bifurcation of periodic orbits of periodic equations with multiple parameters by averaging method abstract we investigate a class of n dimensional periodic differential equations with multiple small parameters by averaging method and obtain sufficient conditions for the existence of multiple periodic orbits as applications of the main results a cubic system is studied

upper estimates for the number of periodic solutions to multi dimensional systems abstract the maximal number of zeros of multi dimensional real analytic maps with small parameter is studied by means of the multi dimensional generalization of rouche s theorem the obtained result is applied to study the maximal number of periodic solutions to multi dimensional differential systems an application to a class of three dimensional autonomous systems is given

averaging theory at any order for computing periodic orbits abstract we provide a recurrence formula for the coefficients of the powers of e in the series expansion of the solutions around e 0 of the perturbed first order differential equations using it we give an averaging theory at any order in e for the following two kinds of analytic differential equation d x d k 1 e k f k x d x d k 0 e k f k x a planar polynomial differential system with a singular point at the origin can be transformed using polar coordinates to an equation of the previous form thus we apply our results for studying the limit cycles of a planar polynomial differential systems

a characterization of centres via first integrals on considere l existence d une integrale premiere lisse et l existence d une mesure invariante equivalente a la mesure de lebesgue pour des systemes differentiels dans le plan

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

compactification of the moduli spaces of vortices and coupled vortices vortices and coupled vortices arise from yang mills higgs theories and can be viewed as generalizations or analogues to yang mills connections and in particular hermitianyang mills connections we proved an analytic compactification of the moduli spaces

twisted holomorphic chains and vortex equations over non compact kahler manifolds abstract in this paper we study twisted holomorphic chains and related gauge equations over non compact kahler manifolds we use the heat flow method to solve the dirichlet boundary problem for the related gauge equations and prove a hitchin kobayashi type correspondence for twisted holomorphic chain over some non compact kahler manifolds

convergence of yang mills higgs fields in this paper we study the convergence of yang mills higgs ymh fields defined on fiber bundles over riemann surfaces where the fiber is a compact symplectic manifold and the conformal structure of the underlying surface is allowed to vary we show that away from the nodes the ymh fields converges up to gauge to a smooth ymh field modulo finitely many harmonic spheres while near the nodes where the conformal structure degenerates the ymh fields converges to a pair consisting of a flat connection and a twisted geodesic with potential in particular we generalize the recent compactness results on both harmonic maps from surfaces and twisted holomorphic curves to general ymh fields

the self duality equations on a riemann surface in this paper we shall study a special class of solutions of the self dual yang mills equations the original self duality equations which arose in mathematical physics were defined on euclidean 4 space the physically relevant solutions were the ones with finite action the so called instantons the same equations may be dimensionally reduced to euclidean 3 space by imposing invariance under translation in one direction these equations also have physical relevance the solutions which have finite action in three dimensions are the magnetic monopoles if we take the reduction process one step further and consider solutions which are invariant under two translations we obtain a set of equations in the plane here however there is no clear physical meaning and indeed attempts to find finite action solutions have failed nevertheless these are the equations we shall consider despite the lack of interesting solutions in r2 the equations have the important property conformal invariance which allows them to be defined on manifolds modelled on r2 by conformal maps namely riemann surfaces we shall consider here solutions of the self duality equations defined on a compact riemann surface there are in fact solutions as we shall show and the moduli space of all solutions turns out to be a manifold with an extremely rich geometric structure which will be the focus of our study it brings together in a harmonious way the subjects of riemannian geometry topology algebraic geometry and symplectic geometry illuminating all these facets of the same object accounts for the length of this paper the self duality equations are equations from gauge theory geometrically they are defined in terms of connections on principal bundles while the group of the principal bundle may be chosen arbitrarily for the equations to make sense we restrict attention here to the simplest case of su 2 or so 3 there are two reasons for this the first and most obvious is that it simplifies calculations and avoids the use of inductive processes which are inherent in the consideration of a general lie group of higher rank the second reason is that solutions for su 2 have an intimate relationship with the internal structure of the riemann surface as a consequence of results we shall prove about solutions to the self duality equations we learn something about the moduli space of complex structures on the surface itself namely teichmuller space

averaging methods of arbitrary order periodic solutions and integrability in this paper we provide an arbitrary order averaging theory for higher dimensional periodic analytic differential systems this result extends and improves results on averaging theory of periodic analytic differential systems and it unifies many different kinds of averaging methods applying our theory to autonomous analytic differential systems we obtain some conditions on the existence of limit cycles and integrability for polynomial differential systems with a singularity at the origin having a pair of pure imaginary eigenvalues we prove that there always exists a positive number n such that if its first n averaging functions vanish then all averaging functions vanish and consequently there exists a neighborhood of the origin filled with periodic orbits consequently if all averaging functions vanish the origin is a center for n 2 furthermore in a punctured neighborhood of the origin the system is c infty completely integrable for n 2 provided that each periodic orbit has a trivial holonomy finally we develop an averaging theory for studying limit cycle bifurcations and the integrability of planar polynomial differential systems near a nilpotent monodromic singularity and some degenerate monodromic singularities

