algorithm for direct numerical simulation of emulsion flow through a granular material a multipole accelerated 3d boundary integral algorithm capable of modelling an emulsion flow through a granular material by direct multiparticle multidrop simulations in a periodic box is developed and tested the particles form a random arrangement at high volume fraction rigidly held in space including the case of an equilibrium packing in mechanical contact deformable drops with non deformed diameter comparable with the particle size squeeze between the particles under a specified average pressure gradient the algorithm includes recent boundary integral desingularization tools especially important for drop solid and drop drop interactions the hebeker representation for solid particle contributions and unstructured surface triangulations with fixed topology multipole acceleration with two levels of mesh node decomposition entire drop solid surfaces and patches is a significant improvement over schemes used in previous purely multidrop simulations it remains efficient at very high resolutions 10 4 10 5 triangular elements per surface and has no lower limitation on the number of particles or drops such resolutions are necessary in the problem to alleviate lubrication difficulties especially for near critical squeezing conditions as well as using 10 4 time steps and an iterative solution at each step both for contrast and matching viscosities examples are shown for squeezing of 25 40 drops through an array of 9 14 solids with the total volume fraction of 70 for particles and drops the flow rates for the drop and continuous phases are calculated extensive convergence testing with respect to program parameters triangulation multipole truncation etc is made

regular dynamics in a delayed network of two neurons with all or none activation functions we consider a delayed network of two neurons with both self feedback and interaction described by an all or none threshold function the model describes a combination of analog and digital signal processing in the network and takes the form of a system of delay differential equations with discontinuous nonlinearity we show that the dynamics of the network can be understood in terms of the iteration of a one dimensional map and we obtain simple criteria for the convergence of solutions the existence multiplicity and attractivity of periodic solutions

examples for obstructions to action angle coordinates on donne des exemples de varietes symplectiques qui sont aussi des fibres toriques principaux avec des fibres lagrangiennes ces fibres sont des exemples d espaces avec une obstruction a l existence globale des variables action angle

an efficient algorithm for calculating multiparticle thermal interaction in a concentrated dispersion of spheres abstract the boundary value problem of heat conduction through a multiparticle system in a medium of a different conductivity e is considered the particle set is a random arrangement of n equisized spheres of conductivity in a cubic cell continued periodically into all space the mean temperature gradient is given the problem is reduced to an infinite set of equations for the coefficients of the temperature expansion into spherical harmonics on the interfaces the algorithm exploits the idea that only the interaction of the low frequency harmonics is long ranged to construct the economical truncation as well as the rotational transformations of spherical harmonics to promote the efficient iterative solution the method is capable of providing high resolution for arbitrary conductivity ratio e and enabling large scale simulations of the effective conductivity of highly concentrated dispersions with various microstructures even on a small computer

convergence to equilibria in scalar nonquasimonotone functional differential equations abstract we consider a class of scalar functional differential equations generating a strongly order preserving semiflow with respect to the exponential ordering introduced by smith and thieme it is shown that the boundedness of all solutions and the stability properties of an equilibrium are exactly the same as for the ordinary differential equation which is obtained by ignoring the delays the result on the boundedness of the solutions combined with a convergence theorem due to smith and thieme leads to explicit necessary and sufficient conditions for the convergence of all solutions starting from a dense subset of initial data under stronger conditions guaranteeing that the functional differential equation is asymptotically equivalent to a scalar ordinary differential equation a similar result is proved for the convergence of all solutions

accelerated boundary integral method for multiphase flow in non periodic geometries an accelerated boundary integral method for stokes flow of a suspension of deformable particles is presented for an arbitrary domain and implemented for the important case of a planar slit geometry the computational complexity of the algorithm scales as o n or o n log n where n is proportional to the product of number of particles and the number of elements employed to discretize the particle this technique is enabled by the use of an alternative boundary integral formulation in which the velocity field is expressed in terms of a single layer integral alone even in problems with non matched viscosities the density of the single layer integral is obtained from a fredholm integral equation of the second kind involving the double layer integral acceleration in this implementation is provided by the use of general geometry ewald like method ggem for computing the velocity and stress fields driven by a set of point forces in the geometry of interest for the particular case of the slit geometry a fourier chebyshev spectral discretization of ggem is developed efficient implementations employing the ggem methodology are presented for the resulting single and the double layer integrals the implementation is validated with test problems on the velocity of rigid particles and drops between parallel walls in pressure driven flow the taylor deformation parameter of capsules in simple shear flow and the particle trajectories in pair collisions of capsules in simple shear flow the computational complexity of the algorithm is verified with results from several large scale multiparticle simulations

asymptotic periodicity monotonicity and oscillation of solutions of scalar neutral functional differential equations abstract we consider the periodic scalar neutral functional differential equation d dt x t c t x t h t x t h t x t where c is continuously differentiable h is increasing in its second argument and both c and h are 1 periodic in the t variable the two time lags and are not required to be the same it is shown that under certain conditions i the set of 1 periodic solutions is an ordered arc and each solution is convergent to a periodic solution ii the asymptotic and oscillatory behaviors of each solution are completely classified in terms of the value of the first integral at the initial condition

