melnikov method for discontinuous planar systems abstract this paper treats the occurrence of homoclinic solutions in planar systems with discontinuous right hand side more precisely we deal with a t periodic perturbed system such that the unperturbed system is an autonomous possessing homoclinic orbit by means of the so called non smooth melnikov function there is shown the existence of a homoclinic solution for a perturbed system the non smooth melnikov function is derived and the method of how to find it in concrete problems is also introduced

a linear estimate of the number of limit cycles for some planar piecewise smooth quadratic differential system abstract in this paper we study a class of planar piecewise smooth quadratic integrable non hamiltonian systems which have a center by using the averaging method we give an estimation of the number of limit cycles which bifurcate from the above periodic annulus under the polynomial perturbation of degree n our estimation is linear depending on n and it is at least twice the associated estimation of smooth systems

the focus center limit cycle bifurcation in symmetric 3d piecewise linear systems the birth of limit cycles in 3d three dimensional piecewise linear systems for the relevant case of symmetrical oscillators is considered a technique already used by the authors in planar systems is extended to cope with 3d systems where a greater complexity is involved under some given nondegeneracy conditions the corresponding theorem characterizing the bifurcation is stated in terms of the deviation from the critical value of the bifurcation parameter expressions in the form of power series for the period amplitude and the characteristic multipliers of the bifurcating limit cycle are also obtained the results are applied to accurately predict the birth of symmetrical periodic oscillations in a 3d electronic circuit genealogically related to the classical van der pol oscillator

on the implementation of singly implicit runge kutta methods a modified newton method is often used to solve the algebraic equations that arise in the application of implicit runge kutta methods when the runge kutta method has a coefficient matrix a with a single point spectrum with eigenvalue a the efficiency of the modified newton method is much improved by using a similarity transformation of a each iteration involves vector transformations in this article an alternative iteration scheme is obtained which does not require vector transformations and which is simpler in other respects also both schemes converge in a finite number of iterations when applied to linear systems of differential equations but the new scheme uses the nilpotency of a ai to achieve this numerical results confirm the predicted convergence for nonlinear systems and indicate that the scheme may be a useful alternative to the modified newton method for low dimensional systems the scheme seems to become less effective as the dimension increases however it has clear advantages for parallel computation making it competitive for high dimensional systems

on the number of limit cycles in general planar piecewise linear systems of node node types abstract we investigate the existence and number of limit cycles in a class of general planar piecewise linear systems constituted by two linear subsystems with node node dynamics using the lienard like canonical form with seven parameters some sufficient and necessary conditions for the existence of limit cycles are given by studying the fixed points of proper poincare maps in particular we prove the existence of at least two nested limit cycles and describe some parameter regions where two limit cycles exist the main results are applied to the pwl morris lecar neural model to determine the existence and stability of the limit cycles

dynamics and bifurcations of nonsmooth systems a survey abstract in this survey we discuss current directions of research in the dynamics of nonsmooth systems with emphasis on bifurcation theory an introduction to the state of the art also for non specialists is complemented by a presentation of the main open problems we illustrate the theory by means of elementary examples the main focus is on piecewise smooth systems which have recently attracted a lot of attention but we also briefly discuss other important classes of nonsmooth systems such as nowhere differentiable ones and differential variational inequalities this extended framework allows us to put the diverse range of papers and surveys in this special issue in a common context a dedicated section is devoted to concrete applications that stimulate the development of the field this survey is concluded by an extensive bibliography

galerkin runge kutta discretizations for parabolic equations with time dependent coefficients a new class of fully discrete galerkin runge kutta methods is constructed and analyzed for linear parabolic initial boundary value problems with time dependent coefficients unlike any classical counterpart this class offers arbitrarily high order convergence while significantly avoiding what has been called order reduction in support of this claim error estimates are proved and computational results are presented additionally since the time stepping equations involve coefficient matrices changing at each time step a preconditioned iterative technique is used to solve the linear systems only approximately nevertheless the resulting algorithm is shown to preserve the original convergence rate while using only the order of work required by the base scheme applied to a linear parabolic problem with time independent coefficients furthermore it is noted that special runge kutta methods allow computations to be performed in parallel so that the final execution time can be reduced to that of a low order method

