dynamics of differential equations on invariant manifolds abstract the simplification resulting from reduction of dimension involved in the study of invariant manifolds of differential equations is often difficult to achieve in practice appropriate coordinate systems are difficult to find or are essentially local in nature thus complicating analysis of global dynamics this paper develops an approach which avoids the selection of coordinate systems on the manifold conditions are given in terms compound linear differential equations for the stability of equilibria and periodic orbits global results include criteria for the nonexistence of periodic orbits and a discussion of the nature of limit sets as an application a global stability criterion is established for the endemic equilibrium in an epidemiological model

absolute value equations abstract we investigate existence and nonexistence of solutions for np hard equations involving absolute values of variables ax x b where a is an arbitrary n n real matrix by utilizing an equivalence relation to the linear complementarity problem lcp we give existence results for this class of absolute value equations aves as well as a method of solution for special cases we also give nonexistence results for our ave using theorems of the alternative and other arguments

geometrical aspects of stability theory for hill s equations where a and b are real parameters apparently the first stability diagram was drawn in the classical paper by b van der pol m j o sti utt 15 since then many papers and textbooks have appeared on the subject we mention j j stoker 13 j meixner f w sch fke 12 d m levy 8 j b keller 11 m i weinstein j b keller 16 17 j poschel e trubowltz 9 and v i arnold 3 5 6 this literature contains extensive estimates on the order of tangency of resonance tongues and the behavior of stability boundaries at infinity a full understanding of the stability diagrams is still lacking for instance the appearance of instability pockets in some cases see figure 1 cf 15 calls for a geometric explanation it should be noted that this phenomenon does not occur in the classical mathieu case cf 12 for a preliminary study near resonances using singularity theory we refer to afsharnejad 1 related bifurcational aspects of nonlinear perturbations of mathieu s equation near resonances were studied by broer vegter 7 the global geometry of the symplectic group was used for a stability proof of a hill equation in levi 10 we introduce the main object of study of this paper

on bendixson s criterion abstract for autonomous differential equations in r n criteria are developed which preclude the existence of invariant closed curves such as periodic or homoclinic trajectories the technique is based on the study of functionals on 2 surfaces results generalize to higher dimensions a criterion of bendixson for the nonexistence of nonconstant periodic solutions in the case n 2 as an example an application to the lorenz system in r 3 is given

complex dynamics of an seir epidemic model with saturated incidence rate and treatment abstract in this paper we describe the dynamics of an seir epidemic model with saturated incidence treatment function and optimal control rigorous mathematical results have been established for the model the stability analysis of the model is investigated and found that the model is locally asymptotically stable when r 0 1 the model is locally as well as globally asymptotically stable at endemic equilibrium when r 0 1 the proposed model may possess a backward bifurcation the optimal control problem is designed and obtained their necessary results numerical results have been presented for justification of theoretical results

synchronized hopf bifurcation analysis in a neural network model with delays we consider the synchronized periodic oscillation in a ring neural network model with two different delays in self connection and nearest neighbor coupling employing the center manifold theorem and normal form method introduced by hassard et al we give an algorithm for determining the hopf bifurcation properties using the global hopf bifurcation theorem for fde due to wu and bendixson s criterion for high dimensional ode due to li and muldowney we obtain several groups of conditions that guarantee the model have multiple synchronized periodic solutions when the transfer coefficient or time delay is sufficiently large

asymptotic behavior of stability regions for hill s equation the asymptotic behavior of the solutions of hill s equation u e lambda 2 q x u 0 is determined for large positive real values of the coupling constant lambda 2 and large real values of the energy e the locations and widths of the stability bands and instability gaps are found the band widths are shown to decrease exponentially as lambda increases when lambda 2 e lies between the minimum and maximum values of the periodic potential q x the gap widths decrease exponentially with lambda when lambda 2 e is greater than the maximum of q x for lambda 2 e asymptotically equal to the maximum of q x the width of the nth band is asymptotically half the width of the nth gap the exponentially small band and gap widths are related to the exponentially small transmission and reflection coefficients associated with one period of q x the present results extend previous ones of meixner and schafke harrell and the authors in which lam

