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

birational rigidity of complete intersections we prove that every smooth complete intersection x x d 1 ldots d s subset mathbb p sum i 1 s d i defined by s hypersurfaces of degree d 1 ldots d s is birationally superrigid if 5s 1 le frac 2 sum i 1 s d i 1 sqrt prod i 1 s d i in particular x is non rational and mathrm bir x mathrm aut x we also prove birational superrigidity of singular complete intersections with similar numerical condition these extend the results proved by tommaso de fernex

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

on the hopf index and the conley index the following generalization of the poincare hopf index theorem is proved if s is an isolated invariant set of a flow on a manifold m then the sum of the hopf indices on s is equal up to a sign to the euler characteristic of the homology conley index of 5 one of the classical results in dynamical systems relating local and global information is the poincare hopf index theorem if sf is a vectorfield on a compact manifold m with isolated zeros then the intersection number of with the zero section of tm is defined to be the sum of the hopf indices of the zeros i ist f yhlx s if is without boundary or if af is outward on dm the poincare hopf theorem asserts that the intersection number of sf is equal to the homological euler characteristic of m i see 4 7 this has the restriction that the local information the hopf indices is only related to the global information the betti numbers when the vectorfield is outward on the boundary pugh 8 removes this restriction in the case of generic contact of with m by relating the intersection number to a sum of euler characteristics dm j where r is the codimensional exit region and is its boundary in this paper we further develop this result by relating the hopf index and intersection numbers to the conley index for dynamical systems suppose that x is a smooth vectorfield on m with s c m an isolated invariant set of the induced flow then the intersection number of a on a neighborhood of s is up to a sign the euler characteristic of the homology conley index of s in m the conley euler number 5 in particular if all zeros on af on s are nondegenerate then the conley index and the hopf index are related by the formula nx m s 2ix there are two aspects to this result first it gives the hopf index and intersection numbers a homological interpretation in a broad setting second it shows that the properties of the hopf index such as additivity stability etc are also properties of conley euler numbers to develop the theorem we first received by the editors february 8 1988 1980 mathematics subject classification 1985 revision primary 34c35 55m20 secondary 54h20 58f25 work supported in part by oarpa applied and computational mathematics program 1989 american mathematical society 0002 9947 89 1 00 5 25 per page

a predator prey system involving group defense a connection matrix approach the purpose of this paper is to give a concrete demonstration of how the connection matrix can be used to analyse a l parameter family of differential equations in particular we have four goals 1 to show how to compute connection matrices 2 to use these matrices to classify the structure of solutions to our set of differential equations for various parameter values 3 to prove the existence of local and global bifurcations 4 to demonstrate how to ignore certain bifurcations strange as it may seem we want to emphasize the importance of this last goal even in seemingly simple systems it is possible for global bifurcations to occur which are difficult to detect or exclude in our case global bifurcations which lead to the existence of multiple periodic orbits nevertheless one wants to be able to make significant statements about the general structure of the flow for all parameter values thus it is important to have techniques which are able to ignore subtle or difficult to detect phenomenon and still work the set of equations which we have chosen to study arise from a predator prey model in which the prey exhibits group defense i e the more prey the better their chances for protecting themselves from the predator this set of equations has been studied before see freedman and wolkowicz 3 wolkowicz 13 and mischaikow and wolkowicz 9 and we do not claim any new results rather it is our techniques which are novel to be more specific the system we consider is two dimensional and hence phase plane techniques can be employed to obtain the results which we shall present herein this is the approach taken in 13 of course phase plane techniques are more difficult if not impossible to apply in the case of higher dimensional systems the advantage of our approach is that the connection matrix is dimension independent and hence the tehniques we are describing here are in principle applicable to higher dimensional

some second order nonlinear systems arising in natural convection heat transfer abstract solutions to the second order nonlinear systems arising in natural convection flow and heat transfer in vertical channels are obtained furthermore existence uniqueness and concavity results are established by using green s function and degree theory

on global dynamics of reaction diffusion systems at resonance abstract in this paper we use the homotopy invariants methods to study the global dynamics of the reaction diffusion systems that are at resonance at infinity considering degrees of the resonance for the nonlinear perturbation we establish landesman lazer type conditions and use them to express the rybakowski conley index of the invariant set consisting of all bounded solutions obtained results are applied to study the existence of solutions connecting stationary points for the system of nonlinear heat equations

