analytic torsion and holomorphic determinant bundles in this paper we prove that in the case of holomorphic locally kahler fibrations the analytic and algebraic geometry constructions of determinant bundles for direct images coincide we calculate the curvature of the holomorphic hermitian connection for the quillen metric on the determinant bundle we study the behavior of the quillen metric under change of metrics in the fibers and also on the twisting vector bundles we thus generalize the conformal anomaly formula to kahler manifolds of arbitrary dimension we also study the quillen metrics on determinants associated with exact sequences of vector bundles we prove that the quillen metric is smooth on the grothendieck knudsen mumford determinant for arbitrary holomorphic fibrations

analytic torsion and holomorphic determinant bundles i bott chern forms and analytic torsion we attach secondary invariants to any acyclic complex of holomorphic hermitian vector bundles on a complex manifold these were first introduced by bott and chern bot c our new definition uses quillen s superconnections we also give an axiomatic characterization of these classes these results will be used in bgs2 and bgs3 to study the determinant of the cohomology of a holomorphic vector bundle

spherical prediction variance properties of central composite and box behnken designs for quadratic regression on the hypercube a single number criterion such as a g efficiency that is based on the prediction variance is often included as one of the criteria when selecting a response surface design as an alternative to the single number criterion approach the variance dispersion graph presented by giovannitti jensen and myers is a graphical technique for evaluating prediction variance properties throughout the experimental region three properties of interest are the maximum minimum and average spherical prediction variances given the spherical radius as an alternative to the computer based approach requiring an optimization algorithm to evaluate these properties the maximum minimum and spherical prediction variances for central composite and box behnken designs can be determined analytically and are functions only of the radius and the design parameters these functions yield the exact values of the spherical prediction variance properties of central composite and box behnke

universal scaling behaviour for iterated maps in the complex plane according to the theory of schroder and siegel certain complex analytic maps possess a family of closed invariant curves in the complex plane we have made a numerical study of these curves by iterating the map and have found that the largest curve is a fractal when the winding number of the map is the golden mean the fractal curve has universal scaling properties and the scaling parameter differs from those found for other types of maps also for this winding number there are universal scaling functions which describe the behaviour asn of theq n th iterates of the map whereq n is thenth fibonacci number

orbifolds and analytic torsions dans cet article on calcule le comportement de metrique de quillen par immersions d orbifold on etend ainsi une formule de bismut lebeau au cas d orbifold

a measure of rotatability for response surface designs a measure that quantifies the amount of rotatability in a given response surface design is introduced in this article the measure which is expressible as a percentage takes the value 100 if and only if the design is rotatable one of the main advantages of this measure is that it can be used to repair a nonrotatable design by the addition of experimental runs that maximize the percent rotatability over a spherical region of interest four numerical examples are given to illustrate the applications of this measure

on the asymptotic expansion of bergman kernel we study the asymptotic of the bergman kernel of the spin c dirac operator on high tensor powers of a line bundle

a variety of period doubling universality classes in multi parameter analysis of transition to chaos abstract in multi parameter analysis of the onset of chaos non feigenbaum period doubling behavior may occur at some special paths in the parameter space there are two possibilities i the dynamics at the onset of chaos remains essentially one dimensional but the one dimensional map is distorted in such a way that leaves feigenbaum s universality class ii a new mode comes to the threshold of instability and increases the effective dimension of the dynamics we submit a list of one dimensional and two dimensional maps which represent distinct classes of the period doubling universality discuss the properties of the associated types of critical behavior at the border of chaos and demonstrate pictures of the universal parameter space arrangement near the critical points

generalized bergman kernels on symplectic manifolds we study the near diagonal asymptotic expansion of the generalized bergman kernel of the renormalized bochner laplacian on high tensor powers of a positive line bundle over a compact symplectic manifold we show how to compute the coefficients of the expansion by recurrence and give a closed formula for the first two of them as a consequence we calculate the density of states function of the bochner laplacian and establish a symplectic version of the convergence of the induced fubini study metric we also discuss generalizations of the asymptotic expansion for non compact or singular manifolds as well as their applications our approach is inspired by the analytic localization techniques of bismut and lebeau

