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

shape and morse theory of attractors we study the shape in the sense of borsuk of attractors of continuous semi dynamical systems on general metric spaces we show in particular that the natural inclusion of the global attractor into the state space is a shape equivalence this and other results of the paper are used to develop an elementary morse theory of an attractor 2000 john wiley sons inc

a natural order in dynamical systems based on conley markov matrices abstract we introduce a natural order to study properties of dynamical systems especially their invariant sets the new concept is based on the classical conley index theory and transition probabilities among neighborhoods of different invariant sets when the dynamical systems are perturbed by white noises the transition probabilities can be determined by the fokker planck equation and they form a matrix called a markov matrix in the limiting case when the random perturbation is reduced to zero the markov matrix recovers the information given by the conley connection matrix the markov matrix also produces a natural order from the least to the most stable invariant sets for general dynamical systems in particular it gives the order among the local extreme points if the dynamical system is a gradient like flow of an energy functional consequently the natural order can be used to determine the global minima for gradient like systems some numerical examples are given to illustrate the markov matrix and its properties

connected simple systems transition matrices and heteroclinic bifurcations given invariant sets a b and c and connecting orbits a b and b c we state very general conditions under which they bifurcate to produce an a c connecting orbit in particular our theorem is applicable in settings for which one has an admissible semiflow on an isolating neighborhood of the invariant sets and the connecting orbits and for which the conley index of the invariant sets is the same as that of a hyperbolic critical point our proof depends on the connected simple system associated with the conley index for isolated invariant sets

oscillation of certain neutral delay parabolic equations abstract sufficient conditions are given to ensure the solutions of a class of neutral parabolic equations with continuous distributed deviating arguments to oscillate

urn models replicator processes and random genetic drift to understand the relative importance of natural selection and random genetic drift in finite but growing populations the asymptotic behavior of a class of generalized polya urns is studied using the method of ordinary differential equation ode of particular interest is the replicator process two balls individuals are chosen from an urn the population at random with replacement and balls of the same colors strategies are added or removed according to probabilities that depend only on the colors of the chosen balls under the assumption that the expected number of balls being added always exceeds the expected number of balls being removed whenever balls are in the urn the probability of nonextinction is shown to be positive on the event of nonextinction three results are proven i the number of balls increases asymptotically at a linear rate ii the distribution x n of strategies at the nth update is a noisy cauchy euler approximation to the mean limit ode of the process and iii the limit set of x n is almost surely a connected internally chain recurrent set for the mean limit ode under a stronger set of assumptions it is shown that for any attractor of the mean limit ode there is a positive probability that the limit set for x n lies in this attractor theoretical and numerical estimates for the probabilities of nonextinction and convergence to an attractor suggest that random genetic drift is more likely to overcome natural selection in small populations for which pairwise interactions lead to highly variable outcomes and is less likely to overcome natural selection in large populations with the potential for rapid growth

oscillation of parabolic equations of neutral type in this paper we obtain new sufficient conditions for the oscillation of a class of neutral parabolic differential equations satisfying two kinds of boundary value conditions

integral average method for oscillation of second order partial differential equations with delays using a generalized riccati transformation some new oscillation criteria for second order partial differential equations with delays are found through the method of integral average these results can be considered as generalizations and improvements of the results due to kamenev in ordinary differential equations cases