algebraically stable and implementable runge kutta methods of high order there are three interesting properties of methods for stiff ordinary differential equations order stability and efficiency of implementation this paper constructs runge kutta methods of orders 5 and 6 which possess these properties to a high extent we further classify all algebraically stable methods of an arbitrary order and give various relationships between contractivity and order of implicit methods

bifurcation of limit cycles from generalized homoclinic loops in planar piecewise smooth systems abstract the generalized homoclinic loop appears in the study of dynamics on piecewise smooth differential systems during the past two decades for planar piecewise smooth differential systems there are concrete examples showing that under suitable perturbations of a generalized homoclinic loop one or two limit cycles can appear but up to now there is no a general theory to study the cyclicity of a generalized homoclinic loop that is the maximal number of limit cycles which are bifurcated from it in this paper we provide some sufficient conditions on the cyclicity of some homoclinic loops especially we prove the existence of one or two limit cycles which are bifurcated from a generalized homoclinic loop of an unperturbed piecewise smooth differential systems also we obtain a class of piecewise differential system in which two limit cycles can be bifurcated from a generalized homoclinic loop with multiplicity one the phenomena cannot happen in the smooth differential systems finally we provide five concrete piecewise smooth differential systems showing the applications of our theories on this phenomenon

analysis of energy and quadratic invariant preserving equip methods abstract in this paper we are concerned with the analysis of a class of geometric integrators at first devised in brugnano et al 2010 and brugnano et al 2012 which can be regarded as an energy conserving variant of gauss collocation methods with these latter they share the property of conserving quadratic first integrals but in addition they also conserve the hamiltonian function itself we here reformulate the methods in a more convenient way and propose a more refined analysis than that given in brugnano et al 2012 also providing as a by product a practical procedure for their implementation a thorough comparison with the original gauss methods is carried out by means of a few numerical tests solving hamiltonian and poisson problems

review paper efficient numerical methods for the solution of stiff initial value problems and differential algebraic equations in recent years after a prolonged period of intense activity the study of numerical methods for solving stiff initial value problems for ordinary differential equations and differential algebraic equations has reached a certain maturity there now exist some excellent codes which are both efficient and reliable for solving these particular classes of problems in this paper we sketch some of the main theory which underpins stiff integration methods and we use this to describe and put into context some of the best codes currently available by referencing only codes which have been thoroughly tested and are widely available our aim is to direct users of numerical software to those codes which they should try initially if faced with the problem of solving ordinary differential equations of this type an additional feature is that the codes which we propose serve as benchmarks against which any new codes can be evaluated

type i periodic motions for nonlinear impact oscillators abstract in this paper a general class of nonlinear impact oscillators is considered for subharmonic bifurcation this system can be used to model an inverted pendulum impacting on rigid walls under external periodic excitation and its unperturbed system possesses a pair of homoclinic cycles via the identification given by the impact law and three separate families of periodic orbits inside and outside the homoclinic cycles by discussing the subharmonic orbits inside the homoclinic cycles the subharmonic melnikov method established for smooth dynamical systems is extended to be applicable to the non smooth system

counting rational points on hypersurfaces for any n geq 2 let f in mathbb z x 1 ldots x n be a form of degree d geq 2 which produces a geometrically irreducible hypersurface in mathbb p n 1 this paper is concerned with the number n f b of rational points on f 0 which have height at most b for any varepsilon 0 we establish the estimate n f b o b n 2 varepsilon whenever either n leq 5 or the hypersurface is not a union of lines here the implied constant depends at most upon d n and varepsilon

existence of limit cycles in general planar piecewise linear systems of saddle saddle dynamics abstract the existence and number of limit cycles in a class of general planar piecewise linear systems constituted by two linear subsystems with saddle saddle dynamics are investigated using the lienard like canonical form with seven parameters the parametric regions of the existence of limit cycles are given by constructing proper poincare maps in particular the existence of at least two limit cycles is proved and some parameter regions where two nested limit cycles exist are given

