyclic codes we establish a square root bound on the minimum weight in the quasi cyclic binary codes constructed by bhargava tavares and shiva the proof rests on viewing the codes as ideals in a group algebra over gf 4 theorem 6 answers a question raised by f j macwilliams and n j a sloane in em the theory of error correcting codes theorems 3 4 and 5 provide information about the way the nonzero entries of a codeword of minimum weight are distributed among the coordinate positions

codimension 2 bifurcation of twisted double homoclinic loops a local active coordinates approach is employed to obtain bifurcation equations of twisted double homoclinic loops under the condition of one twisted orbit we obtain the existence and uniqueness and of the 1 1 double homoclinic loop 2 1 double homoclinic loop 2 1 right homoclinic loop 1 1 large homoclinic loop 2 1 large homoclinic loop and 2 1 large period orbit for the case of double twisted orbits we obtain the existence or non existence of 1 1 double homoclinic loop 1 2 double homoclinic loop 2 1 double homoclinic loop 2 2 double homoclinic loop 2 1 large homoclinic loop 1 2 large homoclinic loop 2 2 large homoclinic loop 2 2 right homoclinic loop 2 2 large homoclinic loop 2 2 left homoclinic loop and 2 2 large period orbit moreover the bifurcation surfaces and their existence regions are given besides bifurcation sets are presented on the 2 dimensional subspace spanned by the first two melnikov vectors

fourth power residue double circulant self dual codes quadratic residue codes are a well known class of codes in this paper we consider the constructions of self dual codes by higher power residues especially fourth power residues new infinite families of self dual codes over gf 2 gf 3 gf 4 gf 8 and gf 9 are introduced some of them have better minimum weight than previously known codes we also give general results related to the automorphism group of some of these codes

the monotonicity of period function for codimension four quadratic system q4 abstract in this paper we study the codimension four quadratic system q 4 z iz 4 z 2 2 z 2 b ic z 2 where b and c are real constants i 2 1 z x iy b ic 2 it is proved that the period function of periodic trajectories for q 4 is monotone the content of this paper can be viewed as a contribution to the proof of chicone s conjecture cf mr 94h 58072

limit cycle bifurcations by perturbing a quadratic integrable system with a triangle abstract finding cyclicity of homoclinic or heteroclinic loops in quadratic systems is an open problem which is related to the second part of hilbert s 16th problem this paper mainly considers the problem of limit cycle bifurcation for quadratic polynomial systems near a triangle it is proved that the cyclicity of the triangle in quadratic systems is at least three

on the transformation formulas of theta series here s is a nondegenerate rational symmetric matrix of signature q s s 3 x iy is a variable in the siegel half space hn of degree n u is a variable in the space cn of all complex n x q matrices runs over the set qq of all rational q x n matrices a is a c valued function on qn that is a finite linear combination of characteristic functions of cosets of qq modulo a z lattice a is a real matrix such that s ta diag lq 1 a and e c exp 2iric for c e c if q 1 s a 1 and a is suitably chosen the series g becomes a classical theta function of jacobi riemann if s is positive definite n 1 u 0 and q is even g is an elliptic modular form of weight q 2 studied by hecke also siegel introduced the series with u 0 in his theory of quadratic forms in si now g satisfies a transformation formula of the following type for every

the cyclicity of period annuli for a class of cubic hamiltonian systems with nilpotent singular points abstract this paper deals with the limit cycles of a class of cubic hamiltonian systems under polynomial perturbations we suppose that the corresponding hamiltonian system which has at least one center has finite singular points and is symmetrical with respect to both x axis and y axis and also the origin is a nilpotent singular point hence the hamiltonian h x y can be written as h x y x 2 a x 4 b x 2 y 2 c y 4 a b c r 3 c 0 for the above h x y the corresponding vector field is x 2 y b x 2 2 c y 2 y 2 x 1 2 a x 2 b y 2 we first obtain that the above vector fields with at least one center can be divided into 8 classes by its topological phase portraits for the following perturbed system x 2 y b x 2 2 c y 2 e f x y y 2 x 1 2 a x 2 b y 2 e g x y where 0 e 1 f x y and g x y are polynomials in x y with degree n we give an estimation of the number of isolated zeros of the corresponding abelian integral the number of limit cycles follows from poincare pontryagin theorem

