on the number of limit cycles for a class of discontinuous quadratic differential systems abstract the present paper is devoted to the study of the maximum number of limit cycles bifurcated from the periodic orbits of the quadratic isochronous center x y 16 3 x 2 4 3 y 2 y x 8 3 x y by the averaging method of first order when it is perturbed inside a class of discontinuous quadratic polynomial differential systems the chebyshev criterion is used to show that this maximum number is 5 and can be realizable in some sense the result and that in paper 6 also answer the questions left in the paper 9

homoclinic loop and multiple limit cycle bifurcation surfaces this paper establishes the existence and analyticity of homoclinic loop bifurcation surfaces z and multiplicity two limit cycle bifurcation surfaces f for planar systems depending on two or more parameters it determines the side of or f on which limit cycles occur and it shows that if x and intersect then typically they do so at a flat contact

the cyclicity of period annuli of a class of quadratic reversible systems with two centers abstract quadratic perturbations of a class of quadratic reversible systems are studied the associated systems can have at most four limit cycles in any compact region of the period annuli and if four limit cycles exist then their distribution is 3 1 in other words three of them are nested and surrounding one focus the other limit cycle is surrounding the different focus

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

invariant algebraic curves and conditions for a centre conditions for the existence of a centre in two dimensional systems are considered along the lines of darboux we show how these methods can be used in the search for maximal numbers of bifurcating limit cycles we also extend the method to include more degenerate cases such as are encountered in less generic systems these lead to new classes of integrals in particular the kukles system is considered and new centre conditions for this system are obtained

linearizability conditions for lotka volterra planar complex quartic systems having homogeneous nonlinearities in this paper we investigate the linearizability problem for the two dimensional lotka volterra complex quartic systems which are linear systems perturbed by fourth degree homogeneous polynomials i e we consider systems of the form email protected x 1 a 3 0x 3 a 2 1x 2y a 1 2xy 2 a 0 3y 3 email protected y 1 b 3 0x 3 b 2 1x 2y b 1 2xy 2 b 0 3y 3 the necessary and sufficient conditions for the linearizability of this system are found from them the conditions for isochronicity of the corresponding real system can be derived

bifurcations and distribution of limit cycles which appear from two nests of periodic orbits abstract in this paper we study the distribution and simultaneous bifurcation of limit cycles bifurcated from the two periodic annuli of the holomorphic differential equation z i z z 3 after a small polynomial perturbation we first show that under small perturbations of the form e p 2 m 1 z z where p 2 m 1 z z is a polynomial of degree 2 m 1 in which the power of z is odd and the power of z is even the only possible distribution of limit cycles is u u for all values of u 0 1 2 m 3 hence the sharp upper bound for the number of limit cycles bifurcated from each two period annuli of z i z z 3 is m 3 for m 4 then we consider a perturbation of the form e p m z z where p m z z is a polynomial of degree m in which the power of z is odd and obtain the upper bound m 5 for m 6 moreover we show that the distribution u v of limit cycles is possible for 0 u m 5 0 v m 5 with u v m 2 and m 9

limit cycles of cubic polynomial vector fields via the averaging theory abstract in this paper we study the maximum number of limit cycles that can bifurcate from the period annulus surrounding the origin of a class of cubic polynomial differential systems using the averaging method more precisely we prove that the perturbations of the period annulus of the center located at the origin of the cubic polynomial differential system x y f x y y x f x y where f x y 0 is a conic such that f 0 0 0 by arbitrary cubic polynomial differential systems provide at least six limit cycles bifurcating from the periodic orbits of the period annulus using only the first order averaging method

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

bifurcation of limit cycles from quadratic isochrones abstract for a one parameter family of plane quadratic vector fields x e depending analytically on a small real parameter e we determine the number and position of the local families of limit cycles which emerge from the periodic trajectories surrounding an isochronous or linearizable center techniques are developed for treating the bifurcations of all orders and these are applied to prove the following results for the linear isochrone the maximum number of continuous families of limit cycles which can emerge is three for one class of nonlinear isochrones at most one continuous family of limit cycles can emerge whereas for all other nonlinear isochrones at most two continuous families of limit cycles can emerge moreover for each isochrone in one of these classes there are small perturbations such that the indicated maximum number of continuous families of limit cycles can be made to emerge from a corresponding number of arbitrarily prescribed periodic orbits within the period annulus of the isochronous center

on stability of interval matrices new sufficient and sometimes necessary and sufficient conditions are obtained for schur and hurwitz stability of interval matrices by relying on the concept of connective stability and m matrices the necessity part is broadened to include interval matrices with mixed signs of the off diagonal elements provided the sign patterns follow that of the morishima matrix the obtained results are extended to cover convex combinations of interval matrices

