h asymptotic stability by fixed point in neutral nonlinear differential equations with delay abstract we use fixed point theory to obtain h asymptotic stability results about the zero solution of nonlinear differential equations with functional delay neither boundedness of the deviations nor a fixed sign on the coefficients functions are asked an explicit estimation of the solutions is obtained and a necessary and sufficient condition is determined classical results are improved and generalized

fixed points and stability of neutral stochastic delay differential equations in this paper we consider a linear scalar neutral stochastic differential equation with variable delays and give conditions to ensure that the zero solution is asymptotically mean square stable by means of fixed point theory these conditions do not require the boundedness of delays nor do they ask for a fixed sign on the coefficient functions an asymptotic mean square stability theorem with a necessary and sufficient condition is proved some well known results are improved and generalized

the complex lorentzian leech lattice and the bimonster ii let d be the incidence graph of the projective plane over f3 the artin group of the graph d maps onto the bimonster and a complex hyperbolic reflection group acting on 13 dimensional complex hyperbolic space y the generators of the artin group are mapped to elements of order 2 resp 3 in the bimonster resp let y y be the complement of the union of the fixed points of reflections in daniel allcock has conjectured that the orbifold fundamental group of y surjects onto bimonster in this article we study the reflection group we show that th e artin group of d maps to the orbifold fundamental group of y thus answering a question in allcock s article a monstrous proposal and taking one step towards the proof of allcock s conjecture the finite group aut d acts on y we make a detailed study of the complex hyperbolic line fixed by the subgroup l3 3 aut d we define meromorphic automorphic forms on y invariant under with poles along mirrors we show that the restriction of these forms to the complex hyperbolic line fixed by l3 3 gives meromorphic modular forms of level 13

reflection group of the quaternionic lorentzian leech lattice we study a second example of the phenomenon studied in the article the complex lorentzian leech lattice and the bimonster we find 14 roots in the automorphism group of the quaternionic lorentzian leech lattice l that form the coxeter diagram given by the incidence graph of projective plane over f2 we prove that the reflections in these roots generate the automorphism group of l the investigation is guided by an analogy with the theory of weyl groups there is a unique point in the quaternionic hyperbolic space fixed by the diagram automorphisms that we call the weyl vector the unit multiples of the 14 roots forming the diagram are the analogs of the simple roots the 14 mirrors perpendicular to the simple roots are the mirrors that are closest to the weyl vector

exponential methods for singularly perturbed ordinary differential difference equations abstract a variety of exponential methods based on piecewise analytical solutions of advection reaction diffusion operators is proposed for the numerical solution of linear ordinary differential difference equations with small delay these methods are shown to be exact in exact arithmetic for linear ordinary differential equations with constant coefficients and right hand side are non standard and include three point non local approximations for the coefficients right hand side and dependent variable also a variety of exponential methods based on the piecewise analytical solution of advection diffusion operators with non local approximations is presented it is shown that exponential methods based on the advection diffusion reaction operator provide more accurate results than those based on the analytical solution of the advection diffusion operator it is also shown that at least for the problems considered here non local approximations for either the coefficients or the right hand side of the differential equations do not result in an increase of accuracy over those based on local approximations

performance of the taylor series method for odes daes this paper revisits the use of the taylor series method for the numerical integration of odes and daes the numerical method is implemented using an efficient variable step variable order scheme several numerical tests comparing with well established numerical codes are presented

the stability of the equilibrium of the damped oscillator with damping changing sign abstract in this paper we prove a sufficient and necessary condition for the stability of the equilibrium x x 0 of the damped oscillator with damping changing sign x h t x a t x 2 n 1 e t x 0 n 1 where a t h t are continuous and 1 periodic with h 0 1 h t d t 0 e t x is continuous 1 periodic in t and dominated by the power x 2 n 2 in a neighborhood of x 0