quantum monodromy and pattern formation hamiltonian monodromy is known to be the first obstruction to the existence of global action coordinates in integrable systems its manifestation in quantum systems can be seen as characteristic defects of the regular lattice formed by the joint eigenvalues of mutually commuting quantum operators the relation between topology of singular fibers of classical integrable fibrations and patterns formed by joint spectrum of corresponding quantum systems is discussed the notion of the sign of elementary monodromy defect is introduced on the basis of cut and glue construction of the lattice defects special attention is paid to non elementary defects which generically appear in phyllotaxis patterns and can be associated with plant morphology

criteria of global attraction in systems of delay differential equations with mixed monotonicity abstract in this paper we extend the classical decomposing embedding method for systems of delay differential equations our extension has two advantages 1 enhancing the range of applicability of the classical method and 2 providing delay dependent criteria of global attraction the leading ideas are on the one hand the extension of the notion of dominance introduced first for difference equations and on the other hand the use of some monotone ideas developed by h l smith and collaborators we apply our results to some classical models of population dynamics mainly models with patch structure

a fast algorithm for simulating vesicle flows in three dimensions vesicles are locally inextensible fluid membranes that can sustain bending in this paper we extend the study of veerapaneni et al s k veerapaneni d gueyffier g biros d zorin a numerical method for simulating the dynamics of 3d axisymmetric vesicles suspended in viscous flows journal of computational physics 228 19 2009 7233 7249 to general non axisymmetric vesicle flows in three dimensions although the main components of the algorithm are similar in spirit to the axisymmetric case spectral approximation in space semi implicit time stepping scheme important new elements need to be introduced for a full 3d method in particular spatial quantities are discretized using spherical harmonics and quadrature rules for singular surface integrals need to be adapted to this case an algorithm for surface reparameterization is needed to ensure stability of the time stepping scheme and spectral filtering is introduced to maintain reasonable accuracy while minimizing computational costs to characterize the stability of the scheme and to construct preconditioners for the iterative linear system solvers used in the semi implicit time stepping scheme we perform a spectral analysis of the evolution operator on the unit sphere by introducing these algorithmic components we obtain a time stepping scheme that circumvents the stability constraint on the time step and achieves spectral accuracy in space we present results to analyze the cost and convergence rates of the overall scheme to illustrate the applicability of the new method we consider a few vesicle flow interaction problems a single vesicle in relaxation sedimentation shear flows and many vesicle flows

on the dynamics of nonautonomous periodic general dynamical systems and differential inclusions abstract we establish the existence and stability results for periodic nonautonomous uniform forward attractors of periodic general dynamical systems set valued dynamical systems we also investigate the dynamical behavior of nonautonomous periodic differential inclusion x t f t x t on r m with only upper semi continuous right hand side by applying the abstract results firstly we show that if the system has a compact uniformly attracting set then it has a periodic nonautonomous uniform forward attractor a secondly we prove that a is robust with respect to both internal and external perturbations finally we apply the robustness result to discuss the effects of small time delays to asymptotic stability of the system

asymptotic constancy for pseudo monotone dynamical systems on function spaces abstract a pseudo monotone dynamical system is a dynamical system which preserves the order relation between initial points and equilibrium points the purpose of this paper is to present some convergence oscillation and order stability criteria for pseudo monotone dynamical systems on function spaces for which each constant function is an equilibrium point some applications to neutral functional differential equations and semilinear parabolic partial differential equations with neumann boundary condition are given

strongly order preserving semiflows generated by functional differential equations the aim of this paper is to apply some recently developed improvements in the theory of monotone dynamical systems due to the authors 12 13 j to certain systems of functional differential equations fde s which do not enjoy the quasimonotone property considered by one of us in 9 the present work extends our previous results in this direction for scalar equations in 111 to systems in the fundamental paper s m w hirsch establishes that most orbits of a strongly monotone scmiflow on a strongly ordered space x tend to the set e of equilibria employing ideas of hirsch and of h matano 7 the authors have extended the theory to obtain some improvements in the results of hirsch and matano under matano s weaker assumption that the semiflow is strongly order preserving recently in 13 we gave sufficient conditions for most orbits of a strongly order preserving semiflow to converge to an equilibrium these conditions require additional smoothness of the semiflow and a certain strong monotonicity condition for the linearized flow determined by the variational equation about each equilibrium p policik s has also given sufficient conditions for most orbits to converge for semilinear parabolic evolution equations possessing certain strong monotonicity properties in this paper wc apply the convergence results developed in 131 to nonquasimonotone fde s in this introduction wc briefly review the application of monotone dynamical systems theory to fde s enjoying the quasimonotone property this research appeared in 9 133 then we state one of the main results of this paper which applies when the quasimonotone property fails but when another property m holds some simple examples are given which