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

higher order bifurcations of limit cycles abstract this paper shows that asymmetrically perturbed symmetric hamiltonian systems of the formx y y x x3 1y 2x2 3xy 4x2y with analytic j e o e have at most two limit cycles that bifurcate for smalle 0 from any period annulus of the unperturbed system this fact agrees with previous results of petrov dangelmayr and guckenheimer and chicone and iliev but shows that the result of three limit cycles for the asymmetrically perturbed exterior duffing oscillator recently obtained by jebrane and oladek is incorrect the proofs follow by deriving an explicit formula for thekth order melnikov function mk h and using a picard fuchs analysis to show that in each case mk h has at most two zeros moreover the method developed in this paper for determining the higher order melnikov functions also applies to more general perturbations of these systems

the hopf cyclicity of the centers of a class of quintic polynomial vector fields abstract we consider families of planar polynomial vector fields having a singularity with purely imaginary eigenvalues for which a basis of its bautin ideal b is known we provide an algorithm for computing an upper bound of the hopf cyclicity less than or equal to the bautin depth of b we also present a method for studying the cyclicity problem for the hamiltonian and the time reversible centers without the necessity of solving previously the dulac complex center problem associated to the larger complexified family as application we analyze the hopf cyclicity of the quintic polynomial family written in complex notation as z i z z z a z 3 b z 2 z c z z 2 d z 3

limit cycles for discontinuous quadratic differential systems with two zones abstract in this paper we study the maximum number of limit cycles given by the averaging theory of first order for discontinuous differential systems which can bifurcate from the periodic orbits of the quadratic isochronous centers x y x 2 y x x y and x y x 2 y 2 y x 2 x y when they are perturbed inside the class of all discontinuous quadratic polynomial differential systems with the straight line of discontinuity y 0 comparing the obtained results for the discontinuous with the results for the continuous quadratic polynomial differential systems this work shows that the discontinuous systems have at least 3 more limit cycles surrounding the origin than the continuous ones

periodic orbits from second order perturbation via rational trigonometric integrals abstract the second order poincare pontryagin melnikov perturbation theory is used in this paper to study the number of bifurcated periodic orbits from certain centers this approach also allows us to give the shape and the period up to the first order we address these problems for some classes of abel differential equations and quadratic isochronous vector fields in the plane we prove that two is the maximum number of hyperbolic periodic orbits bifurcating from the isochronous quadratic centers with a birational linearization under quadratic perturbations of second order in particular the configurations 2 0 and 1 1 are realizable when two centers are perturbed simultaneously the required computations show that all the considered families share the same iterated rational trigonometric integrals

an extension of the kreiss matrix theorem a new condition is shown to be equivalent to the other conditions of the kreiss matrix theorem for power bounded families of matrices this new condition is important for the application of the theory of pseudodifference operators to stability estimates for variable coefficient finite difference equations as an example of the usefulness of this new condition it is used to prove stability of the leapfrog scheme for hyperbolic equations with variable coefficients

on the number of critical points of the period in the study of subharmonic bifurcations from a planar hamiltonian system it is often assumed that the period of a one parameter family of periodic orbits of the unperturbed system is a monotone function of the parameter see e g hale and taboas s chow and hale es however this hypothesis is in many instances difficult to verify for example it is well known that every orbit in the first quadrant of the vollterra lotka equation is periodic it is only in recent years that the monotonicity of the periods of these periodic orbits has been shown hsu 9 and waldvogel 151 such problems also occur in the study of bifurcation of steady state solutions of a reaction diffusion equation in one space variable the steady state solutions satisfy the equation u f u o uer 1 1 on some interval in 14 smoller and wasserman showed that if f u u u u u cc

inhomogeneous fuchs equations and the limit cycles in a class of near integrable quadratic systems we study the bifurcation of limit cycles in general quadratic perturbations of the particular quadratic system which represents one of two codimension five components in the intersection of two strata in the centre manifold and q 4 the study of limit cycles for this degenerate case requires us to investigate not the first but the second variation m 2 of the displacement function we prove that up to three limit cycles can emerge from the period annulus surrounding the centre this implies that the cyclicity of period annuli of nearby systems in and q 4 is at most three as well our approach relies upon the possibility of deriving appropriate picard fuchs equations satisfied by the four independent integrals included in m 2

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

stability theory of difference approximations for multidimensional initial boundary value problems an attempt is made to generalize results for arbitrary multidimensional difference schemes severe obstacles are encountered however for dissipative difference schemes the problem may be resolved

on the number of limit cycles of a perturbed cubic polynomial differential center abstract for e sufficiently small we consider the planar polynomial differential system x y y 2 a x 2 b x c e p x y y x y 2 a x 2 b x c e q x y where p x y and q x y are polynomials of degree n by using the averaging method of first order we bound the number of limit cycles that can bifurcate from period annulus of the unperturbed system