a fixed point formula of lefschetz type in arakelov geometry i statement and proof we consider arithmetic varieties endowed with an action of the group scheme of n th roots of unity and we define equivariant arithmetic k 0 theory for these varieties we use the equivariant analytic torsion to define direct image maps in this context and we prove a riemann roch theorem for the natural transformation of equivariant arithmetic k 0 theory induced by the restriction to the fixed point scheme this theorem can be viewed as an analog in the context of arakelov geometry of the regular case of the theorem proved by p baum w fulton and g quart in bafq we show that it implies an equivariant refinement of the arithmetic riemann roch theorem in a form conjectured by j m bismut cf b2 par l p 353 and also ch soule s question in sabk 1 5 p 162

periodic orbits near bifurcations of codimension two classical mechanics semiclassics and stokes transitions we investigate classical and semiclassical aspects of codimension two bifurcations of periodic orbits in hamiltonian systems a classification of these bifurcations in autonomous systems with two degrees of freedom or time periodic systems with one degree of freedom is presented we derive uniform approximations to be used in semiclassical trace formulae and determine also certain global bifurcations in conjunction with stokes transitions that become important in the ensuing diffraction catastrophe integrals

multiparameter critical situations universality and scaling in two dimensional period doubling maps we review critical situations linked with period doubling transition to chaos which require using at least two dimensional maps as models representing the universality classes each of them corresponds to a saddle solution of the two dimensional generalization of feigenbaum cvitanovic equation and is characterized by a set of distinct universal constants analogous to feigenbaum s and one type of criticality designated h was discovered by several authors in 80 th in the context of period doubling in conservative dynamics but occurs as well in dissipative dynamics as a phenomenon of codimension 2 second is bicritical behavior which takes place in systems allowing decomposition onto two dissipative period doubling subsystems each of which is brought by parameter tuning onto a threshold of chaos types of criticality designated as fq and c occur in non invertible two dimensional maps we present and discuss a number of realistic systems manifesting those types of critical behavior and point out some relevant conditions of their potential observation in physical systems in particular we indicate a possibility for realization of the h type criticality without vanishing dissipation but with its compensation in a self oscillatory system next we present a number of examples coupled henon like maps coupled driven oscillators coupled chaotic self oscillators which manifest bicritical behavior for fq type we indicate possibility to arrange it in non symmetric systems of coupled period doubling subsystems e g in henon like maps and in chua s circuits for c type we present examples of its appearance in a driven rossler oscillator at the period doubling accumulation on the edge of syncronization tongue and in a model map with the neimark sacker bifurcation

the spectrum of the period doubling operator in terms of cycles the eigenvalue spectrum of the period doubling operator is evaluated both in terms of unstable periodic orbits and by a diagonalization at the operator cycles up to length ten yield the twelve leading eigenvalues direct diagonalization yields some 50 eigenvalues the feigenbaum 6 the hausdorff dimension and the escape rate for the period doubling repeller are evaluated to high accuracy in this letter we apply the technique for the extraction of correlation exponents of i to evaluation of the spectrum of the period doubling renormalization operator the basic idea of relating the feigenbaum constant 6 to the scaling or the presentation function is due to sullivan 2 and is implemented as an eigenvalue spectrum calcula tion in 3 51 we refer the reader to 3 41 for an introduction to the cycle expansions in general and their application to the evaluation of the stability the dimension and the escape rate of the period doubling repeller in particular the fredholm determinant evaluation method and the transfer operator diagonalization 11 used here are superior to the method of locating zeros of finite products of dynamical 6 functions 6 of 3 4 7 81 and in that sense the present letter supersedes the above references the repeller we study here is the non wandering set of the period doubling presenta tion function 9