discrete order preserving semigroups and stability for periodic parabolic differential equations on etudie la stabilite globale des solutions des equations paraboliques dans une forme non auto adjointe avec des coefficients periodiques dans le temps la structure mathematique est celle des semigroupes preservant l ordre a temps discret s n n e n sur un reseau de banach

complicated dynamics of scalar reaction diffusion equations with a nonlocal term we consider the dynamics of scalar equations ut uxx f x u c x u 0 x l where denotes some weighted spatial average and dinchlet boundary conditions are assumed prescribing f c appropriately it is shown that complicated dynamics can occur specifically linearisations at equilibria can have any number of purely imaginary eigenvalues moreover the higher order terms of the reduced vector field in an associated centre manifold can be prescribed arbitrarily up to any finite order these results are in marked contrast with the case 0 where bounded solutions are known to converge to equilibrium

diffusive monotonicity and threshold dynamics of delayed reaction diffusion equations abstract for abstract functional differential equations and reaction diffusion equations with delay an exponential ordering is introduced which takes into account the spatial diffusion the induced monotonicity of the solution semiflows is established and applied to describe the threshold dynamics extinction or persistence convergence to positive equilibria for a nonlocal and delayed reaction diffusion population model

existence stability and bifurcation of random complete and periodic solutions of stochastic parabolic equations abstract in this paper we study the existence stability and bifurcation of random complete and periodic solutions for stochastic parabolic equations with multiplicative noise we first prove the existence and uniqueness of tempered random attractors for the stochastic equations and characterize the structures of the attractors by random complete solutions we then examine the existence and stability of random complete quasi solutions and establish the relations of these solutions and the structures of tempered attractors when the stochastic equations are incorporated with periodic forcing we obtain the existence and stability of random periodic solutions for the stochastic chafee infante equation we further establish the multiplicity and stochastic bifurcation of complete and periodic solutions

periodic solutions in an inhibitory two neuron network in this paper we consider a delayed network of two neurons with self feedback and interaction described by an all or none threshold function the discontinuity of signal function makes it difficult to apply directly dynamical system we show that the dynamics of the network can be understood in terms of the iterations of a one dimensional map and we obtain the existence and attractivity of periodic solutions moreover because the network is a limiting case of the corresponding smooth system as the parameter tends to infinity the above results can act as the guide to the rich dynamics of the smooth system therefore our results have important significance in both theory and applications

effects of a degeneracy in the competition model part i classical and generalized steady state solutions abstract we study the competition model where the coefficient functions are strictly positive over the underlying spatial region except b x which vanishes in a nontrivial subdomain of and is positive in the rest of we show that there exists a critical number such that if then the model behaves similarly to the well studied classical competition model where all the coefficient functions are positive constants but when new phenomena occur our results demonstrate the fact that heterogeneous environmental effects on population models are not only quantitative but can be qualitative as well in part i here we mainly study two kinds of steady state solutions which determine the dynamics of the model one consists of finite functions while the other consists of generalized functions which satisfy u v 0 on the part of the domain that b x vanishes but are positive and finite on the rest of the domain and are determined by certain boundary blow up systems the research is continued in part ii where these two kinds of steady state solutions will be used to determine the dynamics of the model

monotone dynamical systems with dense periodic points abstract in this paper we prove a recent conjecture by m hirsch which says that if f is a discrete time monotone dynamical system with f a homeomorphism on an open connected subset of a finite dimensional vector space and the periodic points of f are dense in then f is periodic

global stability for non monotone delay equations with application to a model of blood cell production abstract a set of sufficient conditions for the global stability of the positive equilibrium is established for a large class of non monotone time delayed differential equations by the method of fluctuations and the exponential ordering approach then these analytic results are applied to a mathematical model of white blood cell production numerical simulations are also presented to illustrate the global stability regions of parameters and the long term behavior of solutions

an efficient algorithm for hydrodynamical interaction of many deformable drops an efficient and accurate 3d algorithm for dynamical simulations of many deformable drops with strong hydrodynamical interactions at zero reynolds numbers is developed the drop to medium viscosity ratio and the bond number are arbitrary and the drops are subject to gravity with stationary triply periodic boundary conditions the algorithm at each step is a hybrid of boundary integral and economical multipole techniques with extensive use of rotational transformations and economical truncation of multipole expansions to optimize near field interactions a significant part of the code is the new best paraboloid spline technique for calculating the normal vectors and curvatures on drop surfaces which greatly improves the quality of long time simulations examples show the phenomenon of clustering in a concentrated sedimenting emulsion for 0 25 and 1 which leads to an increase in the average sedimentation velocity with time a high efficiency of the method is demonstrated with two orders of magnitude gains over the standard o n2n2 boundary integral technique for n 102 drops in a periodic cell with n 103 triangular boundary elements per drop so that typical long time dynamical simulations can be performed in a few days or weeks on a standard workstation as compared to the several years which would be required for the same simulations using standard boundary integral techniques the effects of drop triangulation and truncation of multipole expansions on dynamical simulations are assessed

