periodic solutions of a quartic differential equation and groebner bases we consider first order ordinary differential equations with quartic nonlinearities the aim is to find the maximum number of periodic solutions into which a given solution can bifurcate under perturbation of the coefficients it is shown that this number is ten when the coefficients are certain cubic polynomials equations with the maximum number of such periodic solutions are also constructed the paper is heavily dependent on computing groebner bases

matrix resolvent and the discrete kdv hierarchy based on the matrix resolvent approach for an arbitrary solution to the discrete kdv hierarchy we define the tau function of the solution and compare it with another tau function of the solution defined via reduction of the toda lattice hierarchy explicit formulae for generating series of logarithmic derivatives of the tau functions are obtained and applications to enumeration of ribbon graphs with even valencies and to certain special cubic hodge integrals are considered

hodge integrals and tau symmetric integrable hierarchies of hamiltonian evolutionary pdes for an arbitrary semisimple frobenius manifold we construct hodge integrable hierarchy of hamiltonian partial differential equations in the particular case of quantum cohomology the tau function of a solution to the hierarchy generates the intersection numbers of the gromov witten classes and their descendents along with the characteristic classes of hodge bundles on the moduli spaces of stable maps for the one dimensional frobenius manifold the hodge hierarchy is a deformation of the korteweg de vries hierarchy depending on an infinite number of parameters conjecturally this hierarchy is a universal object in the class of scalar hamiltonian integrable hierarchies possessing tau functions mathematics subject classication 2010 primary 53d45 secondary 37k10

a geometric method for detecting chaotic dynamics a new method of detection of chaos in dynamical systems generated by time periodic nonautonomous differential equations is presented it is based on the existence of some sets called periodic isolating segments in the extended phase space satisfying some topological conditions by chaos we mean the existence of a compact invariant set such that the poincare map is semiconjugated to the shift on two symbols and the counterimage by the semiconjugacy of any periodic point in the shift contains a periodic point of the poincare map as an application we prove that the planar equationz 1 ei t z 2 zgenerates chaotic dynamics provided 0 1 288

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

hyperelliptic prym varieties and integrable systems we introduce two algebraic completely integrable analogues of the mumford systems which we call hyperelliptic prym systems because every hyperelliptic prym variety appears as a fiber of their momentum map as an application we show that the general fiber of the momentum map of the periodic volterra lattice is an affine part of a hyperelliptic prym variety obtained by removing n translates of the theta divisor and we conclude that this integrable system is algebraic completely integrable

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

asymptotics for the partition function in two cut random matrix models we obtain large n asymptotics for the random matrix partition function z n v int mathbb r n prod i j x i x j 2 prod j 1 ne nv x j dx j in the case where v is a polynomial such that the random matrix eigenvalues accumulate on two disjoint intervals the two cut case we compute leading and sub leading terms in the asymptotic expansion for log zn v up to terms that are small as n to infty our approach is based on the explicit computation of the first terms in the asymptotic expansion for a quartic symmetric potential v afterwards we use deformation theory of the partition function and of the associated equilibrium measure to generalize our results to general two cut potentials v the asymptotic expansion of log zn v as n to infty contains terms that depend analytically on the potential v and that have already appeared in the literature in addition our method allows us to compute the v independent terms of the asymptotic expansion of log zn v which to the best of our knowledge had not appeared before in the literature we use rigorous orthogonal polynomial and riemann hilbert techniques which had to this point only been successful to compute asymptotics for the partition function in the one cut case

on periodic solutions of planar polynomial differential equations with periodic coefficients abstract we consider the planar equation z ak l t zk z l where ak l is a t periodic complex valued continuous function equal to 0 for almost all k l n we present sufficient conditions imposed on ak l which guarantee the existence of its t periodic solutions and in the case a0 0 0 the conditions for the existence of nonzero ones we use a method which computes the fixed point index of the poincare andronov operator in isolated sets of fixed points generated by so called periodic blocks the method is based on the lefschetz fixed point theorem and the topological principle of wazewski

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

correlation functions of the kdv hierarchy and applications to intersection numbers over m g n we derive an explicit generating function of correlations functions of an arbitrary tau function of the kdv hierarchy in particular applications our formulation gives closed formulae of a new type for the generating series of intersection numbers of classes as well as of mixed and classes in full genera

