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

center manifolds and normal forms for a class of retarded functional differential equations with parameter associated with fold hopf singularity in this paper we present explicit formulas for computing the coefficients of a center manifolds for the fold hopf singularity in autonomous retarded functional differential equations as consequence normal forms associated with the flow on a center manifold up to an arbitrary order are derived the explicit formulas have been implemented using the computer algebra system maple we apply our results to a delayed system in order to show the applicability of the methodology

numerical semi global analysis of a 1 2 resonant hopf hopf bifurcation abstract in this paper a numerical semi global analysis of the dynamics near a 1 2 resonant hopf hopf bifurcation on a four dimensional mathematical model of a simple nonlinear oscillator is performed the 1 2 resonant hopf hopf bifurcation is a codimension three singularity denoted by two pairs of purely imaginary eigenvalues with frequency ratio 1 2 a structure involving 1 1 and 1 2 resonant neimark sacker bifurcations is clearly identified both resonances are coupled by lower codimension singularities such as generalized hopf and period doubling cusp points and cyclic fold curves a three parameter semi global analysis is performed and some of the codimension two singularities unfolded by the 1 2 resonant hopf hopf bifurcation are identified several codimension three points are also detected the obtained results can be useful for further theoretical analysis of the corresponding normal form

on the representation of operators in bases of compactly supported wavelets this paper describes exact and explicit representations of the differential operators d n dx n n 1 2 cdots in orthonormal bases of compactly supported wavelets as well as the representations of the hilbert transform and fractional derivatives the method of computing these representations is directly applicable to multidimensional convolution operators also sparse representations of shift operators in orthonormal bases of compactly supported wavelets are discussed and a fast algorithm requiring o n log n operations for computing the wavelet coefficients of all n circulant shifts of a vector of the length n 2 n is constructed as an example of an application of this algorithm it is shown that the storage requirements of the fast algorithm for applying the standard form of a pseudodifferential operator to a vector see g beylkin r r coifman and v rokhlin comm pure appl math 44 1991 pp 141 183 may be reduced from o n to o log 2 n significant entries

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

fully adaptive multiresolution finite volume schemes for conservation laws the use of multiresolution decompositions in the context of finite volume schemes for conservation laws was first proposed by a harten for the purpose of accelerating the evaluation of numerical fluxes through an adaptive computation in this approach the solution is still represented at each time step on the finest grid resulting in an inherent limitation of the potential gain in memory space and computational time the present paper is concerned with the development and the numerical analysis of fully adaptive multiresolution schemes in which the solution is represented and computed in a dynamically evolved adaptive grid a crucial problem is then the accurate computation of the flux without the full knowledge of fine grid cell averages several solutions to this problem are proposed analyzed and compared in terms of accuracy and complexity

hopf pitchfork bifurcation in van der pol s oscillator with nonlinear delayed feedback abstract first we identify the critical values for hopf pitchfork bifurcation second we derive the normal forms up to third order and their unfolding with original parameters in the system near the bifurcation point by the normal form method and center manifold theory then we give a complete bifurcation diagram for original parameters of the system and obtain complete classifications of dynamics for the system furthermore we find some interesting phenomena such as the coexistence of two asymptotically stable states two stable periodic orbits and two attractive quasi periodic motions which are verified both theoretically and numerically

wavelet based schemes for linear advection dispersion equation abstract in this paper two wavelet based adaptive solvers are developed for linear advection dispersion equation the localization properties and multilevel structure of the wavelets in the physical space are used for adaptive computational methods for solution of equation which exhibit both smooth and shock like behaviour the first framework is based on wavelet galerkin and the second is based on multiscale decomposition of finite element method coiflet wavelet filter is incorporated in both the methods the main advantage of both the adaptive methods is the elimination of spurious oscillations at very high peclet number

dynamics of a limit cycle oscillator under time delayed linear and nonlinear feedbacks abstract we study the effects of time delayed linear and nonlinear feedbacks on the dynamics of a single hopf bifurcation oscillator our numerical and analytic investigations reveal a host of complex temporal phenomena such as phase slips frequency suppression multiple periodic states and chaos such phenomena are frequently observed in the collective behavior of a large number of coupled limit cycle oscillators our time delayed feedback model offers a simple paradigm for obtaining and investigating these temporal states in a single oscillator we construct a detailed bifurcation diagram of the oscillator as a function of the time delay parameter and the driving strengths of the feedback terms we find some new states in the presence of the quadratic nonlinear feedback term with interesting characteristics like birhythmicity phase reversals radial trapping phase jumps and spiraling patterns in the amplitude space our results may find useful applications in physical chemical or biological systems

