generic features of the fluctuation dissipation relation in coarsening systems the integrated response function in phase ordering systems with scalar vector conserved and nonconserved order parameter is studied at various space dimensionalities assuming scaling of the aging contribution ensuremath chi ag t t w t w ensuremath a ensuremath chi stackrel ifmmode hat else fi ensuremath chi t t w we obtain by numerical simulations and analytical arguments the phenomenological formula describing the dimensionality dependence of a ensuremath chi in all cases considered the primary result is that a ensuremath chi vanishes continuously as d approaches the lower critical dimensionality d l this implies that i the existence of a nontrivial fluctuation dissipation relation and ii the failure of the connection between statics and dynamics are generic features of phase ordering at d l

critical behaviour of a fluid in a random shear flow renormalization group analysis of a simplified model critical behaviour of a fluid binary mixture or liquid crystal subjected to strongly anisotropic turbulent mixing is studied by means of the field theoretic renormalization group as a simplified model relaxational stochastic dynamics of a non conserved scalar order parameter coupled to a random velocity field with prescribed statistics is considered the velocity is taken gaussian white in time with a correlation function of the form t t k d where k is the component of the wave vector perpendicular to the distinguished direction direction of the flow the d dimensional generalization of the ensemble introduced by avellaneda and majda 1990 commun math phys 131 381 within the context of passive scalar advection it is shown that depending on the relation between the exponent and the space dimensionality d the system exhibits various types of large scale self similar behaviour associated with different infrared attractive fixed points of the renormalization group equations in addition to well known asymptotic regimes model a of equilibrium critical dynamics and a passively advected scalar with no self interaction the existence of a new non equilibrium and strongly anisotropic type of critical behaviour universality class is established and the corresponding critical dimensions are calculated to the second order of the double expansion in and e 4 d two loop approximation the most realistic values of the model parameters for example d 3 and the kolmogorov exponent 4 3 belong to this class the scaling behaviour appears anisotropic in the sense that the critical dimensions related to the directions parallel and perpendicular to the flow are essentially different the results are in qualitative agreement with the results obtained in experiments and simulations of fluid systems subjected to various kinds of regular and chaotic anisotropic flows

applications of field theoretic renormalization group methods to reaction diffusion problems we review the application of field theoretic renormalization group rg methods to the study of fluctuations in reaction diffusion problems we first investigate the physical origin of universality in these systems before comparing rg methods to other available analytic techniques including exact solutions and smoluchowski type approximations starting from the microscopic reaction diffusion master equation we then pedagogically detail the mapping to a field theory for the single species reaction ka a k we employ this particularly simple but non trivial system to introduce the field theoretic rg tools including the diagrammatic perturbation expansion renormalization and callan symanzik rg flow equation we demonstrate how these techniques permit the calculation of universal quantities such as density decay exponents and amplitudes via perturbative dc d expansions with respect to the upper critical dimension dc with these basics established we then provide an overview of more sophisticated applications to multiple species reactions disorder effects l evy flights persistence problems and the influence of spatial boundaries we also analyse field theoretic approaches to non equilibrium phase transitions separating active from absorbing states we focus particularly on the generic directed percolation universality class as well as on the most prominent exception to this class even offspring branching and annihilating random walks finally we summarize the state of the field and present our perspective on outstanding problems for the future

finite size scaling properties of the damage distance and dynamical critical exponent for the ising model abstract with the damage spreading method scaling properties of the damage distance on the ising model with heat bath dynamics are studied numerically with the parallel flipping scheme the scaling curves fall on two curves which depend on the odd or even lattice sizes the both scaling curves give the consistent dynamical exponent as z 2 16 0 04 for d 2 and z 2 09 0 05 for d 3 respectively by shifting one of them two curves overlap each other perfectly meanwhile all the scaling curves obtained by single spin flipping processes with different odd or even lattice sizes fall on a single curve from which the consistent dynamical critical exponent with the parallel scheme is obtained z 2 18 0 02 for d 2 and z 2 08 0 04 for d 3

dynamic scaling of disordered ising systems the critical dynamics of site disordered three dimensional ising systems has been studied by conventional local dynamics simulations investing 1015 single spin flips the author has verified that the correlation functions of disordered ising systems obey dynamic scaling behaviour the dynamic scaling functions have been calculated the critical exponent z appears to be strongly dependent on the degree of disorder the author has shown that this dependence arises from a crossover phenomenon which has been detected in the static critical behaviour the dynamical exponent has been estimated to be z 2 4 or 0 1 which is surprisingly near to the renormalization group result in o in 1 2

non linearizable algebraic k actions on affine spaces non linearizable faithful algebraic r r actions of torus on the real affine n space r n are given for each r 1 n 3 when n 4

caustics for inner and outer billiards with a plane closed convex curve t we associate two area preserving twist maps the classical inner billiard int and the outer billiard in the exterior oft the invariant circles of these twist maps correspond to certain plane curves the inner and the outer caustics oft we investigate how the shape oft determines the possible location of caustics establish the existence of open regions which are free of caustics and estimate fro below the size of these regions in terms of the geometry oft

an integrable deformation of an ellipse of small eccentricity is an ellipse the classical birkho conjecture claims that the boundary of a strictly convex integrable billiard table is necessarily an ellipse or a circle as a special case in this article we show that a version of this conjecture is true for tables bounded by small perturbations of ellipses of small eccentricity

break up of resonant invariant curves in billiards and dual billiards associated to perturbed circular tables two area preserving twist maps are associated to a smooth closed convex table the classical billiard map and the dual billiard map when the table is circular these maps are integrable and their phase spaces are foliated by invariant curves the invariant curves with rational rotation numbers are resonant and do not persist under generic perturbations of the circle we present a sufficient condition for the break up of these curves this condition is expressed directly in terms of the fourier coefficients of the perturbation it follows from a standard melnikov argument

