criteria for asymptotic stability of linear time delay systems several sufficient conditions which guarantee stability of linear time delay systems are derived each of these results is expressed by a succinct scalar inequality and corresponds to a certain extent to the tradeoff between simplicity and sharpness

stability criteria of linear neutral systems with multiple delays in the present note stability of linear neutral systems with multiple delays is investigated by principle of the argument stability criteria are derived which are necessary and sufficient for asymptotic stability of the neutral systems the stability criteria need only to evaluate the characteristic function and the argument on the boundary of a certain rectangular region several numerical examples are given to illustrate the results

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

an entire solution of a bistable parabolic equation on r with two colliding pulses we consider semilinear parabolic equations of the form equation a ut uxx f u x r t i where i 0 i 0 or i i solutions defined for all x t r2 x t r2 are referred to as entire solutions assuming that f c1 r f c1 r is of a bistable type with stable constant steady states 0 and 0 0 we show the existence of an entire solution u x t u x t of the following form for t t u t u t has two humps or pulses one near the other near as t increases the humps move toward the origin x 0x 0 eventually colliding and forming a one hump final shape of the solution with respect to the locally uniform convergence the solution u t u t is a heteroclinic orbit connecting the stable steady state 0 to the unstable ground state of the equation uxx f u 0uxx f u 0 we find the solution u as the limit of a sequence of threshold solutions of the cauchy problem for equation a

stability of discrete delay systems boundary criteria this paper deals with the stability of linear discrete delay systems two stability criteria that are described by evaluating a corresponding harmonic function on the boundary of a torus region are presented testing these criteria can be proved to be much easier than those reported in the literature

asymptotic stability for the system of neutral delay differential equations abstract this paper deals with the asymptotic stability of the system of neutral delay differential equations in an angle which has fewer reports stability regions are presented geometrically as well as two criteria which are obtained through the evaluation of a harmonic function on the boundary of a certain region numerical experiments on various circumstances are shown to locate the specific region which can make the system unstable so as to exclude it from the whole complex plane

a sequence of order relations encoding heteroclinic connections in scalar parabolic pde abstract we address the problem of heteroclinic connections in the attractor of dissipative scalar semilinear parabolic equations type display u t u xx f x u u x 0 on a bounded interval with neumann conditions introducing a sequence of order relations we prove a new and simple criterion for the existence of heteroclinic connections using only information about nodal properties of solutions to the stationary ode problem this result also allows for a complete classification of possible attractors in terms of the permutation of the equilibria given by their order at the two boundaries of the interval

sturm attractors for quasilinear parabolic equations abstract the goal of this paper is to construct explicitly the global attractors of quasilinear parabolic equations as it was done for the semilinear case by brunovsk and fiedler 1986 8 and generalized by fiedler and rocha 1996 13 in particular we construct heteroclinic connections between hyperbolic equilibria stating necessary and sufficient conditions for heteroclinics to occur such conditions can be computed through a permutation of the equilibria lastly an example is computed yielding the well known chafee infante attractor

delay independent stability criteria for discrete delay systems several sufficient conditions for stability of linear discrete delay systems are derived since these conditions are independent of the delay and possess simple forms they will provide useful tools to check stability of the systems at the first stage

stability of retarded delay differential systems a practical stability test for linear retarded delay differential systems is proposed based on the fact that all unstable zeros of the characteristic quasipolynomial must lie within a restricted bounded region of the complex plane the test provides a readily computable necessary and sufficient condition for asymptotic stability it also copes equally well with both cases of commensurate and non commensurate delays furthermore a method for evaluating the abscissa of stability of the quasi polynomial defined as the largest of the real parts of the zeros of the quasipolynomial is given the method is based on a known technique which makes repeated use of a stability test and thereby avoids the calculation of all the zeros the stability test and the method for computing the abscissa of stability provide useful computational tools for the design of dynamical systems with delays using the method of inequalities numerical examples are given