global stability and wavefronts in a cooperation model with state dependent time delay abstract this article deals with a diffusive cooperative model with state dependent delay which is assumed to be an increasing function of the population density with lower and upper bounds for the cooperative dde system the positivity and boundedness of solutions are firstly given using the comparison principle of the state dependent delay equations obtained the stability criterion of model is analyzed both from local and global points of view when the diffusion is properly introduced the existence of traveling waves is obtained by constructing a pair of upper lower solutions and schauder s fixed point theorem calculating the minimum wave speed shows that the wave is slowed down by the state dependent delay finally the traveling wavefront solutions for large wave speed are also discussed and the fronts appear to be all monotone regardless of the state dependent time delay this is an interesting property since many findings are frequently reported that delay causes a loss of monotonicity with the front developing a prominent hump in some other delay models

adaptive triangulation of evolving closed or open surfaces by the advancing front method the advancing front method is adapted to describe the evolution of open or closed three dimensional surfaces in terms of an unstructured grid consisting of quadratic triangular elements in science and engineering applications the surface may be identified with a material interface a free boundary or a moving front in the numerical method the geometrical properties of the surface and the coordinates of the triangle vertices are computed either in terms of available analytical expressions or by means of interpolation through an underlying coarse grid the shape and size of the curved triangular elements are determined by the maximum magnitude of the mean or directional local surface curvature two algorithms are implemented the first one for simply connected open surfaces bounded by closed lines and for closed surface with plane symmetry and the second for closed surfaces in the case of open surfaces the discretization front advances from a one dimensional boundary grid that traces the bounding curve the boundary grid is generated either by a one dimensional version of the advancing front method or by requiring a set of criteria based on local line representation with circular arcs in the case of closed surfaces the discretization front emanates from the point of the maximum mean curvature in both cases the algorithm also interpolates to generate the values of surface geometrical or physical variables such as temperature or concentration of a surface active agent the method was tested by following the motion of several passive and active surfaces evolving under the action of specified fields of flow while performing occasional regridding to ensure adequate spatial resolution in one test the large deformation of a viscous drop subjected to an infinite simple shear flow at vanishing reynolds number was computed into the regime where a cigar like shape is established thereby extending previous numerical computations for small and moderate deformations and reproducing experimentally observed shapes overall the adaptive front method emerges as an important tool in numerical studies of free boundaries or moving fronts and should be useful in a broad range of applications

an adaptive mesh algorithm for evolving surfaces simulation of drop breakup and coalescence abstract an algorithm is presented for the adaptive restructuring of meshes on evolving surfaces the resolution of the relevant local length scale is maintained everywhere with prescribed accuracy through the minimization of an appropriate mesh energy function by a sequence of local restructuring operations the resulting discretization depends on the instantaneous configuration of the surface but is insensitive to the deformation history application of the adaptive discretization algorithm is illustrated with three dimensional boundary integral simulations of deformable drops in stokes flow the results show that the algorithm can accurately resolve detailed features of deformed fluid interfaces including slender filaments associated with drop breakup and dimpled regions associated with drop coalescence our algorithm should be useful in a variety of fields including computational fluid dynamics image processing geographical information systems and biomedical engineering problems

a high order 3d boundary integral equation solver for elliptic pdes in smooth domains we present a high order boundary integral equation solver for 3d elliptic boundary value problems on domains with smooth boundaries we use nystrom s method for discretization and combine it with special quadrature rules for the singular kernels that appear in the boundary integrals the overall asymptotic complexity of our method is o n3 2 where n is the number of discretization points on the boundary of the domain and corresponds to linear complexity in the number of uniformly sampled evaluation points a kernel independent fast summation algorithm is used to accelerate the evaluation of the discretized integral operators we describe a high order accurate method for evaluating the solution at arbitrary points inside the domain including points close to the domain boundary we demonstrate how our solver combined with a regular grid spectral solver can be applied to problems with distributed sources we present numerical results for the stokes navier and poisson problems

comparison theorems for neutral stochastic functional differential equations abstract the comparison theorems under wu and freedman s order are proved for neutral stochastic functional differential equations with finite or infinite delay whose drift terms satisfy the quasimonotone condition and diffusion term is the same

periodic conduction problems the fast multipole method and convergence of integral equations and lattice sums this paper presents a version of the fast multipole method fmm for integral equations describing conduction through three dimensional periodic heterogeneous media the proposed method is based on the standard rather than periodic fundamental solution and therefore it is very close to the original fmm in deriving the method particular attention is paid to convergence of arising integral equations and lattice sums it is shown that convergence can be achieved without introducing artificial compensatory sources or boundary conditions

