transient measures in the standard map abstract for an area preserving map each chaotic orbit appears numerically to densely cover a region an irregular component of nonzero area surprisingly the measure approximated by a long segment of such an orbit deviates significantly from a constant on the irregular component most prominently there are spikes in the density near the boundaries of the irregular component resulting from the stickiness of its bounding invariant circles we show that this phenomena is transient and therefore numerical ergodicity on the irregular component eventually obtains though the times involved are extremely long 1010 iterates a markov model of the transport shows that the density spikes cannot be explained by the stickiness of a bounding circle of a single class for example a rotational circle however the density spikes do occur in a markov tree model that includes the effects of islands around islands

an investigation of chaotic diffusion in a family of hamiltonian mappings whose angles diverge in the limit of vanishingly action the chaotic diffusion for a family of hamiltonian mappings whose angles diverge in the limit of vanishingly action is investigated by using the solution of the diffusion equation the system is described by a two dimensional mapping for the variables action i and angle theta and controlled by two control parameters i epsilon controlling the nonlinearity of the system particularly a transition from integrable for epsilon 0 to non integrable for epsilon ne 0 and ii gamma denoting the power of the action in the equation defining the angle for epsilon ne 0 the phase space is mixed and chaos is present in the system leading to a finite diffusion in the action characterized by the solution of the diffusion equation the analytical solution is then compared to the numerical simulations showing a remarkable agreement between the two procedures

large excursions of action within the resonance of a degenerate hamiltonian system with two degrees of freedom we investigate the width of the resonance zone in a degenerate hamiltonian system with two degrees of freedom in which the hamiltonian lacks the quadratic term in the taylor expansion this leads to larger excursions of action in the phase space than the nondegenerate one and corresponding resonance frequency widths would become narrower however in contrast to the nonautonomous hamiltonian system with one and half degree of freedom we find that the above case is not generic and only occurs at particular resonances an example relevant to the interaction of resonances is considered analytic results are verified in numerical simulations

threshold to global diffusion in a nonmonotonic map with quadratic nonlinearity abstract in this paper the threshold to global diffusion of a nonmonotonic map with quadratic nonlinearity using a forcing function k sin as in the standard map is analyzed for low values of the stochastic parameter k the breaking of the last kam curve is caused by the period one reconnecting resonance at higher k secondary resonances play an increasingly important role two kinds of kam curves are studied those between reconnecting resonances and those outside a reconnecting resonance pair using a perturbative hamiltonian method to determine the resonance width a weak overlap criterion is used to estimate the breaking of the last kam curve outside of the reconnecting islands for some values of the parameters a nonintegrable reconnecting threshold is found above the threshold of global diffusion in this regime the reconnection increases the diffusion coefficient to a value close to the quasilinear value of the standard map

finite size scaling and the crossover to mean field critical behavior in the two dimensional ising model with medium ranged interactions critical amplitudes in finite size scaling relations show a singular dependence on the range of the interactions r the respective power laws are predicted from phenomenological crossover scaling considerations these predictions are tested by monte carlo simulations for medium ranged ising square lattices it is speculated that some deviations between the simulation results and corresponding predictions may be due to logarithmic corrections

classical to critical crossovers from field theory we extend our previous determinations of nonasymptotic critical behavior of phys rev b 32 7209 1985 and 35 3585 1987 to accurate expressions of the complete classical to critical crossover in three dimensional field theory in terms of the temperaturelike scaling field i e along the critical isochore for 1 the correlation length the susceptibility and the specific heat in the homogeneous phase for the n vector model n 1 to 3 and 2 the spontaneous magnetization coexistence curve the susceptibility and the specific heat in the inhomogeneous phase for the ising model n 1 the present calculations include the seventh loop order of murray and nickel and closely account for the up to date estimates of universal asymptotic critical quantities exponents and amplitude combinations provided by guida and zinn justin j phys a 31 8103 1998

nonmonotonic twist maps abstract we investigate a family of nonmonotonic radial twist maps constructed by unfolding the linear twist function we find several new modes of reconnection and derive conditions for them to occur reconnection and bifurcation of higher order resonances are studied in detail including vortex pairs and triplets the structural instability of some of these exotic figures is mitigated by the presence of chaotic separatrix layers invariant curves of involution pairs are utilized to locate and chart their often very complex metamorphoses

boundary circles for area preserving maps abstract we report on numerical observations of various properties of boundary circles for area preserving maps we find that the continued fraction expansion of their rotation number generally has a special form we describe how their rotation number changes as parameters are varied we observe a statistically self similar structure in the neighbourhood of boundary circles which is probably relevant to the long time tails of the decay of correlations in the irregular components

critical behavior of a kam surface i empirical results kolmogorov arnol d moser kam surfaces are studied in the context of a perturbed two dimensional twist map in particular we ask how a kam surface can disappear as the perturbation parameter is increased following greene we use cycles to numerically construct the kam curve and discover that at the critical coupling it shows structure at all length scales aspects of this structure are fitted by a scaling analysis critical indices and scaling functions are determined numerically some evidence is presented which suggests that the results are universal

locating invariant tori for a family of two dimensional hamiltonian mappings departamento de estatistica matemat apl e computaoinstituto de geociencias e ciencias exatas unesp univ estadual paulista av 24a 1515bela vista cep 13506 900 rio claro sp

