spectral decomposition of the renyi map the authors construct a generalized spectral decomposition of the frobenius perron operator of the general beta adic renyi map using a general iterative operator method applicable in principle to any mixing dynamical system they also explicitly define appropriate rigged hilbert spaces which provide mathematical meaning to the formally obtained spectral decomposition the explicit construction of the eigenvalues and eigenvectors allows one to show that the essential spectral radius of the frobenius perron operator decreases as the smoothness of the domain increases the reason for the change of the spectrum from the unit disk to isolated eigenvalues is the existence of coherent states of the frobenius perron operator which are not infinitely differentiable

diffusion on loopless critical percolation cluster in this paper we calculate the dynamical exponents of diffusion dw df and ds on a percolation cluster at pc with no loops for the square and simple cubic lattices by the method of spectral analysis of the transition probability matrix results show that ds varies significantly with the spatial dimension unlike in conventional percolation but the alexander orbach scaling relation ds 2df dw still holds thus it rules out the possibility that this scaling relation fails for all loopless fractals because of trapping

fick s law and fractality of nonequilibrium stationary states in a reversible multibaker map nonequilibrium stationary states are studied for a multibaker map a simple reversible chaotic dynamical system the probabilistic description is extended by representing a dynamical state in terms of a measure instead of a density function the equation of motion for the cumulative function of this measure is derived and stationary solutions are constructed with the aid of derham type functional equations to select the physical states the time evolution of the distribution under a fixed boundary condition is investigated for an open multibaker chain of scattering type this system corresponds to a diffusive flow experiment through a slab of material for long times any initial distribution approaches the stationary one obeying fick s law at stationarity the intracell distribution is singular in the stable direction and expressed by the takagi function which is continuous but has no finite derivatives the result suggests that singular measures play an important role in the dynamical description of non equilibrium states

on the symbolic dynamics of the henon map the authors study a method for finding all periodic orbits of the dissipative henon map for the canonical values a 1 4 b 0 3 this method is found to give indeed the good symbolic dynamics proposed earlier by two of them but it does not work as proposed for larger values of b this is seen in two ways although the method is thus not universally valid it can be used to study problems related to periodic orbits they find that for the usual values a 1 4 b 0 3 expressions for average quantities are not easily obtained precisely from periodic orbits in agreement with recent claims

fractal fractal dimensions of deterministic transport coefficients if a point particle moves chaotically through a periodic array of scatterers the associated transport coefficients are typically irregular functions under variation of control parameters for a piecewise linear two parameter map we analyse the structure of the associated irregular diffusion coefficient and current by numerically computing dimensions from box counting and from the autocorrelation function of these graphs we find that both dimensions are fractal for large parameter intervals and that both quantities are themselves fractal functions if computed locally on a uniform grid of small but finite subintervals we furthermore show that there is a simple functional relationship between the structure of fractal fractal dimensions and the difference quotient defined on these subintervals

probabilistic approach to homoclinic chaos three dimensional systems possessing a homoclinic orbit associated to a saddle focus with eigenvalues i and giving rise to homoclinic chaos when the shil nikov condition is satisfied are studied the 2d poincare map and its 1d contractions capturing the essential features of the flow are given at homoclinicity these 1d maps are found to be piecewise linear this property allows one to reduce the frobenius perron equation to a master equation whose solution is analytically known the probabilistic properties such as the time autocorrelation function of the state variablex are explicitly derived

spectral decomposition of chebyshev maps abstract we construct a spectral decomposition of the frobenius perron operator for the chebyshev polynomials of the first kind we defined a suitable dual pair or rigged hilbert space which provides mathematical meaning to the spectral decomposition the spectra of the even chebyshev maps do not contain the odd powers of 1 m and the odd eigenfunctions are in the null space of the frobenius perron operator moreover the odd chebyshev maps have degenerate spectra without jordan blocks the eigenvalues in the decompositions are the resonances of power spectrum and have magnitudes less than one as in the case of the family of tent maps

understanding deterministic diffusion by correlated random walks low dimensional periodic arrays of scatterers with a moving point particle are ideal models for studying deterministic diffusion for such systems the diffusion coefficient is typically an irregular function under variation of a control parameter here we propose a systematic scheme of how to approximate deterministic diffusion coefficients of this kind in terms of correlated random walks we apply this approach to two simple examples which are a onedimensional map on the line and the periodic lorentz gas starting from suitable green kubo formulae we evaluate hierarchies of approximations for their parameter dependent diffusion coefficients these approximations converge exactly yielding a straightforward interpretation of the structure of these irregular diffusion coefficients in terms of dynamical correlations

fractality of deterministic diffusion in the nonhyperbolic climbing sine map abstract the nonlinear climbing sine map is a nonhyperbolic dynamical system exhibiting both normal and anomalous diffusion under variation of a control parameter we show that on a suitable coarse scale this map generates an oscillating parameter dependent diffusion coefficient similarly to hyperbolic maps whose asymptotic functional form can be understood in terms of simple random walk approximations on finer scales we find fractal hierarchies of normal and anomalous diffusive regions as functions of the control parameter by using a green kubo formula for diffusion the origin of these different regions is systematically traced back to strong dynamical correlations starting from the equations of motion of the map these correlations are formulated in terms of fractal generalized takagi functions obeying generalized de rham type functional recursion relations we finally analyze the measure of the normal and anomalous diffusive regions in the parameter space showing that in both cases it is positive and that for normal diffusion it increases by increasing the parameter value