stable methods for vortex sheet motion in the presence of surface tension boundary integral techniques provide a convenient way to study the evolution of an interface between inviscid liquids several studies have revealed that standard numerical approximations tend to lead to unstable methods and various remedies have been introduced and tested in this paper we conduct a stability analysis of the linearized equations with a particular objective in mind the determination of how the discrete system fails to capture the physical dispersion relation precisely for the available discrete modes we discover two reasons for the typical failure in numerical discretizations one is the inability of the mesh to represent the vorticity created by surface tension effects on the finest scale and the other is the inaccuracies in the evaluation of the boundary integral for the velocity with the insight gained from our linear analysis we propose a new method that is spectrally accurate and linearly stable further the exact dispersion relation is obtained for all the available discrete modes numerical tests suggest that the method is also stable in the nonlinear regime however our method runs into difficulties generic to methods based on lagrangian motion the markers accumulate near a stagnation point on the interface forcing us to use an ever decreasing time step in our explicit method we introduce a redistribution of markers to overcome this difficulty when we redistribute according to equal arclength we find excellent agreement with a method based on preserving equal spacing in arclength

scalar parabolic pdes and braids the comparison principle for scalar second order parabolic pdes on functions u t x admits a topological interpretation pairs of solutions u 1 t 1 and u 2 t evolve so as to not increase the intersection number of their graphs we generalize to the case of multiple solutions u t n 1 by lifting the graphs to legendrian braids we give a global version of the comparison principle the curves u t evolve so as to weakly decrease the algebraic length of the braid we define a morse type theory on legendrian braids which we demonstrate is useful for detecting stationary and periodic solutions to scalar parabolic pdes this is done via discretization to a finite dimensional system and a suitable conley index for discrete braids the result is a toolbox of purely topological methods for finding invariant sets of scalar parabolic pdes we give several examples of spatially inhomogeneous systems possessing infinite collections of intricate stationary and time periodic solutions

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

generic hyperbolicity of equilibria and periodic orbits of the parabolic equation on the circle in this paper we show that for scalar reaction diffusion equations on the circle s 1 the property of hyperbolicity of all equilibria and periodic orbits is generic with respect to the non linearity in other words we prove that in an appropriate functional space of non linear terms in the equation the set of functions for which all equilibria and periodic orbits are hyperbolic is a countable intersection of open dense sets the main tools in the proof are the property of the lap number and the sard smale theorem

on asymptotic stability of delay difference systems in this paper we obtain some criteria for determining the asymptotic stability of the zero solution for some classes of delay difference systems by using a discrete version of the second lyapunov method our results can be well suited for computational purposes

numbers of zeros on invariant manifolds in reaction diffusion equations on considere l equation a une dimension u t u xx f x u t 0 0 x 1 avec les conditions aux limites de dirichlet u t 0 t 1 0 ou f 0 1 r r est bc 1 c k k 1

a poincare bendixson theorem for scalar reaction diffusion equations for scalar equations u t u xx f x u u x with x e s 1 and f e c 2 we show that the classical theorem of poincare and bendixson holds the limit set of any bounded solution satisfies exactly one of the following alternatives this is surprising because the system is genuinely infinite dimensional

a permutation characterization of sturm global attractors of hamiltonian type abstract we consider neumann boundary value problems of the form u t u x x f on the interval 0 x for dissipative nonlinearities f f u a permutation characterization for the global attractors of the semiflows generated by these equations is well known even in the much more general case f f x u u x we present a permutation characterization for the global attractors in the restrictive class of nonlinearities f f u in this class the stationary solutions of the parabolic equation satisfy the second order ode v f v 0 and we obtain the permutation characterization from a characterization of the set of 2 periodic orbits of this planar hamiltonian system our results are based on a diligent discussion of this mere pendulum equation

the morse smale property for a semi linear parabolic equation on etudie la dynamique d equations aux derivees partielles paraboliques semi lineaires du type suivant u t u xx f x u 0 0 u 0 t u 1 t 0 t 0 u x 0 x

delay independent exponential stability criteria for time varying discrete delay systems derives delay independent exponential stability conditions for linear nonlinear time varying discrete delay systems since these conditions are of delay independence and easily verifiable they may provide handy tools for the stability analysis