simple accurate and efficient revisions to maccormack and saulyev schemes high peclet numbers abstract stream water quality modeling often involves numerical methods to solve the dynamic one dimensional advection dispersion reaction equations adre there are numerous explicit and implicit finite difference schemes for solving these problems and two commonly used schemes are the maccormack and saulyev schemes this paper presents simple revisions to these schemes that make them more accurate without significant loss of computation efficiency using advection dominated high peclet number problems as test cases performances of the revised schemes are compared to performances of five classic schemes forward time centered space ftcs backward time centered space btcs crank nicolson and the traditional maccormack and saulyev schemes all seven of the above numerical schemes were tested against analytical solutions for pulse and step inputs of mass to a steady flow in a channel and performances were considered with respect to stability accuracy and computational efficiency results indicated that both the modified saulyev and the maccormack schemes which are named the saulyevc and maccormackc schemes respectively greatly improved the prediction accuracy over the original ones the computation efficiency in terms of cpu time was not impacted for the saulyevc scheme the maccormackc scheme demonstrated increased time consumption but was still much faster than implicit schemes

on a planar system modelling a neuron network with memory abstract we consider a delay differential equation modelling a network of two neurons with memory the situations without self connections and two delays and with self connections and two equal delays are addressed in terms of local stability and bifurcation analysis in the first case the dynamical behavior is studied by taking one of the delays as the bifurcating parameter namely the hopf bifurcation whose existence was proven in former works occurs as the delay crosses some critical values and is completely described conditions ensuring the stability of the periodic cycles are given when self connections are present hopf and pitchfork bifurcations co exist and the equation describing the flow on the center manifold is derived

on the analysis of the double hopf bifurcation in machining processes via centre manifold reduction the single degree of freedom model of orthogonal cutting is investigated to study machine tool vibrations in the vicinity of a double hopf bifurcation point centre manifold reduction and normal form calculations are performed to investigate the long term dynamics of the cutting process the normal form of the four dimensional centre subsystem is derived analytically and the possible topologies in the infinite dimensional phase space of the system are revealed it is shown that bistable parameter regions exist where unstable periodic and in certain cases unstable quasi periodic motions coexist with the equilibrium taking into account the non smoothness caused by loss of contact between the tool and the workpiece the boundary of the bistable region is also derived analytically the results are verified by numerical continuation the possibility of transient chaotic motions in the global non smooth dynamics is shown

normal forms and hopf bifurcation for partial differential equations with delays the paper addresses the computation of normal forms for some partial functional differential equations pfdes near equilibria the analysis is based on the theory previously developed for autonomous retarded functional differential equations and on the existence of center or other invariant manifolds as an illustration of this procedure two examples of pfdes where a hopf singularity occurs on the center manifold are considered

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed

double hopf bifurcation for van der pol duffing oscillator with parametric delay feedback control the stability and bifurcation of a van der pol duffing oscillator with the delay feedback are investigated in which the strength of feedback control is a nonlinear function of delay a geometrical method in conjunction with an analytical method is developed to identify the critical values for stability switches and hopf bifurcations the hopf bifurcation curves and multi stable regions are obtained as two parameters vary some weak resonant and non resonant double hopf bifurcation phenomena are observed due to the vanishing of the real parts of two pairs of characteristic roots on the margins of the death island regions simultaneously by applying the center manifold theory the normal forms near the double hopf bifurcation points as well as classifications of local dynamics are analyzed furthermore some quasi periodic and chaotic motions are verified in both theoretical and numerical ways

