the complex lorenz equations we have undertaken a study of the complex lorenz equations x x y y r z x ay z bz 12 x y xy where x and y are complex and z is real the complex parameters r and a are defined by r r1 ir2 a 1 ie and and b are real behaviour remarkably different from the real lorenz model occurs only the origin is a fixed point except for the special case e r2 0 we have been able to determine analytically two critical values of r1 namely r1c and r1c the origin is a stable fixed point for 0 r1c a hopf bifurcation to a limit cycle occurs we have an exact analytic solution for this limit cycle which is always stable if 1 then this limit is only stable in the region r1c rlc a transition to a finite amplitude oscillation about the limit cycle occurs the nature of this bifurcation is studied in detail by using a multiple time scale analysis to derive the stuart landau amplitude equation from the original equations in a frame rotating with the limit cycle frequency this latter bifurcation is either a sub or super critical hopf like bifurcation to a doubly periodic motion the direction of bifurcation depending on the parameter values the nature of the bifurcation is complicated by the existence of a zero eigenvalue

numerical approximation for functionals of reflecting diffusion processes the aim of this paper is to approximate the expectation of a large class of functionals of the solution x xi of a stochastic differential equation with normal reflection in a piecewise smooth domain of rd this also yields a monte carlo method for solving partial differential problems of parabolic type with mixed boundary conditions the approximation is based on a modified euler scheme for the stochastic differential equation the scheme can be driven by a sequence of bounded independently and identically distributed i i d random variables or when the domain is convex by a sequence of gaussian i i d random variables the order of weak convergence for both cases is given in the former case the order of convergence is 1 2 and it is shown to be exact by an example in the last section numerical tests are presented the behavior of the error as a function of the final time t for fixed values of the discretization step and as a function of the discretization step for fixed values of the final

a splitting step algorithm for reflected stochastic differential equations in r 1 a new algorithm which is based on the splitting step idea and the penalization method for reflected stochastic differential equation rsde in the upper half space r 1 is presented in this paper after some important estimates about rsdes and penalization odes are obtained the local pathwise error of the suggested algorithm are considered and an estimate of local error in the 2pth moment is proved under the assumptions of lipschitz continuity and linear growth some numerical experiments based on matlab programs are given to support the suggested algorithm

stochastic differential equations with reflecting boundary conditions on resout des equations differentielles stochastiques a conditions aux limites reflechissantes par une approche directe basee sur le probleme de skorokhod

on differences and similarities in the analysis of lorenz chen and lu systems currently it is being actively discussed the question of the equivalence of various lorenz like systems and the possibility of universal consideration of their behavior algaba et al 2013a b 2014b c chen 2013 chen and yang 2013 leonov 2013a in view of the possibility of reduction of such systems to the same form with the help of various transformations in the present paper the differences and similarities in the analysis of the lorenz the chen and the lu systems are discussed it is shown that the chen and the lu systems stimulate the development of new methods for the analysis of chaotic systems open problems are discussed

estimating the ultimate bound and positively invariant set for the lorenz system and a unified chaotic system to estimate the ultimate bound and positively invariant set for a dynamic system is an important but quite challenging task in general in this paper we attempt to investigate the ultimate bound and positively invariant set for two specific systems the lorenz system and a unified chaotic system we derive an ellipsoidal estimate of the ultimate bound and positively invariant set for the lorenz system for all the positive values of its parameters a b and c and obtain the minimum value of volume for the ellipsoid comparing with the best results in the current literature d li j lu x wu g chen estimating the bounds for the lorenz family of chaotic systems chaos solitons fractals 23 2005 529 534 x liao on the global basin of attraction and positively invariant set for the lorenz chaotic system and its application in chaos control and synchronization sci china ser e 34 2004 1404 1419 our new results fill up the gap of the estimate for the cases of 0 a 1 and 0 b 2 x liao on the global basin of attraction and positively invariant set for the lorenz chaotic system and its application in chaos control and synchronization sci china ser e 34 2004 1404 1419 furthermore the estimation derived here contains the results given in d li j lu x wu g chen estimating the bounds for the lorenz family of chaotic systems chaos solitons fractals 23 2005 529 534 and x liao on the global basin of attraction and positively invariant set for the lorenz chaotic system and its application in chaos control and synchronization sci china ser e 34 2004 1404 1419 as special cases along the same line we also provide estimates of cylindrical and ellipsoidal bounds for a unified chaotic system for its parameter range 0 129 and obtain the minimum value of volume for the ellipsoid the estimate is more accurate than and also extends the result of d li j lu x wu g chen estimating the bounds for the lorenz family of chaotic systems chaos solitons fractals 23 2005 529 534 and x liao on the global basin of attraction and positively invariant set for the lorenz chaotic system and its application in chaos control and synchronization sci china ser e 34 2004 1404 1419