axisymmetric vortex sheet motion accurate evaluation of the principal value integral this paper concerns the accurate evaluation of the principal value integral governing axisymmetric vortex sheet motion previous quadrature rules for this integral lose accuracy near the axis of symmetry an approximation by de bernadinis and moore dbm that converges pointwise at the rate of o h3 has maximal errors near the axis that are o h as a result the discretization error is not smooth it contains high wavenumber frequencies that make it difficult to resolve the vortex sheet motion this paper explains the reason for the degeneracy near the axis and proposes a modified quadrature rule that is uniformly o h3 the results are based on an analytic approximation of the integrand whose integral can be precomputed the modification is implemented at negligible additional cost per timestep as an example it is applied to compute the evolution of an initially spherical vortex sheet

transversality of homoclinic orbits and exponential dichotomies for parabolic equations abstract in this paper we discuss the existence of exponential dichotomies on r of linear parabolic equations depending on small parameters and provide a tool of proving the transversality of the homoclinic orbits of parabolic equations and by making use of the results on exponential dichotomies of this paper we investigate the transversality of homoclinic orbits for parabolic equations

some simplifications of the graphical nyquist criterion the nyquist criterion for feedback stability is reformulated in a very simple fashion using the motion of positive and negative crossings a number of encirclements of a point by the nyquist plot can be quickly computed with this reformulation it is a simple matter to determine the value of the feedback gain for which the system is stable

realization of meander permutations by boundary value problems abstract we consider neumann boundary value problems of the form uxx f x u ux 0 on the unit interval 0 x 1 for a certain class of dissipative nonlinearities f associated to these problems we have i meanders in the phase space u ux r 2 which are connected oriented simple curves on the plane intersecting a fixed oriented line the u axis in n points corresponding to the solutions and ii meander permutations f s n obtained by ordering the intersection points first along the u axis and then along the meander the meander permutation f is the permutation defined by the braid of solutions in the space x u ux it was recently shown by fiedler and rocha that f determines the global attractor of the dynamical system generated by the semilinear parabolic differential equation ut uxx f x u ux up to c0 orbit equivalence therefore these permutations are of considerable importance in the classification problem of the morse smale attractors for these dynamical systems in this paper we present a purely combinatorial characterization of the set of meander permutations that are realizable by the above boundary value problems

application of adaptive quadrature to axi symmetric vortex sheet motion studies of the formation of fine structures on free surfaces in liquids such as curvature singularities or interface pinching demand that the motion of the interface must be computed very accurately boundary integral techniques are a popular choice in such studies because they reduce the dimension of the problem by one on the other hand the boundary integrals are singular and their accurate evaluation can prove quite challenging in two dimensional motion the interface is just a curve when this curve is closed or periodic the singularity in the integrand may be removed and the trapezoidal rule may be applied with spectral accuracy unfortunately the nature of the singularity in the integrand for three dimensional motion is much more difficult to treat in this paper we present an accurate adaptive quadrature to compute the motion of a vortex sheet in axi symmetric flow the technique is based on a vector potential formulation which offers some computational advantages over other methods based on the biot savart integral direct numerical computations show that our technique is much more accurate and efficient than existing techniques we apply our technique to study the evolution of an initially spherical vortex sheet we present evidence of the formation of a 3 2 power singularity in the curvature of the vortex sheet

some sufficient conditions for exponential stability of linear neutral functional differential equations abstract in this paper we give some sufficient conditions for exponential stability of linear neutral functional differential equations the obtained results are extensions of the recent results in d q cao p he stability criteria of linear neutral systems with a single delay appl math comput 148 2004 135 143 d q cao p he sufficient conditions for stability of linear neutral systems with a single delay appl math lett 17 2004 139 144 g d hu g d hu x f zou stability of linear neutral systems with multiple delays boundary criteria appl math comput 148 2004 707 715 j h park s won a note on stability of neutral delay differential systems j franklin inst 336 1999 1 9 to linear neutral functional differential equations