gibbs measures and power spectra for type i intermittent maps abstract gibbs measures of dynamical systems are investigated using the corresponding correlation function and power spectrum the method is shown to be equivalent to the concept of order q power spectra which has been developed previously a technique for the computation of these spectra is presented as an application we specially focus on the treatment of type i intermittent model maps and investigate their phase transition in the vicinity of the phase transition point the order q power spectra are shown to obey some typical scaling relation which is captured neither by the topological pressure nor by the ordinary power spectrum

simple deterministic dynamical systems with fractal diffusion coefficients we analyze a simple model of deterministic diffusion the model consists of a one dimensional array of scatterers with moving point particles the particles move from one scatterer to the next according to a piecewise linear expanding deterministic map on unit intervals the microscopic chaotic scattering process of the map can be changed by a control parameter the macroscopic diffusion coefficient for the moving particles is well defined and depends upon the control parameter we calculate the diffusion coefficent and the largest eigenmodes of the system by using markov partitions and by solving the eigenvalue problems of respective topological transition matrices for different boundary conditions we find that the largest eigenmodes of the map match the ones of the simple phenomenological diffusion equation our main result is that the diffusion coefficient exhibits a fractal structure as a function of the control parameter we provide qualitative and quantitative arguments to explain features of this fractal structure

markov chain analysis of random walks in disordered media we study the dynamical exponents d w and d s for a particle diffusing in a disordered medium modeled by a percolation cluster from the regime of extreme disorder i e when the percolation cluster is a fractal at p p c to the lorentz gas regime when the cluster has weak disorder at p p c and the leading behavior is standard diffusion a new technique of relating the velocity autocorrelation function and the return to the starting point probability to the asymptotic spectral properties of the hopping transition probability matrix of the diffusing particle is used and the latter is numerically analyzed using the arnoldi saad algorithm we also present evidence for a new scaling relation for the second largest eigenvalue in terms of the size of the cluster ln lambda max sim s d w d f which provides a very efficient and accurate method of extracting the spectral dimension d s where d s 2d f d w

correlation functions on the border lines of transient chaos chaotic properties of a repeller strongly influence the transient properties of a system close to it in particular the correlations in the transient regime in this paper results are presented for repellers of one dimensional maps having a fixed point whose lyapunov exponent agrees with the escape rate from the repeller it is proven that the corresponding natural measure of the repeller is a ensuremath delta function at the origin eigenfunctions of the frobenius perron operator are computed the correlation function is calculated near the situation of permanent chaos and anomalous decay of the correlations is found scaling properties are given on the route from a weak repeller to a nonrepeller the analytic results are supported by numerical calculations textcopyright 1996 the american physical society

markov matrix analysis of random walks in disordered continuous media abstract we study by a markov matrix analysis of the equivalent random walks the dynamic properties of continuous media consisting of both correlated and uncorrelated equal size spheres we employ a blind ant random walk model using the rule that a walker jumps among centers of the directly connected spherical particles on an infinite network the dominant eigenvalues and eigenvectors of the transition probability matrix of the random walks are calculated yielding estimates of the spectral dimension d s and the fractal dimension d w of random walks on the continuous network we find that for the present model the estimates are very close to the corresponding lattice percolation values though only after the finite size effects have been carefully taken into account we also show that the finite size scaling of the largest nontrivial eigenvalues holds for our model with the same exponents as for the lattice percolation

testing the order of discrete markov chains using surrogate data tests of the null hypothesis that a symbolic sequence is of nth markov order are described these tests are based on resampling methods similar to the method of surrogate data used to test continuous time series for nonlinear structure the size and power of these symbolic surrogate tests are estimated using binary and heptary test sequences of known order conditioning on a sufficient statistic under the null hypothesis yields an estimated true size which closely approximates the nominal size and increases the power of the test with respect to the unconditional test we apply one method to a symbolic sequence derived from the rossler system and to a binary and heptary representation of an interspike interval sequence of an electroreceptor of the weakly electric eel apteronotus leptorhynchus

irregular parameter dependence of generalized diffusion coefficients based on large deviation statistical analysis the nonperturbative non gaussian characteristics of diffusive motion are examined in the framework of the large deviation statistical theory where simple extended mapping models showing chaotic diffusion are taken as an example furthermore by rigorously solving the large deviation statistical quantities it is found that the same type of anomalous complex control parameter dependence as that for the diffusion coefficient reported by klages and dorfman is also observed in the large deviation statistical quantities such as the weighted average the generalized diffusion coefficient and the generalized power spectrum densities

probabilistic models of multidimensional piecewise expanding mappings strongly chaotic systems e g piecewise expanding mappings exhibit diffusion like behavior in the sense of central limit theorems to find more precise statements about the similarity to probabilistic diffusion we study how the evolution of probability densities underd dimensional piecewise expanding mappings can be modeled by markov processes with smooth transition probabilities such as diffusion processes our results can be viewed as a special type of local limit theorem

entropy production in a persistent random walk abstract we consider a one dimensional persistent random walk viewed as a deterministic process with a form of time reversal symmetry particle reservoirs placed at both ends of the system induce a density current which drives the system out of equilibrium the phase space distribution is singular in the stationary state and has a cumulative form expressed in terms of generalized takagi functions the entropy production rate is computed using the coarse graining formalism of gaspard gilbert and dorfman in the continuum limit we show that the value of the entropy production rate is independent of the coarse graining and agrees with the phenomenological entropy production rate of irreversible thermodynamics

