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

upper bounds of limit cycles in abel differential equations with invariant curves abstract new criteria are established for upper bounds on the number of limit cycles of periodic abel differential equations having two periodic invariant curves one of them bounded the criteria are applied to obtain upper bounds of either zero or one limit cycle for planar differential systems

new global exponential stability criteria for nonlinear delay differential systems with applications to bam neural networks we consider a nonlinear non autonomous system with time varying delaysx i t a i t x i h i t j 1mf i jt x j g i j t i 1 mwhich has a large number of applications in the theory of artificial neural networks via the m matrix method easily verifiable sufficient stability conditions for the nonlinear system and its linear version are obtained application of the main theorem requires just to check whether a matrix which is explicitly constructed using the system s parameters is an m matrix comparison with the tests obtained by gopalsamy 2007 and liu 2013 for bam neural networks illustrates novelty of the stability theorems some open problems conclude the paper

critical properties of electron eigenstates in incommensurate systems this paper describes some properties of the eigenvalue equation n 1 n 1 2 cos 2 n n e n this is an example of the more general problem of a hermitian eigenvalue equation in the form of a difference equation with periodic coefficients these equations arise in solid state physics they occur in connection with tight binding models for electrons in one dimensional solids with an incommensurate modulation of the structure and in models for the energy bands of bloch electrons moving in a plane with a perpendicular magnetic field the model studied has a critical point when 1 following some earlier work by azbel azbel m ya phys rev lett 43 1954 1979 an approximate renormalization group transformation is derived this predicts that the spectrum and eigenstates have a remarkable recursive structure at the critical point which is dependent on the expansion of as a continued fraction also when is an irrational number there is a localization transition from extended states to localized states as increases through the critical point this localization transition which was previously discovered by aubry andre aubry s andre g ann israel phys soc 3 133 1979 using the thouless formula for the localization length is explained by the renormalization group transformation derived here

periodic solutions of linear riccati and abel dynamic equations abstract we study the number of periodic solutions of linear riccati and abel dynamic equations in the time scales setting in this way we recover known results for corresponding differential equations and obtain new results for associated difference equations in particular we prove that there is no upper bound for the number of isolated periodic solutions of abel difference equations one of the main tools introduced to get our results is a suitable melnikov function this is the first time that melnikov functions are used for dynamic equations on time scales

on semiclassical dispersion relations of harper like operators we describe some semiclassical spectral properties of harper like operators i e of one dimensional quantum hamiltonians periodic in both momentum and position the spectral region corresponding to the separatrices of the classical hamiltonian is studied for the case of integer flux we derive asymptotic formulae for the dispersion relations the width of bands and gaps and show how geometric characteristics and the absence of symmetries of the hamiltonian influence the form of the energy bands

factorization properties of finite spaces in 1960 schwinger j schwinger 1960 proc natl acad sci 46 570 9 proposed the algorithm for factorization of unitary operators in the finite m dimensional hilbert space according to a coprime decomposition of m using a special permutation operator a we generalize the schwinger factorization to every decomposition of m we obtain the factorized pairs of unitary operators and show that they obey the same commutation relations as schwinger s we apply the new factorization to two problems first we show how to generate two kq like mutually unbiased bases for any composite dimension then using a harper like hamiltonian model in the finite dimension m m1m2 we show how to design a physical system with m1 energy levels each having degeneracy m2

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

limit cycles for continuous and discontinuous perturbations of uniform isochronous cubic centers let p be a uniform isochronous cubic polynomial center we study the maximum number of small or medium limit cycles that bifurcate from p or from the periodic solutions surrounding p respectively when they are perturbed either inside the class of all continuous cubic polynomial differential systems or inside the class of all discontinuous differential systems formed by two cubic differential systems separated by the straight line y 0 in the case of continuous perturbations using the averaging theory of order 6 we show that the maximum number of small limit cycles that can appear in a hopf bifurcation at p is 3 and this number can be reached for a subfamily of these systems using the averaging theory of first order we prove that at most 3 medium limit cycles can bifurcate from the periodic solutions surrounding p and this number can be reached in the case of discontinuous perturbations using the averaging theory of order 6 we prove that the maximum number of small limit cycles that can appear in a hopf bifurcation at p is 5 and this number can be reached for a subfamily of these systems using the averaging method of first order we show that the maximum number of medium limit cycles that can bifurcate from the periodic solutions surrounding p is 7 and this number can be reached we also provide all the first integrals and the phase portraits in the poincare disc for the uniform isochronous cubic centers