study the stability of solutions of functional differential equations via fixed points in this paper we study the stability properties of solutions of a class of functional differential equations with variable delay by using the fixed point theory under an exponentially weighted metric we obtain some interesting sufficient conditions ensuring that the zero solution of the equations is stable and asymptotically stable

non ergodicity of the nose hoover thermostatted harmonic oscillator the nose hoover thermostat is a deterministic dynamical system designed for computing phase space integrals for the canonical gibbs distribution newton s equations are modified by coupling an additional reservoir variable to the physical variables the correct sampling of the phase space according to the gibbs measure is dependent on the nose hoover dynamics being ergodic hoover presented numerical experiments to show that the nose hoover dynamics are non ergodic when applied to the harmonic oscillator in this article we prove that the nose hoover thermostat does not give an ergodynamical system for the one dimensional harmonic oscillator when the mass of the reservoir is large our proof of non ergodicity uses kam theory to demonstrate the existence of invariant tori for the nose hoover dynamical system that separate phase space into invariant regions we present numerical experiments motivated by our analysis that seem to show that the dynamical system is not ergodic even for a moderate thermostat mass

fixed points stability and exact linearization abstract we study the scalar equation x f t x x x b t g x t l 0 by means of contraction mappings conditions are obtained to ensure that each solution x t x t 0 0 as t the conditions allow f to grow as large as t but not as large as t 2 this is parallel to the classical result of smith quart j math oxford ser 12 2 1961 123 for the linear equation without a delay

growth conditions for uniform asymptotic stability of damped oscillators abstract the present paper is devoted to an investigation on the uniform asymptotic stability for the linear differential equation with a damping term x h t x 2 x 0 and its generalization p x h t p x p p x 0 where 0 and p z z p 2 z with p 1 sufficient conditions are obtained for the equilibrium x x 0 0 to be uniformly asymptotically stable under the assumption that the damping coefficient h t is integrally positive the obtained condition for the damped linear differential equation is given by the form of a certain uniform growth condition on h t another representation which is equivalent to this uniform growth condition is also given our results assert that the equilibrium can be uniformly asymptotically stable even if h t is unbounded an example is attached to show this fact in addition easy to use conditions are given to guarantee that the uniform growth condition is satisfied moreover a sufficient condition expressed by an infinite series is presented the relation between the representation of an infinite series and the uniform growth condition is also clarified finally our results are extended to be able to apply to the above mentioned nonlinear differential equation

thermostat algorithm for generating target ensembles we present a deterministic algorithm called contact density dynamics that generates any prescribed target distribution in the physical phase space akin to the famous model of nose and hoover our algorithm is based on a non hamiltonian system in an extended phase space however the equations of motion in our case follow from contact geometry and we show that in general they have a similar form to those of the so called density dynamics algorithm as a prototypical example we apply our algorithm to produce a gibbs canonical distribution for a one dimensional harmonic oscillator

ergodicity of a thermostat family of the nose hoover type one variable thermostats are studied as a generalization of the nos e hoover method which is aimed at achieving gibbs canonical distribution while conserving the time reversibility a condition for equations of motion for the system with the thermostats is derived in the form of a partial differential equation solutions of this equation constitute a family of thermostats including the nos e hoover method as the minimal solution it is shown that the one variable thermostat coupled with the one dimensional harmonic oscillator loses its ergodicity with large enough relaxation time the present result suggests that multivariable thermostats are required to assure the ergodicity and to work as a heat bath

lyapunov stability of elliptic periodic solutions of nonlinear damped equations abstract based on the third order approximation developed by ortega we present the formula of the first twist coefficient for a nonlinear damped equation as an example we consider the lyapunov stability of a superlinear damped differential equation