finite size scaling in the p state mean field potts glass a monte carlo investigation the p state mean field potts glass with bimodal bond distribution j is studied by monte carlo simulations both for p 3 and p 6 states for system sizes from n 5 to n 120 spins considering particularly the finite size scaling behavior at the exactly known glass transition temperature t c it is shown that for p 3 the moments q k of the spin glass order parameter satisfy a simple scaling behavior q k alpha n k 3 tilde f k n 1 3 1 t t c text k 1 2 3 tilde f k being the appropriate scaling function and t the temperature also the specific heat maxima have a similar behavior c v max alpha text const n 1 3 while moments of the magnetization scale as m k alpha n k 2 the approach of the positions t max of these specific heat maxima to t c as n is nonmonotonic for p 6 the results are compatible with a first order transition q k q jump k as n but since the order parameter q jump at t c is rather small a behavior q k n k 3 as n also is compatible with the data thus no firm conclusions on the finite size behavior of the order parameter can be drawn the specific heat maxima c v max behave qualitatively in the same way as for p 3 consistent with the prediction that there is no latent heat a speculative phenomenological discussion of finite size scaling for such transitions is given for small n n 15 for p 3 n 12 for p 6 the monte carlo data are compared to exact partition function calculations and excellent agreement is found we also discuss ratios r x equiv langle x rangle t langle x rangle t text av 2 text av langle x rangle t text av 2 for various quantities x to test the possible lack of self averaging at t c

the birth of twin poincare birkhoff chains near 1 3 resonance for a typical area preserving map we describe the birth process of two twin poincare birkhoff chains i e two rings consisting of center points alternated by saddles wound around an elliptic fixed point these twin chains are not born out of the elliptic fixed point but in the plane from an annular region where the rotation number has a rational extremum this situation generically occurs near a 1 3 resonance we find that the birth of two twin pb chains in such an annular region requires first the birth of two dimerized chains of saddle center pairs by a tangent bifurcation the transition from two dimerized chains to two pb chains involves the breakup of homoclinic saddle connections and the formation of heteroclinic connections it amounts to the reconnection phenomenon of howard and hohs our results can be regarded as a supplement to the poincare birkhoff theorem for the case that the twist condition is not satisfied

topological chaos in spatially periodic mixers topologically chaotic fluid advection is examined in two dimensional flows with either or both directions spatially periodic topological chaos is created by driving flow with moving stirrers whose trajectories are chosen to form various braids for spatially periodic flows in addition to the usual stirrer exchange braiding motions there are additional topologically nontrivial motions corresponding to stirrers traversing the periodic directions this leads to a study of the braid group on the cylinder and the torus methods for finding topological entropy lower bounds for such flows are examined these bounds are then compared to numerical stirring simulations of stokes flow to evaluate their sharpness the sine flow is also examined from a topological perspective

the discrete frenkel kontorova model and its extensions i exact results for the ground states abstract we present a rigorous study of the classical ground states under boundary conditions of a class of one dimensional models generalizing the discrete frenkel kontorova model the extremalization equations of the energy of these models turn out to define area preserving twist maps which exhibits periodic quasi periodic and chaotic orbits for all boundary conditions we select among all the extremum solutions of the energy of the model those which correspond to the ground states of the infinite system we prove that these ground states are either periodic commensurate or quasi periodic incommensurate but are never chaotic we also prove the existence of elementary discommensurations which are minimum energy configuration of the model for certain special boundary conditions the topological structure of the whole set of ground states is described in details in addition to physical applications consequences for twist map homeomorphisms are mentioned part ii s aubry p y ledaeron and g andre will be mostly devoted to exact results on the transition by breaking of analyticity which occurs on the incommensurate ground states when the model parameters vary and on its connection with the stochasticity threshold in the corresponding twist map

critical exponents for a transition from integrability to non integrability via localization of invariant tori in the hamiltonian system critical exponents that describe a transition from integrability to nonintegrability in a two dimensional nonlinear and area preserving map are obtained via localization of the first invariant spanning curve invariant tori in the phase space in a general class of systems the position of the first invariant tori is estimated by reducing the mapping of the system to the standard mapping where a transition takes place from local to global chaos the phase space of the mapping shows a large chaotic sea surrounding periodic islands and limited by a set of invariant tori whose position of the first of them depends on the control parameters the formalism leads us to obtain analytically critical exponents that describe the behaviour of the average variable action along the chaotic sea the result is compared to several models in the literature confirming the approach is of large interest the formalism used is general and the procedure can be extended to many other different systems

normal forms for 4d symplectic maps with twist singularities we derive a normal form for a near integrable four dimensional 4d symplectic map with a fold or cusp singularity in its frequency mapping the normal form is obtained for when the frequency is near a resonance and the mapping is approximately given by the time t mapping of a two degree of freedom hamiltonian flow consequently there is an energy like invariant the fold hamiltonian is similar to the well studied one degree of freedom case but is essentially non integrable when the direction of the singular curve in action does not coincide with curves of the resonance module we show that many familiar features such as multiple island chains and reconnecting invariant manifolds are retained even in this case the cusp hamiltonian has an essential coupling between its two degrees of freedom even when the singular set is aligned with the resonance module using averaging we approximately reduce this case to one degree of freedom as well the resulting hamiltonian and its perturbation with small cusp angle is analyzed in detail

manifold reconnection and diffusion in strong chaos abstract we analyse diffusion across a reconnecting zone in a regime of strong chaos numerical estimates from the angular correlation functions averaged over a suitable set of phase space initial conditions are compared to a local diffusion coefficient obtained using fick s law it emerges that the diffusion coefficient is enhanced from small to larger values as successive reconnection like processes or more usual bifurcations take place this feature is illustrated using a quadratic nontwist map