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

a dynamically adaptive multilevel wavelet collocation method for solving partial differential equations in a finite domain a dynamically adaptive multilevel wavelet collocation method is developed for the solution of partial differential equations the multilevel structure of the algorithm provides a simple way to adapt computational refinements to local demands of the solution high resolution computations are performed only in regions where sharp transitions occur the scheme handles general boundary conditions the method is applied to the solution of the one dimensional burgers equation with small viscosity a moving shock problem and a nonlinear thermoacoustic wave problem the results indicate that the method is very accurate and efficient

technical communique on stability of second order quasi polynomials with a single delay in this note the stability of a second order quasi polynomial with a single delay is studied although there is a vast literature on this problem most available solutions are limited to some particular cases moreover some published results on this subject appear to contain imprecise or even wrong conditions the purpose of this note is to show that by accurate application of known theories a complete explicit characterization of stability regions can be derived in a most general case as a byproduct of the proposed analysis we show that in the high order case the quasi polynomial is delay independent unstable whenever its delay free version has an odd number of unstable roots or equivalently a negative static gain

a frequency sweeping framework for stability analysis of time delay systems for time delay systems the asymptotic behavior analysis of the critical imaginary roots w r t the infinitely many critical delays is an open problem in order to find a general solution we will exploit the link between the asymptotic behavior of critical imaginary roots and the asymptotic behavior of frequency sweeping curves from a new analytic curve perspective as a consequence we will establish a frequency sweeping framework with three main results 1 a finer regularity singularity classification for time delay systems will be obtained 2 the general invariance property will be proved and hence the asymptotic behavior of the critical imaginary roots w r t the infinitely many critical delays can be adequately studied 3 the complete stability problem can be fully solved moreover the frequency sweeping framework is extended to cover a broader class of time delay systems finally the geometric insights of frequency sweeping curves are investigated consequently some deeper properties on the asymptotic behavior of time delay systems and the link to frequency sweeping curves are found

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

minimal determinantal representations of bivariate polynomials abstract for a square free bivariate polynomial p of degree n we introduce a simple and fast numerical algorithm for the construction of n n matrices a b and c such that det a x b y c p x y this is the minimal size needed to represent a bivariate polynomial of degree n combined with a square free factorization one can now compute n n matrices for any bivariate polynomial of degree n the existence of such symmetric matrices was established by dixon in 1902 but up to now no simple numerical construction has been found even if the matrices can be nonsymmetric such representations may be used to efficiently numerically solve a system of two bivariate polynomials of small degree via the eigenvalues of a two parameter eigenvalue problem the new representation speeds up the computation considerably

colored stability crossing sets for siso delay systems this paper considers the stability of a single input single output siso system with an input output i o delay called siso delay system hereafter we first derive an algebraic criterion to solve the complete stability w r t the delay tau next the stability analysis is extended to the case with additional free plant and or controller parameters denoted by a vector x by adopting a mathematical tool for polynomial algebra the discrimination system the complete classification of the critical imaginary roots cirs in the whole x tau parameter space may be obtained furthermore with the above results we present the stability crossing sets scss in a new form called the colored scss for each point on the scs the asymptotic behavior information of the corresponding cir can be coded by a certain color red green or yellow hence the stability in the x tau space can be analyzed easily and intuitively moreover the geometry of the scss can be investigated from a new perspective which will help not only to better understand the stability of the siso delay system but also to considerably simplify the procedure for generating the colored scss

using the refinement equation for evaluating integrals of wavelets the wavelet galerkin method for solving partial differential equations leads to the problem of computing integrals of products of derivatives of wavelets this paper studies the problem from the point of view of stationary subdivision schemes one of the main results is to identify these integrals as components of the unique solution of a certain eigenvector moment problem associated with the coefficients of the refinement equation asymptotic expansions for the corresponding subdivision schemes form an important ingredient of our approach

a nonautonomous predator prey system with stage structure and double time delays in the present paper we study a nonautonomous predator prey model with stage structure and double time delays due to maturation time for both prey and predator we assume that the immature and mature individuals of each species are divided by a fixed age and the mature predator only attacks the immature prey based on some comparison arguments we discuss the permanence of the species by virtue of the continuation theorem of coincidence degree theory we prove the existence of positive periodic solution by means of constructing an appropriate lyapunov functional we obtain sufficient conditions for the uniqueness and the global stability of positive periodic solution two examples are given to illustrate the feasibility of our main results