liouville s theorem and the canonical measure for nonconservative systems from contact geometry standard statistical mechanics of conservative systems relies on the symplectic geometry of the phase space this is exploited to derive hamilton s equations liouville s theorem and to find the canonical invariant measure in this work we analyze the statistical mechanics of a class of nonconservative systems stemming from contact geometry in particular we find out the generalized hamilton s equations liouville s theorem and the microcanonical and canonical measures invariant under the contact flow remarkably the latter measure has a power law density distribution with respect to the standard contact volume form finally we argue on the several possible applications of our results

feedback equivalence of input output contact systems abstract control contact systems represent controlled or open irreversible processes which allow us to represent simultaneously the energy conservation and the irreversible creation of entropy such systems systematically arise in models established in chemical engineering the differential geometric of these systems is a contact form in the same manner as the symplectic 2 form is associated to hamiltonian models of mechanics in this paper we study the feedback preserving the geometric structure of controlled contact systems and render the closed loop system again as a contact system it is shown that only a constant control preserves the canonical contact form hence a state feedback necessarily changes the closed loop contact form for strict contact systems arising from the modelling of thermodynamic systems a class of state feedback that shapes the closed loop contact form and contact hamiltonian function is proposed the state feedback is given by the composition of an arbitrary function and the control contact hamiltonian function the similarity with structure preserving feedback of input output hamiltonian systems leads to the definition of input output contact systems and to the characterization of the feedback equivalence of input output contact systems an irreversible thermodynamic process namely the heat exchanger is used to illustrate the results

global asymptotic stability for damped half linear oscillators abstract a necessary and sufficient condition is established for the equilibrium of the oscillator of half linear type with a damping term p x h t p x p x 0 to be globally asymptotically stable the obtained criterion is given by the form of a certain growth condition of the damping coefficient h t and it can be applied to not only the cases of large damping and small damping but also the case of fluctuating damping the presented result is new even in the linear cases p 2 it is also discussed whether a solution of the half linear differential equation r t p x c t p x 0 that converges to a non zero value exists or not some suitable examples are included to illustrate the results in the present paper

on conditions for asymptotic stability of dissipative infinite dimensional systems with intermittent damping abstract we study the asymptotic stability of a dissipative evolution in a hilbert space subject to intermittent damping we observe that even if the intermittence satisfies a persistent excitation condition if the hilbert space is infinite dimensional then the system needs not being asymptotically stable not even in the weak sense exponential stability is recovered under a generalized observability inequality allowing for time domains that are not intervals weak asymptotic stability is obtained under a similarly generalized unique continuation principle finally strong asymptotic stability is proved for intermittences that do not necessarily satisfy some persistent excitation condition evaluating their total contribution to the decay of the trajectories of the damped system our results are discussed using the example of the wave equation schrodinger s equation and for strong stability also the special case of finite dimensional systems

on the stability of a second order retarded differential equation in this paper we study a second order retarded differential equation with variable delay applying the fixed point theory we obtain a necessary and sufficient condition ensuring that the zero solution of the equation is asymptotically stable our results improve and generalize the known ones in this direction

tsallis dynamics using the nose hoover approach on the basis of the nos e hoover method we developed a deterministic algorithm that produces an arbitrary probability density an ordinary differential equation in the algorithm can realize the tsallis distribution density the tsallis distribution has been considered a candidate of a distribution that represents a physical system in a heat bath the tsallis distribution density employed in this algorithm is defined using a full energy function form e x p along with the tsallis index qg 1 using the current equation numerical simulations were performed for simple systems and the tsallis distributions were observed

non hamiltonian equilibrium statistical mechanics in this paper the equilibrium statistical mechanics of non hamiltonian systems is formulated introducing an algebraic bracket the latter defines non hamiltonian equations of motion in classical phase space according to the approach introduced in phys rev e 64 056125 2001 the jacobi identity is no longer satisfied by the generalized bracket and as a result the algebra of phase space functions is not time translation invariant the presence of a nonzero phase space compressibility spoils also the time reversal invariance of the dynamics the general liouville equation is rederived and the properties of statistical averages are accounted for the features of time correlation functions and linear response theory are also discussed