the effect of the number of simulations on the exponents obtained by finite size scaling relations for the seven dimensional ising model on the creutz cellular automaton abstract the seven dimensional ising model is simulated on the creutz cellular automaton using the finite size lattice with the linear dimension 4 l 8 the exponents in the finite size scaling relations for the magnetic susceptibility the order parameter and the specific heat at the infinite lattice critical temperature are computed to be 3 544 24 3 505 55 3 501 52 1 78 64 1 77 38 1 76 46 and 0 02 11 0 01 17 0 01 16 using 4 l 8 for 7 14 and 21 independent simulations respectively as the number of independent simulations increase exponents are in very good agreement with the theoretical predictions of 7 2 7 4 and 0 respectively the values for the critical temperature of infinite lattice 12 870 12 12 870 10 12 869 3 and 12 871 15 12 871 5 12 871 5 are obtained from the straight line fit of the magnetic susceptibility maxima and the specific heat using 4 l 8 for 7 14 and 21 independent simulations respectively as the number of independent simulations increases the obtained results are in very good agreement with the more precise value of 12 86902 33

on slater s criterion for the breakup of invariant curves abstract we numerically explore slater s theorem in the context of dynamical systems to study the breakup of invariant curves slater s theorem states that an irrational translation over a circle returns to an arbitrary interval in at most three different recurrence times expressible by the continued fraction expansion of the related irrational number the hypothesis considered in this paper is that slater s theorem can be also verified in the dynamics of invariant curves hence we use slater s theorem to develop a qualitative and quantitative numerical approach to determine the breakup of invariant curves in the phase space of area preserving maps

the defocusing mechanism of isochronous resonances abstract we present a numerical study concerning the defocusing mechanism of isochronous resonance island chains in the presence of two permanent robust tori the process is initialized and concluded through bifurcations of fixed points located on the robust tori our approach is based on a hamiltonian system derived from the resonant normal form choosing a convenient parameter in this system we are able to depict a comprehensive analysis of the dynamics of the problem

crossover behavior of the susceptibility and the specific heat near a second order phase transition abstract simple crossover equations for the susceptibility and the specific heat in zero field have been obtained on the basis of the renormalization group method and e expansion the equations contain the ginzburg number as a parameter at temperatures near the critical temperature scaling behavior including the first wegner corrections is reproduced at temperatures far away from the critical temperature the classical landau expansion with square root corrections is recovered for small values of the ginzburg number the crossover equations approach a universal form the equations are applied to represent experimental specific heat data for ch 4 c 2 h 6 ar o 2 and co 2 along the critical isochore in a universal form

ergodic and statistical properties of piecewise linear hyperbolic automorphisms of the 2 torus we study generic piecewise linear hyperbolic automorphisms of the 2 torus we explain why the resulting dynamical system is ergodic and mixing and prove the exponential decay of correlations

a degenerate bifurcation to chaotic scattering in a multicentre potential many scattering systems can be described as scattering of a point particle off a multicentre potential in this paper we present a two centre system which shows either regular or chaotic scattering depending on the kinetic energy i e the velocity of the incoming particle the transition points to chaotic scattering can be derived analytically by linearization of the poincare map at one of these transition velocities there is a degenerate bifurcation where the invariant set contains a parabolic surface and where the time delay statistics is algebraic with power sigma 2

the occupation statistics for indistinguishable dumbbells on a rectangular lattice space i the method of hock and mcquistan used recently to solve the occupation statistics for indistinguishable dumbbells or dimers on a 2 2 n lattice is extended further to obtain for the l m n lattice general expressions for the normalization expectation and dispersion of the statistics and their limit as n becomes very large in particular an explicit expression of the partition function in the thermodynamic limit x is obtained for any value of the absolute activity x of dimers the developed mathematical formalism is then applied to planar lattices 1 m n with m 1 2 3 and 4 the known results for m 1 and 2 are recovered and some new ones are obtained the recurrence relation for the number a q n of arrangements of q dumbbells on a 1 m n lattice which has 3 and 5 terms when m 1 and 2 respectively is found to have 15 and 65 terms for m 3 and 4 analysis and extrapolation of the results enable one to predict the expectation 1mn on a planar 1 m n lattice to be 63 4 in the limit as both m a

resonant islands without separatrix chaos an alternative type of hamiltonian nonlinear resonant island is analyzed in the usual case where the resonant island is pendulumlike chains bifurcated out of the central elliptic point undergo infinite cascades of period doubling bifurcations as they approach the island boundary in the present case we find that those chains undergo either period doubling or inverse saddle node bifurcations depending on the strength of perturbing terms in the saddle node case we show that just after a reconnection process external chains cross the island boundary to collapse against the bifurcated internal chains

control of transport in a chaotic lattice abstract we examine a mechanism for the elimination of vertical transport in a square isotropic chaotic lattice we find that applying a shear jet along a line of symmetry in the lattice can completely impede chaotic transport across a specified barrier this is accomplished through a change in allegiance of heteroclinic connections within the lattice this mechanism is relatively insensitive to details such as amplitude or shape of the applied shear it depends quite sensitively however on the precise location of the shear this suggests that the placement of flow control devices in physical situations may be much more critical to transport behavior than are other control parameters in addition we develop a necessary condition for the existence of boundary circles in non twist maps using this condition we find that transport across a barrier can be eliminated for all practical purposes without evidence for an invariant boundary circle

a jordan algebraic approach to potential reduction algorithms we consider the linear monotone complementarity problem for domains obtained as the intersection of an affine subspace and the cartesian product of symmetric cones a primal dual potential reduction algorithm is described and its complexity estimates are established with the help of the jordan algebraic technique