monte carlo simulation of classical spin models with chaotic billiards it has recently been shown that the computing abilities of boltzmann machines or ising spin glass models can be implemented by chaotic billiard dynamics without any use of random numbers in this paper we further numerically investigate the capabilities of the chaotic billiard dynamics as a deterministic alternative to random monte carlo methods by applying it to classical spin models in statistical physics first we verify that the billiard dynamics can yield samples that converge to the true distribution of the ising model on a small lattice and we show that it appears to have the same convergence rate as random monte carlo sampling second we apply the billiard dynamics to finite size scaling analysis of the critical behavior of the ising model and show that the phase transition point and the critical exponents are correctly obtained third we extend the billiard dynamics to spins that take more than two states and show that it can be applied successfully to the potts model we also discuss the possibility of extensions to continuous valued models such as the xy model

entropy extropy and information potential in stochastic systems far from equilibrium the relations between information entropy and energy which are well known in equilibrium thermodynamics are not clear far from equilibrium moreover the usual expression of the classical thermodynamic potentials is only valid near equilibrium in previous publications we showed for a chemical system maintained far from equilibrium that a new thermodynamic potential the information potential can be de4ned by using the stochastic formalism of the master equation here we extend this theory to a completely general discrete stochastic system for this purpose we use the concept of extropy which is de4ned in classical thermodynamics as the total entropy produced in a system and in the reservoirs during their equilibration we show that the statistical equivalent of the thermodynamic extropy is the relative information if a coarse grained description by means of the thermodynamic extensive variables is available which is the case for many macroscopic systems the coarse grained statistical extropy allows one to de4ne the information potential in the thermodynamic limit using this potential we study the evolution of such systems towards a non equilibrium stationary state we derive a general thermodynamic inequality between energy dissipation and information dissipation which sharpens the law of the maximum available work for non equilibrium systems c 2002 published by elsevier science b v

grammatical complexity for two dimensional maps we calculate the grammatical complexity of the symbol sequences generated from the henon map and the lozi map using the recently developed methods to construct the pruning front when the map is hyperbolic the language of symbol sequences is regular in the sense of the chomsky hierarchy and the corresponding grammatical complexity takes finite values it is found that the complexity exhibits a self similar structure as a function of the system parameter and the similarity of the pruning fronts is discussed as an origin of such self similarity for non hyperbolic cases it is observed that the complexity monotonically increases as we increase the resolution of the pruning front

recurrence time statistics in chaotic dynamics i discrete time maps the dynamics of transitions between the cells of a finite phase space partition in a variety of systems giving rise to chaotic behavior is analyzed with special emphasis on the statistics of recurrence times in the case of one dimensional piecewise markow maps the recurrence problem is cast into a renewal process in the presence of intermittency transitions between cells define a non markovian non renewal process reflected in the presence of power law probability distributions and of divergent variances and mean values

fractal dimension of transport coefficients in a deterministic dynamical system in many low dimensional dynamical systems transport coefficients are very irregular and are perhaps even fractal functions of control parameters to analyse this phenomenon we study a dynamical system defined by a piecewise linear map and investigate the dependence of transport coefficients on the slope of the map we present analytical arguments supported by numerical calculations showing that both the minkowski bouligand and hausdorff fractal dimension of the graphs of these functions is 1 with a logarithmic correction and we find that the exponent controlling this correction is bounded from above by 1 or 2 depending on some detailed properties of the system using numerical techniques we show local self similarity of the graphs the local self similarity scaling transformations turn out to be dependent irregularly on the values of the system control parameters

computing the measure of nonattracting chaotic sets abstract recently a numerical method called the pim triple algorithm has been formulated for generating long orbits on nonattracting chaotic sets we investigate under which conditions the measure for a nonattracting invariant set generated by the pim triple algorithm agrees with the natural measure

irregular diffusion in the bouncing ball billiard abstract we call a system bouncing ball billiard if it consists of a particle that is subject to a constant vertical force and bounces inelastically on a one dimensional vibrating periodically corrugated floor here we choose circular scatterers that are very shallow hence this billiard is a deterministic diffusive version of the well known bouncing ball problem on a flat vibrating plate computer simulations show that the diffusion coefficient of this system is a highly irregular function of the vibration frequency exhibiting pronounced maxima whenever there are resonances between the vibration frequency and the average time of flight of a particle in addition there exist irregularities on finer scales that are due to higher order dynamical correlations pointing towards a fractal structure of this curve we analyze the diffusive dynamics by classifying the attracting sets and by working out a simple random walk approximation for diffusion which is systematically refined by using a green kubo formula

on exponential long time evolution in statistical mechanics penrose coveney 1994 recently introduced an invertible discrete time dynamical system called the pastry cook s transformation for which they constructed a canonical non equilibrium ensemble in the present paper we apply the brussels formalism of non equilibrium statistical mechanics to this system the use of the formalism is justified rigorously and the operators which arise in the theory are calculated exactly the set of ensembles for which the theory is valid is a banach space of functions satisfying a certain smoothness condition this condition ensures that ensembles show a decay towards equilibrium in agreement with the time asymmetry observed in thermodynamics we also calculate the decay of time correlation functions using ruelle s theory of dynamical resonances we find that all three methods furnish essentially the same description of the exponential decay to equilibrium in this system

dynamical entropy of c algebras and von neumann algebras the definition of the dynamical entropy is extended for automorphism groups ofc algebras as an example the dynamical entropy of the shift of a lattice algebra is studied and it is shown that in some cases it coincides with the entropy density

locating resonances for axiom a dynamical systems for a class of differentiable dynamical systems called axiom a systems it has been shown by pollicott and the author that correlation functions have fourier transforms which are meromorphic in a strip the poles or resonances are however not easy to locate this note reviews the results which are known and discusses a simple model where the position of resonances can be estimated the effect of noise is also discussed

