the connection matrix theory for morse decompositions the connection matrix theory for morse decompositions is introduced the connection matrices are matrices of maps between the homology indices of the sets in the morse decomposition the connection matrices cover in a natural way the homology index braid of the morse decomposition and provide information about the structure of the morse decomposition the existence of connection matrices of morse decompositions is established and examples illustrating applications of the connection matrix are provided

existence of homoclinic solutions for a class of second order systems abstract in this paper we study the existence of nontrivial homoclinic solutions for a second order differential equation of the form q a q l t q w q t q 0 here we assume that a is a antisymmetric constant matrix l t is a continuous positive definite symmetric matrix valued function depending periodically on t and assume that potential w is asymptotically quadratic as q 0 and q

homoclinic solutions for a class of the second order hamiltonian systems abstract we study the existence of homoclinic orbits for the second order hamiltonian system q v q t q f t where q r n and v c 1 r r n r v t q k t q w t q is t periodic in t a map k satisfies the pinching condition b 1 q 2 k t q b 2 q 2 w is superlinear at the infinity and f is sufficiently small in l 2 r r n a homoclinic orbit is obtained as a limit of 2 kt periodic solutions of a certain sequence of the second order differential equations

special functions arising from discrete painleve equations a survey this article is a survey on recent studies on special solutions of the discrete painleve equations especially on hypergeometric solutions of the q painleve equations the main part of this survey is based on the joint work k kajiwara t masuda m noumi y ohta y yamada hypergeometric solutions to the q painleve equations imrn 2004 47 2004 2497 2521 k kajiwara t masuda m noumi y ohta y yamada construction of hypergeometric solutions to the q painleve equations imrn 2005 24 2005 1439 1463 with kajiwara masuda ohta and yamada after recalling some basic facts concerning painleve equations for comparison we give an overview of the present status of studies on difference discrete painleve equations as a source of special functions

a note on convergence under dynamical thresholds with delays we complement the study of the asymptotic behaviour of the dynamical threshold neuron model with delay introduced by gopalsamy and leung 1997 by providing a description of the dynamics of the system in the remaining parameters range we characterize the regions of harmless delays and those in which delay induced oscillations appear

algebraic isomonodromic deformations of logarithmic connections on the riemann sphere and finite braid group orbits on character varieties we study algebraic isomonodromic deformations of flat logarithmic connections on the riemann sphere with n ge 4 poles for arbitrary rank we introduce a natural property of algebraizability for the germ of universal deformation of such a connection we relate this property to a peculiarity of the corresponding monodromy representation to yield a finite braid group orbit on the appropriate character variety under reasonable assumptions on the deformed connection we may actually establish an equivalence between both properties we apply this result in the rank two case to relate finite branching and algebraicity for solutions of garnier systems for general rank a byproduct of this work is a tool to produce regular flat meromorphic connections on vector bundles over projective varieties of high dimension

an algorithmic approach to the construction of homomorphisms induced by maps in homology this paper is devoted to giving the theoretical background for an algorithm for computing homomorphisms induced by maps in homology the principal idea is to insert the graph of a given continuous map f into ag raph of a multi valued representable map f the multi valued representable maps have well developed continuity properties and admit a nite coding that per mits treating them by combinatorial methods we provide the construction of the homomorphism f induced by f such that f f the presented construction does not require subsequent barycentric subdivisions and simpli cial approximations of f the main motivation for this paper comes from the project of computing the conley index for discrete dynamical systems

homology computation by reduction of chain complexes abstract a new algorithm for computing the homology module of a finitely generated chain complex is given it is based on local one step reductions of the size of the initial chain complex and it has a clear geometrical interpretation the complexity of the algorithm is discussed in special cases

existence and instability of steady states for a triangular cross diffusion system a computer assisted proof abstract in this paper we present and apply a computer assisted method to study steady states of a triangular cross diffusion system our approach consist in an a posteriori validation procedure that is based on using a fixed point argument around a numerically computed solution in the spirit of the newton kantorovich theorem it allows to prove the existence of various non homogeneous steady states for different parameter values in some situations we obtain as many as 13 coexisting steady states we also apply the a posteriori validation procedure to study the linear stability of the obtained steady states proving that many of them are in fact unstable

