morse theory for periodic solutions of hamiltonian systems and the maslov index in this paper we prove morse type inequalities for the contractible 1 periodic solutions of time dependent hamiltonian differential equations on those compact symplectic manifolds m for which the symplectic form and the first chern class of the tangent bundle vanish over q m the proof is based on a version of infinite dimensional morse theory which is due to floer the key point is an index theorem for the fredholm operator which plays a central role in floer homology the index formula involves the maslov index of nondegenerate contractible periodic solutions this maslov index plays the same role as the morse index of a nondegenerate critical point does in finite dimensional morse theory we shall use this connection between floer homology and maslov index to establish the existence of infinitely many periodic solutions having integer periods provided that every i periodic solution has at least one floquet multiplier which is not equal to 1

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

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

effect of degree correlations on networked traffic dynamics abstract in order to enhance the transport capacity of scale free networks we study the relation between the degree correlation and the transport capacity of the network we calculate the degree degree correlation coefficient the maximal betweenness and the critical value of the generating rate r c traffic congestion occurs for r r c numerical experiments indicate that both assortative mixing and disassortative mixing can enhance the transport capacity we also reveal how the network structure affects the transport capacity assortative disassortative mixing changes distributions of nodes betweennesses and as a result the traffic decreases through nodes with the highest degree while it increases through the initially idle nodes

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

extremum statistics in scale free network models we investigate the statistics of the most connected node in scale free networks for a scale free network model with homogeneous nodes we show by means of extensive simulations that the exponential truncation due to the finite size of the network of the degree distribution governs the scaling of the extreme values and that the distribution of maxima follows the gumbel statistics for a scale free network model with heterogeneous nodes we show that scaling no longer holds and that the truncation of the degree distribution no longer controls the maxima distribution

estimating order statistics of network degrees abstract we model the order statistics of network degrees of big data sets by a range of generalised beta distributions a three parameter beta distribution due to libby and novick 1982 is shown to give the best overall fit for at least four big data sets the fit of this distribution is significantly better than the fit suggested by olhede and wolfe 2012 across the whole range of order statistics for all four data sets

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

random walk access times on partially disordered complex networks an effective medium theory an analytic effective medium theory is constructed to study the mean access times for random walks on hybrid disordered structures formed by embedding complex networks into regular lattices considering transition rates f that are different for steps across lattice bonds from the rates f across network shortcuts the theory is developed for structures with arbitrary shortcut distributions and applied to a class of partially disordered traversal enhanced networks in which shortcuts of fixed length are distributed randomly with finite probability numerical simulations are found to be in excellent agreement with predictions of the effective medium theory on all aspects addressed by the latter access times for random walks on these partially disordered structures are compared to those on small world networks which on average appear to provide the most effective means of decreasing access times uniformly across the network

spectra of random graphs with given expected degrees in the study of the spectra of power law graphs there are basically two competing approaches one is to prove analogues of wigner s semicircle law whereas the other predicts that the eigenvalues follow a power law distribution although the semicircle law and the power law have nothing in common we will show that both approaches are essentially correct if one considers the appropriate matrices we will prove that under certain mild conditions the eigenvalues of the normalized laplacian of a random power law graph follow the semicircle law whereas the spectrum of the adjacency matrix of a power law graph obeys the power law our results are based on the analysis of random graphs with given expected degrees and their relations to several key invariants of interest are a number of new values for the exponent where phase transitions for eigenvalue distributions occur the spectrum distributions have direct implications to numerous graph algorithms such as for example randomized algorithms that involve rapidly mixing markov chains