negative and nonlinear response in an exactly solved dynamical model of particle transport we consider a simple model of particle transport on the line defined by a dynamical map f satisfying f x 1 1 f x for all x and f x ax b for x 1 and therein mainly to the part a 1 we study along the lines of previous investigations r klages and j r dorfman phys rev lett 74 387 1995 on the restricted symmetric b 0 one parameter version of the present model the parameter dependence of the transport properties i e not only of the diffusion coefficient d a b but this time also of the current j a b a major difference however is that this time an important tool for such a study has been available in the form of exact expressions for j and d obtained recently by one of the authors these expressions allow for a quite efficient numerical implementation which is important because the functions encountered typically have a fractal character the main results of our present preliminary survey of the parameter plane of the model are presented in several plots of these functions j a b and d a b and in an over all chart displaying in the parameter plane in principle all possibly relevant information on the system including e g the dynamical phase diagram as well as by way of illustration values of some topological invariants kneading numbers which according to the formulas determine the singularity structure of j a b and d a b what we regard as our most significant findings are 1 nonlinear response the parameter dependence of these transport properties is throughout the ergodic part of the parameter plane i e outside the infinitely many arnol d tongues fractally nonlinear 2 negative response inside certain regions with an apparently fractal boundary the current j and the bias b have opposite signs

some relations between dimension and lyapounov exponents we consider differentiable maps and compact invariant sets we introduce dimensional quantities related to the ergodic invariant measures and prove some simple relations

properties of border states of transient chaos transient chaos with absolutely continuous conditionally invariant measure is studied and a general solution of the frobenius perron equation is presented for complete one dimensional maps in particular properties of borderline situations exhibiting phase transitions within the framework of thermodynamical formalism are investigated

dynamical entropy for systems with stochastic perturbation dynamics of deterministic systems perturbed by random additive noise is characterized quantitatively since for such systems the kolmogorov sinai ks entropy diverges if the diameter of the partition tends to zero we analyze the difference between the total entropy of a noisy system and the entropy of the noise itself we show that this quantity is finite and non negative and we call it the dynamical entropy of the noisy system in the weak noise limit this quantity is conjectured to tend to the ks entropy of the deterministic system in particular we consider one dimensional systems with noise described by a finite dimensional kernel for which the frobenius perron operator can be represented by a finite matrix

hierarchy of random chaotic maps with an invariant measure hierarchy of one and many parameter families of random chaotic maps and one parameter random elliptic maps of cn type with an invariant measure have been introduced using the invariant measure sinai ruelle bowen measure the kolmogorov sinai entropy of the random chaotic maps have been calculated analytically where the numerical simulations support the results

dynamical chaos in the lorentz lattice gas this paper provides an introduction to the applications of dynamical systems theory to nonequilibrium statistical mechanics in particular to a study of nonequilibrium phenomena in lorentz lattice gases with stochastic collision rules using simple arguments based upon discussions in the mathematical literature we show that such lattice gases belong to the category of dynamical systems with positive lyapunov exponents this is accomplished by showing how such systems can be expressed in terms of continuous phase space variables expressions for the lyapunov exponent of a one dimensional lorentz lattice gas with periodic boundaries are derived other quantities of interest for the theory of irreversible processes are discussed

spectrum and eigenfunctions of the frobenius perron operator of the tent map the spectrum and eigenfunctions of the frobenius perron operator induced by the tent map are discussed in detail special attention is paid to the case where the critical point of the map lies on an aperiodic trajectory and the differences from maps with a periodic critical trajectory are stressed it is shown that the relevant eigenvalues of the spectrum are not sensitive to the aperiodicity of the critical trajectory all other parts of the spectrum and all eigenfunctions in particular are changed drastically if the critical trajectory becomes aperiodic the intimate connection between the point spectrum and the kneading invariant is established and the critical slowing down as well as the band splitting are a consequence of its properties the structure of the infinite sequence of null spaces and its implications on the spectrum of the operator are discussed it is shown that any initial distributionp 0 x of bounded variation can be projected uniquely onto the eigenfunctions of the relevant eigenvalues and that the time dependence ofp n x is determined by this expansion up to an erroro n from this the stationary and the asymptotic behavior of the correlation function x n x can be derived exactly

dynamical system models of transport chaos characteristics the macroscopic limit and irreversibility the escape rate formalism and the thermostating algorithm describe relaxation towards a decaying state with absorbing boundaries and a steady state of periodic systems respectively it has been shown that the key features of the transport properties of both approaches if modeled by low dimensional dynamical systems can conveniently be described in the framework of multibaker maps in the present paper we discuss in detail the steps required to reach a meaningful macroscopic limit the limit involves a sequence of coarser and coarser descriptions projections until one reaches the level of irreversible macroscopic advection diffusion equations the influence of boundary conditions is studied in detail only a few of the chaos characteristics possess a meaningful macroscopic limit but none of these is sufficient to determine the entropy production in a general non equilibrium state 2003 elsevier b v all rights reserved

transport in polygonal billiards we present in this work a numerical study of the dynamics of ensembles of point particles within a polygonal billiard chain this billiard is a system with no exponential instability our numerical results suggest that some members of the family exhibit normal diffusive behavior while others present anomalous diffusion our conclusions are drawn from the numerical evaluation of the mean square displacement the velocity autocorrelation function and its spectral analysis furthermore we analyze the properties of the incoherent scattering function the super burnett coefficient seems to be ill defined in all systems the multifractal analysis of the spectrum of the velocity autocorrelation functions is also reported finally we study the heat conduction in our polygonal chain