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

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

on the quadratic two parameter eigenvalue problem and its linearization we introduce the quadratic two parameter eigenvalue problem and linearize it as a singular two parameter eigenvalue problem this together with an example from model updating shows the need for numerical methods for singular two parameter eigenvalue problems and for a better understanding of such problems there are various numerical methods for two parameter eigenvalue problems but only few for nonsingular ones we present a method that can be applied to singular two parameter eigenvalue problems including the linearization of the quadratic two parameter eigenvalue problem it is based on the staircase algorithm for the extraction of the common regular part of two singular matrix pencils

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

polynomials with and without determinantal representations abstract the problem of writing real zero polynomials as determinants of linear matrix polynomials has recently attracted a lot of attention helton and vinnikov 9 have proved that any real zero polynomial in two variables has a determinantal representation branden 2 has shown that the result does not extend to arbitrary numbers of variables disproving the generalized lax conjecture we prove that in fact almost no real zero polynomial admits a determinantal representation there are dimensional differences between the two sets the result follows from a general upper bound on the size of linear matrix polynomials we then provide a large class of surprisingly simple explicit real zero polynomials that do not have a determinantal representation we finally characterize polynomials of which some power has a determinantal representation in terms of an algebra with involution having a finite dimensional representation we use the characterization to prove that any quadratic real zero polynomial has a determinantal representation after taking a high enough power taking powers is thereby really necessary in general the representations emerge explicitly and we characterize them up to unitary equivalence

cubic spline wavelets with complementary boundary conditions abstract we propose a new construction of a stable cubic spline wavelet basis on the interval satisfying complementary boundary conditions of the second order it means that the primal wavelet basis is adapted to homogeneous dirichlet boundary conditions of the second order while the dual wavelet basis preserves the full degree of polynomial exactness we present quantitative properties of the constructed bases and we show superiority of our construction in comparison to some other known spline wavelet bases in an adaptive wavelet method for the partial differential equation with the biharmonic operator

hopf bifurcating periodic orbits in a ring of neurons with delays in this paper we consider a ring of neurons with self feedback and delays the linear stability of the model is investigated by analyzing the associated characteristic transcendental equation based on the normal form approach and the center manifold theory we derive the formula for determining the properties of hopf bifurcating slowly oscillating periodic orbits for a ring of neurons with delays including the direction of hopf bifurcation stability of the hopf bifurcating slowly oscillating periodic orbits and so on moreover by means of the symmetric bifurcation theory of delay differential equations coupled with representation theory of standard dihedral groups we not only investigate the effect of synaptic delay of signal transmission on the pattern formation but also obtain some important results about the spontaneous bifurcation of multiple branches of periodic solutions and their spatio temporal patterns

normal forms of non resonance and weak resonance double hopf bifurcation in the retarded functional differential equations and applications the norm forms of the double hopf bifurcation for a general dde is established the van der pol duffing oscillator with delayed position and velocity feedback is investigated the dynamical classification near the double hopf bifurcation point is studied the stable periodic doubling solutions and unstable three dimensional torus are found in this paper we firstly present the general framework of calculation of normal forms of non resonance and weak resonance double hopf bifurcation for the general retarded functional differential equations by using the normal form theory of delay differential equations due to faria and magalh a es then the dynamical behavior of van der pol duffing oscillator with delayed position and velocity feedback is considered specifically the dynamical classification near the double hopf bifurcation point is investigated by analyzing the obtained normal form finally the numerical simulations support the theoretical results and present some interesting phenomena

bogdanov takens singularity in tri neuron network with time delay this brief reports a retarded functional differential equation modeling tri neuron network with time delay the bogdanov takens b t bifurcation is investigated by using the center manifold reduction and the normal form method we get the versal unfolding of the norm forms at the b t singularity and show that the model can exhibit pitchfork hopf homoclinic and double limit cycles bifurcations some numerical simulations are given to support the analytic results and explore chaotic dynamics finally an algorithm is given to show that chaotic tri neuron networks can be used for encrypting a color image

