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

existence of dynamical hull functions with two variables for the ac driven frenkel kontorova model abstract in this paper we are concerned with the dynamics of incommensurate structures in the overdamped limit case of the frenkel kontorova model with periodic driving force we show in the infinite dimensional phase space there exists an ordered circle on which both the poincare map and the space shift map induce orientation preserving circle homeomorphisms with rotation numbers and respectively the latter characterizing the mean spacing of particles furthermore we prove that the average velocity for the particle chain with irrational exists and t where t denotes the period of the driving force and it is continuous and nondecreasing with respect to f the average of the external driving force if the shift map on the invariant ordered circle is ergodic we demonstrate the existence of two variable dynamical hull functions and that there are no mode locking behaviors

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

the periodic orbits of an area preserving twist map we study the oscillation properties of periodic orbits of an area preserving twist map the results are inspired by the similarity between the gradient flow of the associated action function and a scalar parabolic pde in one space dimension the conley zehnder morse theory is used to construct orbits with prescribed oscillatory behavior

a model for the growth of a population exhibiting stage structure cannibalism and cooperation abstract a model of a stage structured population with fixed maturity time for the immature stage and interaction terms that may be interpreted as cooperation or cannibalism is proposed the existence and stability of the equilibrium set are discussed in the case of cannibalism it is shown by a numerical example how a hopf bifurcation could result in a stable periodic solution

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

curvature measures and random sets ii in choosing models of stochastic geometry three general problems play a role which are closely connected with each other in the present paper second order local geometric properties of random subsets of r d are of interest these properties are described by signed curvature measures in a measure geometric context the theory of point processes on general spaces here on the space of subsets with positive reach provides an appropriate framework for solving construction and measurability problems mean value relations for random curvature measures associated with such set processes are derived by means of invariance properties ergodic interpretations of the curvature densities are also given the appendix provides auxiliary results for random signed radon measures in locally compact separable hausdorff spaces

the discrete frenkel kontorova model and its extensions i exact results for the ground states abstract we present a rigorous study of the classical ground states under boundary conditions of a class of one dimensional models generalizing the discrete frenkel kontorova model the extremalization equations of the energy of these models turn out to define area preserving twist maps which exhibits periodic quasi periodic and chaotic orbits for all boundary conditions we select among all the extremum solutions of the energy of the model those which correspond to the ground states of the infinite system we prove that these ground states are either periodic commensurate or quasi periodic incommensurate but are never chaotic we also prove the existence of elementary discommensurations which are minimum energy configuration of the model for certain special boundary conditions the topological structure of the whole set of ground states is described in details in addition to physical applications consequences for twist map homeomorphisms are mentioned part ii s aubry p y ledaeron and g andre will be mostly devoted to exact results on the transition by breaking of analyticity which occurs on the incommensurate ground states when the model parameters vary and on its connection with the stochasticity threshold in the corresponding twist map

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

interfacial waves and deformations in pre stressed elastic media the propagation of interfacial stoneley waves along the boundary between two half spaces of pre stressed incompressible isotropic elastic material is examined the underlying deformation in each half space corresponds to a pure homogeneous strain with one principal axis of strain normal to the interface and the others having a common orientation the secular equation governing the wave speed for propagation along a principal axis is obtained in respect of general strain energy functions detailed analysis of the secular equation reveals general sufficient conditions for the existence of a wave and in particular cases necessary and sufficient conditions for the existence of a unique interfacial wave it is also shown that when an interfacial wave exists its speed is greater than that of the least of the rayleigh wave speeds for the separate half spaces paralleling a result from the linear theory for the special case of quasi static interfacial deformations corresponding to vanishing wave speed an existence criterion is found moreover it is shown that inequalities that exclude surface deformations in each half space also exclude interfacial deformations dependence of the above results on the underlying homogeneous deformations and on material parameters is illustrated by numerical results for the neo hookean strain energy function

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

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

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

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

ghost circles for twist maps abstract completely ordered invariant circles are found for the gradient of the energy flow in the state space containing the critical sets corresponding to the birkhoff orbits of all rotation number in particular these ghost circles contain the aubry mather sets and map invariant circles as completely critical sets when these exist we give a criterion for a sequence of rational ghost circles to converge to a completely critical one

rational quadratic flux minimizing circles for area preserving twist maps abstract by considering families of periodic pseudo orbits on the cylindrical phase space of an area preserving twist map we construct curves of rational rotation number that are critical points of a non negative error norm the mean square action gradient or quadratic flux two alternative angle parametrizations of the curves are presented and numerical evidence is given that the curves possess the property of non intersection thus making them candidates for a generalized action angle representation for nonintegrable systems

carrying simplices in nonautonomous and random competitive kolmogorov systems abstract the purpose of this paper is to investigate the asymptotic behavior of positive solutions of nonautonomous and random competitive kolmogorov systems via the skew product flows approach it is shown that there exists an unordered carrying simplex which attracts all nontrivial positive orbits of the skew product flow associated with a nonautonomous random competitive kolmogorov system