infinite dimensional linear programming approach to singularstochastic control we consider a multidimensional singular stochastic control problem with state dependent diffusion matrix and drift vector and control cost depending on the position and direction of displacement of the controlled process the objective is to minimize the total expected discounted cost we write an equivalent infinite dimensional linear programming problem on a subspace of the space conjugate to cal c bbb r n times cal c bbb r n times b where b is the unit sphere in bbb r n we write a dual linear program and prove absence of duality gap the dual program characterizes the optimal cost function as a maximal solution to the variational inequality with gradient constraints

a note on hopf bifurcation analysis and ultimate bound estimation of a new 4 d quadratic autonomous hyper chaotic system in appl math comput 291 2016 323 339 by amin zarei and saeed tavakoli abstract in the recent paper entitled hopf bifurcation analysis and ultimate bound estimation of a new 4 d quadratic autonomous hyper chaotic system in appl math comput 291 2016 323 339 by amin zarei and saeed tavakoli they proposed the following new four dimensional 4 d quadratic autonomous hyper chaotic system x 1 a x 2 x 1 x 2 b x 1 x 2 e x 4 x 1 x 3 x 3 c x 3 x 1 x 2 x 1 2 x 4 d x 2 which generates double wing chaotic and hyper chaotic attractors with only one equilibrium point combining theoretical analysis and numerical simulations they investigated some dynamical properties of that system like lyapunov exponent spectrum bifurcation diagram phase portrait hopf bifurcation etc in particular they formulated a conclusion that the system has the ellipsoidal ultimate bound by employing the method presented in the paper entitled ultimate bound estimation of a class of high dimensional quadratic autonomous dynamical systems int j bifurc chaos 21 09 2011 2679 2694 by p wang et al however by means of detailed theoretical analysis we show that both the conclusion itself and the derivation of its proof in appl math comput 291 2016 323 339 are erroneous furthermore we point out that the method adopted for studying the ultimate bound of that system is not applicable at all therefore the ultimate bound estimation of that system needs further studying in future work

boundedness solutions of the complex lorenz chaotic system this paper is concerned with the boundedness of solutions of the complex lorenz system we have obtained the global exponential attractive set j l m and the ultimate bound w l m for this system furthermore we confirm that the rate of the trajectories of the system going from the exterior of the set j l m to the interior of the set j l m is an exponential rate the rate of the trajectories is also obtained numerical simulations are presented to show the effectiveness of the proposed scheme

analysis of a new chaotic system abstract this paper reports a new three dimensional continuous quadratic autonomous chaotic system modified from the lorenz system in which each equation contains a single quadratic cross product term which is different from the lorenz rossler chen lu systems basic properties of the new system are analyzed by means of lyapunov exponent spectrum and bifurcation diagrams analysis results show that this system has complex dynamics with some interesting characteristics

on the boundedness of solutions of the chen system by constructing a suitable lyapunov function we show that for the system parameters in some specified regions the solutions of the chen system are globally bounded