on the homotopy index for infinite dimensional semiflows absrac r in this paper we consider semiflows whose solution operator is eventually a conditional a contraction such semiflows include solutions of retarded and neutral functional differential equations of parabolic and certain other classes of partial differential equations we prove existence of nonsmooth isolating blocks and index pairs for such semiflows via the construction of special lyapunov functionals we show that index pairs enjoy all the properties needed to define the notion of a homotopy index thus generalizing earlier results of conley 2 finally using a result of maii 9 we prove that under additional smoothness assumptions on the semiflow the homotopy index is essentially a finite dimensional concept this gives a formal justification of the applicability of waiewski s principle to infinite dimensional problems several examples illustrate the theory

computation of axisymmetric suction flow through porous media in the presence of surface tension abstract the effect of small surface tension on a class of axisymmetric flows with suction is studied numerically the dynamic evolution of a blob of incompressible viscous fluid surrounded by air and drawn into an interior sink is considered the velocity field of the viscous fluid is assumed to satisfy darcy s law and thus the motion is that of a flow through porous media the fluid interface motion is computed using a highly accurate boundary integral method this method combines recent numerical techniques to achieve efficient and high order space and time discretizations for this type of flow through accurate computations it is shown that in the presence of small surface tension the dynamic behavior of the axisymmetric flows is very similar to that of the two dimensional hele shaw counterparts a long finger develops in the fluid interface and forms a cone singularity when it reaches the sink before all the fluid is sucked out the finger bulges and develops a neck for sufficiently small surface tension the bulge neck formation is enhanced by the additional component of the mean curvature in the three dimensional flow however its effect is not strong enough to cause the interface to pinch off at the neck for the data considered here

simple stability criteria for nonlinear time varying discrete systems abstract we present two sufficient conditions for asymptotic stability of nonlinear time varying discrete systems our main results generalize mori s result for linear discrete time invariant systems to nonlinear time varying systems

stability of neutral delay differential systems boundary criteria this paper deals with the delay dependent stability of linear invariant neutral delay differential systems with a positive constant delay six criteria that are described by evaluating a corresponding harmonic function on the boundary of a certain rectangular region are presented

algebraic criteria for stability of linear neutral systems with a single delay abstract the stability of linear neutral delay differential systems with a single delay via routh hurwitz and schur cohn criteria is investigated some algebraic criteria for delay independent stability are presented these criteria may complement those reported in the literature finally two examples illustrate the criteria

spatially monotone homoclinic orbits in nonlinear parabolic equations of super fast diffusion type this work deals with positive classical solutions of the degenerate parabolic equation u t u p u xx quad quad star when p 2 which via the substitution v u1 p transforms into the super fast diffusion equation v t v m 1 v x x with m frac 1 p 1 in 1 0 it is shown that star possesses some entire positive classical solutions defined for all t in mathbb r and x in mathbb r which connect the trivial equilibrium to itself in the sense that u x t 0 both as t and as t locally uniformly with respect to x in mathbb r moreover these solutions have quite a simple structure in that they are monotone increasing in space the approach is based on the construction of two types of wave like solutions one of them being used for t 0 and the other one for 0 t both types exhibit wave speeds that vary with time and tend to zero as t and t respectively the solutions thereby obtained decay as x uniformly with respect to t in mathbb r but they are unbounded as x it is finally shown that within the class of functions having such a behavior as x there does not exist any bounded homoclinic orbit

vortex blob methods applied to interfacial motion we develop a boundary integral method for computing the motion of an interface separating two incompressible inviscid fluids the velocity integral is regularized so that the vortex sheet on the interface is replaced by a sum of blobs of vorticity the regularization allows control of physical instabilities we design a class of high order blob methods and analyze the errors numerical tests suggest that the blob size should be scaled with the local spacing of the interfacial markers for a vortex sheet in one fluid with a first order kernel we obtain a spiral roll up similar to krasny j comput phys 65 1986 292 but the higher order kernels lead to more detailed structure we verify the accuracy of the new method by computing a liquid gas interface with rayleigh taylor instability we then apply the method to the more difficult case of rayleigh taylor flow separating two fluids of positive density a case for which the regularization appears to be essential as found by kerr and tryggvason both j comput phys 76 1988 48 75 1988 253 we use a blob regularization in certain local terms in the evolution equations as well as in the velocity integral we find strong evidence that improved spatial resolution with fixed blob size leads to a converged regularized solution without numerical instabilities however it is not clear that there is a weak limit as the regularization is decreased