finding critical exponents for two dimensional hamiltonian maps two dimensional nonlinear area preserving mappings have been considered in the study of dynamical systems for many years applications of this formalism can be used in study of channel flows waveguide transport properties fermi acceleration and also for the study of magnetic field lines in toroidal plasma devices with reversed shear such as tokamaks1 10 generally such kind of mappings have one or more control parameters they can control the intensity of the nonlinearity controlling also a transition from integrability to nonintegrability the phase space of these mappings exhibit quite often mixed forms one can observe a large chaotic sea that eventually surrounds kolmogorov arnold moser islands and is limited by a set of invariant tori the size of the chaotic sea is strongly influenced by the control parameters since they play the rule of controlling the intensity of the nonlinearity in this paper we address specifically a transition from integrability to nonintegrability present in a special class of two dimensional nonlinear area preserving mappings we look at average properties along the chaotic sea and therefore we find critical exponents that describe the behavior of average quantities of the chaotic sea such quantities are described in terms of scaling arguments leading to a generic and robust description of a scaling observed for the chaotic sea the critical exponents are obtained for several values of the control parameters the knowledge of such exponents allow us to define and compare classes of universality for the two dimensional hamiltonian mappings the paper is organized as follows in sec ii we present the model and discuss the variables and control parameters used some properties of the phase space are also discussed including a characterization of fixed points the chaotic component of the phase space is characterized by lyapunov exponents in sec iii we obtain and describe the critical exponents that define classes of universality a merger of all the curves of the average quantity is obtained finally in the sec iv we present our concluding remarks ii definition of the problem and the mapping in this section we present and discuss some dynamical properties for a set of two dimensional hamiltonian mappings we assume that there is a two dimensional integrable system that is slightly perturbed the hamiltonian function that in principle describes the system is written as 11

monte carlo investigations of phase transitions status and perspectives abstract using the concept of finite size scaling monte carlo calculations of various models have become a very useful tool for the study of critical phenomena with the system linear dimension as a variable as an example several recent studies of ising models are discussed as well as the extension to models of polymer mixtures and solutions it is shown that using appropriate cluster algorithms even the scaling functions describing the crossover from the ising universality class to the mean field behavior with increasing interaction range can be described additionally the issue of finite size scaling in ising models above the marginal dimension d 4 is discussed

exact results for an approximate renormalisation scheme and some predictions for the breakup of invariant ori abstract an approximate renormalisation scheme is derived for the breakup of invariant tori of arbitrary winding ratio in hamiltonian systems of one and a half degrees of freedom similar to that of escande and doveil it is a free semi group with two generators this scheme is solved exactly for its orbits stable manifolds unstable manifolds and critical set various results are found including a cantor set of universal fractal diagrams the robustness of noble tori and a scaling law for areas near critical circles

failure of universal finite size scaling above the upper critical dimension abstract previous theories have predicted that o n symmetric systems in a finite cubic geometry with periodic boundary conditions have universal finite size scaling functions near criticality in d 4 dimensions on the basis of exact results for the o n symmetric 4 model in the large n limit we show that universal finite size scaling does not hold in the predicted form because of significant cut off and lattice effects for d 4 it is shown that finite size scaling is valid with two reference lengths which turn out to be identical with the amplitudes of the bulk correlation length for the 4 field theory the finite size scaling functions are shown to be non universal i e to depend explicitly on the cut off and on the bare four point coupling constant whereas for a 4 lattice model the finite size scaling functions have a different form that is independent of the lattice spacing and the four point coupling constant

escape exponent for transient chaos and chaotic scattering in non hyperbolic hamiltonian systems it is shown that in non hyperbolic hamiltonian systems where correlations decay as a power law transient chaos and chaotic scattering demonstrate different power laws numerical evidence of this effect for the stadium billiard is presented

mixing enhancement in enzymatic chemical reactions by optimal tuning of flows abstract we consider the problem of mixing enhancement in enzymatic chemical reactions in moving fluids which are modeled by a reaction convection diffusion equation the problem is formulated as an optimal control problem using the flow field as a control variable by variational principles we prove the existence of an optimal flow and derive an optimality system that consists of nonlinear reaction convection diffusion equations and laplace s equations the uniqueness of the optimal flow is left open

rotational chaos in dissipative systems abstract an investigation is made into chaotic attractors arising from a quasiperiodic transition to chaos using a quantity called the rotation interval the rotation interval describes the short term rotation rates available to the attractor we present algorithms to calculate it given an appropriate map differential equation or time series we find that the rotation interval has a very robust parameter dependence its endpoints are almost always phase locked our numerical ideas are based on the theory of dissipative twist maps which is reviewed this theory is also used to prove a theorem about the non existence of certain strange attractors in nearly conservative systems finally an investigation is made into the relationship between the rotation interval and topological entropy and the breakup of invariant circles

self scaled barriers for irreducible symmetric cones self scaled barrier functions are fundamental objects in the theory of interior point methods for linear optimization over symmetric cones of which linear and semidefinite programming are special cases in this article we classify the special class of self scaled barriers which are defined on irreducible symmetric cones together with a decomposition theorem for general self scaled barriers this concludes the algebraic classification theory of these functions

critical function for the standard map abstract the long time behavior of hamiltonian systems is determined by the broken kam tori which lie near the border of chaos the critical function tells us when these tori disintegrate and indirectly the extent to which they interfere with the motion of orbits across them i use lie transforms to find an expression for the critical function from this expression i recover the critical scaling found by renormalization techniques more importantly i find the deviations from scaling and show that they are crucial for understanding long time behavior in hamiltonian systems this resolves a long standing problem raised by the numerical experiments of karney and others in addition i find a relation between the values of the critical function at different frequencies this derivation clarifies the numerical results found by buric percival and vivaldi