global stability of a stage structured epidemic model with a nonlinear incidence in this paper a stage structured epidemic model with a nonlinear incidence with a factor s p is investigated by using limit theory of differential equations and theorem of busenberg and van den driessche global dynamics of the model is rigorously established we prove that if the basic reproduction number r 0 is less than one the disease free equilibrium is globally asymptotically stable and the disease dies out if r 0 is greater than one then the disease persists and the unique endemic equilibrium is globally asymptotically stable numerical simulations support our analytical results and illustrate the effect of p on the dynamic behavior of the model

global dynamics of nicholsonʼs blowflies equation revisited onset and termination of nonlinear oscillations abstract we revisit nicholson s blowflies model with natural death rate incorporated into the delay feedback we consider the delay as a bifurcation parameter and examine the onset and termination of hopf bifurcations of periodic solutions from a positive equilibrium we show that the model has only a finite number of hopf bifurcation values and we describe how branches of hopf bifurcations are paired so the existence of periodic solutions with specific oscillation frequencies occurs only in bounded delay intervals the bifurcation analysis and the matlab package dde biftool developed by engelborghs et al guide some numerical simulations to identify ranges of parameters for coexisting multiple attractive periodic solutions

some techniques for solving absolute value equations in this paper we introduce and analyze two new methods for solving the np hard absolute value equations ave a x x b where a is an arbitrary n n real matrix and b rn in the case singular value of a exceeds 1 the comparison with other known methods is carried to show the effectiveness of the proposed methods for a variety of randomly generated problems the ideas and techniques of this paper may stimulate further research

some examples describing the need of modifying the bendixson criterion for piecewise c1 systems abstract some numerical examples to clarify the use of distributions in ordinary differential equations with discontinuous right hand sides

solving absolute value equation using complementarity and smoothing functions in this paper we reformulate the np hard problem of the absolute value equation ave as a horizontal linear complementarity one and then solve it using a smoothing technique this approach leads to a new class of methods that are valid for general absolute value equation an asymptotic analysis proves the convergence of our schemes and provides some interesting error estimates this kind of error bound or estimate had never been studied for other known methods the corresponding algorithms were tested on randomly generated problems and applications these experiments show that in the general case one observes a reduction of the number of failures

a negative bendixson like criterion for a class of hybrid systems a condition which ensures the absence of periodic orbits for nonsmooth dynamical systems is presented the condition is a higher dimensional generalization of bendixson s criterion applicable to differential inclusions that are useful in the description of hybrid systems the main argument is based on contraction analysis of the d measured volume along the system trajectories a connection to methods for estimating the hausdorff dimension is emphasized for a class of hybrid systems described by a linear system with relay feedback the conditions are presented in the form of linear matrix inequalities a simple but illustrative example is analyzed

a numerical method based on the complementarity and optimal control formulations for solving a family of zero sum pursuit evasion differential games abstract this paper presents an efficient numerical method for a class of zero sum pursuit evasion differential games zspedgs the aim of the presented method is to resolve the drawbacks of the indirect methods in solving zspedgs in the indirect methods the solution of zspedg is found by solving a two point boundary value problem tpbvp derived from the necessary conditions the indirect methods are accurate and fast however they are very sensitive to initial guess and when the control is bounded some non smooth equations appear in the resulting tpbvp these drawbacks restrict the use of indirect methods for solving zspedgs to overcome these drawbacks at first we reformulate the discontinuous equations of tpbvp by complementarity conditions and as a result a differential complementarity system dcs is obtained then the resulting dcs is reformulated to an optimal control problem which can be solved by a well developed direct or indirect method the efficiency and robustness of the method are reported by means of two benchmarks and two real life zspedg problems