existence of at most 1 2 or 3 zeros of a melnikov function and limit cycles abstract we investigate the existence of at most one two or three limit cycles bifurcated from a periodic annulus of a hamiltonian system under a class of perturbations and obtain some sufficient conditions which ensure that the corresponding melnikov function has at most one two or three zeros in an open interval we also give applications to some systems which appear in codimension two bifurcations and to some lienard systems

asymptotics and stability of the delayed duffing equation abstract we examine how insertion of a delay in a system of ordinary differential equations can change the asymptotic dynamics and stabilize the structure of the system under further change

hopf takens bifurcations and centres the paper deals with hopf takens bifurcations both in generic families and in families containing centres attention goes to the relation between normal forms lyapunov coefficients and the bautin ideal in case the bautin ideal has only one generator we pay attention to the first non zero melnikov function

motives and siegel modular forms for a motive m over q we define the fundamental periods of m using invariant theory our definition generalizes deligne s periods we show that if a motive m is constructed from motives m 1 m 2 m n by a standard algebraic operation then the fundamental periods of m can be expressed as monomials of the fundamental periods of m 1 m 2 m n applying this theory we discuss two hypothetical motives attached to a siegel modular form we show that a siegel modular form of degree m has at most m 1 period invariants

double circulant quadratic residue codes we give a lower bound for the minimum distance of double circulant binary quadratic residue codes for primes p spl equiv spl plusmn 3 mod8 this bound improves on the square root bound obtained by calderbank and beenker using a completely different technique the key to our estimates is to apply a result by helleseth to which we give a new and shorter proof combining this result with the weil bound leads to the improvement of the calderbank and beenker bound for large primes p their bound is of order spl radic 2p while our new improved bound is of order 2 spl radic p the results can be extended to any prime power q and the modifications of the proofs are briefly indicated

an explicit formula for siegel series combining induction formulas for local densities with a functional equation for the siegel series we give an explicit formula for the siegel series by this formula we also give an explicit formula for the fourier coefficients of the siegel eisenstein series

bifurcation of limit cycles for a quartic near hamiltonian system by perturbing a nilpotent center hopf bifurcation which produces oscillations is a very important phenomena in the theory and application of dynamical systems almost all works available about hopf bifurcations are related to a non degenerate focus or center for the case of a degenerate focus or center the study of the bifurcations becomes challenge in this paper we consider the bifurcation of limit cycles for a quartic near hamiltonian system by perturbing a nilpotent center we take coefficients as parameters then we can get six limit cycles

two dimensional fuchsian systems and the chebyshev property let x t y t be a solution of a fuchsian system of order two with three singular points the vector space of functions of the form p t x t q t y t where p q are real polynomials has a natural filtration of vector spaces according to the asymptotic behavior of the functions at infinity we describe a two parameter class of fuchsian systems for which the corresponding vector spaces obey the chebyshev property the maximal number of isolated zeros of each function is less than the dimension of the vector space up to now only a few particular systems were known to possess such a non oscillation property it is remarkable that most of these systems are of the type studied in the present paper we apply our results in estimating the number of limit cycles that appear after small polynomial perturbations of several quadratic or cubic hamiltonian systems in the plane

perturbed rigidly isochronous centers and their critical periods abstract this paper investigates the bifurcation of critical periods from a cubic rigidly isochronous center under any small polynomial perturbations of degree n it proves that for n 3 4 and 5 there are at most 2 and 4 critical periods induced by periodic orbits of the unperturbed cubic system respectively and in each case this upper bound is sharp moreover for any n 5 there are at most n 1 2 critical periods induced by periodic orbits of the unperturbed cubic system an example is given to show that the upper bound in the case of n 11 can be reached

application of circulant matrices to the construction and decoding of linear codes the fourier transform technique is used to analyze and construct several families of double circulant codes the minimum distance of the resulting codes is lower bounded by 2 square root r and can be decoded easily employing the standard bch decoding algorithm or the majority logic decoder of reed muller codes a decoding procedure for reed solomon codes is presented based on a representation of the parity check matrix by circulant blocks the decoding procedure inherits both the relatively low time complexity of the berlekamp massey algorithm and the hardware simplicity characteristic of blahut s algorithm the procedure makes use of the encoding circuit together with a reduced version of blahut s decoder