fast flat histogram method for generalized spin models we present a monte carlo method that efficiently computes the density of states for spin models having any number of interaction per spin by combining a random walk in the energy space with collective updates controlled by the microcanonical temperature our method yields dynamic exponents close to their ideal random walk values reduced equilibrium times and very low statistical error in the density of states the method can host any density of states estimation scheme including the wang landau algorithm and the transition matrix method our approach proves remarkably powerful in the numerical study of models governed by long range interactions where it is shown to reduce the algorithm complexity to that of a short range model with the same number of spins we apply the method to the q state potts chains 3 ensuremath leqslant q ensuremath leqslant 12 with power law decaying interactions in their first order regime we find that conventional local update algorithms are outperformed already for sizes above a few hundred spins by considering chains containing up to 2 16 spins which we simulated in fairly reasonable time we obtain estimates of transition temperatures correct to five figure accuracy finally we propose several efficient schemes aimed at estimating the microcanonical temperature

the vanishing twist in the restricted three body problem abstract this paper demonstrates the existence of twistless tori and the associated reconnection bifurcations and meandering curves in the planar circular restricted three body problem near the lagrangian equilibrium l 4 a twistless torus is created near the tripling bifurcation of the short period family decreasing the mass ratio leads to twistless bifurcations which are particularly prominent for rotation numbers 3 10 and 2 7 this scenario is studied by numerically integrating the regularised hamiltonian flow and finding rotation numbers of invariant curves in a two dimensional poincare map to corroborate the numerical results the birkhoff normal form at l 4 is calculated to eighth order truncating at this order gives an integrable system and the rotation numbers obtained from the birkhoff normal form agree well with the numerical results a global overview for the mass ratio 4 3 is presented by showing lines of constant energy and constant rotation number in action space

an approximate solution of the monomer dimer problem on a square lattice ii the mathematical method developed in paper i is applied to obtain the partition function and thermodynamical properties of the monomer dimer problem for a square lattice in terms of the absolute activity x we also obtain by extrapolation an approximate expression of the partition function which is accurate to better than 0 1 in the range 0 x 10 the expectation of the statistics x is calculated in two different ways and numerical results agree to better than 3 thus showing the consistency of the underlying mathematical method consistent with earlier studies there is no phase transition approximation methods used in earlier work are also found to be in good agreement with our analytic study

effective dynamics in hamiltonian systems with mixed phase space an adequate characterization of the dynamics of hamiltonian systems at physically relevant scales has been largely lacking here we investigate this fundamental problem and we show that the finite scale hamiltonian dynamics is governed by effective dynamical invariants which are significantly different from the dynamical invariants that describe the asymptotic hamiltonian dynamics the effective invariants depend both on the scale of resolution and the region of the phase space under consideration and they are naturally interpreted within a framework in which the nonhyperbolic dynamics of the hamiltonian system is modeled as a chain of hyperbolic systems

another look at the saddle centre bifurcation vanishing twist abstract in the saddle centre bifurcation a pair of periodic orbits is created out of nothing in a hamiltonian system with two degrees of freedom it is the generic bifurcation with multiplier one we show that out of nothing should be replaced by out of a twistless torus more precisely we show that invariant tori of the normal form have vanishing twist right before the appearance of the new orbits vanishing twist means that the derivative of the rotation number with respect to the action for constant energy vanishes we explicitly derive the position of the twistless torus in phase and in parameter space near the saddle centre bifurcation the theory is applied to the area preserving henon map

ergodic properties of the discontinuous sawtooth map we prove the ergodicity for the discontinuous sawtooth map adapting a technique previously used in billiard theory the core of the proof is the construction of a hopf chain passing through a countable dense set of discontinuity lines

scaling properties for a family of discontinuous mappings abstract scaling exponents that describe a transition from integrability to non integrability in a family of two dimensional nonlinear and discontinuous mappings are obtained the mapping considered is parameterized by the exponent in the action variable the scaling exponents describing the behavior of the average square action along the chaotic orbits are obtained for different values of therefore classes of universality can be defined for specific values of our mapping acts as the discontinuous map representation of well known nonlinear systems thus making our study broadly applicable also the formalism used is general and the procedure can be extended to characterize many other dynamical systems

calculation of cantori for hamiltonian flows cantori are the invariant sets remaining after the destruction of kam surfaces and create partial barriers to transport in chaotic regions cantori may be approximated by high order periodic orbits however field line tracing methods for locating periodic orbits perform poorly in chaotic regions to approximate cantori for continuous flow dynamics high order periodic orbits are determined by lagrangian variational methods the method is robust to chaos converges quadratically and the computational cost scales linearly with the periodicity length of the orbit minimizing periodic orbits with periodicities in the tens of thousands that closely approximate cantori have been constructed

critical properties of the three dimensional equivalent neighbor model and crossover scaling in finite systems accurate numerical results are presented for the three dimensional equivalent neighbor model on a cubic lattice for 12 different interaction ranges coordination number between 18 and 250 these results allow the determination of the range dependences of the critical temperature and various critical amplitudes which are compared to renormalization group predictions in addition the analysis yields an estimate for the interaction range at which the leading corrections to scaling vanish for the spin 1 2 model and confirms earlier conclusions that the leading wegner correction must be negative for the three dimensional nearest neighbor ising model by complementing these results with monte carlo data for systems with coordination numbers as large as 52 514 the full finite size crossover curves between classical and ising like behavior are obtained as a function of a generalized ginzburg parameter also the crossover function for the effective magnetic exponent is determined