homoclinic solutions for a class of subquadratic second order hamiltonian systems in this paper we study the existence of infinitely many homoclinic solutions for a class of subquadratic second order hamiltonian systems by using the variant fountain theorem we obtain a new criterion for guaranteeing that second order hamiltonian systems has infinitely many homoclinic solutions recent results from the literature are generalized and significantly improved an example is also given in this paper to illustrate our main results

infinitely many homoclinic solutions for nonautonomous p t laplacian hamiltonian systems by using the symmetric mountain pass theorem we establish some existence criteria which guarantee that the second order ordinary p t laplacian systems of the form ddt email protected t p t email protected t a t u t p t 2u t email protected w t u t 0 have infinitely many homoclinic solutions where email protected r email protected r n email protected c r r and p t 1 email protected c r r and email protected c 1 rxr n r are non periodic in t

infinitely many homoclinic orbits for hamiltonian systems with indefinite sign subquadratic potentials abstract in this paper we deal with the existence and multiplicity of homoclinic solutions of the second order hamiltonian system u t l t u t w t u t 0 where l t and w t x are neither autonomous nor periodic in t under the assumption that w t x is indefinite sign and subquadratic as x and l t is a n n real symmetric positive definite matrices for all t r we establish some existence criteria to guarantee that the above system has at least one or infinitely many homoclinic solutions by using the genus properties in critical theory

canonical structure and symmetries of the schlesinger equations the schlesinger equations s n m describe monodromy preserving deformations of order m fuchsian systems with n 1 poles they can be considered as a family of commuting time dependent hamiltonian systems on the direct product of n copies of m m matrix algebras equipped with the standard linear poisson bracket in this paper we present a new canonical hamiltonian formulation of the general schlesinger equations s n m for all n m and we compute the action of the symmetries of the schlesinger equations in these coordinates

a shooting approach to the lorenz equations we announce and outline a proof of the existence of a homoclinic orbit of the lorenz equations in addition we develop a shooting technique and two key conditions which lead to the existence of a one to one correspondence between a set of solutions and the set of all infinite sequences of 1 s and 3 s

existence of homoclinic solutions for a class of time dependent hamiltonian systems n 1 n n 2 where q s q q g r w g c r r r and l t g c r r 1 n is a positive definite symmetric matrix for all t g r the case where l t and w t q are either periodic in t or independent of t were studied by several authors see for instance ambrosetti and w x w x bertotti 1 coti zelati ekeland and sere 4 coti zelati and rabinowitz w x w x w x 5 rabinowitz 15 17 sere 19 and their references in this kind of problem the function l t plays an important role for example if l t s ai where a 0 and i denotes the identity matrix n n w x felmer and silva 8 have showed the existence of homoclinic orbits as the limit of subharmonic solutions provided that hs is autonomous see also w x w x rabinowitz and tanaka 18 serra tarallo and terracini 20 using an w x w x approach introduced by 4 see also 19 treated the hs problem

homoclinic orbits for second order hamiltonian systems possessing superquadratic potentials received by the editors november 27 1990 1991 mathematics subject classification primary 34c99 58e99 58f99 the first author was supported by ministero p i gruppo 40 calcolo delle variazioni the second author s research was sponsored in part by the u s army research office under contract daal03 87 k 0043 the national science foundation under grant mcs 8110556 and the office of naval research under grant n00014 88 ko134 any reproduction for the purpose of the united states govemment is permitted

the connection map for attractor repeller pairs on montre que l application de connexion est additive les proprietes de l application connexion conduisent a une caracterisation des ensembles invariants isoles avec des points critiques hyperboliques comme une paire attracteur repulseur

connections between morse sets for delay differential equations on etudie la dynamique globale de l equation differentielle a retard scalaire x t f x t x t 1 avec des conditions appropriees sur f

homoclinic orbits for a class of hamiltonian systems consider the second order hamiltonian system where q n and v c 1 n is t periodic in t suppose v q t 0 0 0 is a local maximum for v t and v t x x under these and some additional technical assumptions we prove that hs has a homoclinic orbit q emanating from 0 the orbit q is obtained as the limit as k of 2 kt periodic solutions i e subharmonics q k of hs the subharmonics q k are obtained in turn via the mountain pass theorem