flow rule of dense granular flows down a rough incline we present experimental findings on the flow rule for granular flows on a rough inclined plane using various materials including sand and glass beads of various sizes and four types of copper particles with different shapes we characterize the materials by measuring h s the thickness at which the flow subsides as a function of the plane inclination ensuremath theta on various surfaces measuring the surface velocity u of the flow as a function of flow thickness h we find that for sand and glass beads the pouliquen flow rule u sqrt gh ensuremath sim ensuremath beta h h s provides reasonable but not perfect collapse of the u h curves measured for various ensuremath theta and mean particle diameter d improved collapse is obtained for sand and glass beads by using a recently proposed scaling of the form u sqrt gh ensuremath beta h phantom rule 0 2em 0ex mathrm tan 2 phantom rule 0 2em 0ex ensuremath theta h s phantom rule 0 2em 0ex mathrm tan 2 phantom rule 0 2em 0ex ensuremath theta 1 where ensuremath theta 1 is the angle at which the h s ensuremath theta curves diverge measuring the slope ensuremath beta for ten different sizes of sand and glass beads we find a systematic strong increase of ensuremath beta with the divergence angle ensuremath theta 1 of h s copper materials with different shapes are not well described by either flow rule with u ensuremath sim h 3 2

the connection matrix theory for semiflows on not necessarily locally compact metric spaces abstract the index theory of rybakowski for isolated invariant sets and attractor repeller pairs in the setting of a semiflow on a not necessarily locally compact metric space is extended to include a connection matrix theory for morse decompositions partially ordered morse decompositions and attractor semifiltrations of invariant sets are defined and shown to be equivalent the definition and proof of existence of index filtrations for an ordered morse decomposition is provided via the index filtration the homology index braid and the connection matrices of the morse decomposition are defined

lattice structures for attractors i we describe the basic lattice structures of attractors and repellers in dynamical systems the structure of distributive lattices allows for an algebraic treatment of gradient like dynamics in general dynamical systems both invertible and noninvertible we separate those properties which rely solely on algebraic structures from those that require some topological arguments in order to lay a foundation for the development of algorithms to manipulate these structures computationally

nonlocal rheological properties of granular flows near a jamming limit we study the rheology of sheared granular flows close to a jamming transition we use the approach of partially fluidized theory pft with a full set of equations extending the thin layer approximation derived previously for the description of the granular avalanches phenomenology this theory provides a picture compatible with a local rheology at large shear rates g d r midi eur phys j e 14 341 2004 and it works in the vicinity of the jamming transition where a description in terms of a simple local rheology comes short we investigate two situations displaying important deviations from local rheology the first one is based on a set of numerical simulations of sheared soft two dimensional circular grains the next case describes previous experimental results obtained on avalanches of sandy material flowing down an incline both cases display close to jamming significant deviations from the now standard pouliquen s flow rule o pouliquen phys fluids 11 542 1999 11 1956 1999 this discrepancy is the hallmark of a strongly nonlocal rheology and in both cases we relate the empirical results and the outcomes of pft the numerical simulations show a characteristic constitutive structure for the fluid part of the stress involving the confining pressure and the material stiffness that appear in the form of an additional dimensionless parameter this constitutive relation is then used to describe the case of sandy flows we show a quantitative agreement as far as the effective flow rules are concerned a fundamental feature is identified in pft as the existence of a jammed layer developing in the vicinity of the flow arrest that corroborates the experimental findings finally we study the case of solitary erosive granular avalanches and relate the outcome with the pft analysis

the effect of sidewall friction on dense granular flows we present novel results on flow properties using molecular dynamics simulation as well as c densitometry experiments we show that fundamentally different types of granular flow can be observed in one unique system using both frictional and frictionless confining walls we show that the differences existing between these regimes originate in the frictional properties of the walls the salient features of each regime are described in detail in particular the packing fraction and streamwise velocity profiles are shown to hinge on the frictional properties of the confining walls finally we propose a dimensionless number independent of the grain size which helps to delineate the different regimes

poincare lefschetz duality for the homology conley index the conley index for continuous dynamical systems is defined for one sided semiflows for two sided flows there are two indices defined one for the forward flow and one for the reverse flow in general the two indices give different information about the flow but for flows on orientable manifolds there is a duality isomorphism between the homology conley indices of the forward and reverse flows this duality preserves the algebraic structure of many of the constructions of the conley index theory sums and products continuation attractor repeller sequences and connection matrices