transition curves for the quasi periodic mathieu equation in this work we investigate an extension of mathieu s equation the quasi periodic qp mathieu equation given by ddot psi delta eps cos t cos omega t psi 0 for small eps and irrational omega of interest is the generation of stability diagrams that identify the points or regions in the delta omega parameter plane for fixed eps for which all solutions of the qp mathieu equation are bounded numerical integration is employed to produce approximations to the true stability diagrams both directly and through contour plots oflyapunov exponents in addition we derive approximate analytic expressions for transition curves using two distinct techniques 1 a regular perturbation method under which transition curves delta delta omega eps are each expanded in powers of eps and 2 the method of harmonic balance utilizing hill s determinants both analytic methods deliver results in good agreement with those generated numerically in the sense that pre

obtaining certificates for complete synchronisation of coupled oscillators abstract in this paper we provide a novel reformulation of sufficient conditions that guarantee global complete synchronisation of coupled identical oscillators to make them computationally implementable to this end we use semidefinite programming techniques for the first time we can efficiently search for and obtain certificates for synchronisability and additionally also optimise associated cost functions in this paper a lyapunov like function certificate is used to certify that all trajectories of a networked system consisting of coupled dynamical systems will eventually converge towards a common one which implies synchronisation moreover we establish new conditions for complete synchronisation which are based on the so called bendixson s criterion for higher dimensional systems this leads to major improvements on the lower bound of the coupling constant that guarantees global complete synchronisation importantly the certificates are obtained by analysing the connection network and the model representing an individual system only in order to illustrate the strength of our method we apply it to a system of coupled identical lorenz oscillators and to coupled van der pol oscillators

equivalent embeddings of the dynamics on an invariant manifold a dynamical system admitting an invariant manifold can be interpreted as a single element of an infinite class of dynamical systems that all exhibit the same behaviour on the invariant manifold this observation is used in the context of autonomous ordinary differential equations to generalize a global stability result of li and muldowney the new result is demonstrated on an epidemiological model

an efficient method for optimal correcting of absolute value equations by minimal changes in the right hand side our goal in this work is to give an optimum correction of the infeasible absolute value equations ave in order to make the mentioned system feasible we apply the minimal correction using the l 2 norm by changing just the right hand vector we will show that this problem can be formulated as an unconstrained optimization problem with a quadratic objective function we propose an extension of newton s method for solving unconstrained objective optimization some examples are provided to illustrate the efficiency and validity of our proposed method

dynamic model of worms with vertical transmission in computer network an e epidemic seirs model for the transmission of worms in computer network through vertical transmission is formulated it has been observed that if the basic reproduction number is less than or equal to one the infected part of the nodes disappear and the worm dies out but if the basic reproduction number is greater than one the infected nodes exists and the worms persist at an endemic equilibrium state numerical methods are employed to solve and simulate the system of equations developed we have analyzed the behavior of the susceptible exposed infected and recovered nodes in the computer network with real parametric values

bifurcation analysis of a population model and the resulting sis epidemic model with delay this paper deals with the model for matured population growth proposed in cooke et al interaction of matiration delay and nonlinear birth in population and epidemic models j math biol 39 1999 332 352 and the resulting sis epidemic model the dynamics of these two models are still largely undetermined and in this paper we perform some bifurcation analysis to the models by applying the global bifurcation theory for functional differential equations we are able to show that the population model allows multiple periodic solutions for the sis model we obtain some local bifurcation results and derive formulas for determining the bifurcation direction and the stability of the bifurcated periodic solution

using fourier coefficients to determine the unstable points of hill equation abstract the main purpose of this paper is to determine the unstable points of the hill equation x q t x 0 the approach is based on fourier coefficients combining with a novel form discriminant of hill equation the method is much different from the previous methods employed in the literature

three dimensional competitive lotka volterra systems with no periodic orbits the following conjecture of m l zeeman is proved if three interacting species modeled by a competitive lotka volterra system can each resist invasion at carrying capacity then there can be no coexistence of the species indeed two of the species are driven to extinction it is also proved that in the other extreme if none of the species can resist invasion from either of the others then there is stable coexistence of at least two of the species in this case if the system has a fixed point in the interior of the positive cone in r3 then that fixed point is globally asymptotically stable representing stable coexistence of all three species otherwise there is a globally asymptotically stable fixed point in one of the coordinate planes of r3 representing stable coexistence of two of the species