visibility graph network analysis of gold price time series abstract mapping time series into a visibility graph network the characteristics of the gold price time series and return temporal series and the mechanism underlying the gold price fluctuation have been explored from the perspective of complex network theory the network degree distribution characters which change from power law to exponent law when the series was shuffled from original sequence and the average path length characters which change from l ln n into ln l ln n as the sequence was shuffled demonstrate that price series and return series are both long rang dependent fractal series the relations of hurst exponent to the power law exponent of degree distribution demonstrate that the logarithmic price series is a fractal brownian series and the logarithmic return series is a fractal gaussian series power law exponents of degree distribution in a time window changing with window moving demonstrates that a logarithmic gold price series is a multifractal series the power law average clustering coefficient demonstrates that the gold price visibility graph is a hierarchy network the hierarchy character in light of the correspondence of graph to price fluctuation means that gold price fluctuation is a hierarchy structure which appears to be in agreement with elliot s experiential wave theory on stock price fluctuation and the local rule growth theory of a hierarchy network means that the hierarchy structure of gold price fluctuation originates from persistent short term factors such as short term speculation

rigorous computation of the global dynamics of integrodifference equations with smooth nonlinearities topological tools such as conley index theory have inspired rigorous computational methods for studying dynamics these methods rely on the construction of an outer approximation a combinatorial representation of the system that incorporates small bounded error in this work we present an automated approach to constructing outer approximations for systems in a class of integrodifference operators with smooth nonlinearities chebyshev interpolants and galerkin projections form the basis for the construction while analysis and interval arithmetic are used to incorporate explicit error bounds this represents a significant advance to the approach given by day junge and mischaikow siam j appl dyn syst 3 2004 pp 117 160 extending the nonlinearities that may be studied from low degree polynomials to smooth functions and the studied portion of phase space from a simulated attracting region to the global maximal invariant set as a demonstration of the techniques a morse decomposition of the

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

on the behavior of the lorenz equation backward in time abstract the sets of solutions to the lorenz equations that exist backward in time and are bounded at an exponential rate determined by the eigenvalues of the linear part of the equation are examined the set associated with the middle eigenvalue is shown to project surjectively onto a plane thereby providing a lower estimate for its dimension specific bounds are also found for a cone containing this set

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

symmetric and antisymmetric pulses in parallel coupled nerve fibres travelling wave solutions for equations that model two parallel coupled nerve fibres are found the travelling wave here represents the action potential it is shown that the introduction of weak coupling between the fibres induces either symmetry or antisymmetry of the action potentials a symmetric pulse is a solution where both fibres fire simultaneously and the action potentials propagate locked in phase at the same wave speed along the length of each fibre an antisymmetric pulse is a solution where one fibre fires resulting in an action potential propagating along it while the other remains at rest geometric singular perturbation theory and the exchange lemma are used to prove the existence of solutions in addition a technique which involves the use of differential forms for detecting transversalities of small order is introduced

effective medium approximation for lattice random walks with long range jumps we consider the random walk on a lattice with random transition rates and arbitrarily long range jumps we employ bruggeman s effective medium approximation ema to find the disorder averaged coarse grained dynamics the ema procedure replaces the disordered system with a cleverly guessed reference system in a self consistent manner we give necessary conditions on the reference system and discuss possible physical mechanisms of anomalous diffusion in the case of a power law scaling between transition rates and distance lattice variants of l evy flights emerge as the effective medium and the problem is solved analytically bearing the effective anomalous diffusivity finally we discuss several example distributions and demonstrate very good agreement with numerical simulations

weak subordination breaking for the quenched trap model we map the problem of diffusion in the quenched trap model onto a different stochastic process brownian motion that is terminated at the coverage time s nx with nx being the number of visits to site x here 0 t t g 1 is a measure of the disorder in the original model this mapping allows us to treat the intricate correlations in the underlying random walk in the random environment the operational time s is changed to laboratory time t with a l evy time transformation investigation of brownian motion stopped at time s yields the diffusion front of the quenched trap model which is favorably compared with numerical simulations in the zero temperature limit of 0 we recover the renormalization group solution obtained by monthus phys rev e 68 036114 2003 our theory surmounts the critical slowing down that is found when 1 above the critical dimension 2 mapping the problem to a continuous time random walk becomes feasible though still not trivial

