hopf transcritical bifurcation in toxic phytoplankton zooplankton model with delay abstract in this paper a phytoplankton zooplankton model with toxic liberation delay is considered firstly the critical values of hopf bifurcation transcritical bifurcation and hopf transcritical bifurcation are given and to give more detailed information about the periodic oscillations the direction and stability of hopf bifurcation is studied by using the normal form theory and center manifold theorem then we give the detailed bifurcation set by calculating the universal unfoldings near the hopf transcritical bifurcation point finally we show that the plankton system may exhibit quasi periodic oscillations which are verified both theoretically and numerically and explain the experimental observed fluctuation phenomenon of plankton population

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

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

existence regularity and properties of generalized apparent horizons we prove a conjecture of tom ilmanen and hubert bray on the existence of the outermost generalized apparent horizon in an initial data set and that it is outer area minimizing

bifurcations stability and monotonicity properties of a delayed neural network model abstract a delay differential equation modelling an artificial neural network with two neurons is investigated a linear stability analysis provides parameter values yielding asymptotic stability of the stationary solutions these can lose stability through either a pitchfork or a hopf bifurcation which is shown to be supercritical at appropriate parameter values an interaction takes place between the pitchfork and hopf bifurcations conditions are also given for the set of initial conditions that converge to a stable stationary solution to be open and dense in the functional phase space analytic results are illustrated with numerical simulations

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

van der pol s oscillator under delayed feedback the effect of delayed feedback on oscillatory behaviour is investigated for the classical van der pol oscillator it is shown how the presence of delay can change the amplitude of limit cycle oscillations or suppress them altogether the result is compared to the conventional proportional and derivative feedback the derivative like effect of delay is also demonstrated in a modified equation where a delayed term provides the damping

oscillations in a van der pol equation with delayed argument we consider the van der pol equation d2dt2u t eddtu t eu2 t r ddtu t r u t 0 with the delayed time r where r and e are real parameters with e 0 small and 0 r 2 and prove the existence and stability of a periodic solution with period near 2 and amplitude near 2 cosr with these properties this solution is unique up to translations in time

stability and bifurcation analysis in van der pol s oscillator with delayed feedback abstract the classical van der pol equation with delayed feedback and a modified equation where a delayed term provides the damping are considered linear stability of the equations is investigated by analyzing the associated characteristic equations it is found that there exist the stability switches when delay varies and the hopf bifurcation occurs when the delay passes through a sequence of critical values the bifurcation diagram is drawn in e k plane and the stability and direction of the hopf bifurcation are determined by applying the normal form theory and the center manifold theorem

on the riemannian penrose inequality in dimensions less than eight the positive mass theorem states that a complete asymptotically flat manifold of nonnegative scalar curvature has nonnegative mass and that equality is achieved only for the euclidean metric the riemannian penrose inequality provides a sharp lower bound for the mass when black holes are present more precisely this lower bound is given in terms of the area of an outermost minimal hypersurface and equality is achieved only for schwarzschild metrics the riemannian penrose inequality was first proved in three dimensions in 1997 by g huisken and t ilmanen for the case of a single black hole see hi in 1999 bray extended this result to the general case of multiple black holes using a different technique see br in this article we extend the technique of br to dimensions less than eight part of the argument is contained in a companion article by lee l the equality case of the theorem requires the added assumption that the manifold be spin

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

limit cycle oscillators subject to a delayed feedback the coexistence of two stable limit cycles exhibiting different periods is examined for a nonlinear oscillator subject to a delayed feedback for the case of a weakly nonlinear oscillator we discuss the validity of a previously determined phase equation for the case of a strongly nonlinear oscillator we derive a phase equation and analyze its bifurcation diagram our analysis is motivated by previous experimental studies of chemical oscillators controlled by a delayed feedback