stability of linear neutral systems with multiple delays boundary criteria the stability of linear neutral delay differential systems with multiple delays is investigated the delay dependent stability criteria are presented through the evaluation of the corresponding harmonic functions on the boundary of a certain half circular region and such criteria are extensions of some results in literature an example is given to illustrate the stability criteria

layers and spikes in non homogeneous bistable reaction diffusion equations we study e 2 u f u x au 1 u u where a a u x 0 x 0 1 and e 0 is sufficiently small on an interval 0 l with boundary conditions u 0 at x 0 l all solutions with an e independent number of oscillations are analyzed existence of complicated patterns of layers and spikes is proved and their morse index is determined it is observed that the results extend to f a u x u u u with x x x and also to an infinite interval

the dynamics of rotating waves in scalar reaction diffusion equations on etudie l attracteur compact maximal pour l equation u t u xx f u u x avec des conditions aux limites periodiques on montre qu un ensemble limite contient une onde de rotation

the zero set of a solution of a parabolic equation on etudie l ensemble nul d une solution u t x de l equation u t a x t u xx b x t u x c x t u sous des hypotheses tres generales sur les coefficients a b et c

the numerical solution of discrete delay systems in this note we consider the numerical solution of initial value discrete delay systems an interpolation procedure is introduced to compute the numerical solution the convergence and stability of the interpolation procedure for the discrete delay systems is discussed

some simple criteria for stability of neutral delay differential systems this paper deals with the stability of linear neutral delay differential systems some simple criteria for delay independent stability of this class of problems which are described by evaluating the measure and norm of corresponding matrices are presented the main results proposed in this study are sharper than those reported in the literature

the numerical solution of discrete delay systems abstract in this note we consider the numerical solution of initial value discrete delay systems an interpolation procedure is introduced to compute the numerical solution the stability of the interpolation procedure for linear discrete delay systems is discussed

heteroclinic orbits between rotating waves of semilinear parabolic equations on the circle abstract we investigate heteroclinic orbits between equilibria and rotating waves for scalar semilinear parabolic reaction advection diffusion equations with periodic boundary conditions using zero number properties of the solutions and the phase shift equivariance of the equation we establish a necessary and sufficient condition for the existence of a heteroclinic connection between any pair of hyperbolic equilibria or rotating waves

a new simple approach to the derivation of exact analytical formulae for the zeros of analytic functions a very simple method for the construction of the polynomial whose zeros coincide with the zeros of an analytic function inside and along a simple closed contour in the complex plane based on an appropriate application of the cauchy theorem in complex analysis is proposed the present method was motivated by the classical burniston siewert method based on the theory of the riemann hilbert boundary value problem for the construction of the aforementioned polynomial but although essentially equivalent to the burniston siewert method it is much simpler

stability analysis for delay differential equations with multidelays and numerical examples in this paper we are concerned with the asymptotic stability of the delay differential equation x t a 0 x t a k x t k where a 0 a k c d d are constant complex matrices and x t k x 1 t k1 x 2 t k2 x d t kd t kl 0 stand for n x d constant delays k 1 n i 1 d we obtain two criteria for stability through the evaluation of a harmonic function on the boundary of a certain region we also get similar results for the neutral delay differential equation x t lx t m i x t i n j x t j where l m i and nj e c d d are constant complex matrices and i j 0 stands for constant delays i 1 m j 1 n numerical examples on various circumstances are shown to check our results which are more general than those already reported