extensions of effective medium theory of transport in disordered systems the effective medium theory of transport in disordered systems whose basis is the replacement of spatial disorder by temporal memory is extended in several practical directions restricting attention to a one dimensional system with bond disorder for specificity a transformation procedure is developed to deduce explicit expressions for the memory functions from given distribution functions characterizing the system disorder it is shown how to use the memory functions in the laplace domain forms in which they first appear and in the time domain forms which are obtained via numerical inversion algorithms to address time evolution of the system beyond the asymptotic domain of large times normally treated an analytic but approximate procedure is provided to obtain the memories in addition to the inversion algorithm good agreement of effective medium theory predictions with numerically computed exact results is found for all time ranges for the distributions used except near the percolation limit as expected the use of ensemble averages is studied for normal as well as correlation observables the effect of size on effective medium theory is explored and it is shown that even in the asymptotic limit finite size corrections develop to the well known harmonic mean prescription for finding the effective rate a percolation threshold is shown to arise even in one dimension for finite but not infinite systems at a concentration of broken bonds related to the system size spatially long range transfer rates are shown to emerge naturally as a consequence of the replacement of spatial disorder by temporal memories in spite of the fact that the original rates possess nearest neighbor character pausing time distributions in continuous time random walks corresponding to the effective medium memories are calculated

using machine learning to predict catastrophes in dynamical systems nonlinear dynamical systems which include models of the earth s climate financial markets and complex ecosystems often undergo abrupt transitions that lead to radically different behavior the ability to predict such qualitative and potentially disruptive changes is an important problem with far reaching implications even with robust mathematical models predicting such critical transitions prior to their occurrence is extremely difficult in this work we propose a machine learning method to study the parameter space of a complex system where the dynamics is coarsely characterized using topological invariants we show that by using a nearest neighbor algorithm to sample the parameter space in a specific manner we are able to predict with high accuracy the locations of critical transitions in parameter space

conley index of poincare maps in isolating segments abstract we calculate the discrete time conley index of the poincare map of a time periodic ordinary differential equation in an isolated invariant set generated by a periodic isolating segment as an application we present results on the existence of bounded solutions of some planar equations

oscillations in multi stable monotone systems with slowly varying feedback the study of dynamics of gene regulatory networks is of increasing interest in systems biology a useful approach to the study of these complex systems is to view them as decomposed into feedback loops around open loop monotone systems key features of the dynamics of the original system are then deduced from the input output characteristics of the open loop system and the sign of the feedback this paper extends these results showing how to use the same framework of input output systems in order to prove existence of oscillations if the slowly varying strength of the feedback depends on the state of the system

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

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

computer assisted proof of chaos in the lorenz equations abstract in this paper we prove with computer assistance the existence of chaos in a suitable poincare map generated by the lorenz system of equations by chaos we mean the existence of symbolic dynamics with infinite number of periodic trajectories the proof combines abstract results based on the fixed point index and finite rigorous computer calculations discussion concerning numerical algorithms is also included

traversal times for random walks on small world networks we study the mean traversal time for a class of random walks on newman watts small world networks in which steps around the edge of the network occur with a transition rate that is different from the rate for steps across small world connections when f f the mean time to traverse the network exhibits a transition associated with percolation of the random graph i e small world part of the network and a collapse of the data onto a universal curve this transition was not observed in earlier studies in which equal transition rates were assumed for all allowed steps we develop a simple self consistent effective medium theory and show that it gives a quantitatively correct description of the traversal time in all parameter regimes except the immediate neighborhood of the transition as is characteristic of most effective medium theories

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

global dynamics of neural nets with infinite gain abstract we consider a model of neural and gene networks where the nonlinearities in the system of differential equations are discontinuous and piecewise constant we develop a framework for study of such systems as a first step we associate to the system a graph g on a hypercube and show how the collection of strongly connected components of g relates to the dynamics of the flow on the set of rays through the origin in the second step we discuss the relationship between the invariant sets of the ray flow and the invariant sets of the original flow we provide a sufficient condition for a one to one correspondence between these sets finally we study the class of binary networks within this framework under certain conditions we can determine the structure of an invariant set corresponding to the lowest strongly connected component of the hypercube graph