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

symmetric functional differential equations and neural networks with memory we establish an analytic local hopf bifurcation theorem and a topological global hopf bifurcation theorem to detect the existence and to describe the spatial temporal pattern the asymptotic form and the global continuation of bifurcations of periodic wave solutions for functional differential equations in the presence of symmetry we apply these general results to obtain the coexistence of multiple large amplitude wave solutions for the delayed hopfield cohen grossberg model of neural networks with a symmetric circulant connection matrix

phase translation group actions on strongly monotone skew product semiflows we establish a convergence property for pseudo bounded forward orbits of strongly monotone skew product semiflows with invariant phasetranslation group actions the results are then applied to obtain global convergence of certain chemical reaction networks whose associated systems in reaction coordinates are monotone as well as the dynamics of certain reactiondiffusion systems in time recurrent structure including periodicity almost periodicity and almost automorphy

asymptotic behavior of strongly monotone time periodic dynamical processes with symmetry given a strongly monotone discrete time dynamical system tn x x n z in an open and order convex subset x of a separable strongly ordered banach space v and a compact connected metrizable group g whose action on x is monotone and commutes with t we prove under some reasonable additional hypotheses that the limit set x of every stable point x x consists of symmetric points only i e g w w for all w x and g g moreover the set of all unstable points is contained in a union of at most countably many lipschitz manifolds of codimension one in v where each manifold is invariant under t and the action of g this result is applied to the time periodic spatially independent irreducible cooperative system of n reaction diffusion equations u t d t u f t u for t x r1 rn with spatially periodic boundary conditions in rn and with an initial distribution u0 which is continuous and satisfies the boundary conditions if u0 is stable then u0 contains spatially constant functions only and the dynamics on u0 is given by the irreducible cooperative system of n odes dudt f t u for t r1 here t is the poincare map if n 1 then u0 is a single fixed point of t if n 2 then v0 is a single fixed point of t for each v0 u0

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

asymptotic behaviour of comparable skew product semiflows with applications the almost 1 cover lifting property of omega limit sets is established for nonmonotone skew product semiflows which are comparable to uniformly stable eventually strongly monotone skew product semiflows these results are then applied to study the asymptotic behavior of solutions to the nonmonotone comparable systems of odes reaction diffusion systems differential systems with time delays and semilinear parabolic equations

monotone semiflows in scalar non quasi monotone functional differential equations abstract by introducing a stronger than pointwise ordering conditions are found under which scalar functional differential equations generate monotone semiflows even if they are not quasi monotone typically the maximum delay must be the smaller the more quasi monotonicity is violated the theory of monotone semiflows is used to show that most solutions converge to equilibrium and that stability of equilibria is essentially the same as for ordinary differential equations

dynamics integrability and topology for some classes of kolmogorov hamiltonian systems in r 4 abstract in this paper we first give the sufficient and necessary conditions in order that two classes of polynomial kolmogorov systems in r 4 are hamiltonian systems then we study the integrability of these hamiltonian systems in the liouville sense finally we investigate the global dynamics of the completely integrable lotka volterra hamiltonian systems in r 4 as an application of the invariant subsets of these systems we obtain topological classifications of the 3 submanifolds in r 4 defined by the hypersurfaces a x y b z w c x 2 y d x y 2 e z 2 w f z w 2 h where a b c d e f w and h are real constants

travelling waves for delayed reaction diffusion equations with global response we develop a new approach to obtain the existence of travelling wave solutions for reaction diffusion equations with delayed non local response the approach is based on an abstract formulation of the wave profile as a solution of an operational equation in a certain banach space coupled with an index formula of the associated fredholm operator and some careful estimation of the nonlinear perturbation the general result relates the existence of travelling wave solutions to the existence of heteroclinic connecting orbits of a corresponding functional differential equation and this result is illustrated by an application to a model describing the population growth when the species has two age classes and the diffusion of the individual during the maturation process leads to an interesting non local and delayed response for the matured population

a fast solver for the stokes equations with distributed forces in complex geometries we present a new method for the solution of the stokes equations the main features of our method are 1 it can be applied to arbitrary geometries in a black box fashion 2 it is second order accurate and 3 it has optimal algorithmic complexity our approach to which we refer as the embedded boundary integral method ebi is based on anita mayo s work for the poisson s equation the fast solution of poisson s and the biharmonic equations on irregular regions siam journal on numerical analysis 21 1984 285 299 we embed the domain in a rectangular domain for which fast solvers are available and we impose the boundary conditions as interface jump conditions on the velocities and tractions we use an indirect boundary integral formulation for the homogeneous stokes equations to compute the jumps the resulting equations are discretized by nystrom s method the rectangular domain problem is discretized by finite elements for a velocity pressure formulation with equal order interpolation bilinear elements q1 q1 stabilization is used to circumvent the inf sup condition for the pressure space for the integral equations fast matrix vector multiplications are achieved via an n logn algorithm based on a block representation of the discrete integral operator combined with kernel independent singular value decomposition to sparsify low rank blocks the regular grid solver is a krylov method conjugate residuals combined with an optimal two level schwartz preconditioner for the integral equation we use gmres we have tested our algorithm on several numerical examples and we have observed optimal convergence rates