homoclinic solutions for a class of second order hamiltonian systems abstract some new results for the existence of homoclinic orbits are obtained for the second order hamiltonian systems u t v t u t f t where t r u r n and v c 1 r r n r v t x k t x w t x is t periodic with respect to t t 0 and f r r n is a continuous and bounded function our assumptions on k and f are rather loose

homoclinic solutions for a class of second order hamiltonian systems abstract a new result for existence of homoclinic orbits is obtained for the second order hamiltonian systems u t l t u t w 1 t u t w 2 t u t f t where t r u r n and w 1 w 2 c 1 r r n r and f c r r n are not necessary periodic in t this result generalizes and improves some existing results in the literature

a proof of wright s conjecture abstract wright s conjecture states that the origin is the global attractor for the delay differential equation y t y t 1 1 y t for all 0 2 when y t 1 this has been proven to be true for a subset of parameter values we extend the result to the full parameter range 0 2 and thus prove wright s conjecture to be true our approach relies on a careful investigation of the neighborhood of the hopf bifurcation occurring at 2 this analysis fills the gap left by complementary work on wright s conjecture which covers parameter values further away from the bifurcation point furthermore we show that the branch of slowly oscillating periodic orbits originating from this hopf bifurcation does not have any subsequent bifurcations and in particular no folds for 2 2 6 830 10 3 when combined with other results this proves that the branch of slowly oscillating solutions that originates from the hopf bifurcation at 2 is globally parametrized by 2

homoclinics for asymptotically quadratic and superquadratic hamiltonian systems abstract in this paper we find new conditions which are different from those used in previous related studies to ensure the existence of infinitely many homoclinic orbits for the second order hamiltonian systems of the form q v q t q here we assume that v t q depends periodically on t and assume on q that v t q is asymptotically quadratic at q 0 and is as q either asymptotically quadratic or superquadratic as well as the new conditions

existence of even homoclinic orbits for second order hamiltonian systems some existence theorems for even homoclinic orbits are obtained for a class of second order nonautonomous hamiltonian systems with symmetric potentials under a class of new superquadratic conditions a homoclinic orbit is obtained as a limit of solutions of a certain sequence of nil boundary value problems which are obtained by the minimax methods

simplicial models for the global dynamics of attractors abstract given an unknown attractor a in a continuous dynamical system how we can discover the topology and dynamics of a as a practical matter how can we do so from only a finite amount of information one way of doing so is to produce a semi conjugacy from a onto a model system m whose topology and dynamics are known the complexity of m then provides a lower bound for the complexity of a in this paper we use the techniques of the conley index to construct a simplicial model and a surjective semi conjugacy for a large class of attractors the essential features of this construction are that the model m can be explicitly described and that the finite amount of information needed to construct it is computable

morse decompositions for delay differential equations abstract the attractor for the scalar delay differential equation x t f x t x t 1 is shown to possess a morse decomposition under the hypothesis of a negative feedback condition in the delay term using this decomposition we prove results on the asymptotic rate of oscillation of solutions of the initial value problem existence of both slowly and rapidly oscillating periodic solutions is also proved

rigorous verification of chaos in a molecular model the thiele wilson system a simple model of a linear triatomic molecule has been studied extensively in the past the system exhibits complex molecular dynamics including dissociation periodic trajectories and bifurcations in addition it has for a long time been suspected to be chaotic but this has never been proved with mathematical rigor in this paper we present numerical results that using interval methods rigorously verify the existence of transversal homoclinic points in a poincar e map of this system by a theorem of smale the existence of transversal homoclinic points in a map rigorously proves its mixing property i e the chaoticity of the system

existence of homoclinic solution for the second order hamiltonian systems abstract an existence theorem of homoclinic solution is obtained for a class of the nonautonomous second order hamiltonian systems u t l t u t w t u t 0 t r by the minimax methods in the critical point theory specially the generalized mountain pass theorem where l t is unnecessary uniformly positively definite for all t r and w t x satisfies the superquadratic condition w t x x 2 as x uniformly in t and need not satisfy the global ambrosetti rabinowitz condition

