measuring the strangeness of strange attractors we study the correlation exponent v introduced recently as a characteristic measure of strange attractors which allows one to distinguish between deterministic chaos and random noise the exponent v is closely related to the fractal dimension and the information dimension but its computation is considerably easier its usefulness in characterizing experimental data which stem from very high dimensional systems is stressed algorithms for extracting v from the time series of a single variable are proposed the relations between the various measures of strange attractors and between them and the lyapunov exponents are discussed it is shown that the conjecture of kaplan and yorke for the dimension gives an upper bound for v various examples of finite and infinite dimensional systems are treated both numerically and analytically

legendre tau approximations for functional differential equations in this paper we consider the numerical approximation of solutions to linear retarded functional differential equations using the so called legendre tau method the functional differential equation is first eformulated as a partial differential equation with a nonlocal boundary condition involving time differentiation the approximate solution is then represented as a truncated legendre series with time varying coefficients which satisfy a certain system of ordinary differential equations the method is very easy to code and yields very accurate approximations convergence is established various numerical examples are presented and comparison between the latter and cubic spline approximations is made

onset of chaos in differential delay equations abstract in the present paper we study the onset of chaos in the differential delay equation x dot t x t bx t 2 1 x 10 t 2 b 0 we develop a new numerical technique and apply it to study the global behavior of the flow on the attractor using this technique we find unstable periodic orbits connections between different periodic orbits transverse homoclinic orbits and try to explain the underlying reason for chaos

stress resultant geometrically nonlinear shell theory with drilling rotations part i a consistent formulation abstract in this work we present a consistent theoretical framework for a novel stress resultant geometrically nonlinear shell theory the main feature of the present shell theory development which stands in contrast with the classical developments in the shell theory is the presence of a rotation component around the shell normal so called drilling rotation in the description of shell finite rotations the relationship of the proposed theory with a finite deformation theory of a three dimensional continuum with independent rotation field is clearly identified the latter is shown to be an important link which facilitates the proper choice of the shell constitutive model and a proper construction of the regularized form of the theory capable of supporting the drilling rotations the corresponding linearized form of the present shell theory is discussed in the closure

stress resultant geometrically nonlinear shell theory with drilling rotations part ii computational aspects abstract in this work we discuss some details of the numerical implementation of the geometrically nonlinear shell theory presented in part i two possibilities to represent finite rotations with an orthogonal matrix and with a rotation vector are examined in detail along with their mutual relationship in both spatial and material description the issues pertinent to the consistent linearization procedure corresponding to these rotation parameterizations are also carefully considered the geometrically nonlinear method of incompatible modes is extended to the nonlinear shell theory under consideration and used to provide a 4 node shell element with enhanced performance a rather extensive set of numerical examples in nonlinear elastostatics is solved in order to corroborate the non locking performance of the incompatible mode based shell elements the examples include not only analyses of simple shell structures undergoing very large displacements and rotations but also cases of strong practical interest such as non smooth shell structures and shell structures with stiffeners

postbuckling behavior and imperfection sensitivity of elastic structures by the lyapunov schmidt koiter approach beginning with the work of koiter in 1945 valuable insights into the postbuckling behavior of structures have been gained by lyapunov schmidt decomposition of the displacements followed by an asymptotic expansion about the bifurcation point here this methodology is generalized to include nonlinear prebuckling behavior as well as multiple not necessarily coincident buckling modes the expansion of the reduced equilibrium equations is performed about a reference point which need not coincide with any of the bifurcation points and applies no matter whether the modes are coincident closely spaced or well separated from a variety of possible decompositions of the admissible space of displacements two are incorporated into a finite element program theoretical considerations and numerical examples in which asymptotic results are compared to exact results indicate that one of the decompositions has some important advantages over the other examples include a shallow arch and a beam on elastic foundation problem exhibiting symmetry breaking modal interaction

on the limit cycles of planar polynomial system with non rational first integral via averaging method at any order in this paper we consider a class of cubic planar polynomial differential system with non rational first integral using the averaging method at any order we bound the maximum number of limit cycles which bifurcate from the periodic annulus of the origin when we perturb them inside the class of all polynomial systems of degree n