convergence and stability for essentially strongly order preserving semiflows abstract this paper is concerned with a class of essentially strongly order preserving semiflows which are defined on an ordered metric space and are generalizations of strongly order preserving semiflows for essentially strongly order preserving semiflows we prove several principles which are analogues of the nonordering principle for limit sets the limit set dichtomy and the sequential limit set trichotomy for strongly order preserving semiflows then under certain compactness hypotheses we obtain some results on convergence quasiconvergence and stability in essentially strongly order preserving semiflows finally some applications are made to quasimonotone systems of delay differential equations and reaction diffusion equations with delay and the main advantages of our results over the classical ones are that we do not require the delicate choice of state space and the technical ignition assumption

morse decompositions for general dynamical systems and differential inclusions with applications to control systems the morse decomposition theory for single valued dynamical systems is extended to general dynamical systems and differential inclusions relations between chain recurrent sets chain recurrent components chain control sets and morse decompositions for open loop systems are discussed the robustness of feedback laws of closed loop systems with respect to small time delays and sample hold controls is also addressed

new generic quasi convergence principles with applications in this paper essentially strongly order preserving and conditionally set condensing semiflows are considered obtained is a new type of generic quasi convergence principles implying the existence of an open and dense set of stable quasi convergent points when the state space is order bounded the generic quasi convergence principles are then applied to essentially cooperative and irreducible systems in the forms of ordinary differential equations and delay differential equations giving some results of theoretical and practical significance

an fmm for periodic boundary value problems for cracks for helmholtz equation in 2d this paper presents an fmm fast multipole method for periodic boundary value problems for helmholtz equation in 2d the periodic green function is an important ingredient in our formulation which is computed efficiently with the help of the fourier analysis we validate the proposed method by comparing the obtained numerical results with those computed with the conventional approach we then apply the proposed method to the analysis of scattering problems for periodic array of cracks and plot the energy transmittance vs wave numbers the stopband and related phenomena are observed clearly copyright 2007 john wiley sons ltd

a system of coupled oscillators can have arbitrary prescribed attractors a system of coupled nonlinear oscillators is considered if the size n of this oscillator network is large enough then the model can have arbitrary prescribed attractors which can be defined by some finite dimensional flows one can effectively find a simple interaction between the oscillators which gives these prescribed attractors

complicated dynamics in scalar semilinear parabolic equations in higher space dimension abstract we study the dynamics of the boundary value problem ut lu g x u u x 1 u 0 2 where l is a second order uniformly elliptic operator and r n is diffeomorphic to the ball in r n n 2 the main result asserts that given any ck vector field v on r n 1 with v 0 0 one can adjust coefficients of l and the function g such that the corresponding problem 1 2 has an n 1 dimensional invariant manifold through the equilibrium u 0 and the taylor expansion at u 0 of the vector field representing the flow on this manifold coincides in appropriate coordinates with the taylor expansion of v up to k th order terms this result implies that a hyperbolic invariant n torus can be found in 1 2 if l and g are appropriately chosen this result also indicates that chaotic dynamics is likely to occur for some choices of l and g

a multipole accelerated algorithm for close interaction of slightly deformable drops an efficient multipole accelerated boundary integral algorithm is developed to study close contact three dimensional interaction of two drops at zero reynolds numbers and very small but non zero capillary numbers ca when the drops are nearly spherical the numerical difficulties compared to the case of larger deformations include severe stability limitations on the time step and a singular perturbation for ca 1 requiring very high surface resolution both in the small gap and in the outer region the mesh triangle vertices are grouped into a large number of non overlapping patches with economical rotation based multipole reexpansions to handle patch to patch interactions and limit the use of expensive direct summations a novel concept of a dynamical projective mesh is developed to maintain fixed topology gap adaptive surface triangulations for o 10 5 boundary elements per drop in close contact the algorithm has at least an order of magnitude advantage over the standard boundary integral method making such dynamical calculations feasible in gravity induced and shear induced motion exact results are obtained for the dynamics of the surface clearance h m i n which attains values less than 0 001 of the drop radii and for the separation angle b s e p determining the configuration when two drops in apparent contact start to separate the shear flow problem is studied in the wide range of drop to medium viscosity ratios 0 25 0

a fast algorithm for simulating multiphase flows through periodic geometries of arbitrary shape this paper presents a new boundary integral equation bie method for simulating particulate and multiphase flows through periodic channels of arbitrary smooth shape in two dimensions the authors consider a particular system multiple vesicles suspended in a periodic channel of arbitrary shape to describe the numerical method and test its performance rather than relying on the periodic green s function as classical bie methods do the method combines the free space green s function with a small auxiliary basis and imposes periodicity as an extra linear condition as a result we can exploit existing free space solver libraries quadratures and fast algorithms and handle a large number of vesicles in a geometrically complex channel spectral accuracy in space is achieved using the periodic trapezoid rule and product quadratures while a first order semi implicit scheme evolves particles by treating the vesicle channel interactions explicitly new constraint correction formulas are introduced that pre

on the finite dimensional dynamical systems with limited competition the asymptotic behavior of dynamical systems with limited competition is investigated we study index theory for fixed points permanence global stability convergence everywhere and coexistence it is shown that the system has a globally asymptotically stable fixed point if every fixed point is hyperbolic and locally asymptotically stable relative to the face it belongs to a nice result is the necessary and sufficient conditions for the system to have a globally asymptotically stable positive fixed point it can be used to establish the sufficient conditions for the system to persist uniformly and the convergence result for all orbits applications are made to time periodic ordinary differential equations and reaction diffusion equations

