the second expansion of the solution for a singular elliptic boundary value problem abstract in this paper we analyze the second expansion of the unique solution near the boundary to the singular dirichlet problem u b x g u u 0 x u 0 where is a bounded domain with smooth boundary in r n g c 1 0 0 g is decreasing on 0 with lim s 0 g s and g is normalised regularly varying at zero with index 1 b c is positive in may be vanishing on the boundary

the boundary behavior of the unique solution to a singular dirichlet problem abstract in this paper we analyze the boundary behavior of the unique solution to the singular dirichlet problem for a lane emden fowler equation u b x g u u 0 x u 0 where is a bounded domain with smooth boundary in r n g c 1 0 0 lim s 0 g s g is decreasing on 0 which is rapidly varying or normalized regularly varying at zero b c loc is positive in may be vanishing or singular on the boundary and belongs to the kato class k

global attractivity in monotone and subhomogeneous almost periodic systems abstract a global attractivity theorem is first proved for a class of skew product semiflows then this result is applied to monotone and subhomogeneous almost periodic reaction diffusion equations ordinary differential systems and delay differential equations for their global dynamics

invariant grids for reaction kinetics abstract in this paper we construct low dimensional manifolds of reduced description for equations of chemical kinetics from the standpoint of the method of invariant manifold mim mim is based on a formulation of the condition of invariance as an equation and its solution by newton iterations a grid based version of mim is developed the method of invariant grids we describe the newton method and the relaxation method for the invariant grids construction the problem of the grid correction is fully decomposed into the problems of the grid s nodes correction the edges between the nodes appear only in the calculation of the tangent spaces this fact determines high computational efficiency of the method of invariant grids the method is illustrated by two examples the simplest catalytic reaction michaelis menten mechanism and the hydrogen oxidation the algorithm of analytical continuation of the approximate invariant manifold from the discrete grid is proposed generalizations to open systems are suggested the set of methods covered makes it possible to effectively reduce description in chemical kinetics

second order expansion for the solution to a singular dirichlet problem in this paper we analyze the second order expansion for the unique solution near the boundary to the singular dirichlet problem u b x g u u 0 x u 0 where is a bounded domain with smooth boundary in r n g c1 0 0 g is decreasing on 0 with lim s 0 g s and g is normalized regularly varying at zero with index 1 b c l o c 0 1 is positive in may be vanishing or singular on the boundary and belongs to the kato class k our analysis is based on the sub supersolution method and karamata regular variation theory

attractor minimal sets for non autonomous delay functional differential equations with applications for neural networks the dynamics of a class of non autonomous convex or concave and monotone delay functional differential systems is studied in particular we provide an attractivity result when two completely st

cooperative systems of differential equations with concave nonlinearities on considere le systeme dynamique discret engendre par une application monotone concave t r n r n on etudie ensuite des systemes d equations differentielles cooperatifs periodiques x f t x a non linearites concaves

attractor minimal sets for cooperative and strongly convex delay differential systems abstract we study the skew product semiflow induced by a family of convex and cooperative delay differential systems under some monotonicity assumptions we obtain an ergodic representation for the upper lyapunov exponent of a minimal subset in addition when eventually strong convexity at one point is assumed and there exist two completely strongly ordered minimal subsets k1 ck2 we show that k1 is an attractor subset which is a copy of the base the long time behaviour of every trajectory strongly ordered with k2 is then deduced some examples of application of the theory are shown

on a singular nonlinear semilinear elliptic problem where k x c2 b v9 a p 0 1 and l is a real parameter such singular elliptic problems arise in the contexts of chemical heterogeneous catalysts nonnewtonian fluids and also the theory of heat conduction in electrically conducting materials see 3 5 8 9 for a detailed discussion obviously 1 1 cannot have a solution u c2 v9 if k x is not vanishing near v however under various appropriate assumptions on k x we will obtain classical solutions of 1 1 for l belonging to a certain range and we will also obtain some uniqueness criteria here a classical solution is a solution u of 1 1 which belongs to c2 v mc v9 with u 0 in v we also study the boundary behaviour of solutions of 1 1 and we will show that the solution u of 1 1 lies in a certain holder class the special case when k x is negative and l 0 has been studied by several authors the existence and uniqueness of the solution were established by crandall rabinowitz and tartar 6 del pino 7 gomes 10 and lazer and mckenna