stability of collocation at gaussian points symmetric runge kutta schemes are particularly useful for solving stiff two point boundary value problems such a stable schemes perform well in many cases but it is demonstrated that in some instances the stronger property of algebraic stability is required a characterization of symmetric algebraically stable runge kutta schemes is given the class of schemes thus defined turns out not to be very rich the only collocation schemes in it are those based on gauss points and other schemes in the class do not seem to offer any advantage over collocation at gaussian points

melnikov method for homoclinic bifurcation in nonlinear impact oscillators based on an inverted pendulum impacting on rigid walls under external periodic excitation a class of nonlinear impact oscillators is discussed for its homoclinic bifurcation the melnikov method established for smooth dynamical systems is extended to be applicable to the nonsmooth one for nonlinear impact systems closed form solutions between impacts are generally unavailable the absence of closed form solutions makes difficulties in estimation of the gap between the stable manifold and unstable manifold in this paper we give a method to compute the melnikov functions up to the n t h order so as to obtain conditions of parameters for the persistence of homoclinic cycles which are formed via the identification given by the impact rule

vinogradov s mean value theorem via efficient congruencing we obtain estimates for vinogradov s integral that for the rst time approach those conjectured to be the best possible several applications of these new bounds are provided in particular the conjectured asymptotic formula in waring s problem holds for sums of s kth powers of natural numbers whenever s 2k 2 2k 3

manin s conjecture for certain biprojective hypersurfaces using the circle method we count integer points on complete intersections in biprojective space in boxes of different side length provided the number of variables is large enough depending on the degree of the defining equations and certain loci related to the singular locus having established these asymptotics we deduce asymptotic formulas for rational points on such varieties with respect to the anticanonical height function in particular we establish a conjecture of manin for certain smooth hypersurfaces in biprojective space of sufficiently large dimension

limit cycles with chattering in relay feedback systems relay feedback has a large variety of applications in control engineering several interesting phenomena occur in simple relay systems in the paper scalar linear systems with relay feedback are analyzed it is shown that a limit cycle where part of the limit cycle consists of fast relay switchings can occur this chattering is analyzed in detail and conditions for approximating it by a sliding mode are derived a result on existence of limit cycles with chattering is given and it is shown that the limit cycles can have arbitrarily many relay switchings each period limit cycles with regular sliding modes are also discussed examples illustrate the results

efficiently implementable algebraically stable runge kutta methods recent work by butcher bit 16 1976 pp 237 240 on the implementation of implicit runge kutta methods has shown that if his technique is used the most efficiently implementable methods are those whose runge kutta matrix has a single real eigenvalue in this paper therefore we analyze the algebraic stability of two families of methods with this property namely singly implicit and diagonally implicit methods

melnikov theory for a class of planar hybrid systems abstract in this paper we extend the well known melnikov theory for smooth systems to a class of two zonal planar hybrid piecewise smooth systems in this class each zone of differentiability is separated by a straight line and in these zones the dynamics are governed by a smooth system when an orbit reaches the separation line then a reset map applies before entering the orbit in the other zone the family of non generic systems of this class which has a continuum of periodic orbits is considered then we study the periodic orbits of the continuum that persist under a perturbation to achieve this objective first we build an appropriate poincare map after that we introduce a function of one variable whose zeros provide us the periodic orbits that remain after the perturbation this function will be called the melnikov function because it is very close to the so called melnikov function for smooth systems finally we apply the obtained results to some piecewise linear systems continuous as well as discontinuous ones