connected simple systems and the conley index of isolated invariant sets the object of this paper is to present new and simplified proofs for most of the basic results in the index theory for flows simple explicit formulae are derived for the maps which play a central role in the theory the presentation is self contained

homoclinic solutions of some second order non autonomous systems abstract in this paper we investigate the existence of homoclinic solutions for the following second order non autonomous system q a q l t q w q t q 0 ds where a is an antisymmetric constant matrix l c r r n 2 is a symmetric and positive definite matrix for all t r w t q a t v q such that a r r is a continuous function and v c 1 r n r assuming that v q is subquadratic as q and some technical assumptions on a and l we establish two existence criteria to guarantee that ds has at least one nontrivial homoclinic solution by using a standard minimizing argument besides that in some particular case for the first time the uniqueness of homoclinic solutions of ds is also obtained recent results in the literature are generalized and significantly improved

chaotic attractors of an infinite dimensional dynamical system abstract we study the chaotic attractors of a delay differential equation the dimension of several attractors computed directly from the definition agrees to experimental resolution with the dimension computed from the spectrum of lyapunov exponents according to a conjecture of kaplan and yorke assuming this conjecture to be valid as the delay parameter is varied from computations of the spectrum of lyapunov exponents we observe a roughly linear increase from two to twenty in the dimension while the metric entropy remains roughly constant these results are compared to a linear analysis and the asymptotic behavior of the lyapunov exponents is derived

homotopy invariants of repeller attractor pairs ii continuation of r a pairs on considere une famille continue a 1 parametre de flots dans le plan parametrise par 0 1 on etablit un cadre qui rend precise la notion de famille continue de paires de repulseur attracteurs on montre qu il y a des equivalences naturelles entre les suites d indices de 2 paires r a le long d un parcours de paires r a

finite branch solutions to painleve vi around a fixed singular point every finite branch local solution to the sixth painleve equation around a fixed singular point is an algebraic branch solution in particular a global solution is an algebraic solution if and only if it is finitely many valued globally the proof of this result relies on algebraic geometry of painleve vi riemann hilbert correspondence geometry and dynamics on cubic surfaces resolutions of kleinian singularities and power geometry of algebraic differential equations in the course of the proof we are also able to classify all finite branch solutions up to backlund transformations

non periodic damped vibration systems with sublinear terms at infinity infinitely many homoclinic orbits abstract in this paper we study a class of non periodic damped vibration systems with subquadratic terms we obtain infinitely many nontrivial homoclinic orbits by a variant fountain theorem developed recently by zou to the best of our knowledge there is no result published concerning the existence of infinitely many homoclinic orbits for this class of non periodic damped vibration systems with subquadratic terms at infinity

homoclinic solutions for a class of non autonomous subquadratic second order hamiltonian systems abstract in this paper we consider the existence of homoclinic solutions for the following second order non autonomous hamiltonian system hs q l t q w q t q 0 where l t c r r n 2 is a symmetric and positive definite matrix for all t r w t q a t q with a t r r is a positive continuous function and 1 2 is a constant adopting some other reasonable assumptions for l and w we obtain a new criterion for guaranteeing that hs has one nontrivial homoclinic solution by use of a standard minimizing argument in critical point theory recent results from the literature are generalized and significantly improved

monodromy of certain painleve vi transcendents and reflection groups we study the global analytic properties of the solutions of a particular family of painleve vi equations with the parameters 0 1 2 and 2 2 1 2 with arbitrary 2 we introduce a class of solutions having critical behaviour of algebraic type and completely compute the structure of the analytic continuation of these solutions in terms of an auxiliary reflection group in the three dimensional space the analytic continuation is given in terms of an action of the braid group on the triples of generators of the reflection group we show that the finite orbits of this action correspond to the algebraic solutions of our painleve vi equation and use this result to classify all of them we prove that the algebraic solutions of our painleve vi equation are in one to one correspondence with the regular polyhedra or star polyhedra in the three dimensional space