universal small scale structure near the boundary of siegel disks of arbitrary rotation number abstract we present numerical evidence that for analytic maps of the complex plane with a siegel disk of any irrational rotation number and a quadratic critical point on its boundary the small scale structure near the critical point can be described asymptotically by a universal two parameter family we propose an explanation in terms of an ergodic attractor for a renormalisation operator

what makes chaos border sticky a microscopic investigation of long time correlations near the chaos border of a particular model system has been carried out apparent stickiness of chaos border seems to be due to the influence of nearby hyperbolic orbits which are convergent to the invariant curve which constitutes the chaos border the diffusion near the chaos border can be described by a one dimensional biased random walk problem with pausing time which is inversely proportional to the distance from the border monte carlo simulation yields that the first passage time distribution has an algebraic decay tail of the form t with its exponent 2 80 0 05

the statistics of interacting dumbbells on a 2 n lattice space a set theoretic argument is utilizied to determine a 31 term recursion that describes exactly the composite nearest neighbor degeneracy for indistinguishable dumbbell particles distributed on a rectangular 2 n lattice space the associated generating functions the expectation of the lattice coverage and the density of occupied nearest neighbor pairs are also determined

thermodynamics and molecular freedom of dimers on plane triangular lattices this is an extension of three previous papers dealing with dimers on rectangular lattices one two and three dimensions the technique presented in the first paper in this series continues to be fruitful for dimers on plane triangular lattices entropy isothermal compressibility constant pressure heat capacity and molecular freedom per dimer at close packing are obtained exactly for lattices infinite in one direction and finite in the other observations made in the third paper of the series concerning molecular freedom per dimer at close packing on rectangular lattices are used to extrapolate our results to infinite plane triangular lattices at close packing the molecular freedom per dimer on an infinite plane triangular lattice is calculated to be 2 356 527 in agreement with the value obtained by nagle based on our earlier findings the value of 2 356 527 was used to obtain the analytic fit for the thermodynamic quantities in terms of the normalized number density

unstable manifolds for the mackay approximate renormalisation abstract for a renormalisation of a critical phenomenon in some class of dynamical systems it is the unstable manifolds of the critical set which give the universal families which describe the critical transition in this paper we study such manifolds for the mackay 1988 approximate renormalisation scheme which models the breakup of invariant circles of arbitrary rotation number in area preserving twist maps we derive a number of properties of the unstable manifolds and then discuss the implications of these to the full renormalisation in particular a we suggest the importance of continuity in the definition of unstable manifolds and b show that their structure is similar to that observed numerically by mackay and percival 1987 in a related renormalisation finally we show that the residues of approximating periodic orbits are bounded above on the critical set

a renormalization approach to invariant circles in area preserving maps abstract kadanoff and shenker introduced a renormalisation approach to invariant circles in area preserving maps this paper makes more precise the connection between invariant circles and the renormalisation operator restricting attention to noble rotation numbers the stability of a simple fixed point of the renormalisation is analysed corresponding to a linear twist map it is found to be essentially attracting so that noble circles persist under perturbation giving a new view on kam theory shenker and kadanoff found evidence for another fixed point corresponding to a map with a non smooth noble circle further evidence is given in this paper it has essentially only one unstable direction and its stable manifold is believed to give the boundary of the set of twist maps with a noble circle finally noble circles are shown to be locally most robust in an important sense

a long step primal dual algorithm for the symmetric programming problem abstract based on the techniques of euclidean jordan algebras we prove complexity estimates for a long step primal dual interior point algorithm for the optimization problem of the minimization of a linear function on a feasible set obtained as the intersection of an affine subspace and a symmetric cone this result provides a meaningful illustration of a power of the technique of euclidean jordan algebras applied to problems under consideration

converse kam theory and practice we unify extend reinterpret and apply criteria of birkhoff 1 herman 9 mather 2 3 aubry et al 4 5 and newman and percival 6 for the nonexistence of invariant circles for area preserving twist maps the criteria enable one to establish regions of phase space through which no rotational invariant circles pass for families of maps the same can be done for regions of the combined space of phase points and parameters the criteria can be implemented rigorously on a computer and give a practical method of proving quite strong results as an example we present a computer program which proved that the standard map has no rotational invariant circles for any parameter value k 63 64

primal dual algorithms and infinite dimensional jordan algebras of finite rank we consider primal dual algorithms for certain types of infinite dimensional optimization problems our approach is based on the generalization of the technique of finite dimensional euclidean jordan algebras to the case of infinite dimensional jb algebras of finite rank this generalization enables us to develop polynomial time primal dual algorithms for infinite dimensional second order cone programs we consider as an example a long step primal dual algorithm based on the nesterov todd direction it is shown that this algorithm can be generalized along with complexity estimates to the infinite dimensional situation under consideration an application is given to an important problem of control theory multi criteria analytic design of the linear regulator the calculation of the nesterov todd direction requires in this case solving one matrix differential riccati equation plus solving a finite dimensional system of linear algebraic equations on each iteration the number of equations and unknown variables of this algebraic system is m 1 where m is a number of quadratic performance criteria

decoupling of a system of partial difference equations with constant coefficients and application consider d multi variable functions aj n j 1 to d where n stands for the evaluation point in the associated multi dimensional space of coordinates n1 n2 let us assume that the aj s satisfy a system of d linearly coupled finite difference equations the value of each function ai at the evaluation point n is given as a linear combination of the values of this function and others at shifted evaluation points by introducing d suitable generating functions gj j 1 to d one is able to replace the d coupled difference equations by a system of d linear equations where the gj s play the role of the d unknowns after solving this new system of equations it is then possible to construct a difference equation for each of the aj s relating the value of ai at the evaluation point n to the values of ai itself at shifted arguments the solution of such a decoupled equation can then be handled using the multi dimensional combinatorics function technique