semiclassical asymptotics of perturbed cat maps we derive an exact representation for tr u n where u is the quantum propagator associated with an anosov perturbed cat map this takes the form of a sum over the fixed points of the n th iterate of the classical transformation the contribution of each one being given by an n fold multiple integral we focus in particular on the case when n 1 an asymptotic evaluation of the integral in question then leads to a complete semiclassical series expansion the first term of which corresponds to the gutzwiller tabor trace formula it is demonstrated that this series diverges but that summing it down to its least term provides an approximation to the quantum trace that is exponentially accurate in 1 a simple universal approximation to the late terms is then derived this explains the divergence of the semiclassical expansion in terms of complex tunnelling periodic orbits and implies the existence of unusual relations between different orbit actions it also allows us to recover the semiclassical contributions from the complex orbits explicitly using borel resummation these exponentially subdominant terms are shown to exhibit the stokes phenomenon which causes them to depend sensitively on the size of the perturbation parameter finally we develop an alternative expansion based on the orbits of the unperturbed cat map rather than diverging this is shown to converge absolutely thus making possible an exact calculation of the quantum trace using only classical mechanics its properties are however distinctly anti semiclassical

semiclassical wigner functions for quantum maps on a torus a semiclassical formula is derived for the wigner representation of the quasi energy states of a quantum map on a torus it is expressed as a finite sum over the classical fixed points of the map a criterion for the appearance of scars is presented semiclassical wigner functions of the cat map are calculated and shown to reproduce the exact quantum mechanical results

crossover from dissipative to conservative behaviour in period doubling systems abstract we investigate the crossover properties between the conservative and dissipative limit for period doubling bifurcations in two dimentional iterative maps as a generic example we use the standard form of the henon map the approximants to the feigenbaum constant lie on a universal crossover curve which turns out to be non monotonic a similar curve is found for the scaling factor more generally we discuss the crossover of the trajectory scaling function 1 and its properties in the conservative limit

on universality for area preserving maps of the plane abstract we present detailed evidence that one parameter families of area preserving maps exhibit cascades of period doubling with universal geometric scaling in the parameter we relate this behaviour to a fixed point equation of the form 1 and det d 1 r 2 r 2 in particular we argue that the scaling transformation r 2 r 2 is conjugate to the transformation 0 x y x y with 2 and in fact 2 we present some numerical evidence that 8 721 1 4 018 1 16 36 where is the asymptotic ratio of the differences of the parameter values corresponding to the successive periods 2k described above

the ƒ a formalism applied to the crossover from conservative to dissipative scaling behaviour in period doubling systems abstract we use the a spectral dimension formalism to give a global description of the crossover from conservative to dissipative scaling behaviour in period doubling systems as a generic model we consider the two dimensional henon map for which the jacobian 0 b 1 measures the amount of dissipation and we calculate the successive approximants k a b to the generalized spectral dimension for small i e experimentally feasible values of k the crossover properties from the extreme dissipative case b 0 to the conservative case b 1 are discussed

towards a semiclassical theory of the quantum baker s map abstract we develop semiclassical 0 treatments of the quantum baker s map paying special attention to a the discrete and finite nature of its hilbert space and b the asymptotic analysis of the true quantum objects themselves new techniques are described at the theoretical level exact symbolic decomposition of the propagator and at the numerical level color graphics display of operators in the kirkwood representation both of which are of general interest for other quantized chaotic maps periodic orbit theory is tested numerically and we find that log corrections need to be included before an accurate calculation of the spectrum from periodic orbits can be achieved however the asymptotic analysis of the traces of the propagator reveals unexpected departures from standard semiclassical theory those anomalies which we compute explicitly in a few cases are traced ultimately to the discontinuity of the map and to the compactness of the phase space the analysis is far from complete but the results point towards the necessity of careful assessment of semiclassical folklore when applied to chaotic maps