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

uniqueness of limit cycles for polynomial first order differential equations abstract we study the uniqueness of limit cycles periodic solutions that are isolated in the set of periodic solutions in the scalar ode x k 1 m a k sin i k t cos j k t x n k in terms of i k j k n k our main result characterizes under some additional hypotheses the exponents i k j k n k such that for any choice of a 1 a m r the equation has at most one limit cycle the obtained results have direct application to rigid planar vector fields thus planar systems of the form x y x r x y y x y r x y where r x y k 1 m a k x i k y j k concretely when the set i k j k k 1 m has at least three elements or exactly one and another technical condition is satisfied we characterize the exponents i k j k such that the origin of the rigid system is a center for any choice of a 1 a m r and also when there are no limit cycles surrounding the origin for any choice of a 1 a m r

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

topological properties of quantum periodic hamiltonians we consider periodic quantum hamiltonians on the torus phase space harper like hamiltonians we calculate the topological chern index which characterizes each spectral band in the generic case this calculation is made by a semiclassical approach with the use of quasi modes as a result the chern index is equal to the homotopy of the path of these quasi modes on phase space as the floquet parameter of the band is varied it is quite interesting that the chern indices defined as topological quantum numbers can be expressed from simple properties of the classical trajectories

doubling feature of the wigner function finite phase space the wigner function has a doubling feature which is expressed in the fact that an x p point in the wigner function leads to a 2x 2p point in the wavefunctions this feature is of no special consequence in the continuous phase plane but in the finite phase plane it is of crucial importance this doubling feature enables one to define the finite phase plane wigner function directly from the wigner function in the continuous phase plane bearing in mind the doubling feature of the wigner function we define the inversion operator not only around the points of the discrete phase plane lattice but also around their mid points our approach makes it clear why there is a difference in constructing the wigner function in odd and even dimensional phase spaces

planar nonautonomous polynomial equations ii coinciding sectors abstract we give a few sufficient conditions for the existence of periodic solutions of the equation z j 0 n a j t z j where a j s are complex valued we prove the existence of one up to two periodic solutions and heteroclinic ones

on the structure of quantum phase space the space of labels characterizing the elements of schwinger s basis for unitary quantum operators is endowed with a structure of symplectic type this structure is embodied in a certain algebraic cocycle whose main features are inherited by the symplectic form of classical phase space in consequence the label space may be taken as the quantum phase space it plays in the quantum case the same role played by phase space in classical mechanics some differences coming inevitably from its nonlinear character

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

existence of closed solutions for a polynomial first order differential equation we find new criteria for the existence of closed solutions in a first order polynomial differential equation which contains the abel equation as a particular case such results are applied to the problem of the existence of limit cycles in planar polynomial vector fields

a positivity based approach to delay dependent stability of systems with large time varying delays abstract in this paper we propose new explicit tests for positivity and exponential stability of systems with large time varying delays our approach is based on nonoscillation of solutions of the corresponding diagonal scalar delay differential equations numerical examples illustrate the efficiency of the results

tomographically complete sets of orthonormal bases in finite systems quantum systems where the position and momentum are in the ring d is an odd integer are considered a tomographically complete set of bases i e a set of bases such that probabilities from tomography experiments corresponding to these bases can be used for the calculation of an arbitrary density matrix is considered such a set of bases is constructed by considering a factorization of a system with dimension d p1p2 where p1 p2 are prime numbers in terms of two subsystems with dimensions p1 and p2 appropriate combination of the mutually unbiased bases in the subsystems leads to a tomographically complete set of bases in the full system this set leads to probabilities along all the lines through the origin 0 0 in the phase space with exactly d points it is shown that the number of these lines is d the dedekind function the general theory is exemplified with examples

