the sine gordon equation with perturbation terms of power jacobian elliptic functions in this paper the sine gordon equation sge with some solvable perturbations is discussed in detail as well as physical conclusions are drawn unlike the usual form of perturbations in terms of trigonometric and hyperbolic functions here it is assumed that inhomogeneous terms appear as elliptic functions in quadratic powers one suggests employing the analytic means for the investigation of some properties of special perturbed sge which now has to be renamed into jacobian gordon equation jge new classes of jge are introduced

bifurcation analysis on a survival red blood cells model in this paper we consider a model described the survival of red blood cells in animal its dynamics are studied in terms of local and global hopf bifurcations we show that a sequence of hopf bifurcations occur at the positive equilibrium as the delay crosses some critical values using the reduced system on the center manifold we also obtain that the periodic orbits bifurcating from the positive equilibrium are stable in the center manifold and all hopf bifurcations are supercritical further particular attention is focused on the continuation of local hopf bifurcation we show that global hopf bifurcations exist after the second critical value of time delay

bifurcation analysis in van der pol s oscillator with delayed feedback in this paper a classical van der pol s equation with generally delayed feedback is considered it is shown that there are bogdanov takens bifurcation triple zero and hopf zero singularities by analyzing the distribution of the roots of the associated characteristic equation in the situation that the zero is as a simple eigenvalue the normal forms of the reduced equations are obtained by the center manifold theory and normal form method for functional differential equation and hence the stability of the fixed point is determined and transcritical and pitchfork bifurcations are found

control of unstable steady states by extended time delayed feedback time delayed feedback methods can be used to control unstable periodic orbits as well as unstable steady states we present an application of extended time delay autosynchronization introduced by socolar et al phys rev e 50 3245 1994 to an unstable focus this system represents a generic model of an unstable steady state which can be found for instance in hopf bifurcation in addition to the original controller design we investigate effects of control loop latency and a bandpass filter on the domain of control furthermore we consider coupling of the control force to the system via a rotational coupling matrix parametrized by a variable phase we present an analysis of the domain of control and support our results by numerical calculations

random flights connecting porous medium and euler poisson darboux equations in this paper we consider the porous medium equation and establish a relationship between its kompanets zel dovich barenblatt solution u xd t xd rd t 0 and random flights the time rescaled version of u xd t is the fundamental solution of the euler poisson darboux equation which governs the distribution of random flights performed by a particle whose displacements have a dirichlet probability distribution and choosing directions uniformly on a d dimensional sphere we consider the space fractional version of the euler poisson darboux equation and present the solution of the related cauchy problem in terms of the probability distributions of random flights governed by the classical euler poisson darboux equation furthermore this research is also aimed at studying the relationship between the solutions of a fractional porous medium equation and the fractional euler poisson darboux equation a considerable part of this paper is devoted to the analysis of the probabilistic tools of the solutions of the fractional equations the extension to the higher order euler poisson darboux equation is considered and the solutions are interpreted as compositions of laws of pseudoprocesses

bifurcations for a predator prey system with two delays in this paper a predator prey system with two delays is investigated by choosing the sum of two delays as a bifurcation parameter we show that hopf bifurcations can occur as crosses some critical values by deriving the equation describing the flow on the center manifold we can determine the direction of the hopf bifurcations and the stability of the bifurcating periodic solutions in addition special attention is paid to the global continuation of local hopf bifurcations using a global hopf bifurcation result of j wu symmetric functional differential equations and neural networks with memory trans amer math soc 350 1998 4799 4838 we may show the global existence of periodic solutions

stability and bifurcation for a delayed predator prey model and the effect of diffusion abstract we consider a predator prey system with one or two delays and a unique positive equilibrium e its dynamics are studied in terms of the local stability of e and of the description of the hopf bifurcation that is proven to exist as one of the delays taken as a parameter crosses some critical values we also consider a reaction diffusion system with neumann conditions resulting from adding one spatial variable and diffusion terms in the previous model the spectral and bifurcation analysis in the neighborhood of e now as a stationary point of this latter system is addressed and the results obtained for the case without diffusion are applied

multiple bifurcations in a delayed predator prey system with nonmonotonic functional response abstract a delayed predator prey system with nonmonotonic functional response is studied by using the normal form theory of retarded functional differential equations developed by faria and magalhaes the bifurcation analysis of the model indicates that there is a bogdanov takens singularity for any time delay value a versal unfolding of the model at the bogdanov takens singularity is obtained on the other hand it is shown that small delay changes the stability of the equilibrium of the model for some parameters and the system can exhibit hopf bifurcation as the time delay passes through some critical values