semiclassical dynamics in the strongly chaotic regime breaking the log time barrier we investigate the behavior of the quantum baker s transformation a system whose classical analogue is completely chaotic for time scales where the classical mechanics generates phase space structures on a scale smaller than planck s constant i e past the log time t ln 1 surprisingly we find that a semiclassical theory can accurately reproduce many features of the quantum evolution of a wave packet in this strongly mixing time regime

remarks on the quasi classical propagator of area preserving maps abstract the path integral like expression for the quantum propagator of discrete time area preserving maps is evaluated approximately by neglecting higher than second order terms in an expansion of the action about the classical paths in the resulting quasi classical approximation for the propagator special attention is paid to the recursive nature of its amplitude and the possible appearance of maslov like phases using a further approximation for the amplitude we arrive at explicit expressions which clearly show the differences between the contributions of stable and unstable classical paths an estimate for the range of validity for the quasi classical approximation is also given

eigenvalue correlations in the circular ensembles dyson 1972 introduced two types of brownian motion ensembles of random matrices for studying approximate symmetries in complex quantum systems the magnitude of symmetry breaking plays the role of a fictitious time t or 0 the authors study the eigenvalue correlations in the circular type ensembles which serve as models for the evolution operators of quantum maps with chaotic classical limits in two cases involving time reversal symmetry breaking they evaluate explicitly the eigenangle density correlation functions of all orders for all t and for all values of the matrix dimensionality n the general case is described by a hierarchic set of relations among the correlation functions as a function of t the transition in the correlations is found to be rapid for large n discontinuous for n to infinity as a function of a local parameter lambda which measures the mean square symmetry admixing matrix element in units of the local average spacing the transition is found to be smooth the same lambda dependent results were found earlier for the gaussian type ensembles which serve as models for the hamiltonian operators of autonomous chaotic systems

period doubling for two dimensional non invertible maps renormalization group analysis and quantitative universality two types of universal behavior associated with period doubling in two dimensional non invertible maps are investigated the first called c type may occur as a generic two parameter phenomenon when the map possesses a fold singularity the second kind of universal behavior called fq type is associated with the singularity arising from projecting whitney s umbrella typically this kind of period doubling scaling behavior appears as a phenomenon of codimension three i e at some critical point in the three dimensional parameter space numerical results are presented to illustrate the self similarity of the parameter space topography near the critical points c and fq the possibility of observing these new types of critical behavior in physical systems is discussed

k3 surfaces with involution equivariant analytic torsion and automorphic forms on the moduli space in this paper we introduce an invariant of a k3 surface with 2 action equipped with a 2 invariant kahler metric which we obtain using the equivariant analytic torsion of the trivial line bundle this invariant is shown to be independent of the choice of the kahler metric it can be viewed as a function on the moduli space of k3 surfaces with involution the main result of this paper is that this function can be identified with an automorphic form which characterizes the discriminant locus in particular we show that the ray singer analytic torsion of the trivial line bundle on an enriques surface with ricci flat kahler metric is given by the value of the norm of the borcherds function at its period point

self similarity of period doubling branching 3 d reversible mappings abstract we consider 3 dimensional 3 d mappings which are reversible i e possess a time reversal symmetry in the mappings studied here the reversibility is such that it guarantees the existence of one parameter families curves of symmetric periodic orbits in the phase space it is found that such curves can often intersect one another in particular a curve of n cycles can intersect a curve of 2n cycles which in turn can intersect a curve of 4n cycles etc we show that the tree of branching curves of successively doubled periods in the 3 d phase space possesses some geometric self similarity in particular we identify the scaling factors 4 018 16 363 and 8 721 previously studied 1 parameter 2 d area preserving reversible mappings in which these scalings also occur are special cases of 3 d reversible mappings in that they have an integral of motion the more general mappings studied here have no such integral we discuss the reasons for our result in terms of normal forms