fixed points and stability in differential equations with variable delays abstract in this paper we consider a linear scalar differential equation with variable delays and give conditions to ensure that the zero solution is asymptotically stable by means of fixed point theory these conditions do not require the boundedness of delays nor do they ask for a fixed sign on the coefficient functions an asymptotic stability theorem with a necessary and sufficient condition is proved

improved stability conditions for a class of stochastic volterra levin equations in this paper we study the mean square asymptotic stability of a class of generalized nonlinear stochastic volterra levin equations by using fixed point theory several sufficient conditions are established for ensuring that the equation is mean square asymptotically stable as well as exponentially stable the main results are new which generalize and improve some well known results in burton 2006 4 and luo 2010 18 finally two examples are given to illustrate our results

y groups via transitive extension abstract motivated by an earlier observation by j h fischer around 1980 b conway conjectured that the coxeter diagram y555 on fig 1 together with a single additional so called spider relation ab1c1ab2c2ab3c3 10 1 form a presentation for the wreath product of the monster group m and a group of order 2 this conjecture was proved by s p norton and the author in 1990 the original proof was rather involved relying on simple connectedness results for certain diagram geometries on numerous data obtained by coset enumeration on a computer and on some delicate calculations with subgroups coming from the 26 node theorem in the present work we follow an inductive approach to the identification of y groups by considering larger y groups as transitive extensions of smaller ones along these lines we obtain an alternative identification proof for the y groups which is almost computer free we refer to only one result of double coset enumeration our approach provides a uniform understanding of the y groups particularly of features such as centres and redundant generators

stability analysis of neutral stochastic delay differential equations by a generalisation of banach s contraction principle abstractin this paper we consider a class of neutral stochastic differential systems with variable delays and give sufficient conditions to ensure that the zero solution is asymptotically stable in the mean square by means of a generalisation of banach s contraction principle these conditions do not require the boundedness of delays nor do they ask for a fixed sign on the coefficient functions the results are shown to improve the previous globally stable results derived in the literature

fixed points and exponential stability for stochastic volterra levin equations in this paper we study a stochastic volterra levin equation by using fixed point theory we give some conditions for ensuring that this equation is exponentially stable in mean square and is also almost surely exponentially stable our result generalizes and improves on the results in 14 1 30

26 implies the bimonster the bimonster or wreathed square 2 of the monster group is presented by the coxeter relations of the 26 node incidence graph of the projective plane of order 3 along with the additional relations that all free 12 gons of this diagram generate symmetric groups s we prove this 12 using purely geometric arguments as well as the celebrated ivanov norton theorem 6 7 9 this elementary presentation provides a natural connection from the projective plane of order 3 to the bimonster previously this connection was known in the other direction 2 3 we also consider other groups that can be obtained in a similar fashion a useful reference on coxeter and reflection groups is 5 we use the atlas 1 notation for finite groups

fitted mesh b spline collocation method for singularly perturbed differential difference equations with small delay abstract this paper deals with the singularly perturbed boundary value problem for a linear second order differential difference equation of the convection diffusion type with small delay parameter of o e whose solution has a boundary layer the fitted mesh technique is employed to generate a piecewise uniform mesh condensed in the neighborhood of the boundary layers b spline collocation method is used with fitted mesh parameter uniform convergence analysis of the method is discussed the method is shown to have almost second order parameter uniform convergence the effect of small delay on boundary layer has also been discussed several examples are considered to demonstrate the performance of the proposed scheme and how the size of the delay argument and the coefficient of the delay term affects the layer behavior of the solution

formulas of liapunov functions for systems of linear partial differential equations formulas of explicit quadratic liapunov functions for showing asymptotic stability of the system of linear partial differential equations u t a u t x on 0 are constructed where a is an n n real matrix u u1 u2 un t is a bounded domain in rk with smooth boundary and denotes the laplacian operator on rk with u u1 u2 un t these formulas are also modified and applied to a number of nonautonomous linear and nonlinear systems and models in structural stability traveling wave and navier stokes equations