global attractivity in concave or sublinear monotone infinite delay differential equations abstract we study the dynamical behavior of the trajectories defined by a recurrent family of monotone functional differential equations with infinite delay and concave or sublinear nonlinearities we analyze different sceneries which require the existence of a lower solution and of a bounded trajectory ordered in an appropriate way for which we prove the existence of a globally asymptotically stable minimal set given by a 1 cover of the base flow we apply these results to the description of the long term dynamics of a nonautonomous model representing a stage structured population growth without irreducibility assumptions on the coefficient matrices

applications of the exponential ordering in the study of almost periodic delayed hopfield neural networks abstract this paper studies almost periodic neural networks of hopfield type described by delayed differential equations the authors introduce an exponential ordering to analyze the long term behavior of the solutions they prove some theorems of global convergence and deduce the stabilization role of the fast inhibitory self connections the proof which in the case of the neural networks considered in this paper requires uniform stability uses arguments of comparison of differential equations and methods of the non autonomous monotone theory of dynamical systems when the connections between different neurons are excitatory improved conditions of convergence are obtained and the stabilization role of strongly positive inputs is also shown the applicability of the results is illustrated with several numerical experiments based on two different families of neural networks

regularity of an elliptic problem with a singular nonlinearity we study a semilinear boundary value problem with the feature that the vanishing boundary value makes the equation singular we prove that the positive solution is in general holder continuous up to the boundary and has even better regularity in some special cases

uniform and strict persistence in monotone skew product semiflows with applications to non autonomous nicholson systems abstract we determine sufficient conditions for uniform and strict persistence in the case of skew product semiflows generated by solutions of non autonomous families of cooperative systems of odes or delay fdes in terms of the principal spectrums of some associated linear skew product semiflows which admit a continuous separation our conditions are also necessary in the linear case we apply our results to a noncooperative almost periodic nicholson system with a patch structure whose persistence turns out to be equivalent to the persistence of the linearized system along the null solution

minimal sets in monotone and concave skew product semiflows ii two dimensional systems of differential equations abstract recurrent nonautonomous two dimensional systems of differential equations of ordinary finite delay and reaction diffusion types given by cooperative and concave vector fields define monotone and concave skew product semiflows whose dynamics is analyzed in this paper a complete description of all the minimal sets in a very interesting dynamical situation is provided and some criteria to determine its possible existence and the area on which they lie are given suitable examples prove the optimality of the results

nonlinear analysis based on homogeneous incremental systems abstract the performance of path following techniques depends on so called constraint equations which are used in the augmented incremental systems it is common practice in the literature that these equations are explicitly defined following the arbitrary geometrical or physical ideas in the present paper their role is discussed in order to establish a reliable method of augmentation it is shown that the explicitly defined constraint equation is not necessary the implicit formulation is proposed which is based on the rank analysis performed on the rectangular matrix of the homogeneous incremental system the proposed method allows to solve problems where the well known continuation schemes fail the theoretical considerations are illustrated by the numerical examples

classical and weak solutions of a singular semilinear elliptic problem the singular semilinear elliptic equation u p x f u 0 is shown to have a unique positive classical solution inrnthat decays to zero at ifp x is simply a nontrivial nonnegative continuous function satisfying 0 t max x t p x dt providedfis a non increasing continuously differentiable function on 0 it is also shown that the equation has a unique weakh10 solution on a bounded domain provided e0 f s ds andp x l2