brownian motion in a wedge with oblique reflection this work is concerned with the existence and uniqueness of a strong markov process that has continuous sample paths and the following additional properties i the state space is an infinite two dimensional wedge and the process behaves in the interior of the wedge like an ordinary brownian motion ii the process reflects instantaneously at the boundary of the wedge the angle of reflection being constant along each side iii the amount of time that the process spends at the comer of the wedge is zero i e the set of times for which the process is at the comer has lebesgue measure zero hereafter let be the angle of the wedge 0 2 let 1 and 2 be the angles of reflection on the two sides of the wedge measured from the inward normals the positive angles being toward the corner 1 2 and set 1 2 the question of existence and uniqueness is recast as a submartingale problem in the style used by stroock and varadhan diffusion processes with boundary conditions comm pure appl math 24 1971 pp 147 225 for diffusions on smooth domains with smooth boundary conditions it is shown that no solution exists if 2 in this case there is a unique continuous strong markov process satisfying i ii above it reaches the corner of the wedge almost surely and it remains there if 2 however then there is a unique continuous strong markov process statisfying i iii it is shown that starting away from the corner this process does not reach the corner of the wedge if 0 and does reach the corner if 0 2 the general theory of multi dimensional diffusions does not apply to the above problem because in general the boundary of the state space is not smooth and there is a discontinuity in the direction of reflection at the corner for some values of the process arises from diffusion approximations to storage systems and queueing networks i the state space is an infinite two dimensional wedge and the process behaves in the interior of the wedge like an ordinary brownian motion ii the process reflects instantaneously at the boundary of the wedge and the angle of reflection being constant along each side iii the amount of time that the process spends at the corner of the wedge is zero i e the set of times for which the process is at the corner has lebesgue measure zero

pinning synchronization of delayed complex dynamical networks with nonlinear coupling abstract in this paper we find that complex networks with the watts strogatz or scale free ba random topological architecture can be synchronized more easily by pin controlling fewer nodes than regular systems theoretical analysis is included by means of lyapunov functions and linear matrix inequalities lmi to make all nodes reach complete synchronization numerical examples are also provided to illustrate the importance of our theoretical analysis which implies that there exists a gap between the theoretical prediction and numerical results about the minimum number of pinning controlled nodes

estimating the ultimate bound and positively invariant set for a class of hopfield networks in this paper we investigate the ultimate bound and positively invariant set for a class of hopfield neural networks hnns based on the lyapunov stability criterion and lagrange multiplier method it is shown that a hyperelliptic estimate of the ultimate bound and positively invariant set for the hnns can be calculated by solving a linear matrix inequality lmi furthermore the global stability of the unique equilibrium and the instability region of the hnns are analyzed respectively finally the most accurate estimate of the ultimate bound and positively invariant set can be derived by solving the corresponding optimization problems involving the lmi constraints some numerical examples are given to illustrate the effectiveness of the proposed results

rate of convergeance of a particle method for the solution of a 1d viscous scalar conservation law in a bounded interval in this paper we give a probabilistic interpretation of a viscous scalar conservation law in a bounded interval thanks to a nonlinear martingale problem the underlying nonlinear stochastic process is reflected at the boundary to take into account the dirichlet conditions after proving uniqueness for the martingale problem we show existence thanks to a propagation of chaos result indeed we exhibit a system of n interacting particles the empirical measure of which converges to the unique solution of the martingale problem as n as a consequence the solution of the viscous conservation law can be approximated thanks to a numerical algorithm based on the simulation of the particle system when this system is discretized in time thanks to the euler lepingle scheme d lepingle math comput simulation 38 1995 119 126 we show that the rate of convergence of the error is in o t 1 n where at denotes the time step finally we give numerical results which confirm this theoretical rate

dynkin games via dirichlet forms and singular control of one dimensional diffusions we consider a zero sum game of optimal stopping in which each of the opponents has the right to stop a one dimensional diffusion process there are two types of costs the first is accumulated continuously at the rate h xt where xt is the current position of the process the second is a cost associated with the stopping of the process it is given by the function f1 x for the first player and the function f2 x for the second player where x is the position of the process when the stopping option is exercised we study the solution of the free boundary problem associated with this game via dirichlet forms on the appropriate functional space integrating the value function of the game we get a solution to another free boundary problem which yields the optimal return function for a singular stochastic control problem