systems of differential delay equations floquet multipliers and discrete lyapunov functions abstract we define a discrete integer valued lyapunov function v for cyclic nearest neighbor systems of differential delay equations possessing a feedback condition this extends analogous definitions for cyclic systems of ode s and for scalar differential delay equations we relate the values of v to the real parts of the floquet multipliers for such linear periodic systems and thereby prove all floquet subspaces are at most two dimensional

homoclinic solutions for nonautonomous second order hamiltonian systems with a coercive potential abstract a new existence result of homoclinic orbits is obtained for the second order hamiltonian systems u t f t u t f t where f t x is periodic with respect to t this result generalizes some known results in the literature

computer assisted proofs in analysis and programming in logic a case study in this paper we present a computer assisted proof of the existence of a solution for the feigenbaum equation varphi x frac 1 lambda varphi varphi lambda x there exist by now various such proofs in the literature although the one presented here is new the main purpose of this paper is not to provide yet another version but to give an easy to read and self contained introduction to the technique of computer assisted proofs in analysis our proof is written in prolog programming in logic a programming language which we found to be well suited for this purpose in this paper we also give an introduction to prolog so that even a reader without prior exposure to programming should be able to verify the correctness of the proof

chaos in the lorenz equators a computer assisted proof part ii details details of a new technique for obtaining rigorous results concerning the global dynamics of nonlinear systems is described the technique combines abstract existence results based on the conley index theory with rigorous computer assisted computations as an application of these methods it is proven that for some explicit parameter values the lorenz equations exhibit chaotic dynamics

infinitely many homoclinic solutions for some second order hamiltonian systems abstract in this paper we investigate the existence of infinitely many homoclinic solutions for a class of second order hamiltonian systems by using fountain theorem due to zou we obtain two new criteria for guaranteeing that second order hamiltonian systems have infinitely many homoclinic solutions recent results in the literature are generalized and significantly improved

existence of homoclinic orbits for 2nth order nonlinear difference equations containing both many advances and retardations abstract by establishing a proper variational framework and using the critical point theory we establish some new existence criteria to guarantee the 2nth order nonlinear difference equation containing both many advances and retardations n r t n n u t n q t u t f t u t n u t u t n n z 3 t z has at least one or infinitely many homoclinic orbits where r q f are nonperiodic in t our conditions on the potential are rather relaxed and some existing results in the literature are improved

on the global dynamics of attractors for scalar delay equations a semi conjugacy from the dynamics of the global attractors for a family of scalar delay differential equations with negative feedback onto the dynamics of a simple system of ordinary differential equations is constructed the construction and proof are done in an abstract setting and hence are valid for a variety of dynamical systems which need not arise from delay equations the proofs are based on the conley index theory institute for dynamics university of cincinnati cincinnati ohio 45221 0025 e mail address chris mccorduc edu center for dynamical systems and nonlinear studies georgia institute of tech nology atlanta georgia 30332 e mail address mischaikfmath gatech edu this content downloaded from 157 55 39 27 on wed 07 sep 2016 05 19 39 utc all use subject to http about jstor org terms

leray functor and cohomological conley index for discrete dynamical systems we introduce the leray functor on the category of graded modules equipped with an endomorphism of degree zero and we use this functor to define the cohomological conley index of an isolated invariant set of a homeomorphism on a locally compact metric space we prove the homotopy and additivity properties for this index and compute the index in some examples as one of applications we prove the existence of nonconstant bounded solutions of the euler approximation of a certain system of ordinary differential equations

topological invariants mulitvalued maps and computer assisted proofs in dynamics abstract we introduce the concept of inheritable properties of multivalued maps and show how this concept when combined with interval arithmetic and representable multivalued approximations may be used in adapting topological methods to computer assisted proofs in dynamics as our first example we propose a simple geometric criterion for chaos a more sophisticated example is the conley index theory we show how the computation for the conley index for multivalued representable maps may be reduced to a finite computation this reduction was used in the computer assisted proof of chaos in the lorenz equations represented in 1

some results on connecting orbits for a class of hamiltonian systems abstract the existence of various kinds of connecting orbits is established for a certain hamiltonian system as well as its time dependent analogue for the autonomous case our main assumption is that v has a global maximum e g at x o and we find a various kinds of orbits terminating at o for the time dependent case v has a local but not global maximum at x o and we find a homoclinic orbit emanating from and terminating at o keywords periodic solution