the morse index repeller attractor pairs and the connection index for semiflows on noncompact spaces abstract this paper extends the morse index theory of c c conley to semiflows on a noncompact meric space x is assumed to satisfy a hypothesis related to conditional contraction we collect background material define quasi index pairs and the morse index of a compact isolated invariant set k and prove that the morse index is a connected simple system we study repeller attractor pairs in k define index triples and prove their existence and several properties leading to the concepts of the connection index the connection map and the splitting class finally we consider paths continuous families of pairs k and study continuations of the morse and the connection indices along such paths the present paper is a sequel to the author s previous work on the homotopy index for infinite dimensional semiflows trans amer math soc 269 1982 351 382

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

oscillation of solutions of neutral partial functional differential equations abstract sufficient conditions are established for the oscillation of solutions of neutral partial functional differential equations of the form t p t t u x t i 1 l i t u x t i a t u x t k 1 s a k t u x t k t q x t u x t j 1 m q j x t f j u x t j x t 0 g where is a bounded domain inrnwith a piecewise smooth boundary and is the laplacian in the euclideann spacern

nested sequences of index filtrations and continuation of the connection matrix abstract in this paper we prove the existence of nested sequences of index filtrations for convergent sequences of admissible semiflows on a metric space this result is new even in the context of flows on a locally compact space the nested index filtration theorem implies the continuation of homology index braids which in turn implies the continuation of connection matrices in the infinite dimensional conley index theory

the connection matrix in morse smale flows ii given a connection matrix for a morse smale flow on a compact manifold if there are no periodic orbits of equal or adjacent indices related in the partial order we show that the periodic orbits can be replaced by doubly connected rest points in such a way that the given connection matrix induces the unique connection matrix for the resulting flow it follows that for this class of flows all nonuniqueness in the connection matrix is a consequence of the continuation theorem for connection matrices

dry granular flows down an inclined channel experimental investigations on the frictional collisional regime this paper presents experimental results on dry granular flows down an inclined rough channel different flow regimes were identified depending on the froude number for froude numbers exceeding a critical value function of the channel slope flow was characterized by a fairly linear velocity profile and a discharge equation in the form q ensuremath propto h n with q the flow rate per unit width h the flow depth and n an exponent in the range 2 char21 3 regime a when the froude number was lower than the critical value the flow was characterized by a convex velocity profile and a discharge equation of the form q ensuremath propto h n with n ranging from 0 97 to 1 16 producing the striking result that the mean velocity was constant for a given inclination of the channel regime b experimental data were used to test three theoretical models developed to describe dry granular flows in a frictional collisional regime savage s model provides results that capture experimental trends well and yield the correct magnitude for velocity and discharge for regime a but it reproduces the dependence of the discharge on the channel slope for only a narrow range of slopes s b savage in u s japan seminar on new models and constitutive relations in the mechanics of granular materials ithaca 1982 edited by j t jenkins and m satake elsevier science publishers amsterdam 1982 p 261 in contrast mills et al s model is less refined and requires fitting an input parameter to give the correct magnitude of velocity but it successfully accounts for the variation in the discharge with slope for regime a for a wide range of slopes mills loggia and tixier europhys lett 45 733 1999 eur phys j e 1 5 2000 ancey and evesque s model is also crude in determining the density profile but manages to provide velocity profiles and discharge equations in good agreement with experimental data for regime b c ancey and p evesque phys rev e 62 8349 2000

conley index and tubular neighborhoods ii abstract this paper is a sequel to our previous paper 11 we prove existence of nested index filtrations for tubular singular semiflow perturbations this generalizes a corresponding result from 7 and is applicable to cases in which the phase space of the perturbed semiflow is not necessarily homeomorphic to a product of metric spaces having as a factor the phase space of the limiting semiflow this result implies tubular homology index braid and connection matrix continuation principles

existence of trajectories joining critical points for a symbiotic system via the conley index abstract in this paper we prove the existence of trajectories joining critical points for ecological systems with three species by constructing the index pair for the isolated invariant set and using the conley index theory