an explicit formula for the skorokhod map on 0 a the skorokhod map is a convenient tool for constructing solutions to stochastic differential equations with reflecting boundary conditions in this work an explicit formula for the skorokhod map 0 a on 0 a for any a 0 is derived specifically it is shown that on the space d 0 of right continuous functions with left limits taking values in r 0 a a o 0 where a d 0 d 0 is defined by a t t sups 0 t s a infu s t u and 0 d 0 d 0 is the skorokhod map on 0 which is given explicitly by 0 t t sup s 0 t s in addition properties of a are developed and comparison properties of 0 a are established

traps for reflected brownian motion consider an open set d r d d 2 and a closed ball b d let e x tb denote the expectation of the hitting time of b for reflected brownian motion in d starting from x d we say that d is a trap domain if supx e x tb a domain d is not a trap domain if and only if the reflecting brownian motion in d is uniformly ergodic we fully characterize the simply connected planar trap domains using a geometric condition and give a number of less complete results for d 2

generation of clusters in complex dynamical networks via pinning control many real world networks show community structure i e groups or clusters of nodes that have a high density of links within them but with a lower density of links between them in this paper by applying feedback injections to a fraction of network nodes various clusters are synchronized independently according to the community structure generated by the group partition of the network cluster synchronization this control is achieved by pinning i e applying linear feedback control to a subset of the network nodes those pinned nodes are selected not randomly but according to the topological structure of communities of a given network specifically for a given group partition of a network those nodes with direct connections between groups must be pinned in order to achieve cluster synchronization both the local stability and global stability of cluster synchronization are investigated taking the tree shaped network and the most modular network as two particular examples we illustrate in detail how the pinning strategy influences the generation of clusters the simulations verify the efficiency of the pinning schemes used in this paper

the reflection map with discontinuities we study the multidimensional reflection map on the spacesd 0 t r k andd 0 8 r k of right continuous r k valued functions on 0 t or 0 8 with left limits endowed with variants of the skorohod 1956 m1 topology the reflection map was used with the continuous mapping theorem by harrison and reiman 1981 and reiman 1984 to establish heavy traffic limit theorems with reflected brownian motion limit processes for vector valued queue length waiting time and workload stochastic processes in single class open queueing networks since brownian motion and reflected brownian motion have continuous sample paths the topology of uniform convergence over bounded intervals could be used for those results variants of them1 topologies are needed to obtain alternative discontinuous limits approached gradually by the converging processes as occurs in stochastic fluid networks with bursty exogenous input processes e g with on off sources having heavy tailed on periods or off periods having infinite variance we show that the reflection map is continuous at limits without simultaneous jumps of opposite sign in the coordinate functions provided that the productm1 topology is used as a consequence the reflection map is continuous with the productm1 topology at all functions that have discontinuities in only one coordinate at a time that continuity property also holds for more general reflection maps and is sufficient to support limit theorems for stochastic processes in most applications we apply the continuity of the reflection map to obtain limits for buffer content stochastic processes in stochastic fluid networks

a willmore functional for compact surfaces of complex projective plane where h denotes the mean curvature vector of the immersion k the sectional curvature of m restricted to and da the canonical measure of the induced metric this functional has been extensively studied when m is the euclidean space r or the sphere s or the hyperbolic space rh because w is invariant under conformal transformations of ambient space the type of problems which have been studied for this ambient space r are of a different nature on the one hand are the papers written with the object of obtaining lower bounds for the functional w given that w has 4 as an absolute minimum and is only reached by the round spheres of r willmore proposed to study this functional on tori and conjectured that 2 is the minimum value for these surfaces and it is attained only by the clifford torus this problem which still has not been resolved can be considered as the starting point of a series of important papers in which minimization