a note on new solitary and similarity class of solutions of a fourth order non linear evolution equation abstract in this paper lie s method is used to calculate new class of solutions of a less studied fourth order non linear partial differential equation in our previous paper a huber appl math comp 166 2 2005 464 we have applied the tanh method to generate solutions in this case special class of solutions in form of traveling waves result single soliton solutions as well as class of irregular solutions therefore general families of solutions are of basic interest doing so new results can be found in a huber appl math comp in press using the weierstras transformation derived first by the author moreover a complete characterization of the group properties is given because group properties are unknown up to now we determine the lie point symmetry vector fields and calculate similarity ansatze further we also derive a few non linear transformations and some similarity solutions are obtained explicitly the main purpose for the application of lie s method is of course the fact that we are able to calculate class of general solutions which do not underlie such strong restrictions as in the case of traveling wave ansatze otherwise it is necessary to perform a group analysis in order to improve the solution manifold by an alternative way in addition the studied equation passes the painleve test and seems to be integrable general class of solutions in terms of elliptic functions are derived via lie s approach for the first time representing exact solitary wave propagation and surprisingly cusp solitary class of solutions are obtained

homoclinic orbits and hopf bifurcations in delay differential systems with t b singularity abstract the paper carries the results on takens bogdanov bifurcation obtained in t faria l t magalhaes normal forms for retarded functional differential equations and applications to bogdanov takens singularity j differential equations 122 1995 201 224 for scalar delay differential equations over to the case of delay differential systems with parameters firstly we give feasible algorithms for the determination of takens bogdanov singularity and the generalized eigenspace associated with zero eigenvalue in r n next through center manifold reduction and normal form calculation a concrete reduced form for the parameterized delay differential systems is obtained finally we describe the bifurcation behavior of the parameterized delay differential systems with t b singularity in detail and present an example to illustrate the results

cooperative dynamics in a network of stochastic elements with delayed feedback networks of globally coupled noise activated bistable elements with connection time delays are considered the dynamics of these systems is studied numerically using a langevin description and analytically using 1 a gaussian approximation as well as 2 a dichotomous model the system demonstrates ordering phase transitions and multistability that is for a strong enough feedback it exhibits nontrivial stationary states and oscillatory states whose frequencies depend only on the mean of the time delay distribution function other observed dynamical phenomena include coherence resonance and in the case of nonuniform coupling strengths amplitude death and chaos furthermore an increase of the stability of the trivial equilibrium with increasing nonuniformity of the time delays is observed

bifurcation analysis of a class of first order nonlinear delay differential equations with reflectional symmetry abstract we consider a general class of first order nonlinear delay differential equations ddes with reflectional symmetry and study completely the bifurcations of the trivial equilibrium under some generic conditions on the taylor coefficients of the dde our analysis reveals a hopf bifurcation curve terminating on a pitchfork bifurcation line at a codimension two takens bogdanov point in parameter space we compute the normal form coefficients of the reduced vector field on the centre manifold in terms of the taylor coefficients of the original dde and in contrast to many previous bifurcation analyses of ddes we also compute the unfolding parameters in terms of these coefficients for application purposes this is important since one can now identify the possible asymptotic dynamics of the dde near the bifurcation points by computing quantities which depend explicitly on the taylor coefficients of the original dde we illustrate these results using simple model systems relevant to the areas of neural networks and atmospheric physics and show that the results agree with numerical simulations

restrictions and unfolding of double hopf bifurcation in functional differential equations abstract the normal form of a vector field generated by scalar delay differential equations at nonresonant double hopf bifurcation points is investigated using the methods developed by faria and magalhaes j differential equations 122 1995 181 we show that 1 there exists linearly independent unfolding parameters of classes of delay differential equations for a double hopf point which generically map to linearly independent unfolding parameters of the normal form equations ordinary differential equations 2 there are generically no restrictions on the possible flows near a double hopf point for both general and z 2 symmetric first order scalar equations with two delays in the nonlinearity and 3 there always are restrictions on the possible flows near a double hopf point for first order scalar delay differential equations with one delay in the nonlinearity and in nth order scalar delay differential equations n 2 with one delay feedback

multistability and stable asynchronous periodic oscillations in a multiple delayed neural system abstract we consider a network of three identical neurons with multiple discrete signal transmission delays the model for such a network is a system of nonlinear delay differential equations after some consideration of the absolute synchronization of the system and the global attractivity of the zero solution we present a detailed discussion about the boundaries of the stability region of the trivial solution this allows us to determine the possible codimension one bifurcations which occur in the system in particular we show the existence of standard hopf bifurcations giving rise to synchronized periodic solutions and of d 3 equivariant hopf bifurcations giving rise to three types of periodic solutions phase locked mirror reflecting and standing waves hopf hopf and hopf steady state bifurcations interactions are shown to exist and give rise to coexistence of stable synchronized and desynchronized solutions perturbation techniques coupled with the floquet theory are used to determine the stability of the phase locked oscillations