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

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

oscillation of solutions of systems of neutral type partial functional differential equations abstract sufficient conditions are established for the oscillation of solutions of systems of neutral type partial functional differential equations of the form where is a bounded domain in r n with a piecewise smooth boundary and is the laplacian in euclidean n space r n the obtained results are illustrated by some interesting examples

unstable manifold conley index and fixed points of flows we study dynamical and topological properties of the unstable manifold of isolated invariant compacta of flows we show that some parts of the unstable manifold admit sections carrying a considerable amount of information these sections enable the construction of parallelizable structures which facilitate the study of the flow from this fact many nice consequences are derived specially in the case of plane continua for instance we give an easy method of calculation of the conley index provided we have some knowledge of the unstable manifold and as a consequence a relation between the brouwer degree and the unstable manifold is established for smooth vector fields we study the dynamics of non saddle sets properties of existence or non existence of fixed points of flows and conditions under which attractors are fixed points morse decompositions preservation of topological properties by continuation and classify the bifurcations taking place at a critical point

oscillation of solutions of parabolic differential equations of neutral type in this paper we investigate a class of parabolic differential equations of neutral type and obtain some sufficient conditions of the oscillation for such equations satisfying two kinds of boundary value conditions

an efficient estimate based on fft in topological verification method in this paper localized patterns of the quintic swift hohenberg equation are studied a numerical verification method with the conley index theory developed in zgliczynski and mischaikow rigorous numerics for partial differential equations the kuramoto sivashinsky equation found comput math 1 2001 255 288 is used in order to prove these patterns a new technique to efficiently obtain estimates for nonlinear terms is presented the key idea is based on the pseudo spectral method it is shown that this technique is inevitable for the verification of the localized patterns

morse type inequalities for dynamical systems and the witten laplacian we provide an analytic proof of morse type inequalities for vector flelds determining a morse decomposition with normally hyperbolic dynamics in the demonstration we reduce the problem to the gradient case using a morse bott lyapunov function and

following homology in singularly perturbed systems abstract the fast systems associated to a system of first order equations where a small parameter multiplies the derivatives hereinafter an spmd system are obtained by stretching time by a factor of one over the small parameter and then reducing the small parameter to zero curves of isolated invariant sets associated to the fast system serve as approximate solutions to the spmd system over finite intervals in the following sense a manifold of initial points representing non zero homology in the morse conley index of the invariant set at the left endpoint has a submanifold which is carried by the flow of the spmd system to a manifold representing that non zero homology class in the index of the invariant set at the right endpint which is the continuation of along the curve of invariants sets this is a type of linearization of the approximation problem since the continuation of homology classes along the curve of invariant sets is given by a linear transformation which varies functorially with the curve of invariant sets examples are given for systems in r 2 and r 3 the results are used elsewhere to prove existence theorems for two point boundary value problems to obtain solutions with boundary and interior transition layers

h oscillation of solutions of certain vector hyperbolic differential equations with deviating arguments in this paper sufficient conditions are established for h oscillation of solutions of a class of vector hyperbolic differential equations with deviating arguments where h is a unit vector in r m the main results are illustrated by two examples

connecting orbits for nonlinear differential equations at resonance abstract we study the existence of orbits connecting stationary points for the first order differential equations being at resonance at infinity where the right hand side is the perturbations of a sectorial operator our aim is to prove an index formula expressing the conley index of associated semiflow with respect to appropriately large ball in terms of special geometrical assumptions imposed on the nonlinearity we also prove that the geometrical assumptions are generalization of the well known in literature landesman lazer and strong resonance conditions obtained index formula will be used to derive the criteria determining the existence of orbits connecting stationary points for the heat equation being at resonance at infinity