a trace formula for the semiclassical limit of some hermitian operators the aim of this paper is to provide an algorithm for calculating the leading order contributions for the semiclassical series of a functional of trace class hermitian operators we base our study on the linear space structure of the space of bounded hermitian operators on quantum hilbert space we work with a basis of operators with natural leading order weyl symbols performing stationary phase calculations with no assumption on their smoothness the stationary points are tilted orbits near the periodic ones in phase space we manage to get control of the error in the technique which scales like some power a of 1 n with 1 2 a 1 the calculations are directly applicable to quantum maps and we provide some examples in finite dimension with quantum perturbed cat maps

a local index theorem for non kahler manifolds on considere des conditions suffisantes pour qu un theoreme d indice local soit vrai pour un operateur de dirac associe a une connexion qui ne coincide pas necessairement avec la connexion de levi civita l

universal properties of the transition from quasi periodicity to chaos in dissipative systems abstract an exact renormalization group transformation is developed for dissipative systems which describes how the transition to chaos may occur in a continuous and universal manner if the frequency ratio in the quasi periodic regime is held at a fixed irrational value our approach is a natural extension of k a m theory to strong coupling most of our analysis is for analytic circle maps we have found a strong coupling fixed point where invertibility is lost which describes the universal features of the transition to chaos we find numerically that any two such critical maps with the same winding number are c 1 conjugate it follows that the low frequency peaks in an experimental spectrum are universal and we determine how their envelope scales with frequency when the winding number has a periodic continued fraction our renormalization transform has a fixed point and spectra are self similar in addition for a set of non periodic winding numbers with full measure our renormalization transformation yields an ergodic trajectory in a sub space of all critical maps physically one finds singular and universal spectra that do not scale

quasiperiodicity in dissipative systems a renormalization group analysis abstract dynamical systems with quasiperiodic behavior i e two incommensurate frequencies may be studied via discrete maps which show smooth continuous invariant curves with irrational winding number in this paper these curves are followed using renormalization group techniques which are applied to a one dimensional system circle and also to an area contracting map of an annulus two fixed points are found representing different types of universal behavior a trivial fixed point for smooth motion and a nontrivial fixed point the latter representsthe incipient breakup of a quasiperiodic motion with frequency ratio the golden mean into a more chaotic flow fixed point functions are determined numerically and via an e expansion and eigenvalues are calculated

on the birational invariance of the bcov torsion of calabi yau threefolds fang et al j diff geom 80 2 175 259 2008 sect 4 conj 4 17 conjecture that a certain spectral string theoretic invariant of calabi yau threefolds the bcov torsion is a birational invariant we prove a weak form of this conjecture the proof combines the arithmetic riemann roch theorem in arakelov geometry with some inputs from motivic integration theory

on the full asymptotics of analytic torsion abstract the purpose of this article is to study the asymptotic expansion of ray singer analytic torsion associated with powers p of a given positive line bundle over a compact n dimensional complex manifold as p here we prove that the asymptotic expansion contains only the terms of the form p n i log p p n i for i n for the first two leading terms it was proved by bismut vasserot we calculate the coefficients of the terms p n 1 log p p n 1 in the kahler case and thus answer the question posed in the recent work of klevtsov ma marinescu wiegmann about quantum hall effect our second result concerns the general asymptotic expansion of the analytic torsion for a compact complex orbifold

quantum chaos and random matrix theory some new results abstract new insight into the correspondence between quantum chaos and random matrix theory is gained by developing a semiclassical theory for the autocorrelation function of spectral determinants we study in particular the unitary operators which are the quantum versions of area preserving maps the relevant random matrix ensembles are the circular ensembles the resulting semiclassical expressions depend on the symmetry of the system with respect to time reversal and on a classical parameter tru 1 where u is the classical one step evolution operator for system without time reversal symmetry we are able to reproduce the exact random matrix predictions in the limit 0 for systems with time reversal symmetry we can reproduce only some of the features of random matrix theory for both classes we obtain the leading corrections in the semiclassical theory for integrable systems is also developed resulting in expressions which reproduce the theory for the poissonian ensemble to leading order in the semiclassical limit