conjugate variables in finite phase plane we construct two pairs of quasicoordinates and quasimomenta in a finite phase plane which form sets of conjugate variables in such a plane the coordinate x is quantized with a step c and the momentum p with a step where mc is the size of the phase plane in the x direction the construction depends crucially on the possibility of writing m m1m2 with m1 and m2 relatively prime the conjugate variables are applied to harper like hamiltonians it is shown how to design physical systems with energy spectra containing any desired number of discrete energy levels say m1 each of them having a prescribed degeneracy m2

brief paper on nonoscillation and stability for systems of differential equations with a distributed delay for the system with a distributed delay email protected t email protected k 1 email protected h k t t d sr k t s x s 0 and a nonnegative fundamental matrix solution estimates and stability results are obtained the results are applied to integro differential equations and equations of mixed type including both distributed and concentrated delays

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

singular bohr sommerfeld rules the object of this paper is to present a geometrical formulation of the singular bohr sommerfeld rules given in 8 in a more general context including characteristic manifolds with several saddle points several examples are detailed and numerical checkings of the results are provided

factorizations and physical representations a hilbert space in m dimensions is shown explicitly to accommodate representations that reflect the decomposition of m into prime numbers representations that exhibit the factorization of m into two relatively prime numbers the kq representation zak j 1970 phys today 23 51 and related representations termed q1q2 representations together with their conjugates are analysed as well as a representation that exhibits the complete factorization of m in this latter representation each quantum number varies in a subspace that is associated with one of the prime numbers that make up m

a simple solution of some composition conjectures for abel equations abstract trigonometric abel differential equations appear in the study of the number of limit cycles and the center focus problem for certain families of planar polynomial systems the composition centers are a class of centers for trigonometric abel equations which have been widely studied during last years we characterize this type of centers as the ones given by couples of trigonometric polynomials for which all the generalized moments vanish they also coincide with the strongly and the highly persistent centers our result gives a simple and self contained proof of the so called composition conjecture for trigonometric abel differential equations we also prove a similar version of this result for abel equations with polynomial coefficients

scaling for the discrete mathieu equation the scaling of the total width of the band for the discrete mathieu equation is studied in the critical region near the transition between localized and extended states for the special case in which there is one period of the modulation forp lattice spacings a general expression for the bandwidthw in the critical region is found at the critical point an analytic expression forpw is found which agrees to one part in 108 with the result deduced from numerical work

abel like differential equations with a unique limit cycle abstract for the family of scalar abel like equations x a t x n b b t x m where a t l 1 k a l sin i l t cos j l t b t sin i b t cos i b t a l b r n m k i l j l i b j b z n m 2 and k 1 we characterize the existence of non trivial limit cycles periodic solutions that are isolated in the set of periodic solutions different from the trivial x t 0 in terms of n m k i l j l i b j b

wkb analysis of energy band structure of modulated systems a discrete wkb approximation is applied to harper s equation describing the motion of an electron in a square lattice subject to a magnetic field the treatment is valid when the modulation parameter proportional to the magnetic field is a rational number m n in the limit of large n a uniform asymptotic expression is derived for the floquet wavenumber which characterizes the dynamics and spectral properties the case of energies near the classical separatrix crucial to the properties of the spectrum near the transition between localized and extended states is included an application is a detailed calculation of the total width of the bands in the critical region for m 1 and 2 the results are consistent with thouless conjecture that the bandwidth obeys a universal scaling form independent of m

on the spectrum of a class of second order periodic elliptic differential operators under an additional symmetry condition we prove that the spectrum of a second order self adjoint elliptic differential operator with periodic coefficients is purely absolutely continuous

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

von neumann lattices and magnetic orbitals in finite phase space it is proven that in the finite phase space of dimensionality m the number of independent von neumann states is m r where r is the number of distinct zeros of the kq function when applied to magnetic orbitals this leads to a linear dependence between them for each zero of the corresponding kq function different interesting identities follow from the linear dependence of von neumann states

on the existence of positive solutions for systems of differential equations with a distributed delay for the system with a distributed delay email protected t email protected email protected h k t t d sr k t s x s 0 sufficient conditions for positivity of the fundamental matrix are obtained in the case when the fundamental matrix is positive comparison results and sufficient positivity conditions for solutions are established

computing integrability conditions for a cubic differential system abstract the integrability conditions for certain systems of differential equations p y y q y where p and q are cubic polynomials are obtained using two distinct computational approaches such conditions are required in the investigation of hilbert s 16 th problem and in the development of algorithms for the automatic solution of differential equations

periodic solutions of a quartic nonautonomous equation on considere des equations de la forme z z 4 t z 3 t z 2 t z on donne un ensemble qui a au moins huit solutions periodiques les coefficients sont des polynomes quadratiques en cost et sint