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

cluster synchronization of complex networks via event triggered strategy under stochastic sampling abstract this paper is concerned with the issue of mean square cluster synchronization of non identical nodes connected by a directed network suppose that the nodes possess nonlinear dynamics and split into several clusters then an event triggered control scheme is proposed for synchronization based on the information from stochastic sampling meanwhile an equilibrium is considered to be the synchronization state or the virtual leader for each cluster which can apply pinning control to the following nodes assume that a spanning tree exists in the subgraph consisting of the nodes belonging to the same cluster and the corresponding virtual leader and the instants for updating controllers are determined by the given event triggered strategy then some sufficient conditions for cluster synchronization are presented according to the lyapunov stability theory and linear matrix inequality technique finally a specific numerical example is shown to demonstrate the effectiveness of the theoretical results

a new conformal invariant and its applications to the willmore conjecture and the first eigenvalue of compact surfaces let m be a compact riemannian manifold with a fixed conformal structure then we introduce the concept of conformal volume of m in the following manner for each branched conformal immersion q9 of m into the unit sphere s n we consider the set of all branched conformal immersions obtained by composi t ion of qo with the conformal automorphisms of s we let vc n qg be the max imum volume of these branched immersions the conformal volume of m is defined to be the infimum of v n q0 where qo ranges over all branched conformal immersions of m into the unit sphere s in this paper we study the case when m is a compact surface and we call the conformal volume of m to be the conformal area of m we demonst ra te that this conformal invariant is non trivial in fact we prove that if there exists a minimal immersion of m into s where coordinate functions are first eigenfunctions then the conformal area of m is given by the area of m with respect to the induced metric this enables us to compute the conformal area for several surfaces for example the conformal area of r p z is 6n and the conformal area of the square torus is 27c 2 we believe that the computa t ion of the conformal area for general surfaces will be very impor tant in studying the geometry of compact surfaces we demonstrate this claim by applying the concept of conformal area to two different branches of surface theory the first application is to study the total curvature of a compact surface in r this problem has a long history fenchel and fary 9 proved that for a closed curve o in r slk 27c where k is its curvature then milnor 12

sweeping by a continuous prox regular set abstract the evolution problem known as sweeping process is considered for a class of nonconvex sets called prox regular or convex assuming essentially that such sets contain in the interior a suitable subset and move continuously w r t the hausdorff distance we prove local and global existence as well as uniqueness of solutions which are continuous functions with bounded variation some examples are presented

hopf bifurcation analysis of a bam neural network with multiple time delays and diffusion in this paper a bam neural network with multiple time delays and diffusion under homogeneous neumann boundary conditions is investigated by analyzing the corresponding characteristic equations the local stability of the trivial uniform steady state and the existence of hopf bifurcation under two different cases are established respectively by using the normal form theory and the center manifold reduction of partial functional differential equations pfdes explicit formulae are obtained to determine the direction of bifurcations and the stability of bifurcating periodic solutions numerical simulations are carried out to illustrate the main results

probabilistic approach for nonlinear partial differential equations and stochastic partial differential equations with neumann boundary conditions abstract in this study we consider nonlinear partial differential equations pdes and stochastic pdes spdes with neumann boundary conditions on domains that satisfy the lions sznitmann saisho conditions using the convergence result based on the penalization approximation for stochastic differential equations with normal reflections on nonsmooth and nonconvex domains we establish the existence and comparison principle for the viscosity solutions of nonlinear pdes with the neumann conditions associated with the optimal control problem we also obtain for nonlinear spdes and backward spdes with neumann condition the representations of the stochastic viscosity solutions

construction and decomposition of reflecting diffusions on lipschitz domains with holder cusps we consider a d dimensional euclidean domain d whose boundary is lipschitz continuous but admits locally finite number of outward or inward holder cusp points using a method of stampacchia and moser for pde we first construct a conservative diffusion process on the euclidean closure of d possessing a strong feller resolvent and associated with a second order uniformly elliptic differential operator of divergence form with measurable coefficients a ij the sample path of the constructed diffusion can be uniquely decomposed as a sum of a martingale additive functional and an additive functional locally of zero energy the second additive functional will be proved to be of bounded variation with a skorohod type expression whenever a ij is weakly differentiable and the holder exponent at each outward cusp boundary point is greater than 1 2 regardless the dimension d