resonances in chaotic dynamics we present a discussion and some numerical results on the actual possibility of making accessible by numerical techniques the complex singularities of the power spectrum resonances for a chaotic signal henon s transformation is investigated in detail showing that the position of the leading resonance in the complex frequency plane determines the kind of mixing rate in the time evolution

invariant measurement of strange sets in terms of cycles we argue that extraction of unstable cycles and their eigenvalues is not only experimentally feasible but is also a theoretically optimal measurement of the invariant properties of a dynamical system

analytic study of chaos of the tent map band structures power spectra and critical behaviors chaotic behaviors of the tent map a piecewise linear continuous map with a unique maximum are studied analytically throughout its chaotic region in terms of the invariant density and the power spectrum as the height of the maximum is lowered successive band splitting transitions occur in the chaotic region and accumulate to the transition point into the nonchaotic region the timecorrelation function of nonperiodic orbits and their power spectrum are calculated exactly at the band splitting points and in the vicinity of these points the method of eigenvalue problems of the frobenius perron operator is used 2m 1 critical modes wherem 1 2 3 are found which exhibit the critical slowing down near the 2m 1 band to 2m band transition point after the transition these modes become periodic modes which represent the cycling of nonperiodic orbits among 2m bands together with the periodic modes generated by the preceding band splittings scaling laws near the transition point into the nonchaotic region are investigated and a new scaling law is found for the total intensity of the periodic part of the spectrum

reversible dynamics and the macroscopic rate law for a solvable kolmogorov system the three bakers reaction we investigate a piecewise linear area preserving mapt describing two coupled baker transformations on two squares with coupling parameter 0 c 1 the resulting dynamical system is kolmogorov for anyc 0 for rational values ofc we construct a generating partition on whicht induces a markov chain this markov structure is used to discuss the decay of correlation functions exponential decay is found for a class of functions related to the partition explicit results are given forc 2 n the macroscopic analog of our model is a leaking process between two badly stirred containers according to the markov analysis the corresponding progress variable decays exponentially but the rate coefficients characterizing this decay are not those determined from the one way flux across the cell boundary the validity of the macroscopic rate law is discussed

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

analytic study of power spectra of the tent maps near band splitting transitions successive band splitting transitions occur in the one dimensional map xi 1 g xi i 0 1 2 withg x x 0 x 1 2 x 1 2 x 1 as the parameter is changed from 2 to 1 the transition point fromn 2n bands to 2nbands is given by 2 1 n n 0 1 2 the time correlation function i xi x0 x0 2 xi xi xi is studied in terms of the eigenvalues and eigenfunctions of the frobenius perron operator of the map it is shown that near the transition point 2 i 10 4 2 17 i 0 10 2 8 51 i 1 7 4 2 17 1 ie yi where 2 2 is the damping constant and vanishes at 2 representing the critical slowing down this critical phenomenon is in strong contrast to the topologically invariant quantities such as the lyapunov exponent which do not exhibit any anomaly at 2 the asymptotic expression for i has been obtained by deriving an analytic form of i for a sequence of which accumulates to 2 from the above near the transition point 2 1 n the damping constant of i fori n is given by n 2 n 2 n numerical calculation is also carried out for arbitrary a and is shown to be consistent with the analytic results

simple maps with fractal diffusion coefficients we consider chains of one dimensional piecewise linear chaotic maps with uniform slope we study the diffusive behavior of an initially nonuniform distribution of points as a function of the slope of the map by solving the frobenius perron equation for markov partition values of the slope we relate the diffusion coefficient to eigenvalues of the topological transition matrix the diffusion coefficient obtained shows a fractal structure as a function of the slope of the map this result may be typical for a wide class of maps such as two dimensional sawtooth maps

the origin of diffusion the case of non chaotic systems abstract we investigate the origin of diffusion in non chaotic systems as an example we consider 1d map models whose slope is everywhere 1 therefore the lyapunov exponent is zero but with random quenched discontinuities and quasi periodic forcing the models are constructed as non chaotic approximations of chaotic maps showing deterministic diffusion and represent one dimensional versions of a lorentz gas with polygonal obstacles e g the ehrenfest wind tree model in particular a simple construction shows that these maps define non chaotic billiards in space time the models exhibit in a wide range of the parameters the same diffusive behavior of the corresponding chaotic versions we present evidence of two sufficient ingredients for diffusive behavior in one dimensional non chaotic systems i a finite size algebraic instability mechanism ii a mechanism that suppresses periodic orbits

resonances in a chaotic attractor crisis of the lorenz flow local bifurcations of stationary points and limit cycles have successfully been characterized in terms of the critical exponents of these solutions lyapunov exponents and their associated covariant lyapunov vectors have been proposed as tools for supporting the understanding of critical transitions in chaotic dynamical systems however it is in general not clear how the statistical properties of dynamical systems change across a boundary crisis during which a chaotic attractor collides with a saddle this behavior is investigated here for a boundary crisis in the lorenz flow for which neither the lyapunov exponents nor the covariant lyapunov vectors provide a criterion for the crisis instead the convergence of the time evolution of probability densities to the invariant measure governed by the semigroup of transfer operators is expected to slow down at the approach of the crisis such convergence is described by the eigenvalues of the generator of this semigroup which can be divided into two families referred to as the stable and unstable ruelle pollicott resonances respectively the former describes the convergence of densities to the attractor or escape from a repeller and is estimated from many short time series sampling the state space the latter is responsible for the decay of correlations or mixing and can be estimated from a long times series invoking ergodicity it is found numerically for the lorenz flow that the stable resonances do approach the imaginary axis during the crisis as is indicative of the loss of global stability of the attractor on the other hand the unstable resonances and a fortiori the decay of correlations do not flag the proximity of the crisis thus questioning the usual design of early warning indicators of boundary crises of chaotic attractors and the applicability of response theory close to such crises