centers of quasi homogeneous polynomial differential equations of degree three we characterize the centers of the quasi homogeneous planar polynomial differential systems of degree three such systems do not admit isochronous centers at most one limit cycle can bifurcate from the periodic orbits of a center of a cubic homogeneous polynomial system using the averaging theory of first order

a second order analysis of the periodic solutions for nonlinear periodic differential systems with a small parameter abstract we deal with nonlinear t periodic differential systems depending on a small parameter the unperturbed system has an invariant manifold of periodic solutions we provide the expressions of the bifurcation functions up to second order in the small parameter in order that their simple zeros are initial values of the periodic solutions that persist after the perturbation in the end two applications are done the key tool for proving the main result is the lyapunov schmidt reduction method applied to the t poincare andronov mapping

the center problem via averaging method the averaging method has been used to study the problem of the determination of the number of hyperbolic limit cycles that can bifurcate from the period annulus of a center in this paper we use the averaging method up to any order in the perturbation parameter to determine the center conditions of monodromic singular points of analytic planar vector fields

optimized dissimilarity space embedding for labeled graphs this paper introduces a new general purpose classification system able to face automatically a wide range of classification problems for labeled graphs the proposed graph classifier explicitly embeds the input labeled graphs using the dissimilarity representation framework we developed a method to optimize the dissimilarity space representation estimating the quadratic renyi entropy of the underlying distribution of the generated dissimilarity values the global optimization governing the synthesis of the classifier is implemented using a genetic algorithm and it is carried out by means of two operations that perform prototype selection and extraction on the input set of graphs during the optimization step we adopted a suitable objective function which includes the classification accuracy achieved by the whole classification model on a validation set experimental evaluations have been conducted on both synthetic and well known benchmarking datasets achieving competitive test set classification accuracy results with respect to other state of the art graph embedding based classification systems

period function and normalizers of vector fields in rn with n 1 first integrals we extend the theory that relates period function normalizers and first integrals which has been widely developed in r2 to any dimension in particular we show how to find n 1 independent normalizers of vector fields in rn with n 1 independent first integrals a formula for the derivatives of the period function is obtained and a necessary and sufficient condition for isochronicity is also proved these results generalize previous works of freire gasull and guillamon sabatini and villarini

coupled vortex equations and moduli deformation theoretic approach and kahler geometry we investigate differential geometric aspects of moduli spaces parametrizing solutions of coupled vortex equations over a compact kahler manifold x these solutions are known to be related to polystable triples via a kobayashi hitchin type correspondence using a characterization of infinitesimal deformations in terms of the cohomology of a certain elliptic double complex we construct a hermitian structure on these moduli spaces this hermitian structure is proved to be kahler the proof involves establishing a fiber integral formula for the hermitian form we compute the curvature tensor of this kahler form when x is a riemann surface the holomorphic bisectional curvature turns out to be semi positive it is shown that in the case where x is a smooth complex projective variety the kahler form is the chern form of a quillen metric on a certain determinant line bundle

classification of the centers and their isochronicity for a class of polynomial differential systems of arbitrary degree abstract in this paper we classify the centers localized at the origin of coordinates and their isochronicity for the polynomial differential systems in r 2 of degree d that in complex notation z x i y can be written as z i z a z d n 1 2 z d n 1 2 b z d n 1 2 z d n 1 2 c z d 1 2 z d 1 2 d z d 2 j n 1 2 z d 2 j n 1 2 where j is either 0 or 1 if j 0 then d 5 is an odd integer and n is an even integer satisfying 2 n d 1 2 if j 1 then d 3 is an integer and n is an integer with converse parity with d and satisfying 0 n d 1 3 where denotes the integer part function furthermore r and a b c d c note that if d 3 and j 0 we are obtaining the generalization of the polynomial differential systems with cubic homogeneous nonlinearities studied in k e malkin 1964 17 n i vulpe and k s sibirskii 1988 25 j llibre and c valls 2009 15 and if d 2 j 1 and c 0 we are also obtaining as a particular case the quadratic polynomial differential systems studied in n n bautin 1952 2 h zoladek 1994 26 so the class of polynomial differential systems here studied is very general having arbitrary degree and containing the two more relevant subclasses in the history of the center problem for polynomial differential equations

a unified proof on the weak hilbert 16th problem for n 2 abstract the weak hilbert 16th problem for n 2 was solved by horozov and iliev proc london math soc 69 1994 198 244 zhang and li adv in math 26 1997 445 460 gavrilov invent math 143 2001 449 497 and li and zhang nonlinearity 15 2002 1775 1992 by using different methods for different cases the aim of this paper is to give a unified and easier proof for all cases the proof is restricted to the real domain combines geometric and analytical methods and uses deformation arguments