a proof of jones conjecture abstract in this paper we prove that wright s equation y t y t 1 1 y t has a unique slowly oscillating periodic solution for parameter values 2 1 9 up to time translation this result proves jones conjecture formulated in 1962 that there is a unique slowly oscillating periodic orbit for all 2 furthermore there are no isolas of periodic solutions to wright s equation all periodic orbits arise from hopf bifurcations

multi valued characteristics and morse decompositions abstract a rapid growth of molecular and systems biology in recent years challenges mathematicians to develop robust modeling and analytical tools for this area we combine a theory of monotone input output systems with a classical theory of morse decompositions in the context of ordinary differential equations models of biochemical reactions we show that a multi valued input output characteristic can be used to define non trivial morse decompositions which provide information about a global structure of the attractor the previous work on input output characteristics is shown to apply locally to individual morse sets and is seamlessly incorporated into our global theory we apply our tools to a model of cell cycle maintenance we show that changing the strength of the negative feedback loop can lead to cessation of cell cycle in two different ways it can either lead to globally attracting equilibrium or to a pair of equilibria that attract almost all solutions

non schlesinger deformations of ordinary differential equations with rational coefficients we consider deformations of 2 2 and 3 3 matrix linear odes with rational coefficients with respect to singular points of fuchsian type which do not satisfy the well known system of schlesinger equations or its natural generalization some general statements concerning the reducibility of such deformations for 2 2 odes are proved an explicit example of the general non schlesinger deformation of 2 2 matrix odes of the fuchsian type with four singular points is constructed and application of such deformations to the construction of special solutions of the corresponding schlesinger systems is discussed some examples of isomonodromic and non isomonodromic deformations of 3 3 matrix odes are considered the latter arise as the compatibility conditions with linear odes with non single valued coefficients

index filtrations and the homology index braid for partially ordered morse decompositions abstract on a morse decomposition of an invariant set in a flow there are partial orderings defined by the flow these are called admissable orderings of the morse decomposition the index filtrations for a total ordering of a morse decomposition are generalized in this paper with the definition and proof of existence of index filtrations for admissable partial orderings of a morse decomposition it is shown that associated to an index filtration there is a collection of chain complexes and chain maps called the chain complex braid of the index filtration the homology index braid of the corresponding admissable ordering of the morse decomposition is obtained by passing to homology in the chain complex braid keywords morse decomposition conley index index filtration

computation of highly ramified coverings an almost belyi covering is an algebraic covering of the projective line such that all ramified points except one simple ramified point lie above a set of 3 points of the projective line in general there are 1 dimensional families of these coverings with a fixed ramification pattern that is hurwitz spaces for these coverings are curves in this paper three almost belyi coverings of degrees 11 12 and 20 are explicitly constructed we demonstrate how these coverings can be used for computation of several algebraic solutions of the sixth painleve equation

circular pentagons and real solutions of painleve vi equations we study real solutions of a class of painleve vi equations to each such solution we associate a geometric object a one parametric family of circular pentagons we describe an algorithm that permits to compute the numbers of zeros poles 1 points and fixed points of the solution on the interval 1 infty and their mutual position the monodromy of the associated linear equation and parameters of the painleve vi equation are easily recovered from the family of pentagons

a planar topological horseshoe theory with applications to computer verifications of chaos in this paper we present an elementary theory on the existence and robustness of horseshoes under perturbations in terms of stability of crossing the framework is developed in the setting of 2d euclidean space but can be generalized to metric spaces as an application we give a rigorous verification of the existence of a horseshoe in the ikeda map