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

attractors in strongly monotone flows where f is a c vector field in r and all the off diagonal terms of its jacobian matrix are nonnegative the well known kamke theorem implies that the flow 4 i ir0 generated by 1 has the property it preserves the partial ordering 0 another system which has this property is the semilinear parabolic equation on a compact n dimensional submanifold solutions of its dirichlet or neumann problem determine a monotone flow in appropriate function space see 11 or 51 in general a flow which preserves a partial ordering on the state space is called monotone m w hirsch has studied this flow in a series of papers 14 io he 2 31 has studied the limiting behavior and convergence of solutions of system 1 he 1 4 lo has investigated this flow in the considerably abstract state space in chapter iii of cl he has used two sections to discuss the properties of attractors for monotone flows and obtained some very important results he has shown that a monotone flow cannot have an attracting cycle see l corollary 2 41 and that every attractor must contain an equilibrium see cl theorem 3 11 in particular if the flow is strongly monotone he has shown that every attractor k contains an order stable equilibrium see 1 theorem 4 11 meanwhile if we suppose z is attracted to k but is not quasiconvergent then k contains two order stable equilibria p q such that p o z 4 q see cl theorem 4 31 the purpose of this paper is further to discuss the properties of attractors for strongly monotone flows we shall study when an o limit set can be an 210 0022 247x 91 3 00

stability and convergence in strongly monotone dynamical systems soit x un systeme dynamique x est un espace topologique et le flot t t r est une famille d applications continues de sous ensembles ouverts de x dans x on decrit qualitativement le comportement asymptotique quand t des trajectoires t t x pour les etats initiaux x dans le cas de systemes ayant un principe de comparaison fort

geometric quantization of time dependent completely integrable hamiltonian systems a time dependent completely integrable hamiltonian system is quantized with respect to time dependent action angle variables near an instantly compact regular invariant manifold its hamiltonian depends only on action variables and has a time independent countable energy spectrum

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

stage structured models of intra and inter specific competition within age classes abstract in some species larvae and adults experience competition in completely different ways simple stage structured models without larval competition usually yield a single delay equation for the adults using an age structured system incorporating competition among both larvae and adults we derive a system of distributed delay equations for the numbers of larvae and adults the system is neither cooperative nor reducible to a single equation for either variable positivity boundedness and uniform strong persistence are established linear stability analysis of equilibria is difficult due to the strong coupling but results are proved for small delays using monotone systems theory and exponential ordering for small delay we prove a theorem on generic convergence to equilibria which does not directly follow from standard theory but can be proved indirectly using comparison arguments finally we consider an extension to two strain competition and prove theorems on the linear stability of the boundary equilibria

a front tracking method for viscous incompressible multi fluid flows abstract a method to simulate unsteady multi fluid flows in which a sharp interface or a front separates incompressible fluids of different density and viscosity is described the flow field is discretized by a conservative finite difference approximation on a stationary grid and the interface is explicitly represented by a separate unstructured grid that moves through the stationary grid since the interface deforms continuously it is necessary to restructure its grid as the calculations proceed in addition to keeping the density and viscosity stratification sharp the tracked interface provides a natural way to include surface tension effects both two and three dimensional full numerical simulations of bubble motion are presented

an accurate integral equation method for simulating multi phase stokes flow we introduce a numerical method based on an integral equation formulation for simulating drops in viscous fluids in the plane it builds upon the method introduced by kropinski in 2001 17 but improves on it by adding an interpolatory quadrature approach for handling near singular integrals such integrals typically arise when drop boundaries come close to one another and are difficult to compute accurately using standard quadrature rules adapting the interpolatory quadrature method introduced by helsing and ojala in 2008 11 to the current application very general drop configurations can be handled while still maintaining stability and high accuracy the performance of the new method is demonstrated by some challenging numerical examples

monotone semiflows generated by functional differential equations abstract this obtains sufficient condition for an autonomous functional differential equation to generate a strongly monotone semiflow on a suitable state space this allows the application to functional differential equations of very powerful recent results on strongly monotone semiflows due to hirsch and matano in addition a very striking relationship is established between such functional differential equations and corresponding ordinary differential equations an example involving a biochemical feedback loop is considered keywords stability steady states

convergence of solutions for an equation with state dependent delay abstract a result of smith and thieme shows that if a semiflow is strongly order preserving then a typical orbit converges to the set of equilibria for the equation with state dependent delay t x t f x t r x t where 0 and f and r are smooth real functions with f 0 0 and f 0 we construct a semiflow which is monotone but not strongly order preserving we prove a convergence result under a monotonicity condition different from the strong order preserving property and apply it to the above equation to obtain generic convergence