connectivity and design of planar global attractors of sturm type ii connection graphs abstract based on a morse smale structure we study planar global attractors a f of the scalar reaction advection diffusion equation u t u x x f x u u x in one space dimension we assume neumann boundary conditions on the unit interval dissipativeness of f and hyperbolicity of equilibria we call a f sturm attractor because our results strongly rely on nonlinear nodal properties of sturm type the planar sturm attractor consists of equilibria of morse index 0 1 or 2 and their heteroclinic connecting orbits the unique heteroclinic orbits between adjacent morse levels define a plane graph c f which we call the connection graph its 1 skeleton c f 1 consists of the unstable manifolds separatrices of the index 1 morse saddles we present two results which completely characterize the connection graphs c f and their 1 skeletons c f 1 in purely graph theoretical terms connection graphs are characterized by the existence of pairs of hamiltonian paths with certain chiral restrictions on face passages their 1 skeletons are characterized by the existence of cycle free orientations with certain restrictions on their criticality such orientations are called bipolar in h de fraysseix p o de mendez p rosenstiehl bipolar orientations revisited discrete appl math 56 1995 157 179 in b fiedler c rocha connectivity and design of planar global attractors of sturm type i orientations and hamiltonian paths crelle j reine angew math 2007 in press we have shown the equivalence of the two characterizations moreover we have established that connection graphs of sturm attractors indeed satisfy the required properties in the present paper we show conversely how to design a planar sturm attractor with prescribed plane connection graph or 1 skeleton of the required properties in b fiedler c rocha connectivity and design of planar global attractors of sturm type iii small and platonic examples 2007 submitted for publication we describe all planar sturm attractors with up to 11 equilibria we also design planar sturm attractors with prescribed platonic 1 skeletons

stability in linear delay equations abstract for linear autonomous differential difference equations of retarded or neutral type necessary and sufficient conditions are given for the zero solution to be stable hyperbolic for all values of the delays

numerical simulations of inviscid capillary pinchoff inviscid capillary pinchoff is studied numerically for an axisymmetric model problem in which a bubble pinches at two points on the symmetry plane breaking into two symmetric end bubbles and a satellite bubble in between results are presented for a range of density jumps across the bubble the numerical method uses a formulation in terms of arclength and tangent angle and incorporates a new procedure to redistribute the computational points dynamically in order to maintain resolution in regions of high curvature the results are compared with alternative computations by lepinnen and lister phys fluids 15 2003 568 where available new results include details about the cone crater structure near pinchoff and the dependence of satellite bubble volume and pinchoff time on the density ratio

high order quadratures for the evaluation of interfacial velocities in axi symmetric stokes flows we propose new high order accurate methods to compute the evolution of axi symmetric interfacial stokes flow the velocity at a point on the interface is given by an integral over the surface quadrature rules to evaluate these integrals are developed using asymptotic expansions of the integrands both locally about the point of evaluation and about the poles where the interface crosses the axis of symmetry the local expansions yield methods that converge to the chosen order pointwise for fixed evaluation point the pole expansions yield corrections that remove maximal errors of low order introduced by singular behaviour of the integrands as the evaluation point approaches the poles an interesting example of roundoff error amplification due to cancellation is also addressed the result is a uniformly accurate fifth order method second order pointwise fifth order and uniform fifth order methods are applied to compute three sample flows each of which presents a different computational difficulty an initially bar belled drop that pinches in finite time a drop in a strain flow that approaches a steady state and a continuously extending drop in each case the fifth order methods significantly improve the ability to resolve the flow the examples furthermore give insight into the effect of the corrections needed for uniformity we determine conditions under which the pointwise method is sufficient to obtain resolved results and others under which the corrections significantly improve the results

simple stability criteria for single and composite linear systems with time delays easy ways to test the stability of systems involving time delays have been sought in this paper several stability conditions with an extremely simple form are provided first some criteria for single linear systems with time delays are presented then these results are extended to the composite linear systems with time delays which exist in each subsystem and in the interconnections between subsystems among these criteria those which are expressed by scalar inequalities also permit us to assess the transient behaviour of systems