hopf bifurcation analysis and ultimate bound estimation of a new 4 d quadratic autonomous hyper chaotic system based on lorenz system a new four dimensional quadratic autonomous hyper chaotic attractor is presented in this paper it can generate double wing chaotic and hyper chaotic attractors with only one equilibrium point several properties of the new system are investigated using lyapunov exponents spectrum bifurcation diagram and phase portraits using the center manifold and normal form theories the local dynamics the stability and hopf bifurcation at the equilibrium point are analyzed to obtain the ellipsoidal ultimate bound the ultimate bound of the proposed system is theoretically estimated using lagrange multiplier method lagrangian function and local maximizer point by properly choosing p and q matrices an estimation of the ultimate bound region as a function of the lagrange coefficient is obtained using local maximizer point and reduced hessian matrix to demonstrate the evolution of the bifurcation and to show the ultimate bound region numerical simulations are provided

fluctuations for a φ interface model with repulsion from a wall we consider a interface model on a one dimensional lattice with repulsion from a hard wall we suppose that the repulsion is of the form c 1 where c 0 and denotes the height of the interface from the wall we prove convergence of the equilibrium fluctuations around the hydrodynamic limit to the solution of a spde with singular drift if c 0 the system becomes the funaki olla interface model with reflection at the wall whose equilibrium fluctuations converge to the solution of a spde with reflection we give a new proof of this result using the characterization of such solution as the diffusion generated by an infinite dimensional dirichlet form obtained in a previous paper our method is based on a study of integration by parts formulae w r t the equilibrium measure of the interface model and allows to avoid the proof of the so called boltzmann gibbs principle we also obtain convergence of finite dimensional distributions of non equilibrium fluctuations around the stationary hydrodynamic limit 0

the willmore functional on lagrangian tori its relation to area and existence of smooth minimizers in this paper we prove an existence and regularity theorem for la grangian tori minimizing the willmore functional in euclidean four space r4 with the standard metric and symplectic structure technical difficulties arise because the euler lagrange equation for this problem is a sixth order nonlinear partial differential equation this research was motivated by a study of the seemingly unrelated plateau problem for lagrangian tori and in this paper we illustrate this connection department of mathematics stanford university stanford california 94305 2125 courant institute of mathematical sciences 251 mercer street new york new york 10012 e mail address minicozz0ccims nyu edu this content downloaded from 157 55 39 27 on wed 07 sep 2016 06 38 19 utc all use subject to http about jstor org terms

a system of infinitely many mutually reflecting brownian balls in ℝd an infinite system of skorohod type equations is studied the unique solution of the system is obtained from a finite case by passing to the limit it is a diffusion process describing a system of infinitely many brownian hard balls and has a gibbs state associated with the hard core pair potential as a reversible measure

a stochastic differential game in the orthrant we study a zero sum stochastic differential game in the nonnegative orthrant the state of the system is governed by controlled reflecting diffusions in the nonnegative orthrant we consider discounted and average payoff evaluation criteria we prove the existence of values and optimal strategies for both payoff criteria

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

reflecting brownian motion in a cusp let c be the cusp x y x 0 x y x where 1 set c 1 x y x 0 y x and c 2 x y x 0 y x we study the existence and uniqueness in law of reflecting brownian motion in c the angle of reflection at c j 0 relative to the inward unit normal is a constant j 2 2 and is positive iff the direction of reflection has a negative first component in all sufficiently small neighborhoods of 0 when 1 2 0 existence and uniqueness in law hold when 1 2 0 existence fails we also obtain results for a large class of asymmetric cusps we make essential use of results of warschawski on the differentiability at the boundary of conformal maps

reflected brownian motion with skew symmetric data in a polyhedral domain this paper is concerned with the characterization and invariant measures of certain reflected brownian motions rbm s in polyhedral domains the kind of rbm studied here behaves like d dimensional brownian motion with constant drift in the interior of a simple polyhedron and is instantaneously reflected at the boundary in directions that depend on the face that is hit under the assumption that the directions of reflection satisfy a certain skew symmetry condition first introduced in harrison williams 9 it is shown that such an rbm can be characterized in terms of a family of submartingales and that it reaches non smooth parts of the boundary with probability zero in 9 a purely analytic problem associated with such an rbm was solved here the exponential form solution obtained in 9 is shown to be the density of an invariant measure for the rbm furthermore if the density is integrable over the polyhedral state space then it yields the unique stationary distribution for the rbm in the proofs of these results a key role is played by a dual process for the rbm and by results in 9 for reflected brownian motions on smooth approximating domains