on stability and the worst imperfection shape in solids with nearly simultaneous eigenmodes abstract of interest here is the influence of imperfections on the stability of elastic systems discrete or continuous with nearly simultaneous modes the majority of such structures show an extreme sensitivity of their first instability load to the shape of these small amplitude but unavoidable imperfections the determination of the worst such possible shape is thus a very important issue from the design standpoint following a general analysis of the stability of arbitrary elastic systems with nearly simultaneous bifurcation eigenmodes in the presence of imperfections conditions are given for the determination of the worst imperfection shape which minimizes the first local load maximum for the case of coincident eigenmodes the answer to the worst shape problem is considerably simplified for it is determined from the equilibrium branch of the perfect structure on which the load drops more rapidly

differential delay equations that have periodic solutions of long period if r r is a continuous odd function satisfying xf x 0 x 0 and f x o x 2 as i oo then so called periodic solutions of long period seem to play a prominent role in the dynamics of x t af x t 1 a 0 in this paper we prove the existence of long period periodic solutions of for a class of nonodd functions that decay rapidly to 0 at infinity and satisfy i x 0 these solutions have quite different qualitative features than in the odd case

the poincare bendixson theorem for monotone cyclic feedback systems with delay abstract we consider cyclic nearest neighbor systems of differential delay equations in which the coupling between neighbors possesses a monotonicity property using a discrete integer valued lyapunov function we prove that the poincare bendixson theorem holds for such systems we also obtain results on piecewise monotonicity and stability of periodic solutions of such systems

postbuckling analysis and imperfection sensitivity of general shells by the finite element method abstract buckling and imperfection sensitivity are the primary considerations in analysis and design of thin shell structures the objective here is to develop accurate and efficient capabilities to predict the postbuckling behavior of shells including imperfection sensitivity the approach used is based on the lyapunov schmidt koiter lsk decomposition and asymptotic expansion in conjunction with the finite element method this lsk formulation for shells is derived and implemented in a finite element code the method is applied to cylindrical and spherical shells cases of linear and nonlinear prebuckling behavior coincident as well as non coincident buckling modes and modal interactions are studied the results from the asymptotic analysis are compared to exact solutions obtained by numerically tracking the bifurcated equilibrium branches the accuracy of the lsk asymptotic technique its range of validity and its limitations are illustrated

the stability of special symmetric solutions of x t αf x t 1 with small amplitudes we consider the differential delay equation f x t f x t 1 with odd f c 3 r f 0 0 f 0 0 f 0 0 f 0 0 a theorem of kaplan and yorke 4 implies that 2f 0 is a bifurcation point where a certain class of periodic solutions bifurcates from the trivial zero solutions

center manifolds for periodic functional differential equations of mixed type abstract we study the behaviour of solutions to nonlinear functional differential equations of mixed type mfdes that remain sufficiently close to a prescribed periodic solution under a discreteness condition on the floquet spectrum we show that all such solutions can be captured on a finite dimensional invariant center manifold that inherits the smoothness of the nonlinearity this generalizes the results that were obtained previously in h j hupkes s m verduyn lunel center manifold theory for functional differential equations of mixed type j dynam differential equations 19 2007 497 560 for bifurcations around equilibrium solutions to mfdes

limit cycles of cubic polynomial vector fields via the averaging theory abstract in this paper we study the maximum number of limit cycles that can bifurcate from the period annulus surrounding the origin of a class of cubic polynomial differential systems using the averaging method more precisely we prove that the perturbations of the period annulus of the center located at the origin of the cubic polynomial differential system x y f x y y x f x y where f x y 0 is a conic such that f 0 0 0 by arbitrary cubic polynomial differential systems provide at least six limit cycles bifurcating from the periodic orbits of the period annulus using only the first order averaging method

bifurcation of periodic orbits of periodic equations with multiple parameters by averaging method abstract we investigate a class of n dimensional periodic differential equations with multiple small parameters by averaging method and obtain sufficient conditions for the existence of multiple periodic orbits as applications of the main results a cubic system is studied

domain decomposition method for the global dynamics of delay differential equations with unimodal feedback x t zkmx t c f x tkt with unimodal feedback is studied the existence of the global attractor is shown and bounds of the attractor are given we find attractive invariant intervals and give sufficient conditions that guarantee that all solutions enter the domain where f 0 is negative with respect to a positive equilibrium so the results for delayed monotone feedback can be applied to describe the asymptotic behaviour of solutions in particular the existence of heteroclinic orbits from the trivial equilibrium to a periodic orbit oscillating around the positive equilibrium is established numerical examples using nicholson s blowflies equation and the mackey glass equation are provided to illustrate the main results