analysis of a model representing stage structured population growth with state dependent time delay a stage structured model of population growth is proposed where the time to ma turity is itself state dependent it is shown that under appropriate assumptions all solutions are positive and bounded criteria for the existence of positive equilibria and further conditions for the uniqueness of the equilibria are given the stability of the equilibria are also discussed in addition an attracting region is determined for solutions such that this region collapses to the unique positive equilibrium in the state independent case

on the stability of periodic travelling waves with large spatial period abstract in many circumstances a pulse to a partial differential equation pde on the real line is accompanied by periodic wave trains that have arbitrarily large period it is then interesting to investigate the pde stability of the periodic wave trains given that the pulse is stable using the evans function gardner has demonstrated that every isolated eigenvalue of the linearization about the pulse generates a small circle of eigenvalues for the linearization about the periodic waves in this article the precise location of these circles is determined it is demonstrated that the stability properties of the periodic waves depend on certain decay and oscillation properties of the tails of the pulse as a consequence periodic waves with long wavelength typically destabilize at homoclinic bifurcation points at which multi hump pulses are created that is in contrast to the situation for the underlying pulses whose stability properties are not affected by these bifurcations the proof uses lyapunov schmidt reduction and relies on the existence of exponential dichotomies the approach is also applicable to periodic waves with large spatial period of elliptic problems on r n or on unbounded cylinders r with bounded

exact asymptotic behavior near the boundary to the solution for singular nonlinear dirichlet problems abstract we take up the existence and the exact asymptotic behavior near the boundary of the unique classical solution to a singular dirichlet problem u a x g u x u 0 in u 0 where is a c 1 1 bounded domain in r n n 2 and the function a is singular near the boundary and belongs to the kato class k

asymptotic behavior of positive solutions of a singular nonlinear dirichlet problem abstract let be a c 1 1 bounded domain in r n for n 2 in this paper we are concerned with the asymptotic behavior of the unique positive classical solution to the singular boundary value problem u a x u 0 in u 0 where 0 a is a nonnegative function in c loc 0 1 and there exists c 0 such that 1 c a x x k 1 m log k x k c here 2 k r is a positive constant and x dist x

stability and extensibility results for abstract skew product semiflows abstract in this paper we present new stability and extensibility results for skew product semiflows with a minimal base flow in particular we describe the structure of uniformly stable and uniformly asymptotically stable sets admitting backwards orbits and the structure of omega limit sets as an application the occurrence of almost periodic and almost automorphic dynamics for monotone non autonomous infinite delay functional differential equations is analyzed

continuation and bifurcations numerical techniques and applications bifurcation to rotating waves from non trivial steady states use of approximate inertial manifolds in bifurcation calculations understanding steady state bifurcation diagrams for a model reaction diffusion system bifurcations chaos and self organization in reaction diffusion systems eigenvalue problems with the symmetry of a group and bifurcations steady state steady state mode interaction in nonlinear equations with z2 symmetry symbolic computation and bifurcation methods bifurcation analysis a combined numerical and analytical approach to the numerical solution of symmetry breaking bifurcation problems a computational method and path following for periodic solutions with symmetry global bifurcations and their numerical computation computation of invariant manifold bifurcations a method for homoclinic and heteroclinic continuation in two and three dimensions the global attractor under discretisation the numerical detection of hopf bifurcation points a newton like method for simple bifurcation problems with application to large sparse systems aspects of continuation software interactive system for studies in nonlinear dynamics linlbf a program for continuation and bifurcation analysis of equilibria up to codimension three on the topology of three dimensional separations a guide for classification the construction of cubature formulae using continuation and bifurcation software determining an organizing center for passive optical systems optimization by continuation stability of marangoni convection in a microgravity environment computing with reaction diffusion systems applications in image processing bifurcation of codimension 2 for a discrete map bifurcation of periodic solutions in pde s numerical techniques and applications abstract bifurcation and chaos in chua s circuit family abstract continuation and collocation for parameter dependent boundary value problems abstract porous medium combustion abstract block elimination and the computation of simple turning points abstract bifurcation into gaps in the essential spectrum abstract application of numerical continuation in aerospace problems abstract application of a reduced basis method in structural analysis abstract some applications of bifurcation theory in engineering abstract higher order predictors in numerical path following schemes abstract