ikeda hopf bifurcation revisited abstract a large variety of passive optical systems subject to a delayed feedback have appeared in the literature and are described mathematically by the same class of scalar delay differential equations ddes these equations include ikeda dde and their solutions are determined in terms of a control parameter distinct from the delay we concentrate on the first hopf bifurcation generated by a fixed delay and determine a general expression for its direction of bifurcation we then examine our result in the two limits of small and large delays for small delays we show that a hopf bifurcation to nearly sinusoidal oscillations is possible provided that the feedback rate is sufficiently high bifurcation from infinity for large delays we complement the early work by chow et al proc roy soc edinburgh a 120 1992 223 229 and hale and huang j diff equ 114 1994 1 23 by comparing analytical and numerical bifurcation diagrams as the oscillations progressively change from sine to square wave

singular hopf bifurcation in a differential equation with large state dependent delay we study the onset of sustained oscillations in a classical state dependent delay sdd differential equation inspired by control theory owing to the large delays considered the hopf bifurcation is singular and the oscillations rapidly acquire a sawtooth profile past the instability threshold using asymptotic techniques we explicitly capture the gradual change from nearly sinusoidal to sawtooth oscillations the dependence of the delay on the solution can be either linear or nonlinear with at least quadratic dependence in the former case an asymptotic connection is made with the rayleigh oscillator in the latter van der pol s equation is derived for the small amplitude oscillations sdd differential equations are currently the subject of intense research in order to establish or amend general theorems valid for constant delay differential equation but explicit analytical construction of solutions are rare this paper illustrates the use of singular perturbation techniques and the unusual way in which solvability conditions can arise for sdd problems with large delays

global attractor for weakly damped nonlinear schrodinger equations in l2 r abstract we prove that the weakly damped nonlinear schrodinger flow in l 2 r provides a dynamical system which possesses a global attractor the proof relies on the continuity of the schrodinger flow for the weak topology in l 2 r

fractional schrodinger equation solvability and connection with classical schrodinger equation abstract we consider the dirichlet boundary problem for semilinear fractional schrodinger equation with subcritical nonlinear term local and global in time solvability and regularity properties of solutions are discussed but our main task is to describe the connections of the fractional equation with the classical nonlinear schrodinger equation including convergence of the linear semigroups and continuity of the nonlinear semigroups when the fractional exponent approaches 1

metastability for delayed differential equations in systems at phase transitions two phases of the same substance may coexist for a long time before one of them dominates we show that a similar phenomenon occurs in systems with delayed feedback where short term stable oscillatory patterns can also have very long lifetimes before vanishing into constant or periodic steady states

traveling waves in the nonlocal kpp fisher equation different roles of the right and the left interactions abstract we consider the nonlocal kpp fisher equation u t t x u x x t x u t x 1 k u t x which describes the evolution of population density u t x with respect to time t and location x the non locality is expressed in terms of the convolution of u t with kernel k 0 r k s d s 1 the restrictions k s s 0 and k s s 0 are responsible for interactions of an individual with his left and right neighbors respectively we show that these two parts of k play quite different roles as for the existence and uniqueness of traveling fronts to the kpp fisher equation in particular if the left interaction is dominant the uniqueness of fronts can be proved while the dominance of the right interaction can induce the co existence of monotone and oscillating fronts we also present a short proof of the existence of traveling waves without assuming various technical restrictions usually imposed on k

versal unfoldings for linear retarded functional differential equations we consider parametrized families of linear retarded functional differential equations rfdes projected onto finite dimensional invariant manifolds and address the question of versality of the resulting parametrized family of linear ordinary differential equations a sufficient criterion for versality is given in terms of readily computable quantities in the case where the unfolding is not versal we show how to construct a perturbation of the original linear rfde in terms of delay differential operators whose finite dimensional projection generates a versal unfolding we illustrate the theory with several examples and comment on the applicability of these results to bifurcation analyses of nonlinear rfdes

toroidal normal forms for bifurcations in retarded functional differential equations i multiple hopf and transcritical multiple hopf interaction abstract for finite dimensional bifurcation problems it is well known that it is possible to compute normal forms which possess nice symmetry properties oftentimes these symmetries may allow for a partial decoupling of the normal form into a so called radial part and an angular part analysis of the radial part usually gives an enormous amount of valuable information about the bifurcation and its unfoldings in this paper we are interested in the case where such bifurcations occur in retarded functional differential equations and we revisit the realizability and restrictions problem for the class of radial equations by nonlinear delay differential equations our analysis allows us to recover and considerably generalize recent results by faria and magalhaes t faria l t magalhaes restrictions on the possible flows of scalar retarded functional differential equations in neighborhoods of singularities j dynam differential equations 8 1996 35 70 and by buono and belair p l buono j belair restrictions and unfolding of double hopf bifurcation in functional differential equations j differential equations 189 2003 234 266