exponentially small separatrix splittings and almost invisible homoclinic bifurcations in some billiard tables abstract we present a numerical study of some billiard tables depending on a perturbative parameter 0 and a hyperbolicity parameter h 0 these tables are ellipses for 0 and circumferences in the limit h 0 elliptic billiard tables are integrable and have four separatrices which break up when 0 we conjecture based on numerical experiments that as h 0 the area of the main lobes of the resulting turnstile which can be interpreted as the difference of the lengths of the symmetric primary homoclinic billiard trajectories behaves like an exponential term e 2 h times an asymptotic series j 0 j h 2 j such that 0 0 this series is gevrey 1 of type 1 2 2 so that its borel transform is convergent on a disk of radius 2 2 in the limit 0 the series j 0 j 0 h 2 j is an analytic function which can be explicitly computed with a discrete melnikov method the asymptotic series j 0 j h 2 j associated to the second exponential term e 2 2 h has the same properties finally we have detected some almost invisible homoclinic bifurcations that take place in an exponentially small region of the parameter space our computations have been performed in multiple precision arithmetic namely with several thousands decimal digits and rely strongly on the expansion of the local invariant curves up to very high orders namely with several hundreds taylor coefficients our programs have been written using the pari system

self similar variational perturbation theory for critical exponents we extend field theoretic variational perturbation theory by self similar approximation theory which greatly accelerates convergence this is illustrated by recalculating the critical exponents of o n symmetric ensuremath phi 4 theory from only three loop perturbation expansions in 4 ensuremath dimensions we obtain analytic results for the exponents which are close to those derived recently from ordinary field theoretic variational perturbational theory to seventh order in particular the specific heat exponent is found to be in good agreement with best measured exponent ensuremath alpha ensuremath approx ensuremath 0 0127 of the specific heat peak in superfluid helium found in a satellite experiment in addition our analytic expressions reproduce also the exactly known large n behavior of the exponents

critical exponents for the percolation problem and the yang lee edge singularity gives details of a calculation of critical exponents for a class of field theory models that have an interaction cubic in the fields the results have already been reported give the exponents to third order in epsilon where epsilon 6 d and d is the dimensionality of space the class of models includes the percolation problem and the yang lee edge singularity so the authors give explicit results for the exponents to order epsilon 3 in these cases by resummation methods based on the symptotic behaviour of the epsilon expansion they obtain numerical estimates for these exponents for a number of interesting values of d

exponentially small splitting of separatrices in a weakly hyperbolic case abstract we validate the poincare melnikov method in the singular case of high frequency periodic perturbations of the hamiltonian h 0 x y 1 2 y 2 x 3 x 4 under appropriate conditions which among other things imply that we are considering the bifurcation case when the character of the fixed point changes from parabolic in the unperturbed case to hyperbolic in the perturbed one the splitting is exponentially small

dynamics of 4d symplectic maps near a double resonance abstract we study the dynamics of a family of 4 d symplectic mappings near a doubly resonant elliptic fixed point we derive and discuss algebraic properties of the resonances required for the analysis of a takens type normal form in particular we propose a classification of the double resonances adapted to this problem including cases of both strong and weak resonances around a weak double resonance a junction of two resonances of two different orders both being larger than 4 the dynamics can be described in terms of a simple in general non integrable hamiltonian model the non integrability of the normal form is a consequence of the splitting of the invariant manifolds associated with a normally hyperbolic invariant cylinder we use a 4 d generalisation of the standard map in order to illustrate the difference between a truncated normal form and a full 4 d symplectic map we evaluate numerically the volume of a 4 d parallelotope defined by 4 vectors tangent to the stable and unstable manifolds respectively in good agreement with the general theory this volume is exponentially small with respect to a small parameter and we derive an empirical asymptotic formula which suggests amazing similarity to its 2 d analog different numerical studies point out that double resonances play a key role to understand arnold diffusion this paper has to be seen also as a first step in this direction

field theory of bicritical and tetracritical points i statics we calculate the static critical behavior of systems of o n plus sign in circle o n perpendicular symmetry by the renormalization group method within the minimal subtraction scheme in two loop order summation methods lead to fixed points describing multicritical behavior their stability border lines in the space of the order parameter components n and n perpendicular and spatial dimension d are calculated the essential features obtained already in two loop order for the interesting case of an antiferromagnet in a magnetic field n 1 n perpendicular 2 are the stability of the biconical fixed point and the neighborhood of the stability border lines to the other fixed points leading to very small transient exponents we are also able to calculate the flow of static couplings which allows us to consider the attraction region depending on the nonuniversal background parameters the existence of different multicritical behavior bicritical or tetracritical is possible including a triple point

on the numerical computation of diophantine rotation numbers of analytic circle maps in this paper we present a numerical method to compute diophantine rotation numbers of circle maps with high accuracy we mainly focus on analytic circle diffeomorphisms but the method also works in the case of enough finite differentiability the keystone of the method is that under these conditions the map is conjugate to a rigid rotation of the circle moreover although it is not fully justified by our construction the method turns out to be quite efficient for computing rational rotation numbers we discuss the method through several numerical examples

stability of the phase motion in race track microtrons we model the phase oscillations of electrons in race track microtrons by means of an area preserving map with a fixed point at the origin which represents the synchronous trajectory of a reference particle in the beam we study the nonlinear stability of the origin in terms of the synchronous phase the phase of the synchronous particle at the injection we estimate the size and shape of the stability domain around the origin whose main connected component is enclosed by an invariant curve we describe the evolution of the stability domain as the synchronous phase varies we also clarify the role of the stable and unstable invariant curves of some hyperbolic fixed or periodic points

on the relationship between the bruno function and the breakdown invariant tori we study the ratio epsilon c omega exp eta b omega where epsilon c omega is the breakdown threshold function for invariant tori eta is a parameter and b omega is the bruno function which is purely arithmetic i e it only depends by the continued fractions expansion of number omega we consider the standard map as a model and we focus our analysis on the exponential decay of the chaotic regions close to an invariant torus with diophantine rotation frequency our numerical experiments together with some heuristic consideration show that epsilon c omega exp eta b omega is not a continuous function on diophantine numbers omega for all values of eta

exponentially small splitting of separatrices under fast quasiperiodic forcing we consider fast quasiperiodic perturbations with two frequencies 1 epsiv of a pendulum where is the golden mean number the complete system has a two dimensional invariant torus in a neighbourhood of the saddle point we study the splitting of the three dimensional invariant manifolds associated to this torus provided that the perturbation amplitude is small enough with respect to and some of its fourier coefficients the ones associated to fibonacci numbers are separated from zero it is proved that the invariant manifolds split and that the value of the splitting which turns out to be exponentially small with respect to is correctly predicted by the melnikov function