global and cocycle attractors for non autonomous reaction diffusion equations the case of null upper lyapunov exponent abstract in this paper we obtain a detailed description of the global and cocycle attractors for the skew product semiflows induced by the mild solutions of a family of scalar linear dissipative parabolic problems over a minimal and uniquely ergodic flow we consider the case of null upper lyapunov exponent for the linear part of the problem then two different types of attractors can appear depending on whether the linear equations have a bounded or an unbounded associated real cocycle in the first case e g in periodic equations the structure of the attractor is simple whereas in the second case which occurs in aperiodic equations the attractor is a pinched set with a complicated structure we describe situations when the attractor is chaotic in measure in the sense of li yorke besides we obtain a non autonomous discontinuous pitchfork bifurcation scenario for concave equations applicable for instance to a linear dissipative version of the chafee infante equation

the dynamical systems approach to differential equations bulletin new series of the american mathematical society volume 11 number 1 july 1984 the dynamical systems approach to differential equations by morris w hirsch 1 this harmony that human intelligence believes it discovers in nature does it exist apart from that intelligence no without doubt a reality completely independent of the spirit which conceives it sees it or feels it is an impossibility a world so exterior as that even if it existed would be forever inaccessible to us but what we call objective reality is in the last analysis that which is common to several thinking beings and could be common to all this common part we will see can be nothing but the harmony expressed by mathematical laws h poincare la valeur de la science p 9 ignorance of the roots of the subject has its price no one denies that modern formulations are clear elegant and precise it s just that it s impossible to comprehend how any one ever thought of them m spivak a comprehensive introduction to differential geometry introduction the mathematical subject we call dynamical systems was fathered by poin care developed sturdily under birkhoff and has enjoyed a vigorous new growth for the last twenty years as i try to show in chapter i it is interesting to look at this mathematical development as the natural outcome of a much broader theme which is as old as science itself the universe is a sytem which changes in time every mathematician has a particular way of thinking about mathematics but rarely makes it explicit yet such perspectives in any particular field can be of great value to nonexperts who must apply the results of the field or who want to learn them in some fields the points of view of its practitioners cluster around several comparatively clearly defined concepts and insights dynamical systems theory is one of them the basic concepts go back to poincare the main thrust of chapters i and ii is to explain rather discursively some of received by the editors december 14 1983 1980 mathematics subject classification primary 00a25 00a99 01a45 01a55 01a60 34a40 34c35 34d10 34d30 35b05 35b35 35b40 35b50 35b30 35k55 54h20 58f25 58f40 58f12 58f10 58d25 58d07 46a40 46e10 47h07 47h20 92a15 92a17 secondary 06f30 35b65 58f13 35j60 46e05 92 03 90 03 90a16 supported in part by national science foundation and institute for advanced study 1984 american mathematical society 0273 0979 84 1 00 25 per page

gevrey micro regularity for solutions to first order nonlinear pde abstract let x t r m r and u c 2 r m r we study the gevrey micro regularity of solutions u of the nonlinear equation u t f x t u u x where f x t 0 is a gevrey function of order s 1 and holomorphic in 0 we show that the gevrey wave front set of any c 2 solution u is contained in the characteristic set of the linearized operator l u t j 1 m f j x t u u x x j to achieve this we study the notion of gevrey approximate solutions a concept which we believe is of independent interest and could be applied to much more general situations

problems for elliptic singular equations with a gradient term abstract we investigate the homogeneous dirichlet problem for a class of second order nonlinear elliptic partial differential equations with singular data in particular we study the asymptotic behaviour of the solution near the boundary up to the second order under various assumptions on the growth of the coefficients of the equation