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 proof of a conjecture of marino vafa on hodge integrals where i c1 li is the first chern class of li and j cj e is the j th chern class of the hodge bundle hodge integrals arise in the calculations of gromov witten invariants by localization techniques 14 7 the explicit evaluation of hodge integrals is a difficult problem the hodge integrals involving only classes can be computed recursively by witten s conjecture 26 proven by kontsevich 13 algorithms of computing hodge integrals are described in 2 in 23 m marino and c vafa obtained a closed formula for a generating function of certain open gromov witten invariants some of which has been reduced to hodge integrals by localization techniques which are not fully clarified mathematically this leads to a conjectural formula of hodge integrals to state this formula we introduce some notation following 28 let

multiplicity of periodic solutions for the planar polynomial equation 0 p z where p z n 0 cjz j with cj 1 t t 0 cj t dt and z 1 t t 0 z t dt first we observe that z t is a periodic solutions in the discussed algebra only if z is a root of p i e the t periodic solutions of are only into the bres fl 1 l s over nonzero roots z1 zs of p proposition 2 5 let m be the sum of norms of all coe cients of the main result theorem 2 9 says that if the value mt is estimated from above by a quantity depending on n and

handbook of differential equations chapter headings definitions and concepts transformations exact analytical methods exact methods for odes exact methods for pdes approximate analytical methods numerical methods concepts numerical methods for odes numerical methods for pdes list of tables list of programs list of figures subject index

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

simple maps hurwitz numbers and topological recursion we introduce the notion of fully simple maps which are maps with non self intersecting disjoint boundaries in contrast maps where such a restriction is not imposed are called ordinary we study in detail the combinatorics of fully simple maps with topology of a disk or a cylinder we show that the generating series of simple disks is given by the functional inversion of the generating series of ordinary disks we also obtain an elegant formula for cylinders these relations reproduce the relation between moments and higher order free cumulants established by collins et al 22 and implement the symplectic transformation x leftrightarrow y on the spectral curve in the context of topological recursion we conjecture that the generating series of fully simple maps are computed by the topological recursion after exchange of x and y we propose an argument to prove this statement conditionally to a mild version of the symplectic invariance for the 1 hermitian matrix model which is believed to be true but has not been proved yet our conjecture can be considered as a combinatorial interpretation of the property of symplectic invariance of the topological recursion our argument relies on an unconditional matrix model interpretation of fully simple maps via the formal hermitian matrix model with external field we also deduce a universal relation between generating series of fully simple maps and of ordinary maps which involves double monotone hurwitz numbers in particular ordinary maps without internal faces which are generated by the gaussian unitary ensemble and with boundary perimeters lambda 1 ldots lambda n are strictly monotone double hurwitz numbers with ramifications lambda above infty and 2 ldots 2 above 0 combining with a recent result of dubrovin et al 24 this implies an elsv like formula for these hurwitz numbers

planar nonautonomous polynomial equations the riccati equation abstract we give a few sufficient conditions for the existence of two periodic solutions of the riccati ordinary differential equation in the plane we give also examples of the equation without periodic solutions

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

the euler characteristic of the moduli space of curves let fg 1 g 1 be the mapping class group consisting of all isotopy classes of base point and orientation preserving homeomorphisms of a closed oriented surface f of genus g let 1 be its euler characteristic in the sense of wall that is z f i fgl f l z e f where f is any torsion free subgroup of finite index in f 1 and e is a contractible space on which f acts freely and properly discontinuously an example of such a space is the teichmiiller space 1 and g f can be interpreted as the orbifold euler characteristic of f i join the moduli space of curves of genus g with base point the purpose of this paper is to prove the following formula for f i

the method of holomorphic foliations in planar periodic systems the case of riccati equations abstract we develop a new method of investigation of periodic solutions of planar holomorphic polynomial differential equations with trigonometric coefficients this method relies on analysis of geometry of leaves of certain holomorphic foliations in cp1 cp1 we concentrate our attention on the case of riccati equations we determine the dynamics of the non autonomous equations using some local invariants of holomorphic vector fields defining the foliations in particular new examples of riccati equations without periodic solutions are presented