some remarks on the abundance of stable periodic orbits inside homoclinic lobes abstract we consider a family f of area preserving maps apms with a hyperbolic point h whose invariant manifolds form a figure eight and we study the abundance of elliptic periodic orbits visiting homoclinic lobes epl a domain typically dominated by chaotic behavior to this end we use the chirikov separatrix map sm as a model of the return to a fundamental domain containing lobes we obtain an explicit estimate valid for families f with central symmetry and close to an integrable limit of the relative measure of the set of parameters for which f has epl trajectories to get this estimate we look for epl of the sm with the lowest possible period the analytical results are complemented with quantitative numerical studies of the following families f of apms the sm family and we compare our analytical results with the numerical estimates the standard map stm family and we show how the results referring to the sm model apply to the epl visiting the lobes that the invariant manifolds of the stm hyperbolic fixed point form the conservative henon map family and we estimate the number of a particular type of symmetrical epl related to the separatrices of the 4 periodic resonant islands the results obtained can be seen as the quantitative analogs to those in simo and treschev 2008 9 although here we deal with the a priori stable situation instead

field theory of bicritical and tetracritical points iv critical dynamics including reversible terms this article concludes a series of papers folk holovatch and moser phys rev e 78 041124 2008 78 041125 2008 79 031109 2009 where the tools of the field theoretical renormalization group were employed to explain and quantitatively describe different types of static and dynamic behavior in the vicinity of multicritical points here we give the complete two loop calculation and analysis of the dynamic renormalization group flow equations at the multicritical point in anisotropic antiferromagnets in an external magnetic field we find that the time scales of the order parameters characterizing the parallel and perpendicular ordering with respect to the external field scale in the same way this holds independent whether the heisenberg fixed point or the biconical fixed point in statics is the stable one the nonasymptotic analysis of the dynamic flow equations shows that due to cancellation effects the critical behavior is described in distances from the critical point accessible to experiments by the critical behavior qualitatively found in one loop order although one may conclude from the effective dynamic exponents taking almost their one loop values that weak scaling for the order parameter components is valid the flow of the time scale ratios is quite different and they do not reach their asymptotic values

nonequilibrium critical relaxation with coupling to a conserved density the authors study nonequilibrium critical relaxation properties of model c purely dissipative relaxation of an order parameter coupled to a conserved density starting from a macroscopically prepared initial state with short range correlations using a field theoretic renormalization group approach they show that all the stages of growth of the correlation length display universal behaviour governed by a new critical exponent theta this exponent is calculated to second order in in 4 d where d is the spatial dimension of the system

ageing properties of critical systems in the past few years systems with slow dynamics have attracted considerable theoretical and experimental interest ageing phenomena are observed during this everlasting non equilibrium evolution a simple instance of such a behaviour is provided by the dynamics that takes place when a system is quenched from its high temperature phase to the critical point the aim of this review is to summarize the various numerical and analytical results that have been recently obtained for this case particular emphasis is put on the field theoretical methods that can be used to provide analytical predictions for the relevant dynamical quantities fluctuation dissipation relations are discussed and in particular the concept of fluctuation dissipation ratio fdr is reviewed emphasizing its connection with the definition of a possible effective temperature the renormalization group approach to critical dynamics is summarized and the scaling forms of the time dependent non equilibrium correlation and response functions of a generic observable are discussed from them the universality of the associated fdr follows as an amplitude ratio it is then possible to provide predictions for ageing quantities in a variety of different models in particular the results for models a b and c dynamics of the o n ginzburg landau hamiltonian and model a dynamics of the weakly dilute ising magnet and of the 3 theory are reviewed and compared with the available numerical results and exact solutions the effect of a planar surface on the ageing behaviour of model a dynamics is also addressed within the mean field approximation

non integrability and continuation of fixed points of 2 n dimensional perturbed twist maps abstract in this paper a simple criterion to prove non integrability of symplectic perturbed twist mappings in 2 n dimensions is developed for sufficiently small perturbations in addition an upper bound for the number of isolating integrals the system can possess is provided a criterion for the analytic continuation of isolated periodic orbits in case of a small nonzero perturbation of twist maps is found the evaluation of their linear stability character by obtaining a simplified expression of the eigenvalues of the jacobian matrix concludes the theoretical part the theory is finally applied to a twist map perturbed by the suris potential and also a four dimensional map with physical interest

model c critical dynamics of disordered magnets the critical dynamics of model c in the presence of disorder is considered it is known that in the asymptotics a conserved secondary density decouples from the nonconserved order parameter for disordered systems however couplings between order parameter and secondary density cause considerable effects on non asymptotic critical properties here a general procedure for a renormalization group treatment is proposed already the one loop approximation gives a qualitatively correct picture of the diluted model c dynamical criticality a more quantitative description is achieved using two loop approximation in order to get reliable results resummation technique has to be applied

exponentially small splitting of separatrices beyond melnikov analysis rigorous results in this paper we study the problem of exponentially small splitting of separatrices of one degree of freedom classical hamiltonian systems with a non autonomous perturbation which is fast and periodic in time we provide a result valid for general systems which are polynomials or trigonometric polynomials in the state variables our result consists in obtaining a rigorous proof of the asymptotic formula for the measure of the splitting we have obtained that the splitting has the asymptotic behavior k e a identifying the constants k and a in terms of the features of the system the study of our problem leads us to consider several cases in some cases assuming the per turbation is small enough it turns out that the values of k coincides with the classical melnikov approach we have identified the limit size of the perturbation for which this classical theory holds true however for the limit cases which appear naturally both in averaging theory and bifurcation theory we encounter that generically neither k nor are actually well predicted by melnikov theory