time consistent fluid structure interaction on collocated grids for incompressible flow abstract consistent time integration on collocated grids for incompressible flow has been studied for static grids using the piso method in which the dependencies on time step size and under relaxation has been studied in detail however for moving grids a detailed description is still missing therefore a step by step analysis of a time consistent fluid structure interaction fsi method for incompressible flow on collocated grids is presented the method consist of face normal and area correction for moving grids treatment of velocity boundary conditions for no slip walls time integration of structure equations and fluid force interpolation to structure the basis of the method is the piso method of the incompressible navier stokes equations time consistency on static grids is shown first after which time consistency on moving grids is described and analyzed for moving grids consistent time integration is described in two ways 1 constructing the face velocities from a normal and tangential component and 2 correcting the face flux with a face normal and face area correction for both descriptions the general formulation for the backward differencing schemes bdf are given and the correct behavior of the first second and third order schemes is demonstrated by means of an academic test case circular cavity flow additionally the force coupling from the fluid to the structure is discussed in detail for combining a fourth order explicit runge kutta scheme for the structure with a bdf scheme for the fluid three interpolations for the fluid forces are shown which result in either a first order or second order fsi scheme third order fsi is demonstrated when the third order bdf scheme is applied on both the structure and fluid equations also under relaxation for the fluid equations is considered and it is demonstrated that the order of the three bdf schemes are independent of the under relaxation factor finally the proposed method of time consistent fsi on collocated grids for incompressible flows is demonstrated by applying it to a three dimensional flow over an elastic structure in a channel and the cylinder flap fsi benchmark case of turek and hron

high order algebraically stable multistep runge kutta methods in this paper we examine some of the stability properties of a special class of multivalue methods called in k burrage order properties of multivalue methods ima j numer anal submitted multistep runge kutta methods and construct families of algebraically stable methods of arbitrarily high order

density of rational points on a quadric bundle in mathbb p 3 times mathbb p 3 we establish an asymptotic formula for the number of rational points of bounded anticanonical height which lie on a certain zariski dense subset of the biprojective hypersurface x1y12 x4y42 0 in p3 p3 this confirms the modified manin conjecture for this variety in which the removal of a thin set of rational points is allowed

asymmetric type ii periodic motions for nonlinear impact oscillators abstract in this paper a general class of nonlinear impact oscillators is considered for type ii periodic motions this system can be used to model an inverted pendulum impacting on rigid walls under external periodic excitation the unperturbed system possesses a pair of homoclinic cycles and three separate families of periodic orbits inside and outside the homoclinic cycles via the identification given by the impact law by approximating the poincare map to o e directly a general method of melnikov type for detecting the existence of asymmetric type ii subharmonic orbits outside the homoclinic cycles is presented

limit cycles bifurcating from piecewise quadratic systems separated by a straight line abstract in this paper we are concerned with limit cycles bifurcating from piecewise quadratic systems separated by a straight line by means of the first order averaging theory of discontinuous differential systems and the argument principle we present an estimate of the maximum number of limit cycles bifurcating from the period annulus around the center of piecewise smooth systems

an efficient family of strongly a stable runge kutta collocation methods for stiff systems and daes part i stability and order results for each integer s 3 a new uniparametric family of stiffly accurate strongly a stable s stage runge kutta methods is obtained these are collocation methods with a first internal stage of explicit type the methods are based on interpolatory quadrature rules with precision degree equal to 2s 4 and all of them have two prefixed nodes c 1 0 and c s 1 the amount of implicitness of our s stage method is similar to that involved with the s stage lobattoiiia method or with the s 1 stage radauiia method the new family of runge kutta methods proves to be of interest for the numerical integration of stiff systems and differential algebraic equations in fact on several stiff test problems taken from the current literature two methods selected in our 4 stage family seem to be slightly more efficient than the 3 stage radauiia method and also more robust than the 4 stage lobattoiiia method

a new form of the circle method and its application to quadratic forms if the coefficients r n satisfy suitable arithmetic conditions the behaviour of f will be determined by an appropriate rational approximation a q to with small values of q usually producing large values of f when lies in an interval a q a q with q small a major arc one hopes to estimate f asymptotically while if the corresponding q is large for the minor arcs one hopes that f will be small at least on average one can distinguish two different types of application of the circle method the first type following the above pattern uses one of a variety of mean value techniques hua s inequality for example to give an upper bound for the minor arc contribution results on waring s problem are generally proved this way for example the second type of application on which we shall be focusing our attention avoids the minor arcs completely and treats f asymptotically for all here one cannot take advantage of the mean value ideas used in the first type of problem the savings come instead from the availability of good estimates for complete exponential sums kloosterman s work 12 on quaternary quadratic forms and the papers of the author 5 and hooley 8 and 9 on cubic forms are of this latter type we should point out that such applications do not include any problems in which f is the generating function for a polynomial of degree 4 or more there are