interval oscillation criteria for second order partial differential equations with delays in this paper we present new interval oscillation criteria related to integral averaging technique for second order partial differential equations with delays that are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of t 0 rather than on whole half line our results are sharper than some previous results and handles the cases which are not covered by known criteria

global topological properties of the hopf bifurcation we study the homotopical and homological properties of the attractors evolving from a generalized hopf bifurcation we consider the lorenz equations for parameter values near the hopf bifurcation and study a natural morse decomposition of the global attractor calculating the cech homotopy type of the lorenz attractor the shape indexes of the morse sets and the morse equation of the decomposition

granular surface flow via successive destabilization a continuum approach abstract global and localized granular surface flow is analyzed in a model that takes into account the basic forces between a flowing granular layer and an underlying granular bed starting from a quantitatively correct description of global stick slip avalanches on the surfaces of heaps we are able to describe localized flows of various types as seen in experiments of daerr and douady a daerr s douady two types of avalanche behavior in granular media nature london 399 1999 241 the descriptive level takes into account the free surface and the velocity of the flow solely we limit our model to cases where this surface is effectively one dimensional so that our model equations are consequently one dimensional in space with this kind of description we are able to account for the tight coupling between local grain velocity and local height of the so called lonely waves our model shows the nature of a wave of mobilization and deposition granular material is set into motion in a domino like manner in addition destabilization of material behind an avalanche is taken into account the front velocities are found to be constant and in general head and rear front travel with the same velocity in our model the front velocities depend on the tilt angle of the heap that those localized structures run down on a critical angle is observed above which the rear front velocity abruptly switches sign from positive downslope to negative upslope values furthermore we find global influences of the surface shape of a flowing granular layer on bagnold friction this characteristic collisional friction between flowing layers and beds is shown to cause s shaping of the free surface for rapid flow velocities finally our model also explains the power spectrum of avalanches as experimentally detected by jaeger et al h jaeger c liu s nagel relaxation at the angle of repose phys rev lett 62 1989 40

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

continuum theory of partially fluidized granular flows a continuum theory of partially fluidized granular flows is developed the theory is based on a combination of the equations for the flow velocity and shear stresses coupled with the order parameter equation which describes the transition between the flowing and static components of the granular system we apply this theory to several important granular problems avalanche flow in deep and shallow inclined layers rotating drums and shear granular flows between two plates we carry out quantitative comparisons between the theory and experiment

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

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

symplectic fixed points and holomorphic spheres letp be a symplectic manifold whose symplectic form integrated over the spheres inp is proportional to its first chern class on the loop space ofp we consider the variational theory of the symplectic action function perturbed by a hamiltonian term in particular we associate to each isolated invariant set of its gradient flow an abelian group with a cyclic grading it is shown to have properties similar to the homology of the conley index in locally compact spaces as an application we show that if the fixed point set of an exact diffeomorphism onp is nondegenerate then it satisfies the morse inequalities onp we also discuss fixed point estimates for general exact diffeomorphisms

computations of critical groups and applications to asymptotically linear wave equation and beam equation in this paper by using the morse index theory for strongly indefinite functionals developed in nonlinear anal tma in press we compute precisely the critical groups at the origin and at infinity respectively the abstract theorems are used to study the existence multiplicity of nontrivial periodical solutions for asymptotically wave equation and beam equation with resonance both at infinity and at zero

conley index condition for asymptotic stability in this paper we use conley index theory to develop necessary conditions for stability of equilibrium and periodic solutions of nonlinear continuous time systems the conley index is a topological generalization of the morse theory which has been developed to analyze dynamical systems using topological methods in particular the conley index of an invariant set with respect to a dynamical system is defined as the relative homology of an index pair for the invariant set the conley index can then be used to examine the structure of the system invariant set as well as the system dynamics within the invariant set including system stability properties efficient numerical algorithms using homology theory have been developed in the literature to compute the conley index and can be used to deduce the stability properties of nonlinear dynamical systems