bifurcations of small limit cycles in lienard systems with cubic restoring terms abstract in this paper we study bifurcations of small amplitude limit cycles of lienard systems of the form x y f x y g x where g x is a cubic polynomial and f x is a smooth or piecewise smooth polynomial of degree n by using involutions we obtain sharp upper bounds of the number of small amplitude limit cycles produced around a singular point for some systems of this type

mackey glass model of hematopoiesis with non monotone feedback stability oscillation and control for the blood cell production model with a unimodal hump feedback email protected t email protected q ny email protected q n y n email protected we review the known results and investigate generalizations of this equation permanence oscillation and stability of the positive equilibrium are studied for non autonomous equations including equations with a distributed delay in addition a linear control is introduced and possibilities to stabilize an otherwise unstable positive equilibrium are explored

averaging theory at any order for computing periodic orbits abstract we provide a recurrence formula for the coefficients of the powers of e in the series expansion of the solutions around e 0 of the perturbed first order differential equations using it we give an averaging theory at any order in e for the following two kinds of analytic differential equation d x d k 1 e k f k x d x d k 0 e k f k x a planar polynomial differential system with a singular point at the origin can be transformed using polar coordinates to an equation of the previous form thus we apply our results for studying the limit cycles of a planar polynomial differential systems

dynamics for a type of general reaction diffusion model abstract in this paper we discuss the following reaction diffusion model which is a general form of many population models u t x t u t x u t x f u t x we study the oscillatory behavior of solutions about the positive equilibrium k of system with neumann boundary conditions sufficient and necessary conditions are obtained for global attractivity of the zero solution and acceptable conditions are established for the global attractivity of k these results improve and complement existing results for system without diffusion moreover when these results are applied to the diffusive nicholson s blowflies model and the diffusive model of hematopoiesis some new results are obtained for the latter

exponential fitting of the delayed recruitment renewal equation abstract from respiratory physiology to laser based optical devices the so called delayed recruitment renewal equation 1 e d c t d t c t f c t 1 provides the mathematical model in a diverse spectrum of practical applications here e is inversely proportional to the product of the time delay inherent in the physical system and its rate of decay when this time lag is large relative to the reciprocal of the decay rate e is small and this delay differential equation dde is singularly perturbed when this situation obtains c t can exhibit initial layers and chaotic oscillations in order to accurately capture such solution features numerically one must use an approximation technique tailored to singular perturbation problems in this work we develop such a family of exponentially fitted schemes for the numerical approximation of this fundamental dde application of this new technique is then made to a variety of interesting and important problems not the least of which is the subject of dynamical diseases

delay differential equations with hill s type growth rate and linear harvesting for the equation email protected t r t n t 1 n t c b t n t a t n g t we obtain the following results boundedness of all positive solutions extinction and persistence conditions the proofs employ recent results in the theory of linear delay equations with positive and negative coefficients

on the limit cycles of a class of kukles type differential systems abstract in this paper we study the limit cycles of two families of differential systems in the plane these systems are obtained by polynomial perturbations with arbitrary degree on the second component of the standard linear center the classes under consideration are polynomial generalizations of certain canonical form of a kukles system with an invariant ellipse previously studied in the literature we provide in both cases an accurate upper bound of the maximum number of limit cycles that the perturbed system can have bifurcating from the periodic orbits of the linear center using the averaging theory of first second and third order these upper bounds are presented in terms of the degree of the respective systems moreover the existence of a weak focus with the highest order is also studied

the dynamics of the henon map the purpose of this paper is to develop machinery which is suitable for a study of the long time behavior of systems such as the henon map with expansion combined with strong contraction our study is modeled on the treatment of the one dimensional system x 1 ax 2 and we study the perturbation of a from the value a 2 and b small a surprising number of complications arise when b 0 the computer simulations give some indication of the complicated structure of the attractor but it can safely be stated that the real difficulties do not show up the central aim of the paper is to introduce what we consider the relevant concepts and techniques needed to study mappings such as the henon one

neimark sacker bifurcation for periodic delay differential equations in this paper we study the delay differential equation x t a t x t f t x t 1 where is a real parameter the functions a t f t are c 4 smooth and periodic in the variable t with period 1 varying the parameter eigenvalues of the monodromy operator the derivative of the time one map at the equilibrium 0 cross the unit circle and bifurcation of an invariant curve occurs to detect the critical parameter values we use floquet theory we give an explicit formula to compute the coefficient that determines the direction of the bifurcation we extend the center manifold projection method to our infinite dimensional banach space using spectral projection represented by a riesz dunford integral the neimark sacker bifurcation theorem implies the appearance of an invariant torus in the space c s 1 we apply our results to an equation used in neural network theory