zero hopf bifurcation for van der pol s oscillator with delayed feedback in this paper we study the dynamical behaviors of the following van der pol oscillator with delay x e x 2 1 x x eg x t t in the case that its associated characteristic equation has a simple zero root and a pair of purely imaginary roots zero hopf singularity the normal form is obtained by performing a center manifold reduction and by using the normal form theory developed by faria and magalhaes a critical value e 0 of e in 0 2 is obtained to predict the bifurcation diagrams from which saddle node bifurcation pitchfork bifurcation hopf bifurcation the existence and stability of the periodic solutions and heteroclinic bifurcation are determined some examples are given to confirm the theoretical results

parabolic problems with nonlinear boundary conditions and critical nonlinearities we prove existence uniqueness and regularity of solutions for heat equations with nonlinear boundary conditions we study these problems with initial data in l q omega w 1 w q omega 1 q infinity or measures and with critically growing non linearities

metastable periodic patterns in singularly perturbed state dependent delayed equations abstract we consider the scalar delayed differential equation x t x t f x t r where 0 r r x 1 r x and f represents either a monotone positive feedback d f d x 0 or a monotone negative feedback d f d x 0 plus some other hypotheses when the delay is a constant i e r x 1 this equation can support metastable rapidly oscillating solutions that are transients whose duration is of order exp c for some c 0 in this paper combining analytic and numerical techniques we investigate whether this metastable behavior persists when the delay r x depends non trivially on the state variable x more precisely we proceed in three steps first we explore theoretically and numerically the existence of branches of rapid periodic solutions for these equations then to predict conditions for metastability we introduce transition layer equations that depict the asymptotic shape of transient oscillations when tends to zero and finally these predictions are validated by numerical simulations numerical explorations are also performed for other scalings of with our conclusions are i when 0 0 there are no metastable transients ii when 0 0 and 0 is finite for monotone negative feedback the metastable oscillations exist irrespective of choices of f and r iii for monotone positive feedback they require that the feedback f and the delay r satisfy extra conditions such as f being an odd function and the delay r x an even function one novel result is that state dependent delays may lead to metastable dynamics in equations that cannot support such regimes when the delay is constant

the cavalcante tenenblat equation does the equation admits physical significance the main purpose of the given paper is to analyze a less studied third order non linear partial differential equation the so called cavalcante tenenblat equation cte in the following form u t u x 1 2 x x u x 3 2 since general class of solutions are of basic interest a complete characterization of the group properties is given the traveling wave ansatz restricts the solution manifold to special class of solutions and hence a generalize algorithm is necessary we determine the lie point symmetry vector fields and calculate similarity ansatze further we also derive a few non linear transformations and some similarity solutions are obtained explicitly due to the complexity of some similarity solutions a numerical procedure is of advantage moreover the non classical case potential symmetries is studied to the first time and further we show how the cte leads to approximate symmetries and we apply the method to the first time we call the disturbed equation the cte e equation and we show how to derive new class of solutions finally the equation does not pass the painleve test and is therefore not soluble by the inverse scattering transform method ist hence suitable alternative algebraic approaches are necessary to derive class of solutions explicitly

front dynamics in a delayed feedback system with external forcing a scalar nonlinear delay differential equation of the kind arising in optically bistable systems is reduced in some physically relevant limit to a partial differential equation first and then to a system of ordinary differential equations these simplified equations are studied analytically we describe the continuous deformation of periodic solutions from sine waves to square waves away from the oscillation threshold and determine their stability we also study the effect of a weak periodic modulation it is found that a resonant forcing can stabilize a very large number of oscillation modes taking the form of square waves with missing front pairs

normal forms for retarded functional differential equations with parameters and applications to hopf bifurcation abstract the paper addresses for retarded functional differential equations fdes with parameters the computation of normal forms associated with the flow on a finite dimensional invariant manifold tangent to an invariant space for the infinitesimal generator of the linearized equation at a singularity the analysis is based on the theory previously developed for fdes without parameters by considering an extension of a phase space appropriate to the computation of these normal forms as an application the generic hopf bifurcation for scalar retarded fdes is considered