critical point of tori collision in quasiperiodically forced systems we report on a type of scaling behavior in quasiperiodically forced systems on the parameter plane the critical point appears as a terminal point of the tori collision bifurcation curve its location is found numerically with high precision for two basic models the forced supercritical circle map and the forced quadratic map the hypothesis of universality based on renormalization group arguments is advanced to explain the observed scaling properties for the critical attractor and for the parameter plane arrangement in the neighborhood of the criticality

period doubling in two parameter families abstract we study two aspects of the cross over from dissipative to area preserving behaviour for period doubling cascades first we consider the smoothness of the limiting bifurcation curve at the crossover point we make a conjecture and give supporting numerical evidence second we study the spectrum of the linearised doubling operator acting on arbitrary mappings and show that it can be expressed in terms of the spectrum of its restriction to area preserving mappings

the atiyah singer index theorem for families of dirac operators two heat equation proofs the purpose of this paper is to give two heat equation proofs of the index theorem of atiyah singer for a family of dirac operators

the complete set of hamiltonian intermittency scaling behaviors renormalization group equations describing the phenomenon of intermittency in hamiltonian systems are presented all solutions satisfying certain physical constraints are obtained they are the complete set of simple singularities further considerations lead to precise predictions for scaling behavior at the onset of intermittency

refined analytic torsion given an acyclic representation alpha of the fundamental group of a compact oriented odd dimensional manifold which is close enough to an acyclic unitary representation we define a refinement t alpha of the ray singer torsion associated to alpha which can be viewed as the analytic counterpart of the refined combinatorial torsion introduced by turaev t alpha is equal to the graded determinant of the odd signature operator up to a correction term the metric anomaly needed to make it independent of the choice of the riemannian metric t alpha is a holomorphic function on the space of such representations of the fundamental group when alpha is a unitary representation the absolute value of t alpha is equal to the ray singer torsion and the phase of t alpha is proportional to the eta invariant of the odd signature operator the fact that the ray singer torsion and the eta invariant can be combined into one holomorphic function allows one to use methods of complex analysis to study both invariants in particular using these methods we compute the quotient of the refined analytic torsion and turaev s refinement of the combinatorial torsion generalizing in this way the classical cheeger m uller theorem as an application we extend and improve a result of farber about the relationship between the farber turaev absolute torsion and the eta invariant as part of our construction of t alpha we prove several new results about determinants and eta invariants of non self adjoint elliptic operators

center points in second order response surface designs previous approaches for selecting the number of center points in a second order response surface design are explained and an integrated variance criterion is offered calculations extending results in the literature are made for various composite and box behnken designs for 2 k 8 the results are compared and discussed and indicate that fewer center points than initially recommended are generally appropriate

a semiclassical quantization of area preserving maps abstract a semiclassical path integral formalism is developed for a class of area preserving maps the quasi energy spectrum of these systems is represented as a sum over closed orbits in the manner of gutzwiller the method is illustrated with some numerical tests of the standard map

birth of a new class of period doubling scaling behavior as a result of bifurcation in the renormalization equation it is found that a fixed point of the renormalization group equation corresponding to a system of a unimodal map with extremum of power and a map summarizing values of a function of the dynamical variable of the first subsystem undergoes a bifurcation in the course of increase of it occurs at c 1 984396 and results in a birth of the period 2 stationary solution of the rg equation at 2 this period 2 solution corresponds to the universal period doubling behavior discovered earlier and denoted as the c type criticality kuznetsov and sataev in phys lett a 162 236 242 1992 by combination of analytical methods and numerical computations we obtain and analyze an asymptotic expansion of the period 2 solution in powers of c the developed approach resembles the e expansion in the phase transition theory in which a trivial stationary point of the rg transformation undergoes a bifurcation that gives rise to a new fixed point responsible for the critical behavior with nontrivial critical indices