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

a uniqueness criterion of limit cycles for planar polynomial systems with homogeneous nonlinearities abstract this paper is devoted to study the planar polynomial system x a x y p n x y y x a y q n x y where a r and p n q n are homogeneous polynomials of degree n 2 denote cos q n cos sin sin p n cos sin we prove that the system has at most 1 limit cycle surrounding the origin provided n 1 a 0 furthermore this upper bound is sharp this is maybe the first uniqueness criterion which only depends on a linear condition of for the limit cycles of this kind of systems we show by examples that in many cases the criterion is applicable while the classical ones are invalid the tool that we mainly use is a new estimate for the number of limit cycles of abel equation with coefficients of indefinite signs employing this tool we also obtain another geometric criterion which allows the system to possess at most 2 limit cycles surrounding the origin

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

an explicit bound of the number of vanishing double moments forcing composition we give two new characterizations of pairs of polynomials or trigonometric polynomials that form a composition pair one of them proves that the cancellation of a given number of double moments implies that they form a composition pair this number only depends on the maximum degree of both polynomials this is the first time that composition is characterized in terms of the cancellation of an explicit number of double moments our results allow to recognize the composition centers for polynomial and trigonometric abel differential equations

quantum discrete phase space dynamics and its continuous limit the main aspects of a discrete phase space formalism are presented and the discrete dynamical bracket suitable for the description of time evolution in finite dimensional spaces is discussed a set of operator bases is defined in such a way that the weyl wigner formalism is shown to be obtained as a limiting case in the same form the moyal bracket is shown to be the limiting case of the discrete dynamical bracket the dynamics in quantum discrete phase spaces is shown not to be attained from discretization of the continuous case

a sum rule for the dispersion relations of the rational harper equation the authors derive an exact sum rule satisfied by the dispersion relations of the commensurable case of harper s equation 1955 they use this result to derive a lower bound for the total bandwidth of the spectrum and to provide a stronger analytical justification for a result due to thouless 1990 concerning the total bandwidth when the commensurability is a high order rational

quadratic and cubic systems with degenerate infinity we obtain an affine classification of quadratic systems with degenerate infinity in eight types and prove that they present 13 phase portraits on the poincare sphere for cubic systems with degenerate infinity the situation is much more complicated we show that there are examples of such systems with at least three concentric limit cycles and we give a criterion to ensure that in some cases this number is less than or equal to two

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

some new results on abel equations abstract abel equations of the form r a t r2 b t r3 t t0 t1 are of interest because of their close relation to planar vector fields in this paper three new results on abel equations are presented 1 an asymptotic expansion of the solutions of abel equations in terms of the coefficient functions a and b 2 a simple recurrence relation for bautin quantities of abel equations 3 a center condition for a class of polynomial abel equations

symplectic transformations and quantum tomography in finite quantum systems quantum systems where the position and momentum are in the ring zd d is an odd integer are considered symplectic transformations are studied and the order of sp 2 zd is calculated quantum tomography is also discussed it is shown that measurements used in the inverse radon transform need to be made on j2 d lines where j2 d is the jordan totient function

research on the properties of some planar polynomial differential equations abstract in this article we use the new method of reflecting function to study the behavior of solutions of nonlinear time vary differential equations and give the sufficient conditions for these equations which have the reflecting function in the form of linear and fractional we applied the obtained results to discuss the qualitative behavior of solutions of the higher degree polynomial differential systems and derive the sufficient conditions for a critical point to be a center