zeta functions and eisenstein series on classical groups we construct an euler product from the hecke eigenvalues of an automorphic form on a classical group and prove its analytic continuation to the whole complex plane when the group is a unitary group over a cm field and the eigenform is holomorphic we also prove analytic continuation of an eisenstein series on another unitary group containing the group just mentioned defined with such an eigenform as an application of our methods we prove an explicit class number formula for a totally definite hermitian form over a cm field

a remark on the period of the periodic solution in the lotka volterra system abstract an important consideration in the nonlinear predator prey problem of lotka volterra type is the determination of the period this paper gives a general expression for the period in terms of the given parameters in the lotka volterra system we also discuss the qualitative behavior of the period related to the energy level of the lotka volterra system

the cyclicity and period function of a class of quadratic reversible lotka volterra system of genus one abstract in this paper we investigate a class of quadratic reversible lotka volterra system of genus one z i z z 2 b z 2 with b 3 5 it is proved that the cyclicity of the period annulus under quadratic perturbations is equal to two moreover we prove that the period function of its period trajectories is monotone increasing

quaternary quadratic residue codes and unimodular lattices we construct new self dual and isodual codes over the integers module 4 the binary images of these codes under the gray map are nonlinear but formally self dual the construction involves hensel lifting of binary cyclic codes quaternary quadratic residue codes are obtained by hensel lifting of the classical binary quadratic residue codes repeated hensel lifting produces a universal code defined over the 2 adic integers we investigate the connections between this universal code and the codes defined over z sub 4 the composition of the automorphism group and the structure of idempotents over z sub 4 we also derive a square root bound on the minimum lee weight and explore the connections with the finite fourier transform certain self dual codes over z sub 4 are shown to determine even unimodular lattices including the extended quadratic residue code of length q 1 where q spl equiv 1 mod8 is a prime power when q 23 the quaternary golay code determines the leech lattice in this way this is perhaps the simplest construction for this remarkable lattice that is known

bifurcations of limit cycles in a cubic system with cubic perturbations this paper is concerned with limit cycles on two different cubic systems with nine singular points eleven limit cycles are found and the distributions are studied by using the methods of bifurcation theory and qualitative analysis

zeta functions and eisenstein series on metaplectic groups the main objectives of this paper are 1 to construct an euler product from a hecke eigenform of half integral weight 2 to prove its meromorphic continuation to the whole plane 3 to examine its relationship with the fourier coefficients of the form and 4 to apply these results to the investigations of eisenstein series of half integral weight the automorphic forms are holomorphic ones considered on the metaplectic cover m of g where g n sp n f with a totally real algebraic number field f the first three are generalizations of the corresponding results in the elliptic and hilbert modular cases obtained in 2 and 6 at the same time all four closely parallel similar results in the case of integral weight in 9 and s10 to describe our results let us take for simplicity f to be q in which case the weight k is half an odd positive integer we consider a holomorphic form f of weight k with respect to a principal congruence subgroup of sp n z of level n which is an eigenform in the sense that f it m 2 m f with a complex number 2 m where n is a multiple of 4 and t m is a hecke operator defined for each positive integer m in a natural way as our first main result we shall prove theorem 4 4 that

new configurations of 24 limit cycles in a quintic system this paper concerns with the number and distributions of limit cycles in a z 3 equivariant quintic planar polynomial system 24 limit cycles are found in this system and two different configurations of them are shown by combining the methods of double homoclinic loops bifurcation poincare bifurcation and qualitative analysis the two configurations of 24 limit cycles obtained in this paper are new the results obtained are useful to the study of weakened 16th hilbert problem

cyclicity 1 and 2 conditions for a 2 polycycle of integrable systems on the plane abstract we obtain generic conditions for a 2 polycycle of an integrable system to have cyclicity one and two respectively under multiple parameter perturbations

unfolding of a quadratic integrable system with two centers and two unbounded heteroclinic loops abstract in this paper we present a complete study of quadratic 3 parameter unfoldings of some integrable system belonging to the classqr3 and having two centers and two unbounded heteroclinic loops we restrict to unfoldings that are transverse toqr3 obtain a versal bifurcation diagram and all global phase portraits including the precise number and configuration of the limit cycles it is proved that 3 is the maximal number of limit cycles surrounding a single focus and only the 1 1 configuration can occur in case of simultaneous nests of limit cycles essentially the proof relies on a careful analysis of a related non conservative abelian integral