melnikov method and exponentially small splitting of separatrices abstract we study the splitting of separatrices for high frequency perturbations of a planar hamiltonian system it is well known that in the analytic case the splitting of separatrices is exponentially small with respect to the perturbation period 2 e and that the constant in the exponent is related to the position of singularities of the unperturbed separatrix in the complex time plane on the other hand in many examples the melnikov function can be calculated explicitly and is exponentially small with an exponent very close but larger than the exponent from the upper estimates so the melnikov function appears to be smaller than the upper estimate as well as the standard estimate of the approximation error we found a class of hamiltonians such that the melnikov function gives actual asymptotics for the splitting provided the perturbation amplitude is less than 0ep for some p 0

rigorous lower bound on the dynamic critical exponents of the swendsen wang algorithm we prove the rigorous lower bound z sw v for the dynamic critical exponent of the swendsen wang algorithm for two dimensional q state potts models with q 2 3 4 this implies z sw 0 2 5 1 we present numerical data indicating that z sw 0 55 0 03 0 89 0 05 for q 3 4 95 confidence limits statistical errors only the discrepancy for q 4 appears to be caused by multiplicative logarithmic corrections

effects of mixing and stirring on the critical behaviour stochastic dynamics of a nonconserved scalar order parameter near its critical point subject to random stirring and mixing is studied using the field theoretic renormalization group the stirring and mixing are modelled by a random external gaussian noise with the correlation function t t k4 d y and the divergence free due to incompressibility velocity field governed by the stochastic navier stokes equation with a random gaussian force with the correlation function depending on the relations between the exponents y and y and the space dimensionality d the model reveals several types of scaling regimes some of them are well known model a of equilibrium critical dynamics and linear passive scalar field advected by a random turbulent flow but there are three new non equilibrium regimes universality classes associated with new nontrivial fixed points of the renormalization group equations the corresponding critical dimensions are calculated in the two loop approximation second order of the triple expansion in y y and e 4 d

three dimensional randomly dilute ising model monte carlo results we perform a high statistics simulation of the three dimensional randomly dilute ising model on cubic lattices l 3 with l le 256 we choose a particular value of the density x 0 8 for which the leading scaling corrections are suppressed we determine the critical exponents obtaining nu 0 683 3 eta 0 035 2 beta 0 3535 17 and alpha 0 049 9 in agreement with previous numerical simulations we also estimate numerically the fixed point values of the four point zero momentum couplings that are used in field theoretical fixed dimension studies although these results somewhat differ from those obtained using perturbative field theory the field theoretical estimates of the critical exponents do not change significantly if the monte carlo result for the fixed point is used finally we determine the six point zero momentum couplings relevant for the small magnetization expansion of the equation of state and the invariant amplitude ratio r xi that expresses the universality of the free energy density per correlation volume we find r xi 0 2885 15

on the crossover to universal criticality in dilute ising systems monte carlo simulations of the critical behaviour of disordered ising systems have led to concentration dependent critical exponents which seem to violate universality we apply the renormalization group to investigate the crossover effects which cause the observed behaviour we improve the three loop expansion of the callan symanzik and wilson functions by a pade borel approximation and solve the flow equations for various initial points the exponents found in the simulations can be related to regions in the space of coupling coefficients away from the fixed point

barriers to transport and mixing in volume preserving maps with nonzero flux abstract we identify some geometric structures secondary tori that restrict transport and prevent mixing in perturbations of integrable volume preserving systems with nonzero net flux unlike the customary kam tori secondary tori cannot be continued to the tori present in the integrable system but are generated by resonances and have a contractible direction we also note that secondary tori persist under the addition of a net flux which destroys all customary kam tori we introduce a remarkably simple algorithm to analyze the behavior of volume preserving maps and to obtain quantitative properties of the secondary tori we then implement the algorithm and after running it present assertions regarding the distribution of the escape times of the unbounded orbits the abundance of secondary tori the size of the resonant regions and the robustness of the tori under the addition of a mean flux

universality and criticality of a second order granular solid liquid like phase transition the authors experimentally study a vibrating granular layer that shows a transition between a disordered liquidlike phase and a phase with ordered crystalline clusters they study the properties of this nonequilibrium phase transition and find evidence for universality in the critical exponents

universality in the off equilibrium critical dynamics of the three dimensional diluted ising model we study the off equilibrium critical dynamics of the three dimensional diluted ising model we compute the dynamical critical exponent z and we show that it is independent of the dilution only when we take into account the scaling corrections to the dynamics finally we will compare our results with the experimental data

simulation of static critical phenomena in nonideal fluids with the lattice boltzmann method a fluctuating nonideal fluid at its critical point is simulated with the lattice boltzmann method it is demonstrated that the method employing a ginzburg landau free energy functional correctly reproduces the static critical behavior associated with the ising universality class a finite size scaling analysis is applied to determine the critical exponents related to the order parameter compressibility and specific heat a particular focus is put on finite size effects and issues related to the global conservation of the order parameter

on low dimensional cancellation problems abstract a well known cancellation problem of zariski asks when for two given domains fields k 1 and k 2 over a field k a k isomorphism of k 1 t k 1 t and k 2 t k 2 t implies a k isomorphism of k 1 and k 2 the main results of this article give affirmative answer to the two low dimensional cases of this problem 1 let k be an affine field over an algebraically closed field k of any characteristic suppose k t k t 1 t 2 t 3 then k k t 1 t 2 2 let m be a 3 dimensional affine algebraic variety over an algebraically closed field k of any characteristic let a k x y z w m be the coordinate ring of m suppose a t k x 1 x 2 x 3 x 4 then frac a k x 1 x 2 x 3 where frac a is the field of fractions of a in the case of zero characteristic these results were obtained by kang in ming chang kang a note on the birational cancellation problem j pure appl algebra 77 1992 141 154 ming chang kang the cancellation problem j pure appl algebra 47 1987 165 171 however the case of finite characteristic is first settled in this article that answered the questions proposed by kang in ming chang kang a note on the birational cancellation problem j pure appl algebra 77 1992 141 154 ming chang kang the cancellation problem j pure appl algebra 47 1987 165 171