heteroclinic orbits for retarded functional differential equations abstract suppose is a heteroclinic orbit of a ck functional differential equation x t x i with limit set and limit set being either hyperbolic equilibrium points or periodic orbits necessary and sufficient conditions are given for the existence of an ftf close to in ck with the property that x t tf x t has a heteroclinic orbit gg close to the orbits gg are obtained from the zeros of a finite number of bifurcation functions g f r d r d 1 transversality of is characterized by the nondegeneracy of the derivative d g it is also shown that the tf which have heteroclinic orbits near are on a ck submanifold of finite codimension max 0 ind or on the closure of it where ind is the index of

averaging methods of arbitrary order periodic solutions and integrability in this paper we provide an arbitrary order averaging theory for higher dimensional periodic analytic differential systems this result extends and improves results on averaging theory of periodic analytic differential systems and it unifies many different kinds of averaging methods applying our theory to autonomous analytic differential systems we obtain some conditions on the existence of limit cycles and integrability for polynomial differential systems with a singularity at the origin having a pair of pure imaginary eigenvalues we prove that there always exists a positive number n such that if its first n averaging functions vanish then all averaging functions vanish and consequently there exists a neighborhood of the origin filled with periodic orbits consequently if all averaging functions vanish the origin is a center for n 2 furthermore in a punctured neighborhood of the origin the system is c infty completely integrable for n 2 provided that each periodic orbit has a trivial holonomy finally we develop an averaging theory for studying limit cycle bifurcations and the integrability of planar polynomial differential systems near a nilpotent monodromic singularity and some degenerate monodromic singularities

homoclinic solution and chaos in xf x 0 is followed by a move in the opposite direction i t 0 however there are equations 1 2 which are closely related to applications and where the negative feedback condition 3 is only true in a certain neighbourhood of x 0 furumochi 3 studied equation

the center problem via averaging method the averaging method has been used to study the problem of the determination of the number of hyperbolic limit cycles that can bifurcate from the period annulus of a center in this paper we use the averaging method up to any order in the perturbation parameter to determine the center conditions of monodromic singular points of analytic planar vector fields

on iterations of 1 ax 2 on 1 1 on etudie des iterations de f x a 1 ax u 2 1 x 1 ou a est un parametre dans l intervalle 0 2 on etablit la non existence de cycles attractifs on definit une partition on donne une estimation uniforme de la a derivee

a spectral method for nonlinear wave equations abstract in this paper we introduce a new integration technique to find radially symmetric solutions of nonlinear wave equations defined on a spherical domain with dirichlet boundary conditions and we use this technique to study orbital stability of the standing waves we prove analytically that the wave of lowest energy is stable and we show by numerical computations that the standing waves of higher energy are unstable

existence of chaos in control systems with delayed feedback here f r r denotes some nonlinear function a a positive constant t e r is the independent variable x e r the dependent variable and 1 dx dt the investigation is motivated i by the applications eq 1 1 has found in several areas of applied sciences and ii by the observation of intensely structured periodic and aperiodic oscillations in numerical simulations for a certain class of nonlinearities f we show that these structures are not numerical artefacts but properties of the solution manifold of eq 1 1 in the applications eq 1 1 has been used to explain normal and pathological behavior of control systems in the physiology of blood cell production and respiration 7 8 10 111 and periodic 2 or irregular s activity in neural networks computer simulations to eq 1 1 may be found in 3 the mathematical analysis of this equation so far has been restricted to prove the existence of slowly oscillating simple periodic solutions 1 4 61 a periodic solution is called slowly oscillating if successive