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

the mass of an asymptotically flat manifold we show that the mass of an asymptotically flat n manifold is a geometric invariant the proof is based on harmonic coordinates and to develop a suitable existence theory results about elliptic operators with rough coefficients on weighted sobolev spaces are summarised some relations between the mass scalar curvature and harmonic maps are described and the positive mass theorem for n dimensional spin manifolds is proved

representation rings for fusion systems and dimension functions we define the representation ring of a saturated fusion system mathcal f as the grothendieck ring of the semiring of mathcal f stable representations and study the dimension functions of mathcal f stable representations using the transfer map induced by the characteristic idempotent of mathcal f we find a list of conditions for an mathcal f stable super class function to be realized as the dimension function of an mathcal f stable virtual representation we also give an application of our results to constructions of finite group actions on homotopy spheres

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

jenkins serrin type results for the jang equation let m g k be an initial data set for the einstein equations of general relativity we prove that there exist solutions of the plateau problem for marginally outer trapped surfaces motss that are stable in the sense of motss this answers a question of g galloway and n o murchadha raised in some remarks on the size of bodies and black holes classical quantum gravity 25 2008 no 10 105009 9 mr 2416045 and is an ingredient in the proof of the spacetime positive mass theorem given by l h huang d lee r schoen and the first named author we show that a canonical solution of the jang equation exists in the complement of the union of all weakly future outer trapped regions in the initial data set with respect to a given end provided that this complement contains no weakly past outer trapped regions the graph of this solution relates the area of the horizon to the global geometry of the initial data set in a non trivial way we prove the existence of a scherk type solution of the jang equation outside the union of all weakly future or past outer trapped regions in the initial data set this result is a natural exterior analogue for the jang equation of the classical jenkins serrin theory we extend and complement existence theorems for scherk type constant mean curvature graphs over polygonal domains in m g where m g is a complete riemannian surface we can dispense with the a priori assumptions that a sub solution exists and that m g has particular symmetries also our method generalizes to higher dimensions

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

proof of the positive mass theorem ii the positive mass theorem states that for a nontrivial isolated physical system the total energy which includes contributions from both matter and gravitation is positive this assertion was demonstrated in our previous paper in the important case when the space time admits a maximal slice here this assumption is removed and the general theorem is demonstrated abstracts of the results of this paper appeared in 11 and 13

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

yamabe flow and adm mass on asymptotically flat manifolds in this paper we investigate the behavior of adm mass and einstein hilbert functional under the yamabe flow through studying the yamabe flow by weighted spaces for parabolic operators we show that the asymptotically flat property is preserved under the yamabe flow we also obtain that adm mass is invariant under the yamabe flow and yamabe flow is the gradient flow of einstein hilbert functional on n dimensional n 3 asymptotically flat manifolds with order n 22 for n 3 4 or n 3 for n 4 moreover we show that adm mass and einstein hilbert functional are non increasing under the yamabe flow on n dimensional asymptotically flat manifolds if we only assume the order n 22 for n 4

the boost problem in general relativity we show that any asymptotically flat initial data for the einstein field equations have a development which includes complete spacelike surfaces boosted relative to the initial surface furthermore the asymptotic fall off is preserved along these boosted surfaces and there exists a global system of harmonic coordinates on such a development we also extend former results on global solutions of the constraint equations by virtue of this extension the constraint and evolution parts of the problem fit together exactly several theorems are given which concern the behaviour in the large of general classes of linear and quasilinear differential systems this paper contains in addition a systematic exposition of the functional spaces employed

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

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

stability and bifurcation analysis of micro electromechanical nonlinear coupling system with delay abstract in this paper we study dynamics in delayed micro electromechanical nonlinear coupling system with particular attention focused on hopf and hopf pitchfork bifurcations based on the distribution of eigenvalues we prove that a sequence of hopf and hopf pitchfork bifurcations occur at the trivial equilibrium as the delay increases and obtain the critical values of two types of bifurcations next by applying the multiple time scales method the normal forms near the hopf and hopf pitchfork bifurcations critical points are derived finally bifurcation analysis and numerical simulations are presented to demonstrate the application of the theoretical results we show the regions near above bifurcation critical points in which the micro electromechanical nonlinear coupling system exists stable fixed point or stable periodic solution detailed numerical analysis using matlab extends the local bifurcation analysis to a global picture and stable windows are observed as we change control parameters namely the stable fixed point and stable periodic solution can exist in large regions of unfolding parameters as the unfolding parameters increase away from the critical value