border collision bifurcations including period two to period three for piecewise smooth systems abstract we examine bifurcation phenomena for maps that are piecewise smooth and depend continuously on a parameter in the simplest case there is a surface in phase space along which the map has no derivative or has two one sided derivatives is the border of two regions in which the map is smooth as the parameter is varied a fixed point e may collide with the border and we may assume that this collision occurs at 0 a variety of bifurcations occur frequently in such situations but never or almost never occur in smooth systems in particular e may cross the border and so will exist for 0 but it may be a saddle in one case say 0 for 0 there may be a stable period 3 orbit which similarly shrinks to e0 as 0 hence one observes the following stable periodic orbits a stable period 2 orbit collapses to a point and is reborn as a stable period 3 orbit we also see analogously stable period 2 to stable period p orbit bifurcations with p 5 11 52 or period 2 to quasi periodic or even to a chaotic attractor we believe this phenomenon will be seen in many applications

limit cycles of continuous and discontinuous piecewise linear differential systems in r3 abstract we study the limit cycles of two families of piecewise linear differential systems in r 3 with two pieces separated by a plane in one family the differential systems are continuous on the plane and in the other family they are discontinuous on the plane the usual tool for studying these limit cycles is the poincare map but here we shall use recent results which extend the averaging theory to continuous and discontinuous differential systems all the computations have been done with the algebraic manipulator mathematica

local stability of limit cycles for mimo relay feedback systems abstract this paper concerns with the local stability of limit cycles for decentralized relay feedback systems it presents a sufficient condition for the local stability based on the well known poincare map method the effectiveness of the presented result is illustrated by a numerical example

counting rational points on biquadratic hypersurfaces abstract an asymptotic formula is established for the number of rational points of bounded anticanonical height which lie on a certain zariski open subset of an arbitrary smooth biquadratic hypersurface in sufficiently many variables the proof uses the hardy littlewood circle method

counting rational points on algebraic varieties for any n 2 let z pn be a geometrically integral algebraic variety of degree d this article is concerned with the number nz b of q rational points on z which have height at most b for any e 0 we establish the estimate nz b od e n bdimz e provided that d e as indicated the implied constant depends at most on d e and n

on the representation of integers by quadratic forms let n 4 and let q 2 z x1 xn be a non singular quadratic form when q is indefinite we provide new upper bounds for the least non trivial integral solution to the equation q 0 and when q is positive definite we provide improved upper bounds for the greatest positive integer k for which the equation q k is insoluble in integers despite being soluble modulo every prime power

bifurcations in piecewise smooth feedback systems this paper is concerned with the analysis and classification of non standard bifurcations in piecewise smooth feedback systems bifurcations involving fixed points in maps equilibria and limit cycles in flows are considered with particular attention to border collisions of fixed points in maps non smooth bifurcations of equilibria grazing bifurcations and sliding bifurcations of limit cycles in flows the aim is to describe existing and novel results to form the basis of a consistent theory of bifurcations in such systems in so doing a novel approach to classify non smooth bifurcations of equilibria in flows is presented experimental results on pulse width modulated voltage mode controlled dc dc power converters motivates the relevance to applications of the analysis proposed

on the origin and bifurcations of stick slip oscillations a recently proposed model of macroscopic friction is investigated using methods of dynamical systems analysis particular emphasis is put on the bifurcations associated with the appearance of stick

piecewise smooth dynamical systems persistence of periodic solutions and normal forms abstract we consider an n dimensional piecewise smooth vector field with two zones separated by a hyperplane which admits an invariant hyperplane transversal to containing a period annulus a fulfilled by crossing periodic solutions for small discontinuous perturbations of these systems we develop a melnikov like function to control the persistence of periodic solutions contained in a when n 3 we provide normal forms for the piecewise linear case finally we apply the melnikov like function to study discontinuous perturbations of the given normal forms