classical diffusion on eden trees we study an aggregation process which gives rise to compact clusters with no loops a generalization of the node counting theorem applicable to such branched graphs trees is proved this is used to determine the spectral dimension of these clusters on square and cubic embedding lattices by monte carlo simulations the results are explained in terms of the geometric structure of these trees

quantum physics and human language human languages employ constructions that tacitly assume specific properties of the limited range of phenomena they evolved to describe these assumed properties are true features of that limited context but may not be general or precise properties of all the physical situations allowed by fundamental physics in brief human languages contain excess baggage that must be qualified discarded or otherwise reformed to give a clear account in the context of fundamental physics of even the everyday phenomena that the languages evolved to describe the surest route to clarity is to express the constructions of human languages in the language of fundamental physical theory not the other way around these ideas are illustrated by an analysis of the verb to happen and the word reality in special relativity and the modern quantum mechanics of closed systems

finite size analysis of eigenvalue spectrum for random walks on a critical percolation cluster in four dimensions we study by markov chain analysis the random walks on a critical percolation cluster embedded in a four dimensional hypercubic lattice we calculate the number of dominant eigenvalues of the transition probability matrix and estimate the spectral and fractal dimensions ds and dw of random walks from the eigenvalues and their distribution the estimates of ds and dw obtained from the data for a given size s of the percolation cluster exhibit some s dependence extrapolating the results to s limit we obtain ds 1 330 0 010 close to the previous result by other methods and a new result dw 4 50 0 15 these values are also confirmed by direct monte carlo simulations of random walks on a percolation cluster

diffusion effusion and chaotic scattering an exactly solvable liouvillian dynamics we study diffusion and chaotic scattering in a chain of baker maps coupled together which forms an area preserving mapping of an infinitely extended strip onto itself this exactly solvable mapping sustains chaotic behaviors and diffusion processes the relationship between the diffusion coefficient the lyapunov exponent and the entropy per unit time is derived the long lived classical resonances of the liouville evolution operator are proved to converge toward the eigenvalues of the phenomenological diffusion equation in this sense there is a quasi isomorphism between the resonance spectrum of the liouville evolution and the eigenvalue spectrum of the phenomenological diffusion equation furthermore we show that a fractal repeller is associated to each non equilibrium state in the isolated and finite multibaker chain the nonequilibrium states are all unstable with respect to the equilibrium validating a weak form of the second principle of thermodynamics for the present dynamical system consequences of nonequilibrium fractals on classical measurements are discussed we then describe the open multibaker chain as a scattering system fractal properties of chaotic scattering are here shown to be related to diffusion in the chain

fractal properties of anomalous diffusion in intermittent maps an intermittent nonlinear map generating subdiffusion is investigated computer simulations show that the generalized diffusion coefficient of this map has a fractal discontinuous dependence on control parameters an amended continuous time random walk theory well approximates the coarse behavior of this quantity in terms of a continuous function this theory also reproduces a full suppression of the strength of diffusion which occurs at the dynamical transition from normal to anomalous diffusion similarly the probability density function of this map exhibits a nontrivial fine structure while its coarse functional form is governed by a time fractional diffusion equation a more detailed understanding of the irregular structure of the generalized diffusion coefficient is provided by an anomalous taylor green kubo formula establishing a relation to de rham type fractal functions

spatial chaos in weakly dispersive and viscous media a nonperturbative theory of the driven kdv burgers equation abstract the asymptotic travelling wave solution of the kdv burgers equation driven by the long scale periodic driver is constructed the solution represents a shock train in which the quasi periodic sequence of dispersive shocks or soliton chains is interspersed by smoothly varying regions it is shown that the periodic solution which has the spatial driver period undergoes period doublings as the governing parameter changes two types of chaotic behavior are considered the first type is a weak chaos where only a small chaotic deviation from the periodic solution occurs the second type corresponds to the developed chaos where the solution ignores the driver period and represents a random sequence of uncorrelated shocks in the case of weak chaos the shock coordinate being repeatedly mapped over the driver period moves on a chaotic attractor while in the case of developed chaos it moves on a repeller both solutions depend on a parameter indicating the reference shock position in the shock train the structure of a one dimensional set to which this parameter belongs is investigated this set contains measure one intervals around the fixed points in the case of periodic or weakly chaotic solutions and it becomes a fractal in the case of strong chaos the capacity dimension of this set is calculated

diffusion on a dla cluster in two and three dimensions we present a much improved calculation of the dynamic critical exponents of diffusion dw ds and the combination df dw for the dla diffusion limited aggregate on square and simple cubic lattices using two independent approaches one based on the exact enumeration for the displacement and velocity autocorrelation and the other on the eigenspectrum of random walks on the dla the two methods give consistent results which however clearly rule out the scaling relation ds 2df dw first proposed by alexander and orbach 1982 similarly to the case of the eden tree reported previously

dc currents in hamiltonian systems by levy flights we study the mechanism leading to directed transport in the stochastic layer of ac driven hamiltonian systems we show that current rectification is obtained by breaking the symmetry of levy flights in ballistic channels which exist due to resonance islands with nonzero winding numbers in the framework of the continuous time random walk ctrw approach we construct a generalized asymmetric flights model and derive an expression for the current in terms of the characteristics of the relevant ballistic channels we find very good agreement between the results of direct integration of the dynamical equations and the ctrw formulation