embeddings of hypersurfaces in affine spaces abstract in this paper we address the following two general problems given two algebraic varieties in cn find out whether or not they are 1 isomorphic and 2 equivalent under an automorphism of cn although a complete solution of either of those problems is out of the question at this time we give here some handy and useful invariants of isomorphic as well as of equivalent varieties furthermore and more importantly we give a universal procedure for obtaining all possible algebraic varieties isomorphic to a given one and use it to construct numerous examples of isomorphic but inequivalent algebraic varieties in cn among other things we establish the following interesting fact for isomorphic hypersurfaces p x1 xn 0 and q x1 xn 0 the number of zeros of grad p might be different from that of grad q this implies in particular that although the fibers p 0 and q 0 are isomorphic there are some other fibers p c and q c which are not we construct examples like this for any n 2

a refined formula for the separatrix splitting for the standard map abstract we study the standard map for small values of a parameter e an asymptotic formula is found which describes the size of the splitting of the stable and unstable separatrices of the origin considered as a hyperbolic fixed point the formula contains an exponentially decreasing expression times an asymptotic series on powers of a small parameter the coefficients in the asymptotic expansion are to be found in the process of solving numerically a nonlinear recurrent system of difference equations in the complex domain

rapidly forced planar vector fields and splitting of separatrices abstract we prove that for analytic rapidly forced two dimensional systems the melnikov function is exponentially small with respect to the period we provide an example of non validity of melnikov s method and we obtain upper bounds for conservative and analytic planar systems for big amplitude periodic rapidly forcing

non equilibrium dynamics of the ising model for t less than equal totc the growth of order in ising models with nonconserved order parameter is considered for quenches to final temperatures tf 0 and tf tc the results of numerical simulations in spatial dimension d 2 are presented in all cases a scaling regime is entered for sufficiently long times where the characteristic length scale is the domain size l t approximately t12 for tf 0 and the nonequilibrium correlation length xi t approximately t1z for tf tc the equal time correlation function has the expected scaling forms f r l t and r d 2 eta fc r xi t for tf 0 and tc respectively the scaling function fc x has interesting short distance behaviour which is elucidated using scaling arguments and by in and 1 n expansions the t 0 scaling function f x depends on whether the spin correlations present in the initial conditions are of long or short range as does the exponent which describes the decay of the autocorrelation function a t si t si o approximately l t results for a quench from the equilibrium critical state to tf 0 are consistent with theoretical predictions

hamiltonian maps in the complex plane abstract the standard map is a nonintegrable discrete time analog of the vertical pendulum detailed calculations are presented and illustrated graphically for the standard map at the golden mean frequency the functional dependence of the coordinate q on the canonical angle variable is analtyically continued into the complex plane where natural boundaries are found at constant absolute values of im the boundaries represent the appearance of chaotic motion in the complex plane when the domain of analyticity shrinks to zero the kam invariant curve is destroyed two independent numerical methods with fourier analysis in the angle variable were used one based on a variation annihilation method and the other on a double expansion the results were further checked by direct solution of the complex equations of motion the numerically simpler but intrinsically complex semipendulum and semistandard map are also studied we conjecture that natural boundaries appear in the analogous analytic continuation of the invariant tori or kam surfaces of general nonintegrable systems with analytic hamiltonians

two loop field theory and nonasymptotic properties of the dynamical model for the lambda transition in 3he 4he mixtures model f introduced by siggia and nelson phys rev b 15 1427 1977 describes the critical dynamics of 3he 4he mixtures near the superfluid transition using the minimal subtraction scheme this model is renormalized within dynamical field theory the dynamic zeta functions needed for the nonasymptotic flow properties are presented in two loop order the fixed points are discussed and the stable fixed points are identified the transition to limiting models contained in model f is shown analytically by performing the corresponding limits and numerically by calculating the nonlinear flow these results are the basis for further experimental comparison of the transport coefficents in 3he 4he mixtures at higher concentrations including the tricritical point

a proof of the exponentially small transversality of the separatrices for the standard map in 1984 v f lazutkin laz84 lst89 obtained an asymptotic formula for the separatrix splitting angle for the standard map the difficulty of this problem is related to the exponential smallness of the splitting with respect to a perturbation parameter lazutkin s proof was based on two conjectures probably the original form of those conjectures was incorrect but lazutkin s method was very efficient and inspired a large number of studies on the exponentially small splitting of separatrices the consequent works laz91 laz92 gls94 and gel96 prepared the base to fill all the gaps of the original proof the present paper contains a complete and self contained proof of a refined version of the original formula formula 1 7 of the present paper in this form the formula was obtained in gls94 the proof is inspired by the ideas of lazutkin s original paper laz84

an obstruction method for the destruction of invariant curves abstract an important point in a conservative dynamical system which is a parameter depending perturbation of an integrable system is the destruction of the invariant curves for small perturbations many invariant curves subsist according to kam theorem an increase in the size of perturbations produces eventually the destruction of the curves the stochastic zone is then very large in this paper we present for the standard map a geometrical approach to obtain values of the parameter for which no invariant curves exist this is due to the existence of heteroclinic points as it has been observed using other methods there are several scaling properties in the phenomena which are also discussed here

splitting of separatrices for standard and semistandard mappings abstract we investigate chirikov s standard mapping which is a model for the onset of nonintegrability in nonlinear hamiltonian systems the asymptotic formula when the deviation from an integrable system is small for the angle of intersection of separatrices at a homoclinic point is derived on the basis of approximation by the green percival semistandard mapping in the complex domain computed data on the splitting of separatrices for the semi standard map are presented from which we derive the pre exponential coefficient in the formula for splitting of separatrices in the standard mapping

continuous symmetries of difference equations lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations in this article we review the results of a much more recent program the use of lie groups to study difference equations we show that the mismatch between continuous symmetries and discrete equations can be resolved in at least two manners one is to use generalized symmetries acting on solutions of difference equations but leaving the lattice invariant the other is to restrict them to point symmetries but to allow them to also transform the lattice

non equilibrium relaxation at a tricritical point we study the purely relaxational tricritical dynamics of a non conserved order parameter model a in the upper critical dimension dc 3 and in 3 epsilon dimensions we are especially interested in the relaxation starting from a macroscopically prepared initial state with a small correlation length using the methods of renormalized field theory we obtain the scaling behaviour of the correlation and response functions and study the nonlinear relaxation of the order parameter m t in three dimensions m t displays a crossover from the purely logarithmic short time behaviour m t approximately in t t0 a to a t 14 power law with logarithmic corrections for dimensions d 3 we obtain the exponents which govern the tricritical relaxation at lowest non trivial order in epsilon 3 d the dynamic scaling exponent z is calculated at second order in epsilon