on the stability of double homoclinic and heteroclinic cycles in this paper we give a criterion for the stability of planar double homoclinic and heteroclinic cycles with one or two saddles in some degenerate case

normal linearization and transition map near a saddle connection with symmetric resonances abstract we consider a heteroclinic connection in a planar system between two symmetric hyperbolic saddles of which the eigenvalues are resonant starting with a c normal form defined globally near the connection we normally linearize the vector field by using finitely smooth tags of logarithmic form we furthermore define an asymptotic entry exit relation and we discuss on two examples how to deal with counting limit cycles near a limit periodic set involving such a connection

the cyclicity of the period annulus of the quadratic hamiltonian systems with non morsean point abstract in this paper we investigate the quadratic hamiltonian systems with non morsean point it is proved that in this situation the cyclicity of the period annulus under quadratic perturbations is equal to two

confluent hypergeometric functions on tube domains our problem specialized to the case of the siegel upper half space h of degree m concerns the fourier expansion of a series s z fl det z a det a a l here z is a variable on h l is a lattice in the space v of all real symmetric matrices of size m and fl ec it can be shown that this is convergent if re f l m and also that if re m 2 and r e f l m 2 it has a fourier expansion of the form

limit cycle bifurcations by perturbing a class of integrable systems with a polycycle abstract in this paper we deal with the problem of limit cycle bifurcation near a 2 polycycle or 3 polycycle for a class of integrable systems by using the first order melnikov function we first get the formal expansion of the melnikov function corresponding to the heteroclinic loop and then give some computational formulas for the first coefficients of the expansion based on the coefficients we obtain a lower bound for the maximal number of limit cycles near the polycycle as an application of our main results we consider quadratic integrable polynomial systems obtaining at least two limit cycles

cyclicity of infinite contour around certain reversible quadratic center abstract the paper deals with quadratic vector fields on the poincaire sphere with center of the reversible kind x y 1 2x y x ay2 bx2 in this case there is a 2 cycle with hyperbolic vertices at infinity we show that the cyclicity of this 2 cycle is 2 or 3 the last possibility takes place along some curve in the a b plane

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

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

the period in the lotka volterra system is monotonic abstract periodic solutions in a class of hamiltonian systems with one degree of freedom containing the lotka volterra system are considered their period is represented as an integral appropriate for numerical evaluation as well as for theoretical investigations the monotonicity of the period in the lotka volterra system follows as a simple consequence

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

quadratic centers defining elliptic surfaces abstract let x be a quadratic vector field with a center whose generic orbits are algebraic curves of genus one to each x we associate an elliptic surface a smooth complex compact surface which is a genus one fibration we give the list of all such vector fields and determine the corresponding elliptic surfaces

e factor of representations of classical groups abstract we let g be the isometry group of a nondegenerate symmetric or skew symmetric form h on a space v over some local or global field k i e g u v h we let lg be the l group of go the identity component of g and r be the natural representation of this group we develop here a theory of l functions and e factors attached to this r in particular we develop a theory of local and global zeta integrals for such g we use very special intertwining operators therein we require an analysis of the proper normalization of such operators with such techniques we determine the correct e factors associated to local representations the issue of the correct local l factor is discussed

the cyclicity of quadratic reversible systems with a center of genus one and non morsean point in this paper we investigate a quadratic reversible system with non morsean point it is proved that the cyclicity of the period annulus under quadratic perturbations is less or equal to four

degenerate and non trivial hyperbolic polycycles with two vertices abstract we study some hyperbolic polycycles with two vertices which are degenerate and non trivial and such that the maximal cyclicity in the sufficiently differentiable unfolding depends only on the vector field germ along the polycycle in these cases the maximal cyclicity and versal unfolding of the polycycle are described

on the number of limit cycles of a z4 equivariant quintic polynomial system abstract in this paper we study the number of limit cycles of a near hamiltonian system under z 4 equivariant quintic perturbations using the methods of hopf and heteroclinic bifurcation theory we found that the perturbed system can have 13 limit cycles