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

three dimensional extension of non linear shell formulation based on the enhanced assumed strain concept conventional shell formulations such as 3 or 5 parameter theories or even 6 parameter theories including the thickness change as extra parameter require a condensation of the constitutive law in order to avoid a significant error due to the assumption of a linear displacement field across the thickness this means that the normal stress in thickness direction has to either vanish or be constant in general these extra constraints cannot be satisfied explicitly or they lead to elaborate strain expressions the main objective of the present study is to introduce directly a complete 3 d constitutive law without modification therefore a 7 parameter theory is utilized which includes a linear varying thickness stretch as extra variable allowing also large strain effects

high dimensional chaotic behavior in systems with time delayed feedback abstract the nature of high dimensional chaos exhibited by a class of delay differential equation is investigated by various methods this delay differential equation models systems with time delayed feedback such as nonlinear optical resonators we first describe briefly the bifurcation phenomena exhibited by the system with an increase in a control parameter representing the energy flow rate periodic states bi multi furcate themselves successively forming a hierarchy of multistable periodic states with increasing complexities finally each branch of multistable periodic states makes transition to chaos the possibility of applying such a multistability as a memory device for complicated information is discussed in the chaotic regime each of the bifurcated states in turn merge successively into fewer sets of states with larger attractor dimensions and finally a single developed chaos with a very large attractor dimension is formed lyapunov analysis is introduced to study the high dimensional chaotic states the lyapunov vectors as well as lyapunov spectrum are shown to be very useful to understand the underlying mechanism of the successive merging process mentioned above characteristics of developed chaos are investigated by high pass filtered time series hfts the intermittency characteristics of the hfts changes markedly at a certain frequency the lyapunov analysis reveals that this frequency corresponds to a characteristic lyapunov mode number this characteristic number can be looked upon as the dimension of subspace in which active chaotic information is generated and is different from the attractor dimension in the customary sense

global asymptotic stability of a class of nonautonomous integro differential systems and applications abstract in this paper we study the global asymptotic stability of a class of nonautonomous integro differential systems by constructing suitable lyapunov functionals we establish new and explicit criteria for the global asymptotic stability in the sense of definition 2 1 in the autonomous case we discuss the global asymptotic stability of a unique equilibrium of the system and in the case of periodic system we establish sufficient criteria for existence uniqueness and global asymptotic stability of a periodic solution also explored are applications of our main results to some biological and neural network models the examples show that our criteria are more general and easily applicable and improve and generalize some existing results

instabilities in threshold diffusion equations with delay the introduction of delays into ordinary or partial differential equation models is well known to facilitate the production of rich dynamics ranging from periodic solutions through to spatio temporal chaos in this paper we consider a class of scalar partial differential equations with a delayed threshold nonlinearity which admits exact solutions for equilibria periodic orbits and travelling waves importantly we show how the spectra of periodic and travelling wave solutions can be determined in terms of the zeros of a complex analytic function using this as a computational tool to determine stability we show that delays can have very different effects on threshold systems with negative as opposed to positive feedback direct numerical simulations are used to confirm our bifurcation analysis and to probe some of the rich behaviour possible for mixed feedback

periodic orbits from second order perturbation via rational trigonometric integrals abstract the second order poincare pontryagin melnikov perturbation theory is used in this paper to study the number of bifurcated periodic orbits from certain centers this approach also allows us to give the shape and the period up to the first order we address these problems for some classes of abel differential equations and quadratic isochronous vector fields in the plane we prove that two is the maximum number of hyperbolic periodic orbits bifurcating from the isochronous quadratic centers with a birational linearization under quadratic perturbations of second order in particular the configurations 2 0 and 1 1 are realizable when two centers are perturbed simultaneously the required computations show that all the considered families share the same iterated rational trigonometric integrals

oscillation and global attractivity in hematopoiesis model with periodic coefficients in this paper we shall consider the nonlinear delay differential equation p t email protected t p m email protected 1 p n email protected email protected t p t where k is a positive integer b t and c t are positive periodic functions of period w in the nondelay case we shall show that has a unique positive periodic solution email protected t and we will study the global attractivity of email protected t in the delay case we shall establish some sufficient conditions for oscillation of all positive solutions of about email protected t and establish some sufficient conditions for the global attractivity of email protected t our results in this paper extend as well as improve the results in the autonomous case

chaotic attractors of an infinite dimensional dynamical system abstract we study the chaotic attractors of a delay differential equation the dimension of several attractors computed directly from the definition agrees to experimental resolution with the dimension computed from the spectrum of lyapunov exponents according to a conjecture of kaplan and yorke assuming this conjecture to be valid as the delay parameter is varied from computations of the spectrum of lyapunov exponents we observe a roughly linear increase from two to twenty in the dimension while the metric entropy remains roughly constant these results are compared to a linear analysis and the asymptotic behavior of the lyapunov exponents is derived

multiple periodic solutions with minimal period 4 of the delay differential equation x t f t x t 1 abstract in this paper some sufficient conditions are obtained to guarantee the existence of multiple periodic solutions with minimal period 4 of the delay differential equation x t f t x t 1 by using relative morse index and some considerate analysis techniques