on stochastic sea of the standard map consider a generic one parameter unfolding of a homoclinic tangency of an area preserving surface diffeomorphism we show that for many parameters residual subset in an open set approaching the critical value the corresponding diffeomorphism has a transitive invariant set of full hausdorff dimension the set is a topological limit of hyperbolic sets and is accumulated by elliptic islands

dynamical relaxation and universal short time behavior of finite systems a system belonging to the dynamic universality class of model a is considered in a block v ld geometry with periodic boundary conditions the relaxation of the order parameterm t from an initial value m i is investigated at the bulk critical temperature we demonstrate that a proper scaling description of the problem involves two characteristic times tl lz andtl m i z x wherez is the familiar dynamic bulk exponent while xi is an independent new bulk exponent discovered recently previous analyses of the problem either were restricted tot ti or tacitly used the incorrect assumption thatxi v thus the short time regimet ti with universal dependence on m i was missed as a concrete example we study the exact solution in the large n limit

critical dynamics a field theoretical approach we review the progress made in dynamic bulk critical behaviour in equilibrium in the last 25 years since the review of halperin and hohenberg we unify the presentation of the theoretical background by restricting ourselves to the field theoretic renormalization group method the main results obtained in the different universality classes are presented this contains the critical dynamics near the gas liquid transition in pure fluids model h the plait point and consolute point in mixtures model h the superfluid transition in 4 he model f and 4 he 3 he mixtures model f the curie point model j and neel point model g in heisenberg magnets and the superconducting transition in comparison with experimental results it became clear that in most cases one has to consider apart from the universal asymptotic critical behaviour also the non universal effective behaviour either because it turned out to be inevitable due to a small dynamical transient exponent inhibiting the system to reach the asymptotics e g at the superfluid transition or because one is interested in the region further away from the phase transition like in pure fluids and mixtures at their gas liquid or demixing transition the calculation of the critical dynamics is adequate in most cases only in two loop order we review these results and present the solution to unreasonable features found for some models thus we consider model c where relaxational and diffusive dynamics are coupled and the scaling properties and the limit to a purely relaxational model model a have not been understood in general for models where the order parameter couples to other conserved densities time scale ratios between the kinetic coefficients of the order parameter and the conserved densities play an important role their fixed point values and the approach to the fixed point are changed considerably in two loop order compared to their values in one loop order these considerations are relevant for the explanation of the dynamical critical shape functions of systems such as superfluid helium model f and the isotropic antiferromagnet model g as far as possible the comparison of results obtained by the renormalization group theory with numerical simulations has been made

renormalization group theory for cooling first order phase transitions in potts models we develop a dynamic field theoretic renormalization group rg theory for cooling first order phase transitions in the potts model it is suggested that the well known imaginary fixed points of the q state potts model for q 10 3 in the rg theory are the origin of the dynamic scaling found recently from numerical simulations apart from logarithmic corrections this indicates that the real and imaginary fixed points of the potts model are both physical and control the scalings of the continuous and discontinuous phase transitions respectively of the model our one loop results for the scaling exponents are already not far away from the numerical results further the scaling exponents depend on q only slightly consistent with the numerical results therefore the theory is believed to provide a natural explanation of the dynamic scaling including the scaling exponents and their scaling laws for various observables in the cooling first order phase transition of the potts model

minimal renormalization without epsilon expansion four loop free energy in three dimensions for general n above and below tc we present an analytic four loop calculation of the free energy in three dimensions within the o n symmetric ensuremath varphi 4 theory at infinite cutoff for general n above and below t c it is shown that goldstone singularities arising at intermediate stages of the calculation cancel among themselves the correlation length above t c and an appropriately defined pseudocorrelation length below t c are calculated analytically up to four loop order for general n the method of minimal renormalization at fixed dimension d 3 is used to determine the analytic expressions for the four loop series of the amplitude functions of the free energy correlation length and specific heat above and below t c in terms of the renormalized coupling these expressions provide the basis for future accurate borel resummations of universal amplitude ratios characterizing the asymptotic critical behavior and of crossover functions describing the nonasymptotic critical behavior a brief application is given by a variational calculation of the universal specific heat amplitude ratios a a ensuremath and p ensuremath alpha ensuremath 1 1 ensuremath a a ensuremath and of the universal quantity r ensuremath xi ensuremath xi 0 a 1 d for general n

short time critical dynamics of the baxter wu model we study the early time behavior of the baxter wu model an ising model with three spin interactions on a triangular lattice our estimates for the dynamic exponent z are compatible with results recently obtained for two models which belong to the same universality class of the baxter wu model the two dimensional four state potts model and the ising model with three spin interactions in one direction however our estimates for the dynamic exponent ensuremath theta of the baxter wu model are completely different from the values obtained for those models this discrepancy could be related to the absence of a marginal operator in the baxter wu model

affine varieties with equivalent cylinders abstract a well known cancellation problem asks when for two algebraic varieties v 1 v 2 c n the isomorphism of the cylinders v 1 c and v 2 c implies the isomorphism of v 1 and v 2 in this paper we address a related problem when the equivalence under an automorphism of c n 1 of two cylinders v 1 c and v 2 c implies the equivalence of their bases v 1 and v 2 under an automorphism of c n we concentrate here on hypersurfaces and show that this problem establishes a strong connection between the cancellation conjecture of zariski and the embedding conjecture of abhyankar and sathaye we settle the problem in the affirmative for a large class of polynomials on the other hand we give examples of equivalent cylinders with inequivalent bases those cylinders however are not hypersurfaces another result of interest is that for an arbitrary field k the equivalence of two polynomials in m variables under an automorphism of k x 1 x n n m implies their equivalence under a tame automorphism of k x 1 x 2 n