optical hamiltonians and symplectic twist maps abstract this paper concentrates on optical hamiltonian systems of t t n i e those for which hpp is a positive definite matrix and their relationship with symplectic twist maps we present theorems of decomposition by symplectic twist maps and existence of periodic orbits for these systems the novelty of these results resides in the fact that no explicit asymptotic condition is imposed on the system

construction of invariant measures of lagrangian maps minimisation and relaxation if f is an exact symplectic map on the it d dimensional cylinder mathbb t d times mathbb r d with a generating function h having superlinear growth and uniform bounds on the second derivative we construct a strictly gradient semiflow phi on the space of shift invariant probability measures on the space of configurations mathbb r d mathbb z stationary points of phi are invariant measures of f and the rotation vector and all spectral invariants are invariants of phi using phi and the minimisation technique we construct minimising measures with an arbitrary rotation vector rho in mathbb r d and with an additional assumption that f is strongly monotone we show that the support of every minimising measure is a graph of a lipschitz function using phi and the relaxation technique assuming a weak condition on phi satisfied e g in hedlund s counter example and in the anti integrable limit we show existence of double recurrent orbits of f and f ergodic measures with an arbitrary rotation vector rho in mathbb r d and action arbitrarily close to the minimal action a rho

the birkhoff lewis fixed point theorem and a conjecture of v i arnold abstract the following conjecture of v i arnold is proved every measure preserving diffeomorphism of the torus t2 which is homologeous to the identity and which leaves the center of mass invariant possesses at least 3 fixed points the proof of this global fixed point theorem does not make use of the generating function technique the theorem is a consequence of the statement that a hamiltonian vector field on a torus t2n which depends periodically on time possesses at least 2n 1 forced oscillations these periodic solutions are are found using the classical variational principle by means of two qualitative statements for general flows a second conjecture of v i arnold proved concerns a birkhoff lewis type fixed point theorem for symplectic maps additional keywords periodic functions variational principles equations reprints author

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

generalizations of the poincare birkhoff theorem in this article we prove some generalizations of the classical poincar birkhoff theorem on area preserving homeomorphisms of the annulus which satisfy a boundary twist condition the work of g d birkhoff on this theorem and its applications can be found in b1 b2 and chapter v of b3 a more modern treatment can be found in b n we prove a theorem for the open annulus a s1 x 0 1 since as we will see the theorem for the closed annulus can easily be obtained from this result for the open annulus however it is not immediately obvious what should be the analogue of the twist condition it turns out that the most general hypothesis and the most natural from the point of view of our proof involves the notion of positively and negatively returning disks for some lift of f to the covering space a r x 0 1 more precisely if f a a is a lift of f a a we will say that the e is a positively returning disk for f if there is an open disk u c a such that f u n u 0 and fp u n u k 0 forsome n k 0 here u k denotes the set x k t i x t e u thus u is disjoint from its image but under iteration by f returns to a positive translate of itself a negatively returning disk is defined similarly but with k 0 such that fl u n u 0 in section 2 we prove the following

convergence to equilibrium on invariant d hypersurfaces for strongly increasing discrete time semigroups abstract convergence to equilibrium for every positive semi orbit of strongly increasing discrete time semigroups on strongly ordered spaces is proved the domain of the semigroup is assumed to be slightly more general than a closed order interval in a strongly ordered banach space which is continuously imbedded into a banach lattice the semigroup is assumed to be order compact and every positive semi orbit is assumed to be order bounded the crucial hypothesis is the ljapunov stability of all equilibria it is also proved that the set of equilibria is a simply ordered arc the key tools are invariant d hypersurfaces

instability of discontinuous traveling waves for hyperbolic balance laws abstract it is known that scalar hyperbolic conservation laws with source term and periodic initial value have a property of poincare bendixson type namely the solutions converge either to a constant state or to a periodic traveling wave which is necessarily discontinuous in this paper we show that generically with respect to the l 1 topology the solutions exhibit a behaviour of the former type we also show that while the rate of convergence to a constant state is exponential the convergence to a traveling wave can be arbitrarily slow

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

periodic orbits for hamiltonian systems in cotangent bundles we prove the existence of at least cl m periodic orbits for certain time dependent hamiltonian systems on the cotangent bundle of an arbitrary compact manifold m these hamiltonians are not necessarily convex but they satisfy a certain boundary condition given by a riemannian metric on m we discretize the variational problem by decomposing the time i map into a product of symplectic twist maps a second theorem deals with homotopically non trivial orbits of negative curvature

on support measures in minkowski spaces and contact distributions in stochastic geometry this paper is concerned with contact distribution functions of a random closed set n 1 n in r d where the n are assumed to be random nonempty convex bodies these distribution functions are defined here in terms of a distance function which is associated with a strictly convex gauge body structuring element that contains the origin in its interior support measures with respect to such distances will be introduced and extended to sets in the local convex ring these measures will then be used in a systematic way to derive and describe some of the basic properties of contact distribution functions most of the results are obtained in a general nonstationary setting only the final section deals with the stationary case