double hopf bifurcation in a container crane model with delayed position feedback in this paper we study dynamics in a container crane model with delayed position feedback with particular attention focused on non resonant double hopf bifurcation by using multiple time scales and center manifold reduction methods we obtain the equivalent normal forms near a double hopf critical point in this neutral delayed differential system moreover bifurcations are classified in a two dimensional parameter space near the critical point numerical simulations are presented to demonstrate the applicability of the theoretical results

the plateau problem for marginally outer trapped surfaces we solve the plateau problem for marginally outer trapped surfaces in general cauchy data sets we employ the perron method and tools from geometric measure theory to force and control a blow up of jang s equation substantial new geometric insights regarding the lower order properties of marginally outer trapped surfaces are gained in the process the techniques developed in this paper are flexible and can be adapted to other non variational existence problems

oscillation sliding in a modified van der pol duffing electronic oscillator abstract some periodic behaviour exhibited by a modified van der pol duffing electronic oscillator near a degenerate hopf pitchfork bifurcation have been studied numerical continuation of these periodic orbits leads to the appearance of interesting phenomena after varying one of the characteristics of the oscillator oscillation sliding between two periodic regimes is detected in the region where oscillation sliding is present quasiperiodic oscillations invariant torus breakdown of the torus and the corresponding resonant periodic orbits are also found

effects of time delayed position feedback on a van der pol duffing oscillator abstract the mechanism for the action of time delay in a non autonomous system is investigated in this paper the original mathematical model under consideration is a van der pol duffing oscillator with excitation a delayed system is obtained by adding both linear and nonlinear time delayed position feedbacks to the original system functional analysis is used to change the delayed system into a functional differential equation fde the time delay is taken as a variable parameter to investigate its effect on the dynamics of the system such as the stability and bifurcation of an equilibrium point phase locked periodic and phase shifting solution period doubling quasi periodic motion and chaos a periodic solution expressed in the closed form with the aid of the center manifold and averaging theorem is found to be in good agreement with that obtained by numerical simulation two routes to chaos are found namely period doubling bifurcation and torus breaking the results obtained in this paper show that time delay may be used as a simple but efficient switch to control motions of a system either from order motion to chaos or from chaotic motion to order for different applications

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

blowup of jang s equation at outermost marginally trapped surfaces the aim of this paper is to accurately describe the blowup of jang s equation first we discuss how to construct solutions that blow up at an outermost mots second we exclude the possibility that there are extra blowup surfaces in data sets with non positive mean curvature then we investigate the rate of convergence of the blowup to a cylinder near a strictly stable mots and show exponential convergence with an identifiable rate near a strictly stable mots

the jang equation reduction of the spacetime positive energy theorem in dimensions less than eight we extend the jang equation proof of the positive energy theorem due to schoen and yau commun math phys 79 2 231 260 1981 from dimension n 3 to dimensions 3 n 8 this requires us to address several technical difficulties that are not present when n 3 the regularity and decay assumptions for the initial data sets to which our argument applies are weaker than those in schoen and yau commun math phys 79 2 231 260 1981

dynamics in a ratio dependent predator prey model with predator harvesting abstract the objective of this paper is to study systematically the dynamical properties of a ratio dependent predator prey model with nonzero constant rate predator harvesting it is shown that the model has at most two equilibria in the first quadrant and can exhibit numerous kinds of bifurcation phenomena including the bifurcation of cusp type of codimension 2 i e bogdanov takens bifurcation the subcritical and supercritical hopf bifurcations these results reveal far richer dynamics compared to the model with no harvesting and different dynamics compared to the model with nonzero constant rate prey harvesting in d xiao l jennings bifurcations of a ratio dependent predator prey system with constant rate harvesting siam appl math 65 2005 737 753 biologically it is shown that nonzero constant rate predator harvesting can prevent mutual extinction as a possible outcome of the predator prey interaction and remove the singularity of the origin which was regarded as pathological behavior for a ratio dependent predator prey model in p yodzis predator prey theory and management of multispecies fisheries ecological applications 4 2004 51 58

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