consensus problems with distributed delays with application to traffic flow models this paper focuses on consensus problems for a class of linear systems with distributed delay that are encountered in modeling traffic flow dynamics in the application problems the distributed delay whose kernel is a gamma distribution with a gap represents the human drivers behavior in the average the aim of the paper is to give a characterization of the regions in the corresponding delay parameter space where a consensus is reached for all initial conditions the structure and properties of the system are fully exploited which leads to explicit and computationally tractable expressions as a by product a stability theory for distributed delay systems with a gamma distribution kernel is developed also explicit expressions for the consensus function al of time delay systems with constant and distributed delays that solve a consensus problem are provided several illustrative examples complete the presentation

non autonomous equations related to polynomial two dimensional systems periodic solutions of certain one dimensional non autonomous differential equations are investigated equation 1 4 the independent variable is complex the motivation which is explained in the introductory section is the connection with certain polynomial two dimensional systems several classes of coefficients are considered in each case the aim is to estimate the maximum number of periodic solutions into which a given solution can bifurcate under perturbation of the coefficients in particular we need to know when there is a full neighbourhood of periodic solutions we give a number of sufficient conditions and investigate the implications for the corresponding two dimensional systems

periodic and discrete zak bases weyl s unitary operators for displacement in position and momentum commute with one another if the product of the elementary displacements equals planck s constant then their common eigenstates constitute the zak basis with each state specified by two phase parameters accordingly the transformation function from the position basis to the zak basis maps the hilbert space on the line onto the hilbert space on the torus this mapping is one to one provided that the zak basis states are periodic functions of their phase parameters but then the mapping cannot be continuous on the whole torus with the periodicity of the zak basis enforced the basis has a double fourier series the fourier coefficients identify a discrete basis which complements the periodic zak basis to form a pair of mutually unbiased bases the discrete basis states are the common eigenstates of the two complementary partners to the two unitary displacement operators these partner operators are of angular momentum type with integer eigenvalues and generate the fundamental rotations of the torus conversely the displacement operators are the ladder operators for their partners for each consistent phase convention for the periodic zak basis and thus for the line onto torus mapping there is a corresponding discrete zak basis and a corresponding pair of partner operators examples of particular interest are the conventions that give a continuous mapping in one phase parameter or are symmetric in both phase parameters the latter emphasizes the heisenberg weyl symmetry between position and momentum we discuss briefly the relation between the zak bases and aharonov s modular operators finally as an application of the zak operators for the torus we mention how they can be used to associate with the single degree of freedom of the line a pair of genuine qubits that are potentially entangled

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

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

uniform isochronous cubic and quartic centers revisited abstract in this paper we completed the classification of the phase portraits in the poincare disc of uniform isochronous cubic and quartic centers previously studied by several authors there are three and fourteen different topological phase portraits for the uniform isochronous cubic and quartic centers respectively

rational von neumann lattices a rational von neumann lattice is defined as a lattice in phase space with the constants a and b in the x and p directions given by a ratio of integers zeros of harmonic oscillator functions in the kq representation on such lattices are found it is shown that the number of zeros of the kq function determines the number of states by which a set on a von neumann lattice is overcomplete interesting relations between theta functions are derived on the basis of their connection with the harmonic oscillator states in the kq representation

alien limit cycles in abel equations abstract the aim of this paper is to study the existence of limit cycles for a family of generalized abel equations x a t x m b t x n m n 2 under certain assumptions it is proved that there exists a non trivial limit cycle this limit cycle has the characteristic that it arises from neither a hopf bifurcation nor a perturbation of periodic orbits in a period annulus around the centre at the origin

a geometric criterion for equation x i 0mai t xi having at most m isolated periodic solutions abstract this paper is devoted to the investigation of generalized abel equation x s x t i 0 m a i t x i where a i c 0 1 a solution x t is called a periodic solution if x 0 x 1 in order to estimate the number of isolated periodic solutions of the equation we propose a hypothesis h which is only concerned with s x t on m straight lines there exist m real numbers 1 m such that either 1 i s i t 0 for i 1 m or 1 i s i t 0 for i 1 m by means of lagrange interpolation formula we prove that the equation has at most m isolated periodic solutions counted with multiplicities if hypothesis h holds and the upper bound is sharp furthermore this conclusion is also valid under some weaker geometric hypotheses applying our main result for the trigonometrical generalized abel equation with coefficients of degree one we give a criterion to obtain the upper bound for the number of isolated periodic solutions this criterion is almost equivalent to hypothesis h and can be much more effectively checked