singular dynamics of strongly damped beam equation abstract the paper is devoted to the dynamics of the model for a beam with strong damping pe e 2 u t t e u t u x x x x u t x x x x g 0 l u 2 d e 0 l u t u d u x x 0 x t 0 l 0 where g r r is continuously differentiable r and l e 0 subject to boundary conditions corresponding to hinged or clamped ends we show that for e 0 the dynamics of the equation is close to the dynamics of equation p0 u t u g 0 l u 2 d a 1 2 u where a u u x x x x with the domain determined by one of the above boundary conditions specifically we show that isolated invariant sets of p0 continue to isolated invariant sets of pe e 0 small having the same conley index moreover isolated morse decompositions with respect to p0 continue to isolated morse decompositions of pe e 0 small having isomorphic homology index braids under some additional assumptions we establish existence and upper semicontinuity results for attractors of e 0 small extending previous results by sevcovic

flows on vector bundles and hyperbolic sets this note deals with c conley s topological approach to hyperbolic invariant sets for continuous flows it is based on the notions of isolated invariant sets and morse decompositions and it leads to the concept of weak hyperbolicity

travelling wave solutions to a gradient system soit un systeme d equations de reaction diffusion ou la non linearite derive d un potentiel avec certaines restrictions on utilise l indice de conley et la matrice de connexion pour montrer qu il y a une solution onde de propagation reliant les maxima du potentiel

homotopical dynamics iii real singularities and hamiltonian flows on the space of nondepraved see real isolated singularities we consider the stable equivalence relation induced by smooth deformations whose asymptotic behaviour is controlled by the palais smale condition it is shown that the resulting space of equivalence classes admits a canonical semiring structure and is isomorphic to the semiring of stable homotopy classes of cw complexes in an application to hamiltonian dynamics we relate the existence of bounded and periodic orbits on noncompact level hypersurfaces of palais smale hamiltonians with just one singularity that is nondepraved to the lack of self duality in the sense of e spanier and j whitehead of the sublink of the singularity

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

transversality for local morse homology with symmetries and applications we prove the transversality result necessary for defining local morse chain complexes with finite cyclic group symmetry our arguments use special regularized distance functions constructed using classical covering lemmas and an inductive perturbation process indexed by the strata of the isotropy set a global existence theorem for symmetric morse smale pairs is also proved regarding applications we focus on hamiltonian dynamics and rigorously establish a local contact homology package based on discrete action functionals we prove a persistence theorem analogous to the classical shifting lemma for geodesics asserting that the iteration map is an isomorphism for good and admissible iterations we also consider a chas sullivan product on non invariant local morse homology which plays the role of pair of pants product and study its relationship to symplectically degenerate maxima finally we explore how our invariants can be used to study bifurcation of critical points and periodic points under additional symmetries

forced oscillation for certain systems of hyperbolic differential equations in this paper sufficient conditions are established for the forced oscillation of certain systems of hyperbolic differential equations with deviating arguments the main result is illustrated by an example

dissonant points and the region of influence of non saddle sets abstract the aim of this paper is to study dynamical and topological properties of a flow in the region of influence of an isolated non saddle set we see in particular that some topological conditions are sufficient to guarantee that these sets are attractors or repellers we study in detail the existence of dissonant points of the flow which play a key role in the description of the region of influence of a non saddle set these points are responsible for much of the dynamical and topological complexity of the system we also study non saddle sets from the point of view of the conley index theory and consider among other things the case of flows on manifolds with trivial first cohomology group for flows on these manifolds dynamical robustness is equivalent to topological robustness we carry out a particular study of 2 dimensional flows and give a topological condition which detects the existence of dissonant points for flows on surfaces we also prove that isolated invariant continua of planar flows with global region of influence are necessarily attractors or repellers