the effect of the increase of linear dimensions on exponents obtained by finite size scaling relations for the six dimensional ising model on the creutz cellular automaton the six dimensional ising model is simulated on the creutz cellular automaton using the finite size lattices with the linear dimension 4

vanishing twist in the hamiltonian hopf bifurcation the hamiltonian hopf bifurcation has an integrable normal form that describes the passage of the eigenvalues of an equilibrium through the 1 1 resonance at the bifurcation the pure imaginary eigenvalues of the elliptic equilibrium turn into a complex quadruplet of eigenvalues and the equilibrium becomes a linearly unstable focus focus point we explicitly calculate the frequency map of the integrable normal form in particular we obtain the rotation number as a function on the image of the energy momentum map in the case where the fibres are compact we prove that the isoenergetic non degeneracy condition of the kam theorem is violated on a curve passing through the focus focus point in the image of the energy momentum map this

the occupation statistics for indistinguishable dumbbells on a 2 2 n lattice space a 20 term recursion relation is derived that describes exactly the occupation statistics for indistinguishable dumbbells which occupy two nearest neighbor sites distributed on a 2 2 n lattice space on the basis of this recursion the normalization expectation dispersion and continuous representation of the statistics are also developed when the lattice space is completely filled the recursion relationship reduces to four terms permitting a calculation of the orientational degeneracy of a completely filled 2 2 n lattice space

phase behaviour of heteronuclear dimers adsorbed on square and triangular lattices a monte carlo study abstract monte carlo simulation within the grand canonical ensemble the histogram reweighting technique and finite size scaling analysis are used to explore the phase behaviour of heteronuclear dimers composed of a and b type segments on square and triangular lattices we have considered models with attractive bb and ab nearest neighbour energy u bb u ab 1 and the energy u aa from the interval 0 5 0 5 the phase diagrams and critical parameters have been estimated for both lattices we have also evaluated the dependences of the critical point characteristics on the value of u aa the new ordered structures in liquid phase have been found it has been shown that for repulsive aa interactions the dimers distributed on a triangular lattice belonged to the two dimensional ising class but in the case of a square lattice the system exhibited a non universal critical behaviour

two parameter scaling of correlation functions near continuous phase transitions we discuss the order parameter correlation function in the vicinity of continuous phase transitions using a two parameter scaling form g k k 2 c g k k k c where k is the wave vector and is the correlation length and the interaction dependent nonuniversal momentum scale k c remains finite at the critical fixed point the correlation function describes the entire critical regime and captures the classical to critical crossover one parameter scaling is recovered only in the limit k k c 0 we present an approximate calculation of g x y for the ising universality class using the functional renormalization group

ising universality in three dimensions a monte carlo study we investigate three ising models on the simple cubic lattice by means of monte carlo methods and finite size scaling these models are the spin 1 2 ising model with nearest neighbour interactions a spin 1 2 model with nearest neighbour and third neighbour interactions and a spin 1 model with nearest neighbour interactions the results are in accurate agreement with the hypothesis of universality analysis of the finite size scaling behaviour reveals corrections beyond those caused by the leading irrelevant scaling field we find that the correction to scaling amplitudes are strongly dependent on the introduction of further neighbour interactions or a third spin state in a spin 1 ising model these corrections appear to be very small this is very helpful for the determination of the universal constants of the ising model the renormalization exponents of the ising model are determined as yt 1 587 2 yh 2 4815 15 and yi 0 82 6 the universal ratio q m2 2 m4 is equal to 0 6233 4 for periodic systems with cubic symmetry the critical point of the nearest neighbour spin 1 2 model is kc 0 2216546 10

markov tree model of transport in area preserving maps abstract transport in an area preserving map with a mixture of regular and chaotic regions is described in terms of the flux through invariant cantor sets called cantori a model retaining a discrete set of cantori approaching a boundary circle gives the markov chain description of hanson cary and meiss the inclusion of cantori surrounding island chains and islands about islands etc gives a markov tree model with a slower decay rate the survival probability distribution is shown to decay asymptotically as a power law the decay exponent agrees reasonably well with the computations of karney and of chirikov and shepelyanski

structural ordering of interacting dimers on a square lattice we study the entropy of dimers on a semi infinite m n square lattice n to infinity with lattice interaction energy per dimer mu 1 and nearest neighbour interaction energy mu 2 numerical computations based on recursive transfer matrices show that for m 1 2 3 4 and 5 and under certain conditions the constant mu 2 curves of the entropy as a function of the coverage theta 1 and the fraction theta 2 of the maximum number of possible nearest neighbours merge into a single curve forming arches that meet at cusps at the highest points of these arches the interaction energies are in the ratio of small whole numbers based on the analysis of the configurations at the cusps we extrapolate the results to the infinite two dimensional lattice m infinity and obtain the location of at least two cusps the first cusp occurs at theta 1 1 2 theta 2 0 and zero entropy and the second at theta 1 2 3 theta 2 2 9 and an entropy of 0 102

fractal bifurcation sets renormalization strange sets and their universal invariants the transition structure of the most common routes to chaos are organized by fractal bifurcation sets examples include the quasiperiodic transitions to chaos and the period doubling structure found in arnol d tongues in this paper i discuss the universality of such fractal bifurcation sets and their relation to strange invariant sets of renormalization transformations an important result is that fractal bifurcation sets from within the same universality chaos are lipeomorphic this implies that they have the same fractal structure and in particular the same hausdorff dimension and scaling spectra some other invariants are introduced