bounding the number of limit cycles of discontinuous differential systems by using picard fuchs equations abstract in this paper by using picard fuchs equations and chebyshev criterion we study the upper bounds of the number of limit cycles given by the first order melnikov function for discontinuous differential systems which can bifurcate from the periodic orbits of quadratic reversible centers of genus one r19 x y 12 x 2 16 y 2 y x 16 x y and r20 x y 4 x 2 y x 16 x y and the periodic orbits of the quadratic isochronous centers s 1 x y x 2 y 2 y x 2 x y and s 2 x y x 2 y x x y the systems r19 and r20 are perturbed inside the class of polynomial differential systems of degree n and the system s 1 and s 2 are perturbed inside the class of quadratic polynomial differential systems the discontinuity is the line y 0 it is proved that the upper bounds of the number of limit cycles for systems r19 and r20 are respectively 4 n 3 n 4 and 4 n 3 n 3 counting the multiplicity and the maximum numbers of limit cycles bifurcating from the period annuluses of the isochronous centers s 1 and s 2 are exactly 5 and 6 counting the multiplicity on each period annulus respectively

limit cycles for generalized kukles polynomial differential systems abstract we study the limit cycles of generalized kukles polynomial differential systems using averaging theory of first and second order

a codimension two bifurcation with a third order picard fuchs equation abstract using a third order picard fuchs equation we show that a certain two parameter family of planar vectorfields for parameter values in a certain cone has a unique limit cycle which is born from a hopf bifurcation and dies in a saddle connection this removes a superfluous hypothesis in theorem 3 2 chapter 13 of s n chow and j k hale methods of bifurcation theory springer verlag new york 1982

linear estimation of the number of zeros of abelian integrals for some cubic isochronous centers abstract this paper consists of two parts in the first part we study the relationship between conic centers all orbits near a singular point of center type are conics and isochronous centers of polynomial systems in the second part we study the number of limit cycles that bifurcate from the periodic orbits of cubic reversible isochronous centers having all their orbits formed by conics when we perturb such systems inside the class of all polynomial systems of degree n

limit cycle bifurcations by perturbing piecewise smooth hamiltonian systems with multiple parameters abstract this paper is concerned with the problem of limit cycle bifurcation for piecewise smooth near hamiltonian systems with multiple parameters by the first melnikov function some novel criteria have been established for the existence of multiple limit cycles furthermore an example is included to validate the obtained results by considering the maximum number of limit cycles for a piecewise quadratic system studied in llibre and mereu 2014 12 compared with the result in the above reference one more limit cycle is found by our method

limit cycles by perturbing quadratic isochronous centers inside piecewise polynomial differential systems abstract in this paper we consider the quadratic isochronous centers perturbed inside piecewise polynomial differential systems of arbitrary degree n with the straight line of discontinuity x 0 the main concerns are the number of zeros of the first order melnikov functions and the estimate of the number of limit cycles bifurcating from the period annuli for quadratic isochronous centers s 1 s 2 and s 3 we will provide a sharp upper bound for the number of zeros of the first order melnikov functions for quadratic isochronous center s 4 we give a rough estimate however when the problem is reduced to perturbations inside polynomial differential systems our result for s 4 will improve that in li et al 2000 12 significantly moreover we will reveal out some equivalence between the first order melnikov function method and the first order averaging method for investigating the number of limit cycles of piecewise polynomial differential systems

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

new lower bound for the hilbert number in piecewise quadratic differential systems abstract we study the number of limit cycles bifurcating from a piecewise quadratic system all the differential systems considered are piecewise in two zones separated by a straight line we prove the existence of 16 crossing limit cycles in this class of systems if we denote by h p n the extension of the hilbert number to degree n piecewise polynomial differential systems then h p 2 16 as fas as we are concerned this is the best lower bound for the quadratic class moreover in the studied cases all limit cycles appear nested bifurcating from a period annulus of a isochronous quadratic center

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 center and cyclicity problems for quartic linear like reversible systems abstract in this paper we study a family of quartic linear like reversible polynomial systems having a nondegenerate center at the origin this family has degree one with respect to one of the variables we are interested in systems in this class having two extra nondegenerate centers outside the straight line of symmetry the geometrical configuration of these centers is aligned or triangular we solve the center problem in both situations and in the second case we study the limit cycles obtained from a simultaneous degenerate hopf bifurcation in the quartic polynomials class

simultaneous bifurcation of limit cycles from two nests of periodic orbits abstract let z f z be an holomorphic differential equation having a center at p and consider the following perturbation z f z e r z z we give an integral expression similar to an abelian integral whose zeroes control the limit cycles that bifurcate from the periodic orbits of the period annulus of p this expression is given in terms of the linearizing map of z f z at p the result is applied to control the simultaneous bifurcation of limit cycles from the two period annuli of z i z z 2 after a polynomial perturbation

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

from finite differences to finite elements a short history of numerical analysis of partial differential equations abstract this is an account of the history of numerical analysis of partial differential equations starting with the 1928 paper of courant friedrichs and lewy and proceeding with the development of first finite difference and then finite element methods the emphasis is on mathematical aspects such as stability and convergence analysis