adaptive control of chaotic continuous time systems with delay a simple delay system governed by a first order differential delay equation may behave chaotically but the conditions for the system to have such behaviors have not been well recognized in this paper a set of rules is postulated first for the conditions for the delay system to display chaos a model reference adaptive control scheme is then proposed to control the chaotic system state to converge to an arbitrarily given reference trajectory with certain and uncertain system parameters numerical examples are given to analyze the chaotic behaviors of the delay system and to demonstrate the effectiveness of the proposed adaptive control scheme

smooth bifurcation of symmetric periodic solutions of functional differential equations with odd f c r in contrast to the one dimensional ordinary differential equation this equation can have periodic solutions as the example i t 7r 2 x t 1 with solutions x t z sin rc 2 t z 0 shows numerical bifurcation diagrams e g 6 143 indicate a complicated structure of the period solutions depending on the delay parameter u one finds various intersecting smooth curves one task of mathematics is to prove the existence of periodic solutions in this paper we are mainly interested in symmetric periodic solutions i e solutions x of c which fultil the symmetry property

approximating chaotic saddles for delay differential equations chaotic saddles are unstable invariant sets in the phase space of dynamical systems that exhibit transient chaos they play a key role in mediating transport processes involving scattering and chaotic transients here we present evidence long chaotic transients and fractal basins of attraction of transient chaos in a logistic delay differential equation we adapt an existing method stagger and step to numerically construct the chaotic saddle for this system this is the first such analysis of transient chaos in an infinite dimensional dynamical system and in delay differential equations in particular using poincar e section techniques we illustrate approaches to visualizing the saddle set and confirm that the saddle has the cantor like fractal structure consistent with a chaotic saddle generated by horseshoe type dynamics

systems of differential delay equations floquet multipliers and discrete lyapunov functions abstract we define a discrete integer valued lyapunov function v for cyclic nearest neighbor systems of differential delay equations possessing a feedback condition this extends analogous definitions for cyclic systems of ode s and for scalar differential delay equations we relate the values of v to the real parts of the floquet multipliers for such linear periodic systems and thereby prove all floquet subspaces are at most two dimensional

mackey glass type delay differential equations near the boundary of absolute stability for an equation x t x t f x t h x r f 0 1 0 with c 3 nonlinearity f which has a negative schwarzian derivative and satisfies xf x 0a nd h 1 1 8 are less than some constant independent on h this result gives additional insight to the conjecture about the equivalence between local and global asymptotical stabilities in the mackey glass type delay differential equations 2002 elsevier science usa all rights reserved

on stability of some linear and nonlinear delay differential equations abstract new explicit conditions of exponential stability are obtained for the nonautonomous equation with several delays y t k 1 l a k t y h k t 0 by the following method several delays in the left hand side are chosen and the solution is estimated using an auxiliary ordinary differential equation y t k i a k t y t 0 where i 1 2 l is the chosen set of indices these results are applied to analyze the stability of the nonlinear equation x t k 1 l a k t x h k t f t x t x g 1 t x g m t by the first approximation it is to be noted that coefficients and delays are not assumed to be continuous functions

on super high frequencies in discontinuous 1st order delay differential equations abstract one considers a scalar 1st order nonlinear differential equation with a delayed relay output proportional feedback one shows that under a boundedness condition on the nonlinearity only any solution of this equation has after a finite time a finite number of zeros on compact sets an estimate of the time after which the super high frequency disappears is provided this improves some previous work by shustin as a consequence using some work by fridman et al any solution of the system under study coincides after a finite time with one of the periodic solutions