transient chaos and critical states in generalized baker maps generalized multibaker maps are introduced to model dissipative systems which are spatially extended only in certain directions and escape of particles is allowed in other ones effects of nonlinearity are investigated by varying a control parameter emphasis is put on the appearance of the critical state representing the borderline of transient chaos where anomalous behavior sets in the investigations extend to the conditionally invariant and the related natural measures and to transient diffusion in normal and critical states as well permanent chaos is also considered as a special case

entropy production in open volume preserving systems we describe a mechanism leading to positive entropy production in volume preserving systems under nonequilibrium conditions we consider volume preserving systems sustaining a diffusion process like the multibaker map or the lorentz gas a continuous flux of particles is imposed across the system resulting in a steady gradient of concentration in the limit where such flux boundary conditions are imposed at arbitrarily separated boundaries for a fixed gradient the invariant measure becomes singular for instance in the multibaker map the limit invariant measure has a cumulative function given in terms of the nondifferentiable takagi function because of this singularity of the invariant measure the entropy must be defined as a coarse grained entropy instead of the fined grained gibbs entropy which would require the existence of a regular measure with a density the coarse grained entropy production is then shown to be asymptotically positive and moreover given by the entropy production expected from irreversible thermodynamics

diffusion and spectral dimension on eden tree the authors calculate the eigenspectrum of random walks on the eden tree in two and three dimensions from this they calculate the spectral dimension ds and the walk dimension dw and test the scaling relation ds 2df dw 2d dw for an eden tree finite size induced crossovers are observed whereby the system crosses over from a short time regime where this relation is violated particularly in two dimensions to a long time regime where the behaviour appears to be complicated and dependent on dimension even qualitatively

fluctuation theorem for hidden entropy production elimination of seemingly unnecessary variables in markovian models may cause a difference in the value of irreversible entropy production between the original and reduced dynamics we show that such difference which we call the hidden entropy production obeys an integral fluctuation theorem if all variables are time reversal invariant or if the density function is symmetric with respect to the change of sign of the time reversal antisymmetric variables the theorem has wide applicability since the proposed condition is mostly satisfied in the case where the hidden fast variables are equilibrated the main consequence of this theorem is that the entropy production decreases by the coarse graining procedure by contrast in the case where a stochastic process is obtained by coarse graining a deterministic and reversible dynamics the entropy production may increase implying that the integral fluctuation theorem should not hold for such reductions we reveal with an explicit example that the nonequilibrated time reversal antisymmetric variables play a crucial role in distinguishing these two cases thus guaranteeing the consistency of the presented theorem

the thermodynamic formalism for expanding maps letf x x be an expanding map of a compact space small distances are increased by a factor 1 a generating function z is defined which countsf periodic points with a weight one can express in terms of nonstandard fredholm determinants of certain transfer operators which can be studied by methods borrowed from statistical mechanics in this paper we review the spectral properties of the transfer operators and the corresponding analytic properties of z gibbs distributions and applications to julia sets are also discussed some new results are proved and some natural conjectures are proposed

r adic one dimensional maps and the euler summation formula for the r adic one dimensional maps the author explicitly constructs the decaying eigenstates and adjoint eigenstates associated with the ruelle resonances it is shown that the eigenfunctions of the corresponding frobenius perron operator are the well known bernoulli polynomials the adjoint eigendistributions are obtained as derivatives of the dirac distributions at the end points of the unit interval the resulting expansion of the initial density in terms of the decaying eigenstates is given by the euler summation formula

statistical approach to nonhyperbolic chaotic systems spectral properties of the evolution operator for probability densities are obtained for unimodal maps for which all periodic orbits are unstable and the lyapunov exponent calculated from the first iterate of the critical point converges to a positive constant the method is applied to the logistic map both for parameter values at which finite markov partitions can be found as well as for more typical parameter values a universal behavior is found for the spectral gap in the period doubling inverse cascade of chaotic band merging bifurcations full agreement with numerical simulation is obtained textcopyright 1996 the american physical society

generalized spectral decomposition of the β adic baker s transformation and intrinsic irreversibility abstract we construct a generalized spectral decomposition of the frobenius perron operator of the adic baker s transformation using a general iterative operator method applicable in principle for any mixing dynamical system the eigenvalues in the decomposition are related to the decay rates of the autocorrelation functions and have magnitudes less than one we explicitly define appropriate generalized function spaces which provide mathematical meaning to the formally obtained spectral decomposition the unitary frobenius perron evolution of densities when extended to the generalized function spaces splits into two semigroups one decaying in the future and the other in the past this split which reflects the asymptotic evolution of the forward and backward k partitions shows the instrinsic irreversibility of the baker s transformation

chaotic saddles at the onset of intermittent spatiotemporal chaos in a recent study rempel and chian phys rev lett 98 014101 2007 it has been shown that nonattracting chaotic sets chaotic saddles are responsible for intermittency in the regularized long wave equation that undergoes a transition to spatiotemporal chaos stc via quasiperiodicity and temporal chaos in the present paper it is demonstrated that a similar mechanism is present in the damped kuramoto sivashinsky equation prior to the onset of stc a spatiotemporally chaotic saddle coexists with a spatially regular attractor after the transition to stc the chaotic saddle merges with the attractor generating intermittent bursts of stc that dominate the post transition dynamics

contrary quantum histories and contrary inferences abstract in the context of the quantum histories approach to quantum mechanics we define a contrary relation between quantum histories generalizing the standard definition of contrary quantum properties using this contrary relation we study the possibility of inferring contrary quantum histories we consider the quantum histories approach with three different consistency conditions weak medium and global consistency conditions for the first two conditions we show that it is possible to infer contrary histories from different families of histories in accordance with previous results for global consistency condition we prove that it is not possible to infer contrary histories