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

birational automorphisms of fano hypersurfaces v v y v 0 between smooth three dimensional quartics v v4 v 0 v 0 4 in p is a projective isomorphism this is the original wording of their theorem in fact the arguments of im give a much stronger result namely they actually prove that smooth quartics are birationally superrigid the concept of birational rigidity is not visual being rather technical it makes little impression it is immediate implications of the property of being birationally super rigid that are really impressive for a projective variety x smooth in codimension 1 and a linear system jdj free from xed components we de ne the threshold of canonical adjunction or brie y just threshold of the pair x d setting

unstable attractors in manifolds assume that k is a compact attractor with basin of attraction a k for some continuous flow phi in a space m stable attractors are very well known but otherwise without the stability assumption the situation can be extremely wild in this paper we consider the class of attractors with no external explosions where a mild form of instability is allowed after obtaining a simple description of the trajectories in a k k we study how k sits in a k by performing an analysis of the poincare polynomial of the pair a k k in case m is a surface we obtain a nice geometric characterization of attractors with no external explosions as well as a converse to the well known fact that the inclusion of a stable attractor in its basin of attraction is a shape equivalence finally we explore the strong relations which exist between the shape in the sense of borsuk of k and the shape in the intuitive sense of the whole phase space m much in the spirit of the morse conley theory

a cohomological conley index in hilbert spaces and applications to strongly indefinite problems abstract a cohomological conley index is defined for flows on infinite dimensional real hilbert spaces generated by vector fields of the form f h h f x lx k x where l h h is a bounded linear operator satisfying certain technical assumptions and k is a completely continuous perturbation generalized morse inequalities for morse decompositions of isolated invariant sets are proved simple examples are presented to show how the theory can be applied to strongly indefinite problems

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

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

homotopy invariance of the conley index and local morse homology in hilbert spaces in this paper we introduce a new compactness condition property c for flows in not necessary locally compact metric spaces for such flows a conley type theory can be developed for example regular index pairs always exist for property c flows and a conley index can be defined an important class of flows satisfying the this compactness condition are ls flows we apply e cohomology to index pairs of ls flows and obtain the e cohomological conley index we formulate a continuation principle for the e cohomological conley index and show that all ls flows can be continued to ls gradient flows we show that the morse homology of ls gradient flows computes the e cohomological conley index we use lyapunov functions to define the morse conley floer cohomology in this context and show that it is also isomorphic to the e cohomological conley index

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

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

shape of global attractors in topological spaces in this paper we apply the notion and properties of compactly generated shape to study attractors in topological spaces

conley index for random dynamical systems conley index theory is a very powerful tool in the study of dynamical systems in this paper we generalize conley index theory to discrete random dynamical systems our constructions are basically the random version of franks and richeson in j franks d richeson shift equivalence and the conley index trans amer math soc 352 2000 3305 3322 for maps and the relations of isolated invariant sets between time continuous random dynamical systems and corresponding time h maps are discussed two examples are presented to illustrate results in this paper

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

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

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

regular blocks and conley index of isolated invariant continua in surfaces abstract in this paper we study topological and dynamical features of isolated invariant continua of continuous flows defined on surfaces we show that near an isolated invariant continuum the flow is topologically equivalent to a c 1 flow we deduce that isolated invariant continua in surfaces have the shape of finite polyhedra we also show the existence of regular isolating blocks of isolated invariant continua and we use them to compute their conley index provided that we have some knowledge about the truncated unstable manifold we also see that the ring structure cohomology index of an isolated invariant continuum in a surface determines its conley index in addition we study the dynamics of non saddle sets preservation of topological and dynamical properties by continuation and we give a topological classification of isolated invariant continua which do not contain fixed points and as a consequence we also classify isolated minimal sets

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

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

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

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

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

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

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

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

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

on the fine structure of the global attractor of a uniformly persistent flow abstract we study the internal structure of the global attractor of a uniformly persistent flow we show that the restriction of the flow to the global attractor has duality properties which can be expressed in terms of certain attractor repeller decompositions we also study some natural morse decompositions of the flow and calculate their morse equations these equations provide necessary and sufficient conditions for the existence of attractors with the shape of s 1 or such that their suspension has spherical shape

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

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

a conley index study of the evolution of the lorenz strange set abstract in this paper we study the lorenz equations using the perspective of the conley index theory more specifically we examine the evolution of the strange set that these equations posses throughout the different values of the parameter we also analyze some natural morse decompositions of the global attractor of the system and the role of the strange set in these decompositions we calculate the corresponding morse equations and study their change along the successive bifurcations we see how the main features of the evolution of the lorenz system are explained by properties of the dynamics of the global attractor in addition we formulate and prove some theorems which are applicable in more general situations these theorems refer to poincare andronov hopf bifurcations of arbitrary codimension bifurcations with two homoclinic loops and a study of the role of the traveling repellers in the transformation of repeller attractor pairs into attractor repeller ones

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