splitting of a small separatrix loop near the saddle center bifurcation in area preserving maps abstract when the saddle center bifurcation occurs in an analytic family of area preserving maps a parabolic fixed point first appears at the origin and then this point bifurcates creating an elliptic fixed point and a hyperbolic one separatrices of the hyperbolic fixed point form a small loop around the elliptic point in general the separatrices intersect transversely and the splitting is exponentially small with respect to the perturbation parameter we derive an asymptotic formula which describes the splitting and study the properties of the pre exponential factor

damage spreading and critical exponents for model a ising dynamics using damage spreading and heat bath dynamics we study the ising model in 2 and 3 dimensions with non conservative dynamics our algorithm differs in some important points from previous ones which makes it rather efficient we give estimates for the exponent z which seem to be the most precise published so far 2 172 0 006 for d 2 2 032 0 004 for d 3 we also give precise estimates of the exponent introduced by janssen et al z phys b 73 1989 539 and of analogous but in principle independent exponents we find surprisingly that some of the latter agree with and give an explanation for this

on generalized cancellation problem abstract a well known cancellation problem of zariski asks whether for two given domains fields k 1 and k 2 an isomorphism of k 1 t k t and k 2 t k 2 t implies an isomorphism of k 1 and k 2 in this paper we address a related problem whether the ring field embedding of k 1 t k 1 t into k 2 t k 2 t implies the ring field embedding of k 1 into k 2 our main result is affirmative if k 1 and k 2 are arbitrary domains fields of the finite transcendence degree and k 1 t k 1 t can be embedded into k 2 t k 2 t then k 1 can be embedded into k 2 as a consequence we answer a question of abhyankar eakin and heinzer j algebra 23 1972 310 342

study of the double mathematical pendulum ii investigation of exponentially small homoclinic intersections we consider the double mathematical pendulum in the limit when the ratio of pendulum masses is close to zero and the ratio of pendulum lengths is close to infinity we found that the limit system has a hyperbolic periodic trajectory whose invariant manifolds intersect transversally and the intersections are exponentially small in this case we obtain an asymptotic formula of the homoclinic invariant for the limit system

embeddings of danielewski surfaces a danielewski surface is defined by a polynomial of the form p x nz p y define also the polynomial p x nz r x p y where r x is a non constant polynomial of degree n 1 and r 0 1 we show that when n 2 and deg p y 2 the general fibers of p and p are not isomorphic as algebraic surfaces but that the zero fibers are isomorphic consequently for every non special danielewski surface s there exist non equivalent algebraic embeddings of s in 3 using different methods we also give non equivalent embeddings of the surfaces xz y d n 1 for an infinite sequence of integers d n we then consider a certain algebraic action of the orthogonal group mathcal o 2 on 4 which was first considered by schwarz and then studied by masuda and petrie who proved that this action could not be linearized this was done by comparing the strata of this action to those of the induced tangent space action inequivalent embeddings of a certain singular danielewski surface s in 3 are found we generalize their result and show how this leads to an example of two smooth algebraic hypersurfaces in 3 which are algebraically non isomorphic but holomorphically isomorphic

is there a true model d critical dynamics we show that non locality in the conservation of both the order parameter and a non critical density model d dynamics leads to new fixed points for critical dynamics depending upon the parameters characterizing the non locality in the two fields we find four regions i model a like where both conservations are irrelevant ii model b like with the conservation in the order parameter field relevant and the conservation in the coupling field irrelevant iii model c like where the conservation in the order parameter field is irrelevant but the conservation in the coupling field is relevant and iv model d like where both conservations are relevant while the first three behaviours are already known in dynamical critical phenomena the last one is a novel phenomenon due entirely to the non locality in the two fields

birkhoff periodic orbits for small perturbations of completely integrable hamiltonian systems with convex hamiltonians pour des systemes a n degres de liberte on montre l existence d au moins n orbites periodiques distinctes avec un vecteur frequence rationnel admissible proche du tore correspondant du systeme non perturbe sous l hypothese que le hamiltonien du systeme non perturbe a des surfaces d energie convexes pour les variables action angle

persistence in an antiferromagnetic ising system with conserved magnetisation abstract we obtain the persistence exponents for an antiferromagnetic ising system in which the magnetisation is kept constant this system belongs to model c system with non conserved order parameter with a conserved density and is expected to have persistence exponents different from that of model a system with no conservation but independent of the conserved density our numerical results for both local persistence at zero temperature and global persistence at the critical temperature however indicate that the exponents are dependent on the conserved magnetisation in both two and three dimensions this non universal feature is attributed to the presence of the conserved field and is special to the persistence phenomena

on the critical fluctuations in superconductors the field theoretical gauge model for a superconductor generalized to an n 2 component complex order parameter is considered the question of the order of the phase transition occurring in this model is discussed previous renormalization group calculations suggest that in such a model a fluctuation induced first order phase transition occurs we re examine previously obtained expressions for the renormalization group functions in the two loop approximation in three dimensions special attention is being paid to the fact that the corresponding series might be the asymptotic ones and therefore have zero radius of convergence we discuss the possible ways of analytical continuation of the series obtained comparing results obtained by direct calculations with those obtained by pade analysis and the pade borel resummation technique the conjecture is made that in the model under consideration there still exists a possibility for a second order phase transition

time reversal symmetry in dynamical systems a survey abstract in this paper we survey the topic of time reversal symmetry in dynamical systems we begin with a brief discussion of the position of time reversal symmetry in physics after defining time reversal symmetry as it applies to dynamical systems we then introduce a major theme of our survey namely the relation of time reversible dynamical sytems to equivariant and hamiltonian dynamical systems we follow with a survey of the state of the art on the theory of reversible dynamical systems including results on symmetric periodic orbits local bifurcation theory homoclinic orbits and renormalization and scaling some areas of physics and mathematics in which reversible dynamical systems arise are discussed in an appendix we provide an extensive bibliography on the topic of time reversal symmetry in dynamical systems

break up of invariant surfaces in action angle angle maps and flows abstract an important class of three dimensional volume preserving maps and flows arises as a perturbation from integrable action action angle maps and flows the properties of this class of maps and flows are discussed while action angle angle volume preserving maps admit an analog of the kam theorem general results on nonexistence of two dimensional invariant manifolds of action action angle maps are proven here nonexistence of such two dimensional invariant manifolds means possibility of global transport and a mechanism for such transport the local mechanism of resonance induced dispersion phys rev lett 75 1995 3669 is studied perturbatively resonance induced dispersion is shown to arise from the existence of periodic orbits that survive perturbation at places where two dimensional invariant manifolds break down