the asymptotic behaviour of the unique solution for the singular lane emden fowler equation abstract by karamata regular variation theory and constructing comparison functions we show the exact asymptotic behaviour of the unique classical solution u c 2 c near the boundary to a singular dirichlet problem u k x g u u 0 x u 0 where is a bounded domain with smooth boundary in r n g c 1 0 0 lim t 0 g t g t for each 0 for some 0 and k c loc for some 0 1 is nonnegative on which is also singular near the boundary

minimal sets in monotone and sublinear skew product semiflows i the general case abstract the dynamics of a general monotone and sublinear skew product semiflow is analyzed paying special attention to the long term behavior of the strongly positive semiorbits and to the minimal sets four possibilities arise depending on the existence or absence of strongly positive minimal sets and bounded semiorbits as well as on the coexistence or not of bounded and unbounded strongly positive semiorbits previous results are unified and extended and scenarios which are impossible in the autonomous or periodic cases are described

a method for the numerical computation of hopf bifurcation this paper deals with the computation of hopf bifurcations in parameter dependent ordinary differential equations an algorithm based on an indirect method is presented while the conventional method approaches the hopf bifurcation point from the stationary solution the proposed method detects the hopf bifurcation points from the dynamic solution a computation using the proposed algorithm is implemented for finding the hopf bifurcation point of several cases and the comparison with the results obtained by other methods is given the proposed method is applicable for both autonomous and non autonomous systems

minimal sets in monotone and concave skew product semiflows i a general theory abstract the long term dynamics of a general monotone and concave skew product semiflow is analyzed paying special attention to the region delimited from below by the graph of a semicontinuous subequilibrium or by a minimal set admitting a flow extension different possibilities arise depending on the existence and number or absence of minimal sets strongly above the initially fixed one as well as on the coexistence or not of bounded and unbounded semiorbits on the region previous results are unified and extended and significative differences with the sublinear case are established some scenarios which are impossible in the autonomous or periodic cases may occur in this setting

problems for elliptic singular equations with a quadratic gradient term we investigate the homogeneous dirichlet problem for a class of second order elliptic partial differential equations with a quadratic gradient term and singular data in particular we study the asymptotic behaviour of the solution near the boundary under suitable assumptions on the growth of the coefficients of the equation

efficient gluing of numerical continuation and a multiple solution method for elliptic pdes numerical continuation calculations for ordinary differential equations odes are by now an established tool for bifurcation analysis in dynamical systems theory as well as across almost all natural and engineering sciences although several excellent standard software packages are available for odes there are for good reasons no standard numerical continuation toolboxes available for partial differential equations pdes which cover a broad range of different classes of pdes automatically a natural ach to this problem is to look for efficient gluing computation approaches with independent components developed by researchers in numerical analysis dynamical systems scientific computing and mathematical modeling in this paper we shall study several elliptic pdes lane emden fowler lane emden fowler with microscopic force caginalp via the numerical continuation software pde2path and develop a gluing component to determine a set of starting solutions for the continuation by exploiting the variational structures of the pdes in particular we solve the initialization problem of numerical continuation for pdes via a minimax algorithm to find multiple unstable solution furthermore for the caginalp system we illustrate the efficient gluing link of pde2path to the underlying mesh generation and the fem matlab pdetoolbox even though the approach works efficiently due to the high level programming language and without developing any new algorithms we still obtain interesting bifurcation diagrams and directly applicable conclusions about the three elliptic pdes we study in particular with respect to symmetry breaking in particular we show for a modified lane emden fowler equation with an asymmetric microscopic force how a fully connected bifurcation diagram splits up into c shaped isolas on which localized pattern deformation appears towards two different regimes we conclude with a section on future software development issues that would be helpful to be addressed to simplify interfaces to allow for more efficient time saving gluing computation for dynamical systems analysis of pdes in the near future