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

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

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

on compactly supported spline wavelets and a duality principle let v 1 v 0 v 1 be a multiresolution analysis of l 2 generated by the mth order b spline n m x we exhibit a compactly supported basic wavelet m x that generates the corresponding orthogonal complementary wavelet subspaces w 1 w 0 w 1 consequently the two finite sequences that describe the two scale relations of n m x and m x in terms of n m 2x j j z yield an efficient reconstruction algorithm to give an efficient wavelet decomposition algorithm based on these two finite sequences we derive a duality principle which also happens to yield the dual bases n m x j and 59 m x j relative to n m x j and m x j respectively

wavelets and the numerical solution of partial differential equations we present a new numerical method for the solution of partial differential equations in nonseparable domains the method uses a wavelet galerkin solver with a nontrivial adaptation of the standard capacitance matrix method the numerical solutions exhibit spectral convergence with regard to the order of the compactly supported daubechies wavelet basis furthermore the rate of convergence is found to be independent of the geometry we solve the helmholtz equation since for variations in the parameter the solutions have qualitative properties that well illustrate the applications of our method

non resonant double hopf bifurcation of a class b laser system we investigate the double hopf bifurcation at positive equilibrium point of a class b laser system firstly we give the critical values of hopf and double hopf bifurcations secondly we implement the normal form method and the center manifold theory for a class b laser system and derive the universal unfolding of the double hopf bifurcation thirdly some dynamic behaviors such as periodic and quasi periodic solutions are observed using numerical simulation

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

stability with respect to the delay on a paper of k l cooke and p van den driessche a proposal for the correction of the main result of the title s paper is made some accompanying comments on other points are given so a simpler central result is derived an explicit representation of the set of stable delays is achieved

double hopf bifurcation and quasi periodic attractors in delay coupled limit cycle oscillators abstract we investigate the double hopf bifurcation at zero equilibrium point firstly we give the critical values of hopf and double hopf bifurcations secondly we implement the normal form method and the center manifold theory for delay coupled limit cycle oscillators and derive the universal unfolding and a complete bifurcation diagram of the system thirdly many interesting phenomena such as attractive periodic motion and three dimensional invariant torus are observed using numerical simulation finally the normal forms of several strong resonant cases are listed

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

multiresolution modeling with operator customized wavelets derived from finite elements in this article we describe the construction and application of operator customized wavelets constructed out of general finite element interpolation functions i e not just those of the lagrange family unlike classical hierarchical bases these wavelets are scale orthogonal with respect to a given inner product such as that arising from the weak form of a differential operator this property results in block diagonal stiffness matrices that in turn permits the incremental computation of the solution in an efficient manner especially in the case of adaptive mesh refinement as a specific example we consider the design of such wavelets for the incremental and adaptive solution of fourth order partial differential equations such as those governing the mechanics of thin elastic plates and shells

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

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

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

on computing puiseux series for multiple imaginary characteristic roots of lti systems with commensurate delays obtaining the puiseux series of multiple imaginary characteristic roots mirs is a fundamental issue in the stability analysis of time delay systems however to the best of the authors knowledge this issue has not been fully investigated up to date this note focuses on the puiseux series expansion of mirs of linear time invariant systems including commensurate delays for an mir of any multiplicity we propose an algorithm for defining the structure of the puiseux series as well as the explicit computation of the corresponding coefficients by using the proposed method we can find all the puiseux series corresponding to all the root loci

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

delay independent stability of a special sequence of neutral difference differential equations with one delay abstract the equations of the title for the state x t r l x d a d a n x t b d b d b n x t t 0 d d dt 1 are considered in the parameter space a b b n p where p is defined by re a a is a second real point a a for a r a is the complex conjugate to a c the same notation applies for b by a tedious but elementary analysis the set s0 p for which 1 has for all t 0 only asymptotically stable solutions in the sense of liapunov is explicitly determined in the form b n a b 1 2 the stability boundary decays exponentially with respect to the multiplicity n of the zeros of the coefficient polynomials of 1 we generalize a theorem of f brauer j differential equations 69 1987 185 191 dealing with delay independent stability for characteristic equations of the form h z a z b z exp zt t 0 3 where a z b z are holomorphic in re z 0 in such a way that the neutral case can be dealt with

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

the asymptotic shapes of periodic solutions of a singular delay differential system let f r r be given so that f 0 0 and f x 1 x ax2 bx3 o x3 as x 0 we characterize those small values of e 0 and r for which there are periodic solutions of periods approximately 2k with k n of the following system arising from a network of neurons formula the periodic solutions are synchronized if k is even and phase locked if k is odd we show that as e 0 these periodic solutions approach square waves if a 0 and b 0 or if a 0

nonresonant hopf hopf bifurcation and a chaotic attractor in neutral functional differential equations abstract an algorithm for calculating the third order normal form of a nonresonant hopf hopf singularity in a neutral functional differential equation nfde is established the van der pol equation with extended delay feedback is investigated as an nfde of second order the existence of hopf hopf bifurcation is studied and the unfolding near these critical points is given by applying this algorithm periodic solutions and quasi periodic solutions are found with the aid of the bifurcation diagram and corresponding numerical illustrations are presented with the breaking down of the 3 torus a chaotic attractor appears in this nfde of second order following the ruelle takens newhouse scenario which usually arises for an ordinary differential equation of order at least 4 this transition is shown via both theoretical and numerical approaches