an index of rotatability an index of rotatability is suggested that will enable the experimenter to obtain an immediate appreciation of the overall shape of specified variance contours for symmetrical second order designs values of the index are tabulated for the central composite designs for two to eight factors three designs of roquemore 1976 are assessed via the index comparisons are made with an entirely different index suggested by khuri 1988 it is concluded that both indexes are useful and sensibly consistent

a numerical method for locating stable periodic orbits in chaotic systems abstract a numerical method aimed at locating stable periodic orbits in strongly chaotic systems is presented the method is based on the selection of trajectory segments which are characterized by a relatively low positive local lyapunov exponent once such a selection is made convergence to stable or weakly unstable periodic orbits is obtained by a newton method the algorithm is rather general and can be used for systems with more than two degrees of freedom the proposed approach is tested on the quartic oscillator model and on the potential of the hydrogen atom in a strong magnetic field in the latter case new stable periodic orbits are found in the region of strong chaotic motion the possible quantum localization on these orbits is discussed briefly

a standard perspective on ghosts abstract the semiclassical periodic orbit formula is compared with the asymptotic expansion of exact analytical expressions for the trace of unitary propagators in the case of the quantum standard map the emphasis is on the role played by ghost orbits which are periodic orbits in a complex phase space and hence have complex actions these as well as the real orbits are shown to be necessary to recover the usual fixed point quantization in the semiclassical limit recent results on i ghost orbits arising from different bifurcation schemes ii the role of boundary conditions on the spectral contribution of different orbits both real and ghost iii scarred wavefunctions supported on ghost orbits and iv the relevance of these complex orbits to the analysis of new experiments in mesoscopic devices are also presented

response surface designs in flexible regions abstract when both variance and bias errors are present the choice of an experimental design to fit a response surface depends on the choice of the region of interest box and draper 1959 1963 and draper and lawrence 1965a 1966 investigated spherical and cuboidal regions respectively both of these regions can however be regarded as special cases of a more general flexible region family depending on a single parameter in this manner the sensitivity of the design criterion can be studied and more general results can be obtained the problem is set out here for k predictor variables and specific designs are provided for k 2 predictor variables when variance and bias errors are of roughly the same order a broad rule of thumb that emerges from this work is that for practical situations involving both variance and bias errors and for designs of the composite type used for fitting a second order surface the design points should be near the boundary of the region of interest over which a uni

an exploration of the henon quadratic map abstract the two dimensional mapping introduced by henon in 1976 which yields a strange attractor for the parameter values a 1 4 b 0 3 has been investigated over the extended region 2 a 6 1 b 1 analytical solutions have been obtained for the onset of real stable cycles of periods 1 through 6 the associated bifurcations to stable cycles of twice the period 2 to 12 have also been determined all the resulting boundary curves are displayed in the a b parameter plane finally an interesting three dimensional generalization of the henon map is announced

periodic orbit action correlations in the baker map periodic orbit action correlations are studied for the piecewise linear area preserving baker map semiclassical periodic orbit formulae together with universal spectral statistics in the corresponding quantum baker map suggest the existence of universal periodic orbit correlations the calculation of periodic orbit sums for the baker map can be performed with the help of a perron frobenius type operator this makes it possible to study periodic orbit correlations for orbits with period up to 500 iterations of the map periodic orbit correlations are found to agree quantitatively with the predictions from random matrix theory up to a critical length determined by the semiclassical error exponentially increasing terms dominate the correlations for longer orbits which are due to the violation of unitarity in the semiclassical approximation

period doubling bifurcations for families of maps on ℝ n infinite sequences of period doubling bifurcations in one parameter families 1 pf of maps enjoy very strong universality properties this is known numerically in a multitude of cases and has been shown rigorously for certain 1 pf of maps on the interval these bifurcations occur in 1 pf of analytic maps at values of the parameter tending to a limit with the asymptotically geometric ratio 1 4 6692 in this paper we indicate the main steps of a proof that the same is true for 1 pf of analytic maps from n to n whose restriction to n is real