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

formulation of the normal form of turing hopf bifurcation in partial functional differential equations abstract turing hopf bifurcation is considered as an important mechanism for generating complex spatio temporal patterns in dynamical systems in this work the normal form up to the third order for the hopf steady state bifurcation which includes the turing hopf bifurcation of a general system of partial functional differential equations pfdes in a bounded spatial region of any dimension is derived based on the center manifold and the normal form theories of pfdes the explicit formula for the coefficients in the normal form of hopf steady state bifurcation are presented concisely in a matrix form which makes it more convenient in not only symbolic derivation but also numerical implementation this provides an approach of showing the existence and stability of stationary and time periodic solutions with spatial heterogeneity the derived formulas are also applicable to partial differential equations and functional differential equations

saddle node hopf bifurcation in a modified leslie gower predator prey model with time delay and prey harvesting abstract from the saddle node hopf bifurcation point of view this paper considers a modified leslie gower predator prey model with time delay and the michaelis menten type prey harvesting firstly we discuss the stability of the equilibria obtain the critical conditions for the saddle node hopf bifurcation and give the completion bifurcation set by calculating the universal unfoldings near the saddle node hopf bifurcation point by using the normal form theory and center manifold theorem then we derive the parameter conditions for the existence of monostable coexistence equilibrium and the parameter regions in which both the prey extinction and the coexistence equilibrium or coexistence periodic or quasi periodic solutions are simultaneously stabilized we also investigate the heteroclinic bifurcation and describe the phenomenon that the periodic behavior disappears as through the heteroclinic bifurcation finally some numerical simulations are performed to support our analytic results

bifurcation analysis of a mathematical model for genetic regulatory network with time delays in this paper we aim to investigate the dynamics of a gene regulatory network which is a time delayed version of the model proposed by elowitz and leibler nature 403 2000 335 338 based on the normal form theory and center manifold reduction hopf bifurcations including the bifurcation direction and stability of the bifurcated periodic orbits are investigated we also discuss effects of transcriptional rate and time delay on the amplitude and period of the oscillation of the network it shows that variations of time delay or transcriptional rate can change the period and amplitude of the oscillation more precisely i the amplitude increases with small time delay while the change of amplitude is not sensitive to relatively large time delay however the robustness of amplitudes is not true any more for the case of using the transcriptional rate as parameter where amplitude always increases quickly and linearly with the transcriptional rate ii the period of oscillation increases as the time delay increases but it grows up initially as the transcriptional rate increases and then keeps unchanged to certain constant value which implies that the robustness of period to the transcriptional rate variations occurs our numerical simulations also support the theoretical conclusions namely both suggest that time delay and transcriptional rate can be used as control parameters in genetic regulatory networks

fractional powers approach of operators for abstract evolution equations of third order in time abstract in this paper we consider approximations of a class of third order linear evolution equations in time governed by fractional powers we explicitly calculate the fractional powers of matricial operators associated with evolution equations of third order in time and we characterize the partial scale of the fractional power of order spaces associated with these operators as an application we present parabolic approximations of the moore gibson thompson type equations

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

hopf bifurcations in time delay systems with band limited feedback abstract we investigate the steady state solution and its bifurcations in time delay systems with band limited feedback this is a first step in a rigorous study concerning the effects of ac coupled components in nonlinear devices with time delayed feedback we show that the steady state is globally stable for small feedback gain and that local stability is lost generically through a hopf bifurcation for larger feedback gain we provide simple criteria that determine whether the hopf bifurcation is supercritical or subcritical based on the knowledge of the first three terms in the taylor expansion of the nonlinearity furthermore the presence of double hopf bifurcations of the steady state is shown which indicates possible quasiperiodic and chaotic dynamics in these systems as a result of this investigation we find that ac coupling introduces fundamental differences to systems of ikeda type k ikeda k matsumoto high dimensional chaotic behavior in systems with time delayed feedback physica d 29 1987 223 235 already at the level of steady state bifurcations e g bifurcations exist in which limit cycles are created with periods other than the fundamental period 2 mode found in ikeda type systems

dispersion of small amplitude solutions of the generalized korteweg de vries equation abstract we study the long time behavior of small solutions of the initial value problem for the generalized korteweg de vries equation t u x 3 u x f u 0 gkdv u x 0 g x for the case where f w w s with s 1 4 23 57 3 8625 our results imply that if g l11 g l22 is sufficiently small then sup r 1 t 1 3 u t l in particular the solution tends to zero in the supremum norm the proofs make use of duhamel s formula and dispersion estimates for the linear propagator as well as chain and leibniz rules for fractional derivatives of compositions d f u lp and products d fg lp 0

qualitative analysis in a delayed van der pol oscillator abstract the main purpose of this manuscript is to introduce and compare two methods for discussing the nature of hopf bifurcation namely a research method combining the normal form theory and the center manifold theorem and the other method of multiple scales taking a delayed van der pol oscillator as an example the local stability as well as the occurrence of hopf bifurcation are explored via choosing time delay as the bifurcation parameter on the basis of two different methods respectively the characters of hopf bifurcation which includes the direction of hopf bifurcation as well as the stability of bifurcating periodic solutions are analyzed finally numerical simulations supporting the theoretical findings are given and it is observed that the results of two methods are consistent

the generalized korteweg de vries burgers equation in h2 r abstract the generalized kdv burgers equation u t u x x g u x u x x u f x 0 0 is considered in this paper using the parabolic regularization technique we prove local and global solvability in h 2 r of the cauchy problem for this equation several regularity properties of the approximations stay valid for solutions constructed in such a way next for when 0 we study the asymptotic behavior of the corresponding semigroup on h 2 r constructing the h 2 r h 3 r global attractor