global stability for an hiv 1 infection model including an eclipse stage of infected cells abstract we consider the mathematical model for the viral dynamics of hiv 1 introduced in rong et al 2007 37 one main feature of this model is that an eclipse stage for the infected cells is included and cells in this stage may revert to the uninfected class the viral dynamics is described by four nonlinear ordinary differential equations in rong et al 2007 37 the stability of the infected equilibrium has been analyzed locally here we perform the global stability analysis using two techniques the lyapunov direct method and the geometric approach to stability based on the higher order generalization of bendixson s criterion we obtain sufficient conditions written in terms of the system parameters numerical simulations are also provided to give a more complete representation of the system dynamics

dulac criteria for autonomous systems having an invariant affine manifold abstract for a class of higher dimensional autonomous systems that have an invariant affine manifold conditions are derived to preclude the existence of periodic solutions on the invariant manifold it is established that these conditions are robust under certain types of local perturbations of the vector field as a consequence each bounded semitrajectory on the invariant manifold is shown to converge to a single equilibrium using a c 1 closing lemma for this class applications to autonomous systems that are homogeneous of degree 1 are also considered

global results for an epidemic model with vaccination that exhibits backward bifurcation vaccination of both newborns and susceptibles is included in a transmission model for a disease that confers immunity the interplay of the vaccination strategy together with the vaccine efficacy and waning is studied in particular it is shown that a backward bifurcation leading to bistability can occur under mild parameter constraints compound matrices are used to show that each orbit limits to an equilibrium in the case of bistability this global result requires a novel approach since there is no compact absorbing set

stability bifurcation and global existence of a hopf bifurcating periodic solution for a class of three neuron delayed network models in this paper a system of three delay differential equations representing a hopfield type general model for three neurons with two way bidirectional time delayed connections between the neurons and time delayed self connection from each neuron to itself is studied delay independent and delay dependent sufficient conditions for linear stability instability and the occurrence of a hopf bifurcation about the trivial equilibrium are addressed the partition of the resulting parametric space into regions of stability instability and hopf bifurcation in the absence of self connection is realized to extend the local hopf branches for large delay values a particular bidirectional delayed tri neuron model without self connection is investigated sufficient conditions for global existence of multiple non constant periodic solutions are obtained for such a model using the global hopf bifurcation theorem for functional differential equations due to j wu and the bendixson criterion for higher dimensional ordinary differential equations due to li and muldowney and following the approach developed by wei and li

computing the stability regions of hill s equation a numerical scheme is presented for computing the boundaries between the stable and unstable regions of hill s equation the approach based on formulating an integral representation of hill s equation thus the problem of finding the boundary between the stable and unstable regions in parameter space is then a spectral problem of an integral operator several properties of the operator are derived and an iteration scheme for numerical calculation of the stability boundary is presented the numerical scheme has a number of advantages over the classical methods for locating the stability boundary several examples are presented

the relaxed nonlinear phss like iteration method for absolute value equations finding the solution of the absolute value equation ave a x x b has attracted much attention in recent years in this paper we propose a relaxed nonlinear phss like iterative method which is more efficient than the picard hss iterative method for the ave and is a generalization of the nonlinear hss like iteration method for the ave by using the theory of nonsmooth analysis we prove the convergence of the relaxed nonlinear phss like iterative method for the ave numerical experiments are given to demonstrate the feasibility robustness and effectiveness of the relaxed nonlinear hss like method

hopf bifurcation analysis in a delayed nicholson blowflies equation abstract the dynamics of a nicholson s blowflies equation with a finite delay are investigated we prove that a sequence of hopf bifurcations occur at the positive equilibrium as the delay increases explicit algorithm for determining the direction of the hopf bifurcations and the stability of the bifurcating periodic solutions are derived using the theory of normal form and center manifold global existence of periodic solutions are established using a global hopf bifurcation result of wu trans amer math soc 350 1998 4799 and a bendixson criterion for higher dimensional ordinary differential equations due to li and muldowney j differential equations 106 1994 27