global hopf bifurcation of two parameter flows the behavior of center indices as introduced by j mallet paret j yorke is analyzed for two parameter flows the integer sum of center indices along a one dimensional curve in parameter space is called the h index a nonzero h index implies global hopf bifurcation the index h is not a homotopy invariant this fact is due to the occurrence of stationary points with an algebraically double eigenvalue zero which we call b points to each b point we assign an integer b index such that the h index relates to the b indices by a formula such as occurs in the calculus of residues

period doubling and the lefschetz formula on donne une application du theoreme de point fixe de lefschetz pour demontrer sous certaines hypotheses l existence d une famille d orbites periodiques pour une application lisse

the logarithmic cotangent complex we define the cotangent complex of a morphism of fine log schemes prove that it is functorial and construct under certain restrictions a transitivity triangle we also discuss its relationship with deformation theory

log canonical models for the moduli space of curves the first divisorial contraction in this paper we initiate our investigation of log canonical models for m g as we decrease from 1 to 0 we prove that for the first critical value 9 11 the log canonical model is isomorphic to the moduli space of pseudostable curves which have nodes and cusps as singularities we also show that 7 10 is the next critical value i e the log canonical model stays the same in the interval 7 10 9 11 in the appendix we develop a theory of log canonical models of stacks that explains how these can be expressed in terms of the coarse moduli space

orbits in pr n and equivariant quantum cohomology abstract we compute the g l r 1 equivariant chow class of the g l r 1 orbit closure of any point x 1 x n p r n in terms of the rank polytope of the matroid represented by x 1 x n p r using these classes and generalizations involving point configurations in higher dimensional projective spaces we define for each d n matrix m an n ary operation m on the small equivariant quantum cohomology ring of p r which is the n ary quantum product when m is an invertible matrix we prove that m m is a valuative matroid polytope association like the quantum product these operations satisfy recursive properties encoding solutions to enumerative problems involving point configurations of given moduli in a relative setting as an application we compute the number of line sections with given moduli of a general degree 2 r 1 hypersurface in p r generalizing the known case of quintic plane curves

period doubling cascades of attractors a prerequisite for horseshoes this paper shows that if a horseshoe is created in a natural manner as a parameter is varied then the process of creation involves the appearance of attracting periodic orbits of all periods furthermore each of these orbits will period double repeatedly with those periods going to infinity

global hopf bifurcation for differential equations with state dependent delay abstract we develop a global hopf bifurcation theory for a system of functional differential equations with state dependent delay the theory is based on an application of the homotopy invariance of s 1 equivariant degree using the formal linearization of the system at a stationary state our results show that under a set of mild conditions the information about the characteristic equation of the formal linearization with frozen delay can be utilized to detect the local hopf bifurcation and to describe the global continuation of periodic solutions for such a system with state dependent delay

implicit runge kutta and spectral galerkin methods for the two dimensional nonlinear riesz space fractional diffusion equation abstract a numerical method with high accuracy both in time and in space is proposed for the two dimensional nonlinear riesz space fractional diffusion equation the main idea is based on a spectral galerkin method in spatial direction and an s stage implicit runge kutta method in temporal direction a rigorous stability and error analysis is performed for the proposed method it is shown that the proposed method is stable and convergent the optimal spatial error estimate is also derived numerical experiments are provided to illustrate the theoretical results

stable pair tropical and log canonical compactifications of moduli spaces of del pezzo surfaces we give a functorial normal crossing compactification of the moduli space of smooth cubic surfaces entirely analogous to the grothendieck knudsen compactification m 0 n subset overline m 0 n

geometry of chow quotients of grassmannians we consider kapranov s chow quotient compactification of the moduli space of ordered n tuples of hyperplanes in p r 1 in linear general position for r 2 t his is canonically identified with the grothendieck knudsen compactification of m0 n which has among others the following nice properties 1 modular meaning stable pointed rational curves 2 canonical description of limits of one parameter degenerations 3 natural mori theoretic meaning log canonical compactification we generalize 1 and 2 to all r n but we show that 3 which we view as the deepest fails except possibly in the cases 2 n 3 6 3 7 3 8 where we conjecture that it holds

geometry and combinatorics of julia sets of real quadratic maps for real a correspondence is made between the julia setb forz z 2 in the hyperbolic case and the set of chains with the aid of cremer s theorem it is shown how a number of features ofb can be understood in terms of chains the structure ofb is determined by certain equivalence classes of chains fixed by orders of visitation of certain real cycles and the bifurcation history of a given cycle can be conveniently computed via the combinatorics of chains the functional equations obeyed by attractive cycles are investigated and their relation to chains is given the first cascade of period doubling bifurcations is described from the point of view of the associated julia sets and chains certain julia sets associated with the feigenbaum function and some theorems of lanford are discussed