normal forms for retarded functional differential equations and applications to bogdanov takens singularity abstract the paper addresses for retarded functional differential equations fdes the computation of normal forms associated with the flow on a finite dimensional invariant manifold tangent to invariant spaces for the infinitesimal generator of the linearized equation at a singularity a phase space appropriate to the computation of these normal forms is introduced and adequate nonresonance conditions for the computation of the normal forms are derived as an application the general situation of bogdanov takens singularity and its versal unfolding for scalar retarded fdes with nondegeneracy at second order is considered both in the general case and in the case of differential delay equations of the form t x t x t 1

on perturbations of delay differential equations with periodic orbits abstract equations of retarded type and simple neutral type equations are considered the study concerns both autonomous and non autonomous perturbations of an autonomous equation which possesses a non trivial periodic orbit the main tool is a local coordinate system around the periodic orbit which is obtained from the phase space decomposition via floquet multipliers under the assumption that the perturbation function is lipschitz the existence of an integral manifold with periodic structure for the system in the new coordinates is shown this implies that under autonomous perturbations periodic orbits are continued furthermore we give a description of the flow on the center manifold of the periodic orbit

multiple scale analysis of delayed dynamical systems abstract the model equations of a semiconductor laser with delayed feeback are studied as a prototype of general delayed systems in the case of long delay thermodynamic limit it is shown that its behaviour in the vicinity of the hopf bifurcation can be described by suitable amplitude equations the derivation of the final model follows naturally from the space time representation of the evolution and by the corresponding re formulation of the linear stability problem regions characterized by both direct and inverse hopf bifurcations are found and in particular parameter values are identified for which a chaotic evolution is expected

solitary solutions of some nonlinear evolution equations in this paper the traveling wave reduction combined with the homogeneous balance method hbm is used to find new exact solutions by using less studied nonlinear partial differential equations npdg of higher order the usual starting point is a special transformation converting the npdg in its two variables x and t into an nonlinear ordinary differential equation node in the single variable x using the hyperbolic tangent method new exact solutions for the three npdgs are studied the new feature of this paper is of course the fact that we are dealing with npdgs which cannot be found by studying the relevant literature therefore we believe it is time for publishing some special solutions of this interesting kind of equations clearly we do not use a new formalism but with the aid of the tanh method we are able to calculate at least exact traveling wave solutions solitary waves which can physically interpreted as an action of wave propagation simultaneous we point out the necessity of such sophisticated methods since a general theory of npdgs does not exist the existence of nontrivial noncomplex valued solutions of the ramani equation enables us to generate families of real valued solutions of a nonlinear sixth order system special differential transformations are leading to connections between traveling wave solutions of associated nonlinear evolution equations

retracted stability and hopf bifurcations in a delayed leslie gower predator prey system in this paper we consider the following delayed leslie gower predator prey system x t r 1 x t 1 x t k mx t y t y t r 2 y t 1 y t x t by choosing the delay as a bifurcation parameter we show that hopf bifurcations can occur as the delay crosses some critical values by deriving the equation describing the flow on the center manifold we can determine the direction of the hopf bifurcations and the stability of the bifurcating periodic solutions in addition special attention is paid to the global continuation of local hopf bifurcations using a global hopf bifurcation result of wu j wu symmetric functional differential equations and neural networks with memory trans amer math soc 350 1998 4799 4838 for functional differential equations we may show the global existence of periodic solutions computer simulations illustrate the results

stability and bifurcation in a delayed predator prey system with beddington deangelis functional response we consider a delayed predator prey system with beddington deangelis functional response the stability of the interior equilibrium will be studied by analyzing the associated characteristic transcendental equation by choosing the delay as a bifurcation parameter we show that hopf bifurcation can occur as the delay crosses some critical values the direction and stability of the hopf bifurcation are investigated by following the procedure of deriving normal form given by faria and magalhaes an example is given and numerical simulations are performed to illustrate the obtained results

computation of synchronized periodic solution in a bam network with two delays a bidirectional associative memory bam neural network with four neurons and two discrete delays is considered to represent an analytical method namely perturbation incremental scheme pis the expressions for the periodic solutions derived from hopf bifurcation are given by using the pis the result shows that the pis has higher accuracy than the center manifold reduction cmr with normal form for the values of time delay not far away from the hopf bifurcation point in terms of the pis the necessary and sufficient conditions of synchronized periodic solution arising from a hopf bifurcation are obtained and the synchronized periodic solution is expressed in an analytical form it can be seen that theoretical analysis is in good agreement with numerical simulation it implies that the provided method is valid and the obtained result is correct to the best of our knowledge the paper is the first one to introduce the pis to study the periodic solution derived from hopf bifurcation for a 4 d delayed system quantitatively