the continuation theory for morse decompositions and connection matrices the continuation theory for ordered morse decompositions and the indices defined on them the honmology index braid and the connection matrices is established the equivalence between ordered morse decompositions and consistent attractor filtrations is displayed the spaces of ordered morse decompositions for a product parametrization of a local flow are introduced and the local continuation of ordered morse decompositions is obtained via the above described equivalence and the local continuation of attractors the homology index braid and the connection matrices of an admissible ordering of a morse decomposition are shown to be invariant on path components of the corresponding space of ordered morse decompositions this invariance is used to prove that the collection of connection matrices of a morse decomposition is upper semicontinuous over the space of morse decompositions and over the parameter space under local continuation introduction a theorem that describes the behavior of an index under continuous change in its domain is an important component of any index theory the conley index is defined for isolated invariant sets in a local flow and in 1 it is shown that for a product parameterization of a local flow isolated invariant sets continue locally and the conley index is invariant under the local continuation in 10 kurland establishes a corresponding continuation theory for the index defined on attractor repeller pairs it is the aim of this paper to extend these continuation results to a continuation theory for ordered morse decompositions and the indices the homology index braid and the connection matrices defined on them conley s and kurland s continuation theories actually contain more than the local continuation results described above these local results are a part of a broader global continuation scheme in particular in 1 conley defines the space of isolated invariant sets for a product parameterization of a local flow and shows that this space is locally homeomorphic to the parameter space this local homeomorphism then yields the notion of local continuation of isolated invariant sets next it is shown that the conley index is invariant on path components of the space of isolated invariant sets this yields the invariance of the index under local continuation in this paper we use the same approach we define the spaces of ordered morse decompositions for a product parameterization of a local flow the notion of local continuation of morse decompositions and of admissible orderings of morse received by the editors september 20 1987 1980 mathematics subject classification 1985 revision primary 58f35 secondary 58f12 34c35

on structural stability of ordinary differential equations with respect to discretization methods on compact m dimensional discs morse smale differential systems having no periodic orbits are considered the main result is that they are correctly reproduced by one step discretization methods for methods of order p and stepsize h sufficiently small the time h map of the induced local flow and the h discretized system are joined by a conjugacy o h p near to the identity the paper fits well in the rapidly growing list of results stating that hyperbolic transversal structures are preserved by discretization the proof relies heavily on techniques elaborated by robbin 1971 in establishing his structural stability theorem on self diffeomorphisms of compact c infty manifolds

transformations rs42 3 of the ranks 4 and algebraic solutions of the sixth painleve equation compositions of rational transformations of independent variables of linear matrix ordinary differential equations odes with the schlesinger transformations rs transformations are used to construct algebraic solutions of the sixth painleve equation rs transformations of the ranks 3 and 4 of 2 2 matrix fuchsian odes with 3 singular points into analogous ode with 4 singular points are classified

monodromy preserving deformation of linear ordinary differential equations with rational coefficients iii abstract a unified treatment of monodromy and spectrum preserving deformations is presented in particular a general procedure is described to reduce the latter into the former consistently the concept of the function previously introduced for the former 2 is extended to the isospectral context it is shown that the general monodromy and spectrum preserving deformation equations can be written as hirota s bilinear differential equations by using the functions as dependent variables

homoclinic solutions for ordinary p laplacian systems with a coercive potential abstract a result for the existence of homoclinic orbits is obtained for p laplacian systems d d t u t p 2 u t f t u t f t where p 1 u r n f c 1 r r n r is t periodic with respect to t and f r r n is a continuous and bounded function such that f t x f t 0 b x and r f t 1 d t for some b 0 and 1

algebraic transition matrices in the conley index theory we introduce the concept of an algebraic transition matrix these are degree zero isomorphisms which are upper triangular with respect to a partial order it is shown that all connection matrices of a morse decomposition for which the partial order is a series parallel admissible order are related via a conjugation with one of these transition matrices this result is then restated in the form of an existence theorem for global bifurcations simple exarnples of how these results can be applied are also presented

existence and multiplicity of homoclinic solutions for a class of damped vibration problems abstract the main purpose of this paper is to study the following damped vibration problems 1 1 u t b u t a t u t f t u t a e t r u t 0 u t 0 as t where a a i j t c r r n 2 is an n n symmetric matrix valued function b b i j is an antisymmetry n n constant matrix f c 1 r r n r and f t u u f t u by a symmetric mountain pass theorem and a generalized mountain pass theorem an existence result and a multiplicity result of homoclinic solutions of 1 1 are obtained