algebraic decay in self similar markov chains a continuous time markov chain is used to model motion in the neighborhood of a critical invariant circle for a hamiltonian map states in the infinite chain represent successive rational approximants to the frequency of the invariant circle for the case of a noble frequency the chain is self similar and the nonlinear integral equation for the first passage time distribution is solved exactly the asymptotic distribution is a power law times a function periodic in the logarithm of the time for parameters relevant to the critical noble circle the decay proceeds ast 4 05

resonances and transport in the sawtooth map we study transport in a completely chaotic hamiltonian system the hyperbolic sawtooth map analytical expressions are obtained for its cantori and resonances we show that resonances give a complete partition of phase space the flux leaking out of a resonance is given by its turnstiles whose form and areas are obtained analytically when the total flux out of a resonance becomes one third the area of an island the topology of the turnstiles changes at the same parameter value a horseshoe is formed corresponding to the orbits trapped within the resonance based on this a coding scheme for the trapped orbits is introduced and expressions for trapped ordered orbits are obtained the partial flux transferred from one resonance to another is determined by the degree of overlap of their turnstiles we calculate the survival probability within a resonance using the markov model the results are compared with results obtained numerically and from periodic orbit theory

robust attractor of non twist systems abstract we present a new kind of one dimensional attractor which has not yet been predicted in the non linear dynamics theory we consider a non linear map which presents typical non twist manifestations as isochronous resonances and shearless torus it is known that this torus corresponds to a very sturdy barrier in the phase space of some area preserving systems we show that when dissipation is present in the system the shearless curve carries its robustness to the dissipative scenario it becomes a powerful attractor which we call shearless attractor which is persistent under the variation of the parameters and it exchanges its stability from chaotic to quasi periodic or vice versa depending on the set of parameters

extended chaos and disappearance of kam trajectories abstract in two dimensional area preserving maps kam trajectories serve as natural boundaries between stochastic regions as they disappear points may leak out from one region to the next this escape rate is defined and related to the action generating function analytic and numerical results are presented to describe the behavior of the escape rate near the disappearance of a kam trajectory

crossover scaling in two dimensions we determine the scaling functions describing the crossover from ising like critical behavior to classical critical behavior in two dimensional systems with a variable interaction range since this crossover spans several decades in the reduced temperature as well as in the finite size crossover variable it has up to now largely evaded a satisfactory numerical determination using a dedicated monte carlo method we could obtain accurate results for sufficiently large interaction ranges our data cover the full crossover region both above and below the critical temperature and support the hypothesis that the crossover functions are universal also the so called effective exponents are discussed and we show that these can vary nonmonotonically in the crossover region

subdiffusive transport in stochastic webs we study a one parameter family of piecewise linear continuous area preserving maps of the plane at an infinite discrete set of parameter values the phase space is globally connected the transport proceeds in stochastic webs which are structurally similar to those described recently by zaslavsky and coworkers however the transport is subdiffusive i e the energy grows with time as t where 1 this behaviour is explained in terms of a markov model derived from a detailed analysis of the phase space structure and it is shown that 23 asymptotically for a special case we identify sublattices in phase space which show nongeneric transport with 1

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

linear systems in jordan algebras and primal dual interior point algorithms we discuss a possibility of the extension of a primal dual interior point algorithm suggested recently by alizadeh et al 1994 we consider optimization problems defined on the intersection of a symmetric cone and an affine subspace the question of solvability of a linear system arising in the implementation of the primal dual algorithm is analyzed a nondegeneracy theory for the considered class of problems is developed the jordan algebra technique suggested by faybusovich 1995 plays major role in the present paper

concentration and energy fluctuations in a critical polymer mixture a semi grand canonical monte carlo algorithm is employed in conjunction with the bond fluctuation model to investigate the critical properties of an asymmetric binary ab polymer mixture by applying the equal peak weight criterion to the concentration distribution the coexistence curve separating the a rich and b rich phases is identified as a function of temperature and chemical potential to locate the critical point of the model the cumulant intersection method is used the accuracy of this approach for determining the critical parameters of fluids is assessed attention is then focused on the joint distribution function of the critical concentration and energy which is analysed using a mixed field finite size scaling theory that takes due account of the lack of symmetry between the coexisting phases the essential ising character of the binary polymer critical point is confirmed by mapping the critical scaling operator distributions onto independently known forms appropriate to the 3d ising universality class in the process estimates are obtained for the field mixing parameters of the model which are compared both with those yielded by a previous method and with the predictions of a mean field calculation

the density of occupied nearest neighbor pairs on a rectangular 2 n lattice space utilizing a 15 term recursion that describes exactly the composite nearest neighbor degeneracy for simple indistinguishable particles distributed on a 2 n lattice space 11 the expectation of the normalized number of occupied nearest neighbor pairs is calculated as a function of coverage and the reduced interaction potential

control of mixing in fluid flow a maximum entropy approach in many technological processes a fundamental stage involves the mixing of two or more fluids as a result the design of optimal mixing protocols is a problem of both fundamental and practical importance in this paper the authors formulate a prototypical mixing problem in a control framework where the objective is to determine the sequence of fluid flows that will maximize entropy by developing the appropriate ergodic theoretic tools for the determination of entropy of periodic sequences they derive the form of the protocol which maximizes entropy among all of the possible periodic sequences composed of two shear flows orthogonal to each other the authors discuss the relevance of their results in the interpretation of previous studies of mixing protocols