the density of rational points on curves and surfaces let c be an irreducible projective curve of degree d in mathbb p 3 defined over overline mathbb q it is shown that c has o varepsilon d b 2 d varepsilon rational points of height at most b for any varepsilon 0 uniformly for all curves c this result extends an estimate of bombieri and pila duke math j 59 1989 337 357 to projective curves for a projective surface s in mathbb p 3 of degree d ge 3 it is shown that there are o varepsilon d b 2 varepsilon rational points of height at most b of which at most o varepsilon d b 52 27 varepsilon do not lie on a rational line in s for non singular surfaces one may reduce the exponent to 4 3 16 9d for d 4 or 5 or max 1 3 sqrt d 2 d 1 for d ge 6 even for the surface x 1 d x 2 d x 3 d x 4 d this last result improves on the previous best known as a further application it is shown that almost all integers represented by an irreducible binary form f x y in mathbb z x y have essentially only one such representation this extends a result of hooley j reine angew math 226 1967 30 87 which concerned cubic forms only the results are not restricted to projective surfaces and as an application of other results in the paper it is shown that x 1 x 2 x 3 in mathbb n 3 x 1 d x 2 d x 3 d n ll varepsilon d n theta d varepsilon with theta frac 2 sqrt d frac 2 d 1 when d ge 8 this provides the first non trivial bound for the number of representations as a sum of three d th powers

a and b stability for runge kutta methods characterizations and equivalence using a special representation of runge kutta methods w transformation simple characterizations ofa stability andb stability have been obtained in 9 8 7 in this article we will make this representation and their conclusions more transparent by considering the exact runge kutta method finally we demonstrate by a numerical example that for difficult problemsb stable methods are superior to methods which are only a stable

simultaneous integer values of pairs of quadratic forms we prove that a pair of integral quadratic forms in 5 or more variables will simultaneously represent almost all pairs of integers that satisfy the necessary local conditions provided that the forms satisfy a suitable nonsingularity condition in particular such forms simultaneously attain prime values if the obvious local conditions hold the proof uses the circle method and in particular pioneers a two dimensional version of a kloosterman refinement

the dichotomy of stiffness pragmatism versus theory the problem of stiffness occuring in ordinary differential equations is now well understood and various concepts such as nonlinear stability b convergence and stiff order have been introduced as being relevant to an understanding of the behaviour of numerical methods when applied to stiff problems however it is often the case that some of these properties are not always attainable when methods are implemented efficiently and in this paper we will investigate this paradox in some detail in addition we will give a resume of some of the recent theoretical developments in the applicability of numerical methods to stiff problems and make some comparisons between existing codes based on runge kutta and linear multistep methods we will conclude this discussion with the introduction of a very general family of methods called multivalue methods and show how the important concepts of cheap implementation stage order and stability apply to these methods

limit cycle bifurcations by perturbing piecewise smooth hamiltonian systems with multiple parameters abstract this paper is concerned with the problem of limit cycle bifurcation for piecewise smooth near hamiltonian systems with multiple parameters by the first melnikov function some novel criteria have been established for the existence of multiple limit cycles furthermore an example is included to validate the obtained results by considering the maximum number of limit cycles for a piecewise quadratic system studied in llibre and mereu 2014 12 compared with the result in the above reference one more limit cycle is found by our method

the role of orthogonal polynomials in numerical ordinary differential equations abstract orthogonal polynomials have many applications to numerical ordinary differential equations some of these especially those connected with the quadrature formulae on which many differential equation methods are based are to be expected on the other hand there are many surprising applications quite unlike traditional uses of orthogonal polynomials this paper surveys many of these applications especially those related to accuracy and implementability of runge kutta methods

on hopf bifurcation in non smooth planar systems abstract as we know for non smooth planar systems there are foci of three different types called focus focus ff focus parabolic fp and parabolic parabolic pp type respectively the poincare map with its analytical property and the problem of hopf bifurcation have been studied in coll et al 2001 3 and filippov 1988 6 for general systems and in zou et al 2006 13 for piecewise linear systems in this paper we also study the problem of hopf bifurcation for non smooth planar systems obtaining new results more precisely we prove that one or two limit cycles can be produced from an elementary focus of the least order order 1 for foci of ff or fp type and order 2 for foci of pp type theorem 2 3 different from the case of smooth systems for piecewise linear systems we prove that 2 limit cycles can appear near a focus of either ff fp or pp type theorem 3 3