infinite series in cohomology attractors and conley index in this paper we study the cohomological conley index of arbitrary isolated invariant continua for continuous maps f u subseteq mathbb r d rightarrow mathbb r d by analyzing the topological structure of their unstable manifold we provide a simple dynamical interpretation for the first cohomological conley index describing it completely and relate it to the cohomological conley index in higher degrees a number of consequences are derived including new computations of the fixed point indices of isolated invariant continua in dimensions 2 and 3 our approach exploits a certain attractor repeller decomposition of the unstable manifold reducing the study of the cohomological conley index to the relation between the cohomology of an attractor and its basin of attraction this is a classical problem that in the present case is particularly difficult because the dynamics is discrete and the topology of the unstable manifold can be very complicated to address it we develop a new method that may be of independent interest and involves the summation of power series in cohomology if z is a metric space and k subseteq z is a compact global attractor for a continuous map g z rightarrow z we show how to interpret series of the form sum j ge 0 a j g j as endomorphisms of the cohomology group of the pair z k

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

chaos near a resonant inclination flip abstract horseshoes play a central role in dynamical systems and are observed in many chaotic systems however most points in a neighborhood of the horseshoe escape after finitely many iterations in this work we construct a new model by re injecting the points that escape the horseshoe we show that this model can be realized within an attractor of a flow arising from a three dimensional vector field after perturbation of an inclination flip homoclinic orbit with a resonance the dynamics of this model without considering the re injection often contains a cuspidal horseshoe with positive entropy and we show that for a computational example the dynamics with re injection can have more complexity than the cuspidal horseshoe alone

degree distribution and assortativity in line graphs of complex networks topological characteristics of links of complex networks influence the dynamical processes executed on networks triggered by links such as cascading failures triggered by links in power grids and epidemic spread due to link infection the line graph transforms links in the original graph into nodes in this paper we investigate how graph metrics in the original graph are mapped into those for its line graph in particular we study the degree distribution and the assortativity of a graph and its line graph specifically we show both analytically and numerically the degree distribution of the line graph of an erdos renyi graph follows the same distribution as its original graph we derive a formula for the assortativity of line graphs and indicate that the assortativity of a line graph is not linearly related to its original graph additionally line graphs of various graphs e g erdos renyi graphs scale free graphs show positive assortativity in contrast we find certain types of trees and non trees whose line graphs have negative assortativity 2015 elsevier b v all rights reserved

indices of the iterates of r 3 homeomorphisms at lyapunov stable fixed points abstract given any positive sequence c n n n we construct orientation preserving homeomorphisms f r 3 r 3 such that fix f per f 0 0 is lyapunov stable and lim sup i f m 0 c m we will use our results to discuss and to point out some strong differences with respect to the computation and behavior of the sequences of the indices of planar homeomorphisms

robust linear regression with broad distributions of errors abstract we consider the problem of linear fitting of noisy data in the case of broad say stable distributions of random impacts noise which can lack even the first moment this situation common in statistical physics of small systems in earth sciences in network science or in econophysics does not allow for application of conventional gaussian maximum likelihood estimators resulting in usual least squares fits such fits lead to large deviations of fitted parameters from their true values due to the presence of outliers the approaches discussed here aim onto the minimization of the width of the distribution of residua the corresponding width of the distribution can either be defined via the interquantile distance of the corresponding distributions or via the scale parameter in its characteristic function the methods provide the robust regression even in the case of short samples with large outliers and are equivalent to the normal least squares fit for the gaussian noises our discussion is illustrated by numerical examples

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

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

reaction diffusion on random spatial networks with scale free jumping rates via effective medium theory we study epidemic processes using a metapopulation approach on the line featuring random transport rates between arbitrarily distant sites an average transport network is found using a recently developed variant of the effective medium approximation ema that is capable of dealing with these long range connections using a feynman kac argument in the effective medium we derive an estimate on the size of the infected domain and reproduce the known result of its exponential growth in time we hereby demonstrate the applicability of long range ema to dynamical processes on networks more intricate than simple diffusion

on the morse complex for semilinear parabolic equations abstract a morse smale function on a compact riemannian manifold can be used to define an associated morse complex its homology is isomorphic to the singular homology of the manifold which coincides with the singular homology of the conley index of the manifold in this paper we consider an analogous morse complex for isolated invariant sets of certain semilinear parabolic equations it is shown that its homology is isomorphic to the singular homology of the conley index of the isolated invariant set