generalized markov coarse graining and spectral decompositions of chaotic piecewise linear maps spectral decompositions of the evolution operator for probability densities are obtained for the most general one dimensional piecewise linear markov maps and a large class of repellers the eigenvalues obtained with respect to the space of functions piecewise analytic over the minimal markov partition equal the reciprocals of the zeros of the ruelle zeta functions the logarithms of the zeros correspond to the decay rates of time correlation functions of analytic observables when the system is mixing the space can also be extended to include piecewise analytic observables permitted to have discontinuities at the elements of any given periodic orbit s so that local behavior of observables can be considered the new spectra associated with the extension are surprisingly simple and are related to the relative stability factors of the given orbit s finally arbitrarily slowly decaying periodic and aperiodic nonanalytic eigenmodes are constructed

overview of theoretical models for reaction rates there is enormous recent interest in the development of models for rate processes because rates are an almost universal characterization in the physical and biological sciences in this paper we provide an introduction to several of the problems to be discussed in greater depth by other speakers at a symposium held at the national institutes of health on may 6 8 1985 this review will focus on 1 the smoluchowski model for reaction rates together with its extension by onsager 2 first passage time formalism for discrete and continuous master equations and fokker planck equations 3 the kramers model and its extensions 4 diffusion in the presence of trapping centers

strange objects in the complex plane julia sets are examined as examples of strange objects which arise in the study of long time properties of simple dynamical systems technically they are the closure of the set of unstable cycles of analytic maps physically they are sets of points which lead to chaotic behavior the mapf z z2 p is analyzed for smallp where the julia set is a closed curve and for largep where the julia set is completely disconnected in both cases the hausdorff dimension is calculated in perturbation theory and numerically an expression for the rate at which points escape from the neighborhood of the julia set is derived and tested in a numerical simulation of the escape

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

time reversibility in nonequilibrium thermomechanics abstract time reversibility of dynamical systems has had many different interpretations for the governing differential equations for the numerical approximations to their solutions for the maps which caricature these equations and their solutions and for the phase space distributions describing ensembles of systems here i discuss some of these views taken from recent research literature and relate them to my own experience which is primarily computational i emphasize the usefulness of thermomechanics a simple generalization of conventional molecular dynamics which leads directly inexorably and paradoxically from hamilton s principle of least action to the multifractal time irreversible phase space behavior summed up by the macroscopic second law of thermodynamics

description of diffusion on discrete arrays through system of piecewise linear maps abstract it is shown that diffusion s dynamic on any discrete array including fractals can be described by piecewise linear maps the proposed method which is geometric in its nature allows to construct a system of piecewise linear maps based on the nearest neighbor coupling matrix of cosenza and kapral we propose that the dynamics of the map system describes the diffusion dynamics on the array which in our case is the sierpinski gasket due to the existence of an intrinsic homeomorphism between them it is found that there is a one to one correspondence between the partitions of the map and the cells of the fractal this fact allows to generate specific diffusion trajectories on the fractal in order to study the particle dwell time distribution since the method proposed here is completely deterministic the diffusion on the map is anomalous the mean square displacement scales with time to a power very near to 2 it is also shown that as a consequence of the lattice autosimilarity and the finite precision of the calculations this parameter remains fixed for several levels of construction of the sierpinski gasket

on the long time behaviour of ensembles in a model of deterministic diffusion we describe a large model system based on the baker transformation where deterministic diffusion occurs the model is similar to one recently considered by gaspard and by hasegawa and driebe we point out the close relationship between this system and a simple random walk and analyse the evolution with time of ensembles in the system using the resolvent based version of the subdynamics formalism developed by prigogine and his collaborators we obtain an exact and rigorous description of the long time behaviour of ensembles including the irreversible approach to equilibrium for the case where the system has finite size we also consider the thermodynamic limit where the size of the system becomes infinite and derive a description of the long time behaviour in this case where correlations decay non exponentially with time

complete description of a quantum system at a given time a generalization of the description of a quantum system in the time interval between two measurements is presented a new concept of a generalized state is introduced generalized states yield a complete description of a quantum system when information about the system is available both from the past snd from the future the formalism of generalized states provides a natural language for describing many peculiar situations in particular situations in which one can ascertain the result of a measurement of any one of several non commuting variables are analysed weak measurements on quantum systems described by generalized states are discussed the relation belween weak and strong measurements is investigated

repellers semi attractors and long lived chaotic transients abstract we study the chaotic transients observed in many deterministic systems in general they are related to strange repellers or semi attractors if they are repelling in some and attracting in other directions we propose formulas relating the average life time of the transient to dimensions of the repeller and to lyapunov exponents of the flow on it the formulas are tested numerically in a number of cases

dynamics of coarse grained variables in spatially extended systems abstract some new ways to probe the dynamics of spatially extended systems giving rise to spatio temporal chaos are explored coarse grained observables suitable for describing large scale behavior are introduced by locally averaging the original variables over a spatial region and their stability spatial correlation function and statistical properties are compared to those of the original variables in general these observables do not obey closed form equations truncation schemes in which high order fourier modes or variables other than the coarse grained ones are discarded are developed and their sensitivity to the parameters is assessed

approach to equilibrium for locally expanding maps in ℝ k by using a well known technique from classical statistical mechanics of one dimensional lattice spin systems we prove existence of an absolutely continuous invariant asymptotic measure for certain locally expanding mapst of the unit cube in k we generalize herewith in a certain sense the results of lasota and yorke on piecewise expanding maps of the unit interval to higher dimensions we show a kuzmin type theorem for these systems from which exponential approach to equilibrium and strong mixing properties follow