some global bifurcations of two dimensional endomorphisms by use of critical lines many studies have been devoted to discrete dynamical systems represented by homeomorphisms and diffeomorphisms since the works by smale l and gavrilov and shilnikov 2 31 but comparatively few works have been devoted to endomorphisms that is to maps with a nonunique inverse however the mathematical models derived from the description of evolutive phenomena in several applicative fields are often represented by discrete dynamical systems or maps with a nonunique inverse this is particularly true in the socio economic field 4 61 but examples can also be found in ecology 7 81 biology 9 l l and control systems 121 this work concerns the study of the global dynamics of recurrences represented by twodimensional endomorphisms of applicative interest in the economic context useful notions for the local and global analysis of nonlinear endomorphisms have been introduced by gumowski and mira ll 13 141 for two dimensional endomorphisms in particular this is the notion of critical lines or critical curves representing the twodimensional extension of the critical points in one dimensional maps and the notions of absorbing area and chaotic area bounded by elements of critical lines which determine phase plane domains where chaotic solutions are located these notions which will be recalled in the next section together with some bifurcations involving the chaotic areas are the basic concepts and properties that we shall use in the analysis of the subsequent sections the aim of this paper is to illustrate by an example how powerful the critical lines may be in studying the global dynamics and bifurcations of plane nonlinear endomorphisms particularly when they are used in combination with more traditional techniques it is known that in diffeomorphisms global bifurcations which qualitatively change the structure of a chaotic attractor called interior crises or catastrophes by some authors 15 181 are characterized by the stable manifold of a saddle approaching and touching the chaotic attractor we shall find that this phenomenology also occurs in the endomorphisms of our model with a particular role being played by the critical lines in section 3 we shall describe our mathematical model a two dimensional piecewise linear map which we call map f together with some of its elementary properties and the definition of three classes of endomorphisms into which the analysis can be conveniently subdivided endomorphisms of the first class have been the object of a previous work 191 here we shall show in section 4 how several bifurcations involving chaotic areas can be explained by use of critical lines the main result is the analytical determination of the bifurcation which discriminates between the global attractivity of an absorbing area and the nonglobal attractivity

periodic solutions and chaos in a delay differential equation modelling haematopoesis simple mathematical models have been devised to describe the process of haematopoesis the formation and differentiation of blood cells within the human body l 21 haematopoesis is an example of a biological control system with an inherent delay in it this is due to the time required for primitive blood cells to mature such a delayed response is known to cause oscillatory behaviour in control systems and it has been found that similar oscillatory instabilities in blood cell counts are associated with diseases such as aplastic anaemia 2 this suggest that the study of such equations may be of biological importance mathematically they are of interest because they can exhibit complex aperiodic behaviour a model put forward for consideration by mackey glass 3 is

on the global attractivity controversy for a delay model of hematopoiesis abstract recently particular counterexamples were constructed to some theorems of a previous paper concerning the global attractivity of the positive equilibrium for the delay equation p t p m t 1 p n t p t the purpose of this note is to explore the underlying phenomenon from a more general point of view and to give an explanation of the situation a theorem is proved regarding attractivity properties of the equilibrium zero

mackey glass equation with variable coefficients the mackey glass equation dndt r t n g t 1 n g t c b t n t is considered with variable coefficients and a nonconstant delay under rather natural assumptions all solutions are positive and bounded persistence and extinction conditions are presented for this equation in the case when there exists a constant positive equilibrium local asymptotic stability of the constant solution and oscillation about this equilibrium are analyzed the results are illustrated by numerical examples in particular it is demonstrated that with delay in both terms a solution with positive initial conditions may become negative

chaotic behaviour in simple dynamical systems in this paper a description is given of the chaotic behaviour generated by a transversal homoclinic point of a plane map a proof of smale s theorem via the shadowing property of hyperbolic sets is provided the result is related to certain plane periodic systems of ode s like the periodically perturbed pendulum equation to this end the so called method of melnikov is derived

oscillation and global attractivity in hematopoiesis model with delay time in this paper we shall consider the nonlinear delay differential equations with delay time p t bp m t t 1 p n t t cp t that is proposed as a model of hematopoiesis blood cell production where p t denotes the density of mature cells in blood circulation and the time delay t is the time between the production of immature cells in the bone marrow and their maturation for release in the circulating bloodstreams our aim is to give sufficient condition for oscillation of all positive solutions of about the positive steady state and obtain some sufficient conditions for the global attractivity our results extend and improve the well known oscillation results of when m 0

limit cycles bifurcating from planar polynomial quasi homogeneous centers in this paper we find an upper bound for the maximum number of limit cycles bifurcating from the periodic orbits of any planar polynomial quasi homogeneous center which can be obtained using first order averaging method this result improves the upper bounds given in 7

characteristic multipliers and stability of symmetric periodic solutions of x t g x t 1 on etudie l equation differentielle scalaire a retard x t g x t 1 a retroaction negative on suppose que la fonction non lineaire g est impaire et monotone on demontre que des solutions periodiques x t de type legerement oscillant satisfaisant la condition de symetrie x t x t 2 t r sont non degenerees et ont toutes des multiplicateurs de floquet non triviaux dans le cercle unite