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

element by element construction of wavelets satisfying stability and moment conditions in this paper we construct a class of locally supported wavelet bases for c0 lagrange finite element spaces on possibly nonuniform meshes on n dimensional domains or manifolds the wavelet bases are stable in the sobolev spaces mathcal h s for s frac 3 2 s leq 1 on lipschitz manifolds and the wavelets can in principle be arranged to have any desired order of vanishing moments as a consequence these bases can be used e g for constructing an optimal solver of discretized mathcal h s elliptic problems for s in above ranges the construction of the wavelets consists of two parts an implicit part involves some computations on a reference element which for each type of finite element space have to be performed only once in addition there is an explicit part which takes care of the necessary adaptations of the wavelets to the actual mesh the only condition we need for this construction to work is that the refinements of initial elements are uniform we will show that the wavelet bases can be implemented efficiently

on the adaptive numerical solution of nonlinear partial differential equations in wavelet bases this work develops fast and adaptive algorithms for numerically solving nonlinear partial differential equations of the formut lu nf u where l and n are linear differential operators andf u is a nonlinear function these equations are adaptively solved by projecting the solutionuand the operators l and n into a wavelet basis vanishing moments of the basis functions permit a sparse representation of the solution and operators using these sparse representations fast and adaptive algorithms that apply operators to functions and evaluate nonlinear functions are developed for solving evolution equations for a wavelet representation of the solutionuthat containsnssignificant coefficients the algorithms update the solution usingo ns operations the approach is applied to a number of examples and numerical results are given

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

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

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

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

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

roots of bivariate polynomial systems via determinantal representations we give two determinantal representations for a bivariate polynomial they may be used to compute the zeros of a system of two of these polynomials via the eigenvalues of a two parameter eigenvalue problem the first determinantal representation is suitable for polynomials with scalar or matrix coefficients and consists of matrices with asymptotic order n 2 4 where n is the degree of the polynomial the second representation is useful for scalar polynomials and has asymptotic order n 2 6 the resulting method to compute the roots of a system of two bivariate polynomials is very competitive with some existing methods for polynomials up to degree 10 as well as for polynomials with a small number of terms

absolute stability in delay equations abstract an equilibrium of a delay equation is said to be absolutely stable if it is asymptotically stable for all delays this is equivalent to a certain characteristic equation s having all its roots in the left half plane for all delays we obtain a simple necessary and sufficient condition for absolute stability of a class of delay equations

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

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

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

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

double hopf bifurcation of coupled dissipative stuart landau oscillators with delay in this paper the double hopf bifurcation of delay coupled dissipative stuart landau oscillators models are considered where the time delay is regarded as one of the parameters and the other is the natural frequency it is found that as the values of the two parameters vary a number of double hopf bifurcation points arise consequently it has been found that the system exhibits very rich complex dynamics some resonant and non resonant double hopf bifurcation phenomena can be observed some numerical simulations support our analysis result

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

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

a simple global characterization for normal forms of singular vector fields we derive a new global characterization of the normal forms of amplitude equations describing the dynamics of competing order parameters in degenerate bifurcation problems using an appropriate scalar product in the space of homogeneous vector polynomials we show that the resonant terms commute with the group generated by the adjoint of the original critical linear operator this leads to a very efficient constructive method to compute both the nonlinear coefficients and the unfolding of the normal form explicit examples and results obtained when there are additional symmetries are also presented

symmetric determinantal representation of polynomials abstract we give an elementary proof only using linear algebra of a result due to helton mccullough and vinnikov which says that any polynomial over the reals can be written as the determinant of a symmetric affine linear pencil we give explicit determinantal representation formulas and extend our results to polynomials with coefficients in a ring of characteristic different from 2