dynamics of relay relaxation oscillators relaxation oscillators can usually be represented as a feedback system with hysteresis the relay relaxation oscillator consists of relay hysteresis and a linear system in feedback the objective of this work is to study the existence of periodic orbits and the dynamics of coupled relay oscillators in particular we give a complete analysis for the case of unimodal periodic orbits and illustrate the presence of degenerate and asymmetric orbits we also discuss how complex orbits can arise from bifurcation of unimodal orbits finally we focus on oscillators with an integrator as the linear component and study the entrainment under external forcing and phase locking when such oscillators are coupled in a ring

stability of explicit time discretizations for solving initial value problems stability regions of explicit linear time discretization methods for solving initial value problems are treated if an integration method needsm function evaluations per time step then we scale the stability region by dividing bym we show that the scaled stability region of a method satisfying some reasonable conditions cannot be properly contained in the scaled stability region of another method bounds for the size of the stability regions for three different purposes are then given for general nonlinear ordinary differential systems for systems obtained from parabolic problems and for systems obtained from hyperbolic problems we also show how these bounds can be approached by high order methods

constructive characterization ofa stable approximations to exp z and its connection with algebraically stable runge kutta methods all rational approximations to exp z of order 2m m denotes the maximal degree of nominator and denominator are given by a closed formula involving real parameters using the theory of order stars 9 necessary and sufficient conditions fora stability respectivelyi stability are given on the basis of this characterization relations between the concepts ofa stability and algebraic stability for implicit runge kutta methods are investigated in particular we can partly prove the conjecture that to any irreduciblea stabler z of oderp 0 there exist algebraically stable runge kutta methods of the same order withr z as stability function

bifurcations and chaos in a system with impacts abstract a vibro impact system is considered a body bounces on a flat horizontal surface of the vibro bench hopf bifurcations of the vibro impact system in two kinds of strong resonance cases 03 1 and 04 1 are investigated a poincare map of the system is established the period 1 single impact motion of the system and its stability are studied by analytical methods and hopf bifurcation values and intersecting conditions of the system in strong resonance cases are determined a center manifold theorem technique is applied to reduce the poincare map to a two dimensional one which is put into normal form by theory of normal forms by the theory of hopf and subharmonic bifurcations of fixed points in r 2 strong resonance dynamic behavior of the vibro impact system near points of resonance is analyzed in the resonance case of 03 1 the system generally exhibits unstable period 3 three impact motion in the resonance case of 04 1 the system can exhibit stable period 4 four impact motion and quasi periodic motion the theoretical analyses are verified by numerical solutions routes to chaos in two kinds of strong resonance cases considered are obtained by numerical simulations quasi periodic impacts in non resonance and weak resonance cases doubling periodic bifurcations and chaotic motions are also stated briefly by numerical analyses

bifurcation of periodic orbits in a class of planar filippov systems abstract in this paper we discuss the perturbations of a general planar filippov system with exactly one switching line when the system has a limit cycle we give a condition for its persistence when the system has an annulus of periodic orbits we give a condition under which limit cycles are bifurcated from the annulus we also further discuss the stability and bifurcations of a nonhyperbolic limit cycle when the system has an annulus of periodic orbits we show via an example how the number of limit cycles bifurcated from the annulus is affected by the switching

on the stability of volterra runge kutta methods we investigate the stability of extended runge kutta methods for volterra integral equations of the second kind as test equations we choose linear convolution equations with positive definite l 1 kernels our results are strongly connected with the contractivity of runge kutta methods for nonlinear dissipative differential equations

rational points of bounded height on fano varieties a prime pe7z let v be an algebraic variety defined over f and li a metrized line bundle on v i e a system l consisting of a line bundle l and a family of banach v adic metrics on l f for all v s satisfying well known conditions cf 1 chap vi then we can define a height function hl x f i r by the formula h x is x 1 where s is a local section of l non vanishing