computational scheme of a center manifold for neutral functional differential equations abstract this work addresses a computational algorithm of terms of a center manifold for neutral functional differential equations the bogdanov takens and the hopf singularities are considered finally as an illustration of our scheme we give an example where the second term of a center manifold is explicitly determined

normal forms for semilinear equations with non dense domain with applications to age structured models normal form theory is very important and useful in simplifying the forms of equations restricted on the center manifolds in studying nonlinear dynamical problems in this paper using the center manifold theorem associated with the integrated semigroup theory we develop a normal form theory for semilinear cauchy problems in which the linear operator is not densely defined and is not a hille yosida operator and present procedures to compute the taylor expansion and normal form of the reduced system restricted on the center manifold we then apply the main results and computation procedures to determine the direction of the hopf bifurcation and stability of the bifurcating periodic solutions in a structured evolutionary epidemiological model of influenza a drift and an age structured population model

bogdanov takens singularity in van der pol s oscillator with delayed feedback abstract in this paper we study the classical van der pol s equation with delayed feedback we investigate the bogdanov takens singularity and bifurcation occurring in the system with the variation of the original parameters by using the normal form method and predict the suitable choice in the feedback control in order to obtain the asymptotic stable state

on non linear evolution equations of higher order the introduction and application of a novel computational approach the purpose of the present paper is to introduce a new computational algebraic procedure that can easily be applied to derive class of solutions of non linear partial differential equations npde especially of higher order the crucial step needs an auxiliary variable satisfying some ordinary differential equations ode of first order containing sine cosine and their hyperbolic varieties introducing to the first time general transformations are given to determine class of solutions explicitly the validity and reliability of the method is tested by its application to some important non linear evolution equations leading to new class of solutions with physical significance nevertheless it should be emphasised that this techniques do not need the solution of complicate nodes as in the case of similarity reduction further the algorithm works efficiently is clear structured and can be used in any applications independent of the order of the npde for computational purposes the method is appropriate to rewrite it in any computer languages therefore the given novel algebraic approach is suitable for a wider class of npde in order to augment the solution manifold by a straightforward alternative approach

realizability of the normal form for the triple zero nilpotency in a class of delayed nonlinear oscillators abstract the effects of delayed feedback terms on nonlinear oscillators have been extensively studied and there are important applications in many areas of science and engineering we study a particular class of second order delay differential equations near a point of triple zero nilpotent bifurcation using center manifold and normal form reduction we show that the three dimensional nonlinear normal form for the triple zero bifurcation can be fully realized at any given order for appropriate choices of nonlinearities in the original delay differential equation

bifurcations in a planar system of differential delay equations modeling neural activity abstract a planar system of differential delay equations modeling neural activity is investigated the stationary points and their saddle node bifurcations are estimated by an analysis of the associated characteristic equation hopf bifurcations are demonstrated at the intersection points of the saddle node and hopf bifurcation curves in an appropriate parameter plane the existence of bogdanov takens singularities is shown the properties of the bogdanov takens singularities are studied by applying the center manifold and normal form theory a numerical example illustrates the obtained results

uniform decay rates for the energy of weakly damped defocusing semilinear schrodinger equations with inhomogeneous dirichlet boundary control abstract in this paper we study the open loop stabilization as well as the existence and regularity of solutions of the weakly damped defocusing semilinear schrodinger equation with an inhomogeneous dirichlet boundary control first of all we prove the global existence of weak solutions at the h 1 energy level together with the stabilization in the same sense it is then deduced that the decay rate of the boundary data controls the decay rate of the solutions up to an exponential rate secondly we prove some regularity and stabilization results for the strong solutions in h 2 sense the proof uses the direct multiplier method combined with monotonicity and compactness techniques the result for weak solutions is strong in the sense that it is independent of the dimension of the domain the power of the nonlinearity and the smallness of the initial data however the regularity and stabilization of strong solutions are obtained only in low dimensions with small initial and boundary data

rich dynamics in a delayed htlv i infection model stability switch multiple stable cycles and torus abstract in this paper we investigate the impact of time delay in ctl immune response on a htlv i infection model by defining basic reproduction number for viral infection r 0 and basic reproduction number for ctl response r c t l we characterize the model dynamics according to whether these two threshold values are greater than one especially we obtain the global dynamics if r 0 1 or r c t l 1 r 0 as well as infection persistent result when r 0 1 however the model dynamics become much richer when r c t l 1 in this case we use the time delay as a bifurcation parameter to obtain stability switch result on the positive equilibrium and global bifurcation diagrams for the model system we also conduct higher order normal form analysis and apply center manifold theory to classify the rich model dynamics near the double hopf bifurcation points our analysis indicates that time delay in ctl immune response can induce not only hopf bifurcation and double hopf bifurcation but also quasi periodic orbits torus and coexistence of multiple stable periodic solutions