the boundedness and stability of neutral delay differential equations with damped stochastic perturbations via fixed point theory in this paper a class of neutral stochastic differential equation is studied by using the fixed point method some conditions are established for ensuring the equation is bounded asymptotically stable and j c stable in mean square then with borel cantelli lemma several sufficient conditions for almost surely asymptotic stability and j c stability are given the conditions do not require the boundedness of delays nor do they ask for the continuousness on the coefficient functions the results of this paper are new and they extend and improve previously known results

contact variational integrators we present geometric numerical integrators for contact flows that stem from a discretization of herglotz variational principle first we show that the resulting discrete map is a contact transformation and that any contact map can be derived from a variational principle then we discuss the backward error analysis of our variational integrators including the construction of a modified lagrangian surprisingly this construction presents some interesting simplifications compared to the corresponding construction for symplectic systems throughout the paper we use the damped harmonic oscillator as a benchmark example to compare our integrators to their symplectic analogues

on coxeter diagrams of complex reflection groups we study complex coxeter diagrams of some unitary reflection groups using solely the combinatorics of diagrams we give a new proof of the classification of root lattices defined over e z e 2 i 3 there are only four such lattices namely the e lattices whose real forms are a2 d4 e6 and e8 next we address the issue of characterizing the diagrams for unitary reflection groups a question that was raised by broue malle and rouquier to this end we describe an algorithm which given a unitary reflection group g picks out a set of complex reflections if g is the reflection group of a root lattice defined over e the reflections selected by the algorithm form the known diagram for g for the reflection group of the complex coxeter todd lattice k e 12 we find a new diagram that extends to an affine diagram withz 7z symmetry if g is a weyl group the algorithm immediately yields a set of simple roots otherwise experimental evidences indicate that the algorithm selects a minimal generating set of reflections if g is primitive and g has a set of roots whose z span is a discrete subset of the ambient vector space

new stability conditions for mixed linear levin nohel integro differential equations for the mixed levin nohel integro differential equation we obtain new necessary and sufficient conditions of asymptotic stability these results improve those obtained by becker and burton stability fixed points and inverse of delays proc r soc edinburgh sect a 136 245 275 2006 10 1017 s0308210500004546 and jin and luo stability of an integro differential equation comput math appl 57 7 1080 1088 2009 10 1016 j camwa 2009 01 006 when b t 0 and supplement the 32 stability theorem when a t s 0 in addition the case of the equations with several delays is discussed as well

exact baker campbell hausdorff formula for the contact heisenberg algebra in this work we introduce the contact heisenberg algebra which is the restriction of the jacobi algebra on contact manifolds to the linear and constant functions we give the exact expression of its corresponding baker campbell hausdorff formula we argue that this result is relevant to the quantization of contact systems

a necessary and sufficient condition for global asymptotic stability of time varying lotka volterra predator prey systems abstract the purpose of this paper is to present a necessary and sufficient condition which guarantees that an interior equilibrium of a certain predator prey system is globally asymptotically stable this ecological system is a model of lotka volterra type whose prey population receives time variation of the environment we assume that the time varying coefficient is weakly integrally positive and has a weaker property than uniformly continuous our necessary and sufficient condition is expressed by an improper double integral on the time varying coefficient our work is inspired by the study of the stability theory for damped linear oscillators

stability results for neutral stochastic functional differential equations via fixed point methods in this paper we prove some results on the mean square asymptotic stability of a class of neutral stochastic differential systems with variable delays by using a contraction mapping principle name

uniform asymptotic stability for damped linear oscillators with variable parameters abstract in this paper sufficient conditions for uniform asymptotic stability of the damped linear oscillators with variable coefficients are presented the result of the present study can be applied to even the case of negative damping the conditions presented herein are shown to be essential for uniform asymptotic stability