a parallelizable preconditioner for the iterative solution of implicit runge kutta type methods the main diculty in the implementation of most standard implicit runge kutta irk methods applied to sti ordinary dierential equations odes is to eciently solve the nonlinear system of equations in this article we propose the use of a preconditioner whose decomposition cost for a parallel implementation is equivalent to the cost for the implicit euler method the preconditioner is based on the w transformation of the rk coecient matrices discovered by hairer and wanner for sti odes the preconditioner is by construction asymptotically exact for methods with an invertible rk coecient matrix the methodology is particularly useful when applied to super partitioned additive runge kutta spark methods the nonlinear system can be solved by inexact simplied newton iterations at each simplied newton step the linear system can be approximately solved by an iterative method applied to the preconditioned linear system c 1999 published by elsevier science b v all rights reserved

construction of runge kutta type methods for solving ordinary differential equations in this paper we study runge kutta methods with continuous stage introduced by butcher in 1987 by setting the coefficients of this family of methods and choosing the appropriate numerical integration we derive new classes of runge kutta methods furthermore we extend the w transformation technique by permitting w to be a non square matrix this allows us to construct more high order implicit runge kutta methods with some geometric properties specially we provide runge kutta methods with continuous stage which are conjugate symplectic symmetric or energy preserving for solving hamiltonian systems we also present some numerical experiments to verify our results

feasibility and contractivity in implicit runge kutta methods abstract this paper is concerned with implicit runge kutta methods for the numerical solution of initial value problems in ordinary differential equations for these methods a review is presented of the fundamental concept of algebraic stability introduced in 1979 independently by burrage butcher and by crouzeix furthermore a new framework is given in which algebraic stability becomes equivalent to both contractivity and feasibility of a runge kutta method

multigrade efficient congruencing and vinogradov s mean value theorem we develop a multigrade enhancement of the efficient congru encing method to estimate vinogradov s integral of degree k for moments of order 2s thereby obtaining near optimal estimates for 5 k 2 1 543k 2 the asymptotic formula is also established for sums of 28 fifth powers

generalized hopf bifurcation emanated from a corner for piecewise smooth planar systems abstract in this paper we study the existence of periodic orbits bifurcating from a corner in a piecewise smooth planar dynamical system this phenomenon is interpreted as generalized hopf bifurcation philos trans r soc london ser a 359 1789 2001 2483 2496 hopf bifurcation for nonsmooth planar systems universitat zu koln 2000 northeast math j 17 4 2001 383 386 northeast math j 17 3 2001 261 264 generalized hopf bifurcation for non smooth planar dynamical systems university cologne 2003 in the case of smoothness hopf bifurcation is characterized by a pair of complex conjugate eigenvalues crossing through the imaginary axis this characterization does not apply to a piecewise smooth system due to the lack of linearization in fact the generalized hopf bifurcation is determined by interactions between the geometrical structure of the corner and the eigenstructure of each smooth subsystem we combine a geometrical method and an analytical method to investigate the generalized hopf bifurcation the bifurcating periodic orbits are obtained by studying the fixed points of a return map

fast switches in relay feedback systems relays are common in automatic control systems even linear systems with relay feedback are however far from fully understood new results are given about the behavior of these systems via a state space approach it is proved that there exist multiple fast switches if and only if the sign of the first nonvanishing markov parameter of the linear system is positive fast switches are shown to occur as part of stable limit cycles an analysis is developed for these limit cycles that illustrates how they can be predicted

control of near grazing dynamics in impact oscillators a method is presented for controlling the persistence of a local attractor near a grazing periodic trajectory in a piecewise smooth dynamical system in the presence of discontinuous jumps in the state associated with intersections with system discontinuities in particular it is shown that a discrete linear feedback strategy may be employed to retain the existence of an attractor near the grazing trajectory such that the deviation of the attractor from the grazing trajectory goes to zero as the system parameters approach those corresponding to grazing contact the implementation relies on a local analysis of the near grazing dynamics using the concept of discontinuity mappings numerical results are presented for a linear and a nonlinear oscillator