acceleration waves in elasto plasticity a spectral analysis of the acceleration wave problem in general elasto plastic materials is carried out whereby explicit expressions for the eigenvalues and eigenvectors are obtained in case of nonassociated plasticity all eigenvectors become nonorthogonal and one eigenvalue always remains unchanged and equal to the shear modulus for a very broad class of nonassociated plasticity models it is shown that the eigenvalues are always real implying that so called divergence instability can occur while flutter instability can never occur it is found that a certain value of the hardening modulus exists for which specific propagation directions will always imply that all wave speeds are identical and equal to the elastic distortion wave speed moreover in this situation the eigenvectors are arbitrary corresponding to a state of diffuse wave modes the criteria of von mises and rankine are used to illustrate some of the findings

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

an algorithm and a method to search bifurcation points in non linear problems this paper is devoted to a numerical method able to help the determination of the bifurcation threshold in non linear time independent continuum mechanic problems first some theoretical results about uniqueness are recalled in the framework of the large strain assumption the differences between the classical finite step problem and the rate problem are presented an iterative algorithm able to solve the rate problem is given using different initializations it is seen in some numerical experiments that it is possible with this algorithm to get different solutions when the underlying mathematical problem solved does not enjoy a uniqueness property the constitutive equations used have been chosen to be simple enough to deduce some theoretical knowledge about the corresponding uniqueness problems finally a method is given which is able in some case to give an upper bound of the bifurcation threshold copyright 2001 john wiley sons ltd

surface boundary conditions trigger flutter instability in non associative elastic plastic solids abstract it is shown that surface flutter instability may be triggered by the simultaneous influence of non associativity and boundary conditions even if taken independently neither the elastic plastic constitutive law satisfying deviatoric associativity nor the boundary conditions no applied traction rates would lead to flutter more specifically it is shown that for orthotropic elastic plastic constitutive tensors with an orthotropy axis tangent to the rate traction free boundary the onset of surface flutter instability coincides with the incipience of plasticity for any non associative flow rule whenever the normal to the boundary does not coincide with one of the other two orthotropy axes in the case of deviatoric associativity and coaxiality between the directions of orthotropy and the normal and tangent to the rate traction free boundary surface flutter instability may also occur but only for unusual values of material parameters additional results and discussion are also presented for the onset of stationary surface waves in associative and non associative elastic plastic bodies in contrast to the onset of surface flutter instabilities the condition for the onset of stationary surface waves involves material properties only that is it does not discriminate among different orientations of the rate traction free boundary with respect to the material orthotropy directions

amount of rotation about a point and the morse index for the case of exact area preserving monotone twist diffeomorphisms we give formulas relating the amount of rotation about an orbit and certain morse indices

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

ghost circles in lattice aubry mather theory abstract monotone lattice recurrence relations such as the frenkel kontorova lattice arise in hamiltonian lattice mechanics as models for ferromagnetism and as discretization of elliptic pdes mathematically they are a multi dimensional counterpart of monotone twist maps such recurrence relations often admit a variational structure so that the solutions x z d r are the stationary points of a formal action function w x given any rotation vector r d classical aubry mather theory establishes the existence of a large collection of solutions of w x 0 of rotation vector for irrational this is the well known aubry mather set it consists of global minimizers and it may have gaps in this paper we study the parabolic gradient flow d x d t w x and we will prove that every aubry mather set can be interpolated by a continuous gradient flow invariant family the so called ghost circle the existence of these ghost circles is known in dimension d 1 for rational rotation vectors and morse action functions the main technical result of this paper is therefore a compactness theorem for lattice ghost circles based on a parabolic harnack inequality for the gradient flow this implies the existence of lattice ghost circles of arbitrary rotation vectors and for arbitrary actions as a consequence we can give a simple proof of the fact that when an aubry mather set has a gap then this gap must be filled with minimizers or contain a non minimizing solution

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

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

minkowski functionals of poisson processes this paper treats poisson processes n with the convex ring s as state space having a translation and rotation invariant intensity measure fixing a test set s 0 s denoting by qk the random set of all points covered by n at least k times and collecting the minkowski functionals in a generating function v a simple formula is derived for the expectation of v s 0 q k under an adequate moment condition on this formula reflects the poisson character of the process and extends to intersections of finite families of independent poisson processes

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

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

birkhoff periodic orbits for small perturbations of completely integrable hamiltonian systems with convex hamiltonians pour des systemes a n degres de liberte on montre l existence d au moins n orbites periodiques distinctes avec un vecteur frequence rationnel admissible proche du tore correspondant du systeme non perturbe sous l hypothese que le hamiltonien du systeme non perturbe a des surfaces d energie convexes pour les variables action angle

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

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

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