a second order analysis of the periodic solutions for nonlinear periodic differential systems with a small parameter abstract we deal with nonlinear t periodic differential systems depending on a small parameter the unperturbed system has an invariant manifold of periodic solutions we provide the expressions of the bifurcation functions up to second order in the small parameter in order that their simple zeros are initial values of the periodic solutions that persist after the perturbation in the end two applications are done the key tool for proving the main result is the lyapunov schmidt reduction method applied to the t poincare andronov mapping

periodic orbits in analytically perturbed poisson systems abstract analytical perturbations of a family of finite dimensional poisson systems are considered it is shown that the family is analytically orbitally conjugate in u r n to a planar harmonic oscillator defined on the symplectic leaves as a consequence the perturbed vector field can be transformed in the domain u to the lagrange standard form on the latter use can be made of averaging theory up to second order to study the existence number and bifurcation phenomena of periodic orbits examples are given ranging from harmonic oscillators with a potential and duffing oscillators to a kind of zero hopf singularity analytic normal form

coexistence of infinitely many large stable rapidly oscillating periodic solutions in time delayed duffing oscillators abstract we explore stability and instability of rapidly oscillating solutions x t for the hard spring delayed duffing oscillator x t a x t b x t t x 3 t 0 fix t 0 we target periodic solutions x n t of small minimal periods p n 2 t n for integer n and with correspondingly large amplitudes note how x n t are also marginally stable solutions respectively of the two standard non delayed hamiltonian duffing oscillators x a x 1 n b x x 3 0 stability changes for the delayed duffing oscillator simultaneously for all sufficiently large n n 0 we obtain local exponential stability for 1 n b 0 and exponential instability for 1 n b 0 provided that 0 1 n 1 b t 2 3 2 2 we interpret our results in terms of noninvasive delayed feedback stabilization and destabilization for large amplitude rapidly periodic solutions of the standard duffing oscillators we conclude with numerical illustrations of our results for small and moderate n which also indicate a neimark sacker torus bifurcation at the validity boundary of our theoretical results

bifurcation and chaos in a discrete physiological control system in this paper we introduce a new method to discrete a physiological control system with delay and investigate the dynamical behavior of the discrete model basic properties of the discrete system are analyzed by means of lyapunov exponent spectrum and bifurcation diagrams analysis results show that this system has complex dynamics with some interesting characteristics which may provide some new knowledge for us in fields of physiology

global linearized stability theory for delay differential equations abstract for nonlinear delay differential equations where nonlinearities are approximated by linear functions we discuss a rather general global stability principle several examples which extend and improve known global stability results are presented

examples of transverse homoclinic orbits in delay equations on presente deux exemples specifiques d equations aux differences differentielles scalaires avec un retard de la forme x t x t f x t 1 0 pour lesquelles il y a une orbite periodique avec une orbite homocline transversale q asymptotique a

periodic solutions for an n dimensional cyclic feedback system with delay abstract we study models for n cyclically coupled variables e g neuron activities with overall negative delayed feedback and without symmetry or monotonicity properties our aim is to extract the common parts of similar approaches that are known in dimensions one two and three so far to exhibit how these parts work for general dimension n and to show how this framework includes previous as well as new results we provide a fixed point theorem and a related theorem on periodic orbits for semiflows on banach spaces which then yield periodic solutions of cyclic delayed negative feedback systems for general n we also give criteria for the global asymptotic stability in the same systems which are derived by relating the systems to interval maps

exponentially growing solutions for a delay diffusion equation with negative feedback abstract the scalar partial delay differential equation u t u u t 1 u t is investigated it was known that in one space dimension all nonnegative solutions of the initial boundary value problem stay bounded as t surprisingly this ceases to remain true in two or more dimensions if the delay is large and the diffusion coefficient small there exists a large set of trajectories whose total mass tends exponentially to infinity as t these trajectories escape the negative feedback of the equation by transporting their mass around along periodic paths in space where the paths can be prescribed by the initial data

periodic solutions and chaos in a non linear model for the delayed cellular immune response abstract we model the cellular immune response using a set of non linear delayed differential equations we observe that the stationary solution becomes unstable above a critical immune response time the exponents characterizing the approach to this bifurcation point as well as the critical slow dynamics are obtained in the periodic regime the minimum virus load is substantially reduced with respect to the stationary solution further increasing the delay time the dynamics display a series of bifurcations evolving to a chaotic regime characterized by a set of 2d portraits