rapid fluctuations of chaotic maps on rn the iterates f n of a chaotic map f display heightened oscillations or fluctuations as n i ff is a chaotic interval map in one dimension then it is now known that the total variation of f n on that interval grows exponentially with respect to n g chen t huang y huang chaotic behavior of interval maps and total variations of iterates internat j bifur chaos 14 2004 2161 2186 however the characterization of chaotic behavior of maps in multi dimensional spaces is generally much more challenging here we generalize the definition of bounded variations for vector valued maps in terms of the hausdorff measure and then use it to study what we call rapid fluctuations on fractal sets in multi dimensional chaotic discrete dynamical systems the relations among rapid fluctuations strict turbulence and positive entropy are established for lipschitz continuous systems on general n dimensional euclidean spaces applications to planar monotone or competitive systems and triangular systems on the square are also given

interfacial dynamics for stokes flow theoretical and computational aspects of the method of interfacial dynamics for flow at vanishing reynolds number are considered the mathematical formulation relies on the boundary integral representation that expresses the flow in terms of distributions of stokes flow singularities over the interfaces the densities of the distributions are identified with the jump in the hydrodynamic traction due to interfacial in plane and transverse tensions the interfacial velocity or the strength of a hydrodynamic potential the numerical procedure involves describing the interfaces in terms of interfacial marker points that reproduce the evolving shapes of the interfaces by global or local interpolation solving integral equations of the second kind for the interfacial velocity or for the density of a hydrodynamic potential and computing the motion of the marker points while simultaneously updating interfacial fields relevant to the dynamics including the concentration of a surfactant and the position of interfacial point particles at an equilibrium configuration interfaces exhibiting isotropic tension elastic tensions viscous and incompressible behavior are considered the mathematical modeling of the tensions and bending moments developing over interfaces with a membrane like constitution is discussed in the context of the theory of thin shells to facilitate the numerical implementation the coupling of the interfacial mechanics to the hydrodynamics by means of interface force and torque balances is formulated in global cartesian coordinates recent progress in the implementation of boundary element methods is reviewed and areas for further research are identified

numerical methods for multiple inviscid interfaces in creeping flows we present new highly accurate and efficient methods for computing the motion of a large number of two dimensional closed interfaces in a slow viscous flow the interfacial velocity is found through the solution to an integral equation whose analytic formulation is based on complex variable theory for the biharmonic equation the numerical methods for solving the integral equations are spectrally accurate and employ a fast multipole based iterative solution procedure which requires only o n operations where n is the number of nodes in the discretization of the interface the interface is described spectrally and we use evolution equations that preserve equal arclength spacing of the marker points we assume that the fluid on one side of the interface is inviscid and we discuss two different physical phenomena bubble dynamics and interfacial motion driven by surface tension viscous sintering applications from buoyancy driven bubble interactions the motion of polydispersed bubbles in an extensional flow and the removal of void spaces through viscous sintering are considered and we present large scale fully resolved simulations involving o 100 closed interfaces

fast fourier transform accelerated fast multipole algorithm this paper describes an cal o p 2 log 2 p n implementation of the fast multipole algorithm fma for n body simulations this method of computing the fma is faster than the original which is cal o p 4n where p is the number of terms retained in the truncated multipole expansion representation of the potential field of a collection of charged particles the p term determines the accuracy of the calculation the limiting cal o p 4 computation in the original fma is a convolution like operation on a matrix of multipole coefficients this paper describes the implementation details of a conversion of this limiting computation to linear convolution which is then computed in the fourier domain using the fast fourier transform fft based on a method originally outlined by greengard and rokhlin in addition this paper describes a new block decomposition of the multipole expansion data that provides numerical stability and efficient computation the resulting cal o p 2 log 2 p subroutine has a speedup of 2 on a sequential processor over the original method for p 8 and a speedup of 4 for p 16 the new subroutine vectorizes well and has a speedup of 3 on a vector processor at p 8 and a speedup of 6 at p 16

positive heteroclinics and traveling waves for scalar population models with a single delay the existence of positive heteroclinic solutions is proven for a class of scalar population models with one discrete delay traveling wave solutions for scalar delayed reaction diusion equations are also obtained as perturbations of heteroclinic solutions of the associated equation without diusion as an illustration the results are applied to the nicholson s blowflies equation with diusion n t t x d 2n x2 t x n t x pn t x e an t x in the case of p e for which the nonlinearity is non monotone

asymptotic tracking for constrained monotone systems this note addresses tracking control of nonlinear discrete time monotone systems subject to input and state constraints forcing saturation on a previously designed controller may in general lead to destabilization or at least result in constraint violation and performance losses hereby it is shown that for a certain class of nonlinear monotone systems it is possible to design a static nonlinear output feedback which saturated among suitable state dependent bounds is able to guarantee constraint satisfaction and asymptotic tracking of piecewise constant references with a moderate online computational burden simulation experiments concerning the synthesis of a protein demonstrate the effectiveness of the proposed control strategy