on the robin problem in fractal domains research partially supported by national science foundation nsf grants dms 0244737 and dms 0303310

heavy traffic limits for the g h2 n m queue we establish heavy traffic stochastic process limits for queue length waiting time and overflow stochastic processes in a class ofg gi n m queueing models withn servers andm extra waiting spaces we let the arrival process be general only requiring that it satisfy a functional central limit theorem to capture the impact of the service time distribution beyond its mean within a markovian framework we consider a special class of service time distributions denoted byh2 which are mixtures of an exponential distribution with probabilityp and a unit point mass at 0 with probability 1 p these service time distributions exhibit relatively high variability having squared coefficients of variation greater than or equal to one as in halfin and whitt 1981 heavy traffic limits for queues with many exponential servers oper res 29 567 588 puhalskii and reiman 2000 the multiclassgi ph n queue in the halfin whitt regime adv appl probab 32 564 595 and garnett mandelbaum and reiman 2002 designing a call center with impatient customers manufacturing service oper management 4 208 227 we consider a sequence of queueing models indexed by the number of servers n and letn tend to infinity along with the traffic intensities nso that v n 1 n for 8 8 to treat finite waiting rooms we letm n v n for 0 8 with the specialh2 service time distribution the limit processes are one dimensional markov processes behaving like diffusion processes with different drift and diffusion functions in two different regions above and below zero we also establish a limit for theg m n m m model having exponential customer abandonments

stability switches and hopf bifurcations of an isolated population model with delay dependent parameters the dynamical behaviors of an isolated population model involving delay dependent parameters are investigated it is shown that the positive equilibrium switches from being stable to unstable and then back to stable as the delay increases and the hopf bifurcation occurs finite times between the two critical values of stability changes which can be analytically determined moreover the bifurcating periodic solutions are expressed analytically in an approximate form by the perturbation approach and floquet technique the direction and stability of the bifurcating periodic solutions are also determined finally the validity of the results is shown by the consistency with the numerical simulations

lifschitz tail on hyperbolic space neumann conditions on considere un espace hyperbolique h de dimension n et l operateur laplacien hyperbolique h etant donne une distribution aleatoire de points a l aide d un processus ponctuel de poisson tel que chaque point se situe dans une boule hyperbolique de rayons on etudie la question suivante si est un petit nombre reel combien y a t il de valeurs propres inferieures a pour l operateur 1 2 h dans une boule b de centre o avec des conditions de neumann sur la frontiere et avec des conditions aux limites de dirichlet sur les boules aleatoires formant des intersections avec b si le rayon de b est suffisamment grand

a hyperchaotic system from a chaotic system with one saddle and two stable node foci this paper presents a 4d new hyperchaotic system which is constructed by a linear controller to a 3d new chaotic system with one saddle and two stable node foci some complex dynamical behaviors such as ultimate boundedness chaos and hyperchaos of the simple 4d autonomous system are investigated and analyzed the corresponding bounded hyperchaotic and chaotic attractor is first numerically verified through investigating phase trajectories lyapunove exponents bifurcation path analysis of power spectrum and poincare projections finally two complete mathematical characterizations for 4d hopf bifurcation are rigorous derived and studied

continuous dependence results for non linear neumann type boundary value problems abstract we obtain estimates on the continuous dependence on the coefficient for second order non linear degenerate neumann type boundary value problems our results extend previous work of cockburn et al jakobsen and karlsen and gripenberg to problems with more general boundary conditions and domains a new feature here is that we account for the dependence on the boundary conditions as one application of our continuous dependence results we derive for the first time the rate of convergence for the vanishing viscosity method for such problems we also derive new explicit continuous dependence on the coefficients results for problems involving bellman isaacs equations and certain quasilinear equation