stability of linear neutral differential equations with delays and impulses established by the fixed points method abstract in this paper we consider the impulsive effects on the stability of the zero solution of the linear neutral differential equations with variable delays by transforming the equations into ones without impulses and using fixed point theory some sufficient conditions for asymptotic stability and exponential stability of the zero solution are obtained the paper extends and improves results on sufficient conditions obtained by jin and luo 2008 17 and ardjouni and djoudi 2011 18 which is shown clearly in example 1 this paper also shows that the impulse intensity and the impulse time both influence the decay rate of the convergence to zero of the solutions finally two examples are given to show applications of some results obtained

uniform global asymptotic stability for oscillators with superlinear damping abstract the present paper deals with the damped superlinear oscillator x h t q x 2 x 0 where 0 and q z z q 2 z with q 2 the origin x x 0 0 is the only equilibrium of this oscillator we herein establish a sufficient condition for the equilibrium to be uniformly globally asymptotically stable we conclude that under the assumption that the damping coefficient h t is integrally positive if the integral from to t of a particular solution of the first order nonlinear differential equation u h t q u 1 0 diverges to negative infinity uniformly with respect to then the equilibrium is uniformly globally asymptotically stable the above mentioned result is expressed by an implicit condition we examine when the implicit condition is satisfied and when it is not satisfied we also give explicit sufficient conditions which assure that the equilibrium is uniformly globally asymptotically stable using the obtained result we present an example of which the equilibrium is uniformly globally asymptotically stable even if h t is unbounded

stability analysis for stochastic volterra levin equations with poisson jumps fixed point approach this paper is devoted to investigate a class of stochastic volterra levin equations with poisson jumps to the best of the authors knowledge till now the stability problem for this class of new systems has not yet been solved since poisson jumps are considered the main objective of this paper is to fill the gap by using the fixed point theory we first study the existence and uniqueness of the solution as well as the pth moment exponential stability for the considered system then based on the well known borel cantelli lemma we prove that the solution is almost surely pth moment exponentially stable our results improve and generalize those given in the previous literature finally two simple examples are provided to illustrate the effectiveness of the obtained results

computational method based on reproducing kernel for solving singularly perturbed differential difference equations with a delay abstract in this paper reproducing kernel hilbert space method rkhsm based on collocation scheme is used to solve singularly perturbed second order differential difference equations we implement rkhsm without gram schmidt orthogonalization process for singularly perturbed differential difference equation with boundary layer behavior and also oscillatory behavior with small delay rkhsm in this study is based on the division of the problem domain into two subintervals one with boundary layer and the other one without such a boundary layer several numerical examples are studied to demonstrate the accuracy of the present method results of the present scheme indicate that new algorithm has the following advantages small computational work fast convergence and high precision

a new method of dynamical stability i e fractional generalized hamiltonian method and its applications fractional generalized hamiltonian method of dynamical stability is presented six criterions for the new method of dynamical stability are given the new method s applications are studied five kinds of actual fractional dynamical models are constructed dynamical stability of the five actual fractional dynamical models is explored in the paper we present fractional generalized hamiltonian method of dynamical stability in terms of riesz riemann liouville derivative and study its applications for an actual dynamical system the fractional generalized hamiltonian method of constructing a fractional dynamical model is given and then the six criterions for fractional generalized hamiltonian method of dynamical stability are presented as applications by using the fractional generalized hamiltonian method we construct five kinds of actual fractional dynamical models which include a fractional euler poinsot model of rigid body that rotates with respect to a fixed point a fractional hojman urrutia model a fractional lorentz dirac model a fractional whittaker model and a fractional robbins lorenz model and we explore the dynamical stability of these models respectively this work provides a general method for studying the dynamical stability of an actual fractional dynamical system that is related to science and engineering