asymptotic expansion of homoclinic structures in a symplectic mapping the phenomenon of separatrix splitting and complicated homoclinic structures constitute a symptom of chaos as pointed out by poincare more than a century ago taking a time discrete dynamical system with a double well potential as an example this interesting feature is demonstrated analytically on the basis of asymptotic expansions beyond all orders violent undulations of the unstable manifold in the extreme vicinity of hyperbolic fixed points are recovered excellently comparison with results for standard and henon maps is also made

exponentially small splitting a direct approach abstract in this paper we go beyond what was proposed in theory by melnikov 15 to introduce a practical method to calculate the high order splitting distances of stable and unstable manifold in time periodic equations not only we derive integral formula for splitting distances of all orders but also we develop an analytic theory to evaluate the acquired multiple integrals we reveal that the dominance of the exponentially small poincare melnikov function for equations of high frequency perturbation is caused by a certain symmetry embedded in the kernel functions of high order melnikov integrals this symmetry is beheld by many non hamiltonian equations

chaotic boundary of a hamiltonial map abstract the half plane maps are a class of complex hamiltonial maps whose invariant curves at most fixed irrational frequencies can be obtained as a convergent taylor series expansion for these maps the boundary between regular ordered motion on invariant curves and irregular chaotic motion is given by the radius of convergence of the series the successive terms of the series oscillate wildly due to the presence of small divisors methods are presented for taming the series based on the conversion of the convergenceexponentc ln c into the integral of a continuous but nondifferentiable lambda function whose graph whows a similarity structure on small scales self similarity properties are illustrated for the chaotic boundary function c where v is the frequency

non equilibrium critical relaxation with reversible mode coupling the non equilibrium critical relaxation of systems in which the order parameter couples reversibly to conserved densities is investigated the initial state which may be macroscopically prepared by a quench from a temperature t tc to the critical temperature tc is characterized by short range correlations in the case where the order parameter itself is conserved the relaxation shows universal scaling if on the other hand the order parameter is not conserved the scaling is governed by a new exponent which depends on the width of the initial distribution of the conserved fields however the response function of the conserved fields still shows universal scaling

critical dynamics of stochastic models with energy conservation model c we calculate the field theoretic functions of the generalized dynamical model c in which a conserved secondary density is coupled to a nonconserved complex order parameter in two loop order we show that the fixed points in this extended model are equal to the fixed points obtained in model c with a real order parameter which has been introduced by halperin hohenberg and ma our results correct long standing errors in the field theoretic functions in model c published by several authors leading also to different fixed point values w for the ratio of the two time scales involved the stability regions of the fixed points which remained partially unclear considered in a phase diagram whose axes are the spatial dimension d and number of order parameter components n are now clarified especially an anomalous region found by previous authors in which the scaling properties remained unsolved does not exist there are only two regions one with a finite fixed point w where the dynamical exponent z of the order parameter is z 2 alpha nu and another region where w 0 and z is equal to the model a value

universal aging properties at a disordered critical point we investigate analytically near the dimension d uc 4 and numerically in d 3 the nonequilibrium relaxational dynamics of the randomly diluted ising model at criticality using the exact renormalization group method to one loop we compute the two times t phantom rule 0 2em 0ex t w correlation function and fluctuation dissipation ratio fdr for any fourier mode of the order parameter of finite wave vector q in the large time separation limit the fdr is found to reach a nontrivial value x ensuremath infty independently of small q and coincide with the fdr associated to the total magnetization obtained previously explicit calculations in real space show that the fdr associated to the local magnetization converges in the asymptotic limit to this same value x ensuremath infty through a monte carlo simulation we compute the autocorrelation function in three dimensions for different values of the dilution fraction p at t c p taking properly into account the corrections to scaling we find according to the renormalization group predictions that the autocorrelation exponent ensuremath lambda c is independent of p the analysis is complemented by a study of the nonequilibrium critical dynamics following a quench from a completely ordered state

dynamics in chaotic zones of area preserving maps close to separatrix and global instability zones abstract the purpose of this paper is to study phenomena in chaotic zones of area preserving maps using simpler models which are easier to analyse theoretically and numerically first of all the study of the dynamics in a neighbourhood of the separatrices of a resonant zone is carried out the well known separatrix map defined on a figure eight when needed is used to determine the location of rotational invariant curves r i c inside and outside the resonance the interest in this part is on a quantitative description of the dynamics in a neighbourhood of the separatrices to produce theoretical estimates of the width of the stochastic zone distance to the r i c existence of tiny islands close to the separatrices in every one of the studied items one has tried to complement the limit analytic study with realistic numerical simulations describing the analogy when possible after this study we focus on the formation of larger domains without r i c e g birkhoff domains to this end we introduce the biseparatrix map model although this is a qualitative model the mechanism of destruction of the last r i c and hence the process of creation of zones without r i c is clarified by means of this simple model several numerical examples illustrate the results obtained and are used as a test of the theoretical quantitative predictions

the generalized stable equivalence problem in this note we study the following problem let k be an algebraically closed field and x be an affine variety over k suppose that h1 h2 x are two hypersurfaces such that there exists an automorphism f of x kn satisfying f h1 kn h2 kn for some n 0 does this imply that there exists an automorphism f of x such that f h1 h2 we give an affirmative solution if one hypersurface is not k uniruled and a counterexample in general

late stages of coarsening in model c abstract we present a comprehensive picture of non critical domain growth in model c systems where a non conserved scalar order parameter is coupled to a conserved concentration field for quenches into the region where the ordered and disordered phases coexist we confirm earlier partial numerical results and find a growth exponent z 3 for quenches into the ordered region we confirm the theoretical prediction z 2 finally we discuss the implications of our results for domain growth in the microcanonical 4 model and we offer some criticism of the work of somoza and sagui on the morphology and wetting properties of domains

the strong nagata conjecture it is proved that there exist wild coordinates in the polynomial algebra in three variables over a field of characteristic zero this result implies the famous nagata conjecture