symmetric functional differential equations and neural networks with memory we establish an analytic local hopf bifurcation theorem and a topological global hopf bifurcation theorem to detect the existence and to describe the spatial temporal pattern the asymptotic form and the global continuation of bifurcations of periodic wave solutions for functional differential equations in the presence of symmetry we apply these general results to obtain the coexistence of multiple large amplitude wave solutions for the delayed hopfield cohen grossberg model of neural networks with a symmetric circulant connection matrix

analysis of a time stepping scheme for time fractional diffusion problems with nonsmooth data this paper establishes the convergence of a time stepping scheme for the time fractional diffusion problems with nonsmooth data we first analyze the regularity of the time fractional diffusion pro

pavages des simplexes schemas de graphes recolles et compactification des pglrn 1 pglr some equivariant compactifications of the quotients pglrn 1 pglr are constructed each one is decomposed into locally closed strata which are smooth are indexed by the entire convex pavings of the simplex of dimension n and admit a modular interpretation deduced from that of the grassmann varieties together they form a simplicial scheme which compactifies the classifying simplicial scheme of pglr consisting of all the quotients pglrn 1 pglr n 0

combinatorial geometries convex polyhedra and schubert cells this paper is a continuation of gm which was published in the same journal we will explore a remarkable connection between the geometry of the schubert cells in the grassmann manifold the theory of convex polyhedra and the theory of combinatorial geometries in the sense of crapo and rota cr the results in this paper were obtained simultaneously and independently by gelfand and serganova in moscow and by goresky and macpherson at the i h e s in paris as part of larger programs with different purposes see below the geometry of this simple example is so beautiful that we decided to publish it independently of the applications we believe that combinatorial methods will play an increasing role in the future of geometry and topology we consider the grassmann manifold gi of all n k dimensional subspaces of c by fixing the standard basis in c we obtain an action of the torus h i on g pk which is induced from stretching the coordinate axes in c see also sect 1 we will describe not only the trajectories but also the strata of a new and interesting decomposition of the 301 oool 8708 87 7 50

snakes oriented families of periodic orbits their sources sinks and continuation abstract poincare observed that for a differential equation x x depending on a parameter each periodic orbit generally lies in a connected family of orbits in x space in order to investigate certain large connected sets denoted q of orbits containing a given orbit we introduce two indices an orbit index and a center index defined at certain stationary points we show that genetically there are two types of hopf bifurcation those we call sources 1 and sinks 1 generically if the set q is bounded in x space and if there is an upper bound for periods of the orbits in q then q must have as many source hopf bifurcations as sink hopf bifurcations and each source is connected to a sink by an oriented one parameter snake of orbits a snake is a maximal path of orbits that contains no orbits whose orbit index is 0 see fig 1 1

compactifications of subvarieties of tori we study compactifications of subvarieties of algebraic tori defined by imposing a sufficiently fine polyhedral structure on their non archimedean amoebas these compactifications have many nice properties for example any k boundary divisors intersect in codimension k we consider some examples including m 0 n subset overline m 0 n and more generally log canonical models of complements of hyperplane arrangements and compact quotients of grassmannians by a maximal torus

a canonical partition of the periodic orbits of chaotic maps on montre que les orbites periodiques d une application en fer a cheval qui contracte l aire peuvent se partager en sous ensembles d orbites de periode minimum k 2k 4k 8k pour un k entier positif

a finite difference scheme on graded meshes for time fractional nonlinear korteweg de vries equation abstract in this paper a finite difference scheme for time fractional nonlinear korteweg de vries kdv problems with the caputo type fractional derivative is presented to deal with the weak singularity caused by the fractional derivative that the solution has in the initial layer the well known l1 scheme on graded meshes has been used for time discretization meanwhile a nonlinear finite difference approximation on uniform meshes is proposed for spatial discretization the existence stability and convergence of the numerical solution are studied by the energy method it is proved that the scheme is of min 2 r order convergence in time and of first order convergence in space where is the order of fractional derivative and r is a parameter of graded meshes furthermore in order to increase computational efficiency the corresponding fast algorithm of the presented scheme has been considered in numerical simulation a valid approach based on linear interpolation is designed to test convergence rate for the scheme on temporal graded meshes numerical results demonstrate sharpness of the theoretical convergence estimate and effectiveness of the fast algorithm

on the continuability of periodic orbits of parametrized three dimensional differential equations on considere un systeme autonome a n dimensions d equations differentielles ordinaires a 1 parametre x f x on suppose que f est definie et continue sur un sous ensemble ouvert o de r r n et que f x 0 sur o on etudie les orbites periodiques du systeme contenues dans o

analysis of a temporal discretization for a semilinear fractional diffusion equation abstract this paper considers a temporal discretization of a semilinear fractional diffusion equation with nonsmooth initial data with appropriately graded temporal grids first order temporal accuracy in the norm of l 2 0 t l 2 is derived by a new discrete gronwall s inequality and then first order temporal accuracy in the norm of l 0 t l 2 is established for 1 2 1 finally numerical results are provided to verify the theoretical results