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

a renormalisation group method for iterated maps on the real axis a renormalisation group method for one dimensional iterated maps displaying period doublings is studied the renormalisation transformation involves variable rescaling similar to the doubling transformation of feigenbaum 1978 1979 but is defined on the space of parameters of a given mapping numerical calculations are carried out for maps with different analytic dependence near the extremum

universality in the transition to chaos of dissipative systems the period doubling bifurcation process for two dimensional transforms exhibits a new class of universality when a small dissipation is taken into account the effective jacobian is then defined as a function of both the dissipation and the rank n of the cascade cycle 2n numerical simulations of a simple mechanical system and numerical calculations on the henon mapping show that the decrement lies on a continuous curve as function of the effective jacobian a method using this result to understand experimental data is explained and a first order approximation of the renormalization process yields an analytic expression of the curve among the different transitions to chaos the period doubling bifurcation cascade 1 2 has been extensively studied this transition is characterized by an experimental convergence rate of the bifurcation threshold sequence to the accumulation point the threshold of chaos it is well known that the decrement of this bifurcation cascade can take different values each value corresponds to a specific class of systems which can be characterized by some general features of the system undergoing the transition 3 4 5 we are concerned here with the two values i 4 699 the decrement of the well known one dimensional transform with a quadratic maximum 2 and ii 8 721 the decrement of a two dimensional non dissipative transforms 3 these two classes of systems are generic in physics and the two values i and ii are therefore relevant values of the decrement however these two exponents stand for the infinite dissipation case and the conservative one thus leaving out the general physical situation of a finite dissipation only hints of the effect of a small dissipation in a two dimensional mapping have been given 6 before the work of zisook 7 a thorough study of the effect of dissipation is set forth here the first two sections deal with the physical model used to perform the numerical investigation and the experimental data thus obtained a study of the renormalization process enables to generalise the relation n j n j2 first given by zisook in 7 to all transforms where the jacobian does not depend on the linearization point in the phase space furthermore a first order approximation gives an excellent analytic expression of the universal function displaying the crossover of the decrement between ii and i

ebk quantization of quasi energies the semiclassical einstein brillouin keller vortex tube quantization for the quasi energies in time periodic hamiltonians is studied for typical systems when the vortex tubes of the quantizing tubes may have complicated structures in extended phase space numerical application to a periodically driven duffing quartic oscillator yields semiclassical quasi energies in excellent agreement with the exact quantum mechanical ones for the regular states

renormalization group analysis of bifurcations in area preserving maps abstract two dimensional area preserving maps can be represented by a generating function the action high orders of successive period doubling bifurcations are studied by writing a renormalization group scheme for this action fixed points and eigenvalues of this scheme are found and interpreted

chaotic eigenfunctions in phase space we study individual eigenstates of quantized area preserving maps on the 2 torus which are classically chaotic in order to analyze their semiclassical behavior we use the bargmann husimi representations for quantum states as well as their stellar parametrization which encodes states through a minimal set of points in phase space the constellation of zeros of the husimi density we rigorously prove that a semiclassical uniform distribution of husimi densities on the torus entails a similar equidistribution for the corresponding constellations we deduce from this property a universal behavior for the phase patterns of chaotic bargmann eigenfunctions which is reminiscent of the wkb approximation for eigenstates of integrable systems though in a weaker sense in order to obtain more precise information on chaotic eigenconstellations we then model their properties by ensembles of random states generalizing former results on the 2 sphere to the torus geometry this approach yields statistical predictions for the constellations which fit quite well the chaotic data we finally observe that specific dynamical information e g the presence of high peaks like scars in husimi densities can be recovered from the knowledge of a few long wavelength fourier coefficients which therefore appear as valuable order parameters at the level of individual chaotic eigenfunctions