routes to chaos and multiple time scale dynamics in broadband bandpass nonlinear delay electro optic oscillators the response of a nonlinear optical oscillator subject to a delayed broadband bandpass filtering feedback is studied experimentally numerically and analytically the oscillator loop is characterized by a high cutoff frequency with a response time tau approximately 10 ps and by a low cutoff frequency with a response time theta approximately 1 micros moreover the optoelectronic feedback also consists of a significant delay taud of the order of 100 ns depending on two key physical parameters the loop gain beta and the nonlinearity operating point phi a large variety of multiple time scale regimes are reported including slow or fast periodic oscillations with different waveforms regular or chaotic breathers slow time envelope dynamics complex and irregular self pulsing and fully developed chaos many of these regimes are exhibiting new features that are absent in the classical first order scalar nonlinear delay differential equations ddes which differ in the modeling by the low cutoff only nearly all kinds of solutions are recovered numerically by a new class of integro dde idde that take into account both the high and low cutoff frequencies of the feedback loop for moderate feedback gain asymptotic solutions are determined analytically by taking advantage of the relative values of the time constants tau theta and taud we confirm the experimental observation of two distinct routes to oscillatory instabilities depending on the value of phi one route is reminiscent of the square wave oscillations of the classical first order dde but the other route is quite different and allows richer wave forms for higher feedback gain these two distinct regimes merge leading to complex nonperiodic regimes that still need to be explored analytically and numerically finally we investigate the theoretical limits of our idde model by experimentally exploring phenomena at extreme physical parameter setting namely high frequency locking at strong feedback gain or pulse packages for very large delays the large variety of oscillatory regimes of our broadband bandpass delay electro optic oscillator is attractive for applications requiring rich optical pulse sources with different frequencies and or wave forms chaos based communications random number generation chaos computing and generation of stable multiple ghz frequency oscillations

normal forms for periodic retarded functional differential equations this paper addresses the computation of normal forms for periodic retarded functional differential equations fdes with autonomous linear part the analysis is based on the theory previously developed for autonomous retarded fdes adequate nonresonance conditions are derived as an illustration the bogdanov takens and the hopf singularities are considered

product rule and chain rule estimates for fractional derivatives on spaces that satisfy the doubling condition the purpose of this paper is to prove some classical estimates for fractional derivatives of functions defined on a coifman weiss space of homogeneous type in particular the product rule and chain rule estimates in t kato and g ponce 1988 comm pure appl math 41 891 907 and f m christ and m i weinstein 1991 j funct anal 100 87 109 the fractional calculus of m riesz was extended to these spaces in a e gatto c segovia and s vagi 1996 rev mat iberoamericana12 our main tools are fractional difference quotients and the square fractional derivative of r strichartz in 1967 j math mech 16 9 extended to this context for the particular case of rn our approach unifies the proofs of these estimates and clarifies the role of calderon s formula for these results since the square fractional derivative can be easily discretized we also show that the discrete and continuous triebel lizorkin norms for fractional sobolev spaces on spaces of homogeneous type are equivalent

stability of nonlinear waves in a ring of neurons with delays abstract in this paper we consider a ring of identical neurons with self feedback and delays based on the normal form approach and the center manifold theory we derive some formula to determine the direction of hopf bifurcation and stability of the hopf bifurcated synchronous periodic orbits phase locked oscillatory waves standing waves mirror reflecting waves and so on in addition under general conditions such a network has a slowly oscillatory synchronous periodic solution which is completely characterized by a scalar delay differential equation despite the fact that the slowly oscillatory synchronous periodic solution of the scalar equation is stable we show that the corresponding synchronized periodic solution is unstable if the number of the neurons is large or arbitrary even

exponential asymptotics of localised patterns and snaking bifurcation diagrams abstract localised patterns emerging from a subcritical modulation instability are analysed by carrying the multiple scales analysis beyond all orders the model studied is the swift hohenberg equation of nonlinear optics which is equivalent to the classical swift hohenberg equation with a quadratic and a cubic nonlinearity applying the asymptotic technique away from the maxwell point first it is shown how exponentially small terms determine the phase of the fast spatial oscillation with respect to their slow sech type amplitude in the vicinity of the maxwell point the beyond all orders calculation yields the pinning range of parameters where stable stationary fronts connect the homogeneous and periodic states the full bifurcation diagram for localised patterns is then computed analytically including snake and ladder bifurcation curves this last step requires the matching of the periodic oscillation in the middle of a localised pattern both with an up and a down front to this end a third super slow spatial scale needs to be introduced in which fronts appear as boundary layers in addition the location of the maxwell point and the oscillation wave number of localised patterns are required to fourth order accuracy in the oscillation amplitude