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

isolating blocks and epsilon chains for maps abstract the relationship between the outputs and computer simulations of dynamical systems and their true orbit structures is explored epsilon chains are used as a mathematical representation of computer generated pseudo orbits the question when is a pseudo orbit shadowed by a true orbit is discussed isolated invariant sets are defined and are shown to be contained in special neighborhoods called isolating blocks the conley index is extended and applied to maps

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

realization of all dold s congruences with stability the main goal of this paper is to prove that for each n 2 every sequence of integers satisfying dold s congruences is realized as the sequence of fixed point indices of the iterates of an orientation preserving r n homeomorphism at an isolated stable fixed point

zeta functions periodic trajectories and the conley index abstract we use the conley index to produce a simple sufficient condition stated in terms of the betti numbers of the cohomology conley index for the existence of periodic orbits for flows which admit a cross section

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

transport properties of random walks on scale free regular lattice hybrid networks we study numerically the mean access times for random walks on hybrid disor dered structures formed by embedding scale free networks into regular lattices considering different transition rates for steps across lattice bonds f and across network shortcuts f for fast shortcuts f f 1 and low shortcut densities traversal time data collapse onto a universal curve while a crossover behavior that can be related to the percolation threshold of the scale free network component is identified at higher shortcut densities in analogy to similar observations reported recently in newman watts small world networks further more we observe that random walk traversal times are larger for networks with a higher degree of inhomogeneity in their shortcut distribution and we discuss access time distri butions as functions of the initial and final node degrees these findings are relevant in particular when considering the optimization of existing information networks by the addi tion of a small number of fast shortcut connections

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

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

homology algorithm based on acyclic subspace we present a new reduction algorithm for the efficient computation of the homology of a cubical set the algorithm is based on constructing a possibly large acyclic subspace and then computing the relative homology instead of the plain homology we show that the construction of acyclic subspace may be performed in linear time this significantly reduces the amount of data that needs to be processed in the algebraic way and in practice it proves itself to be significantly more efficient than other available cubical homology algorithms

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

bistable wavefronts in a diffusive and competitive lotka volterra type system with nonlocal delays abstract a diffusive lotka volterra type model with nonlocal delays for two competitive species is considered the existence of a traveling wavefront analogous to a bistable wavefront for a single species is proved by transforming the system with nonlocal delays to a four dimensional system without delay furthermore in order to prove the asymptotic stability up to translation of bistable wavefronts of the system the existence regularity and comparison theorem of solutions of the corresponding cauchy problem are first established for the systems on r by appealing to the theory of abstract functional differential equations the asymptotic stability up to translation of bistable wavefronts are then proved by spectral methods in particular we also prove that the spreading speed is unique by upper and lower solutions technique from the point of view of ecology our results indicate that the nonlocal delays appeared in the interaction terms are not sensitive to the invasion of species of spatial isolation

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

detection and computation of high codimension bifurcations in diffuse predator prey systems abstract in this paper we consider reaction diffusion model with modified leslie gower and holling type ii functional response and by varying simultaneously three model parameters our model exhibits different types of complex dynamics this allowed us to delimit several bifurcation surfaces with higher codimension corresponding to turing turing transcritical turing bogdanov taken turing hopf andronov turing saddle node moreover by varying at least three bifurcation parameters we show that small variations in the ratio of the diffusion coefficients can significantly alter bifurcation structure finally the paper ends with the emergence of spatio temporal patterns via numerical simulations these simultaneous parameter variations leaded to spatio temporal local and global bifurcations which are always catastrophic our results give new insights about how simultaneous changes in environmental and life history parameters drive different distribution dynamics of predator prey populations this only underscores the importance of including global bifurcations in the analysis of food chain models such results can be used to improve ecological decision making on species conservation