on the oblique derivative problem for diffusion processes and diffusion equations with holder continuous coefficients on a c2 domain in a euclidean space we consider the oblique derivative problem for a diffusion equation and assume the coefficients of the diffusion and boundary operators are holder continuous we then prove the uniqueness of diffusion processes and fundamental solutions corresponding to the problem for the purpose obtaining a stochastic representation of some so lutions to the problem plays a key role in our situation a difficulty arises from the absence of a fundamental solution with c2 smoothness up to the boundary it is overcome by showing some stability of a fundamental solution and a dif fusion process respectively under approximation of the domain in particular the stability of the fundamental solution is verified through construction it is done by applying the parametrix method twice to a parametrix with explicit expression

the submartingale problem for a class of degenerate elliptic operators we consider the degenerate elliptic operator acting on c 2 b functions on 0 d mathcal l f x sum i 1 d a i x x i alpha i frac partial 2 f partial x i 2 x sum i 1 d b i x frac partial f partial x i x where the ai are continuous functions that are bounded above and below by positive constants the bi are bounded and measurable and the alpha i in 0 1 we impose neumann boundary conditions on the boundary of 0 d there will not be uniqueness for the submartingale problem corresponding to mathcal l if we consider however only those solutions to the submartingale problem for which the process spends 0 time on the boundary then existence and uniqueness for the submartingale problem for mathcal l holds within this class our result is equivalent to establishing weak uniqueness for the system of stochastic differential equations rm d x t i sqrt 2a i x t x t i alpha i 2 rm d w i t b i x t rm d t rm d l t x i quad x i t geq 0 where w t i are independent brownian motions and l x i t is a local time at 0 for xi

on the semimartingale representation of reflecting brownian motion in a cusp letc be the symmetric cusp x y 2 x y x x 0 where 1 in this paper we decide whether or not reflecting brownian motion inc has a semimartingale representation here the reflecting brownian motion has directions of reflection that make constant angles with the unit inward normals to the boundary our results carry through for a wide class of asymmetric cusps too

stability and hopf bifurcation of a lorenz like system hopf bifurcation is one of the important dynamical behaviors it could often cause some phenomena such as quasiperiodicity and intermittency consequently chaos will happen due to such dynamical behaviors since chaos appears in the lorenz like system to understand the dynamics of such system hopf bifurcation will be explored in this paper first the stability of equilibrium points is presented then hopf bifurcation of the lorenz like system is investigated by applying the normal form theory the conditions guaranteeing the hopf bifurcation are derived further the direction of hopf bifurcation and the stability of bifurcating periodic solutions are also presented finally numerical simulations are given to verify the theoretical analysis it is found that hopf bifurcation could happen when conditions are satisfied the stable bifurcating periodic orbit is displayed chaos will also happen when parameter further increases

reflected brownian motion in a wedge semimartingale property in this paper the object of study is reflected brownian motion in a two dimensional wedge with constant direction of reflection on each side of the wedge the basic question considered here is when is this process a semimartingale it is first shown that a related process defined by specifying the corner of the wedge to be an absorbing state rather than an instantaneous one is a semimartingale conditions for the existence and uniqueness of the process for which the corner is an instantaneous state were given by vardhan and williams brownian motion in a wedge with oblique reflection comm pure appl math to appear under these conditions it is shown that starting away from the corner the process is a semimartingale if and only if there is a convex combination of the directions of reflection that points into the wedge this equivalence is also shown to hold starting from the corner except in one unresolved case for which the wedge angle exceeds and the directions of reflection are exactly opposed

optimal reflection of diffusions and barrier options pricing under constraints we introduce a new class of control problems in which the gain depends on the solution of a stochastic differential equation sde reflected at the boundary of a bounded domain along directions which are controlled by a bounded variation process we provide a pde characterization of the associated value function this study is motivated by applications in mathematical finance where such equations are related to the pricing of barrier options under portfolio constraints

a multidimensional process involving local time on presente une approche probabiliste a un probleme pose par mckean 1967 pour la construction d un systeme de n particules sur r avec un generateur formel

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

generalized bsdes and nonlinear neumann boundary value problems we study a new class of backward stochastic differential equations which involves the integral with respect to a continuous increasing process this allows us to give a probabilistic formula for solutions of semilinear partial differential equations with neumann boundary condition where the boundary condition itself is nonlinear we consider both parabolic and elliptic equations