plug flow and the breakdown of bagnold scaling in cohesive granular flows cohesive granular media flowing down an inclined plane are studied by discrete element simulations previous work on cohesionless granular media demonstrated that within the steady flow regime where gravitational energy is balanced by dissipation arising from intergrain forces the velocity profile in the flow direction scales with depth in a manner consistent with the predictions of bagnold here we demonstrate that this bagnold scaling does not hold for the analogous steady flows in cohesive granular media we develop a generalization of the bagnold constitutive relation to account for our observation and speculate as to the underlying physical mechanisms responsible for the different constitutive laws for cohesive and noncohesive granular media

the conley index for fast slow systems ii multidimensional slow variable we develop a qualitative theory for fast slow systems with a one dimensional slow variable using conley index theory for singularity perturbed systems conditions are given which imply that if one can construct heteroclinic connections and periodic orbits in systems with the derivative of the slow variable set to 0 these orbits persist when the derivative of the slow variable is small and nonzero

detection of renewal system factors via the conley index let n be an isolating neighborhood for a map f if we can decompose n into the disjoint union of compact sets n 1 and n 2 then we can relate the dynamics on the maximal invariant set inv n to the shift on two symbols by noting which component of n each iterate of a point x inv n lies in we examine a method based on work by mischaikow szymczak et al for using the discrete conley index to detect explicit subshifts of the shift associated to n in essence we measure the difference between the conley index of inv n and the sum of the indices of inv n 1 and inv n 2

stochastic flow rule for granular materials there have been many attempts to derive continuum models for dense granular flow but a general theory is still lacking here we start with mohr coulomb plasticity for quasi two dimensional granular materials to calculate average stresses and slip planes but we propose a stochastic flow rule sfr to replace the principle of coaxiality in classical plasticity the sfr takes into account two crucial features of granular materials discreteness and randomness via diffusing spots of local fluidization which act as carriers of plasticity we postulate that spots perform random walks biased along slip lines with a drift direction determined by the stress imbalance upon a local switch from static to dynamic friction in the continuum limit based on a fokker planck equation for the spot concentration this simple model is able to predict a variety of granular flow profiles in flat bottom silos annular couette cells flowing heaps and plate dragging experiments with essentially no fitting parameters although it is only expected to function where material is at incipient failure and slip lines are inadmissible for special cases of admissible slip lines such as plate dragging under a heavy load or flow down an inclined plane we postulate a transition to rate dependent bagnold rheology where flow occurs by sliding shear planes with different yield criteria the sfr provides a general framework for multiscale modeling of plasticity in amorphous materials cycling between continuum limit state stress calculations mesoscale spot random walks and microscopic particle relaxation

morse type index theory for flows and periodic solutions for hamiltonian equations an index theory for flows is presented which extends the classical morse theory for gradient flows on compact manifolds the theory is used to prove a morse type existence statement for periodic solutions of a time dependent periodic in time and asymptotically linear hamiltonian equation

oscillation properties for systems of hyperbolic differential equations of neutral type abstract sufficient conditions are established for the oscillations of systems of hyperbolic differential equations of the form 2 t 2 p t u i x t r 1 d r t u i x t r a i t u i x t j 1 m k 1 s a ijk t u j x k t q i x t u i x t j 1 m h 1 l q ijh x t u j x h t x t 0 g i 1 2 m where is a bounded domain in rn with a piecewise smooth boundary and is the laplacian in euclidean n space rn

uniform persistence and hopf bifurcations in r n we consider parameterized families of flows in locally compact metrizable spaces and give a characterization of those parameterized families of flows for which uniform persistence continues on the other hand we study the generalized poincare andronov hopf bifurcations of parameterized families of flows at boundary points of r n or more generally of an n dimensional manifold and show that this kind of bifurcations produce a whole family of attractors evolving from the bifurcation point and having interesting topological properties in particular in some cases the bifurcation transforms a system with extreme non permanence properties into a uniformly persistent one we study in the paper when this phenomenon happens and provide an example constructed by combining a holling type interaction with a pitchfork bifurcation