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

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

finite fuel problem in nonlinear singular stochastic control we investigate via the dynamic programming approach a finite fuel nonlinear singular stochastic control problem of bolza type we prove that the associated value function is continuous and that its continuous extension to the closure of the domain coincides with the value function of a nonsingular control problem for which we prove the existence of an optimal control moreover such a continuous extension is characterized as the unique viscosity solution of a quasi variational inequality with suitable boundary conditions of mixed type

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

numerical assessment of time domain methods for the estimation of local arterial pulse wave speed a local estimation of pulse wave speed c an important predictor of cardiovascular events can be obtained at arterial locations where simultaneous measurements of blood pressure p and velocity u arterial diameter d and u flow rate q and cross sectional area a or p and d are available using the pu loop sum of squares 2 lndu loop qa loop or new d2p loop methods here these methods were applied to estimate c from numerically generated p u d q and a waveforms using a visco elastic one dimensional model of the 55 larger human systemic arteries in normal conditions theoretical c were calculated from the parameters of the model estimates of c given by the loop methods were closer to theoretical values and more uniform within each arterial segment than those obtained using the 2 the smaller differences between estimates and theoretical values were obtained using the d2p loop method with root mean square errors rmse smaller than 0 18ms 1 followed by averaging the two c given by the pu and lndu loops rmse 2 99ms 1 in general the errors of the pu lndu and qa loops decreased at locations where visco elastic effects were small and nearby junctions were well matched for forward travelling waves the 2 performed better at proximal locations

fractal diffusion coefficient from dynamical zeta functions dynamical zeta functions provide a powerful method to analyse low dimensional dynamical systems when the underlying symbolic dynamics is under control on the other hand even simple one dimensional maps can show an intricate structure of the grammar rules that may lead to a non smooth dependence of global observables on parameters changes a paradigmatic example is the fractal diffusion coefficient arising in a simple piecewise linear one dimensional map of the real line using the baladi ruelle generalization of the milnor thurnston kneading determinant we provide the exact dynamical zeta function for such a map and compute the diffusion coefficient from its smallest zero

chaotic scattering theory of transport and reaction rate coefficients the chaotic scattering theory is here extended to obtain escape rate expressions for the transport coefficients appropriate for a simple classical fluid or for a chemically reacting system this theory allows various transport coefficients such as the coefficients of viscosity thermal conductivity etc to be expressed in terms of the positive lyapunov exponents and kolmogorov sinai entropy of a set of phase space trajectories that take place on an appropriate fractal repeller this work generalizes the previous results of gaspard and nicolis for the coefficient of diffusion of a particle moving in a fixed array of scatterers

propagation of maxima and strong maximum principle for viscosity solutions of degenerate elliptic equations i convex operators we describe the set of propagation of maxima of upper semicontinuous viscosity subsolutions of fully nonlinear degenerate elliptic hamilton jacobi bellman equations in the con cave case i e for operators represented as the minimum of a parametrized family of 2 nd order linear operators we show that the set where an interior maximum propagates contains the reach able set of a suitable deterministic diverential game as well as the unavoidable set of a controlled divusion process we also obtain a strong maximum principle for a class of operators that are not strictly elliptic

the fractality of the relaxation modes in reaction diffusion systems abstract in chaotic reaction diffusion systems with two degrees of freedom the modes governing the exponential relaxation to the thermodynamic equilibrium present a fractal structure which can be characterized by a hausdorff dimension for long wavelength modes this dimension is related to the lyapunov exponent and to a reactive diffusion coefficient this relationship is tested numerically on a reactive multibaker model and on a two dimensional periodic reactive lorentz gas the agreement with the theory is excellent

the classification of topologically expansive lorenz maps we show that topologically expansive lorenz maps can be described up to topological conjugacy by their kneading invariants we also give a simple condition on pairs of symbol sequences which is satisfied if and only if that pair of sequences is the kneading invariant for some topologically expansive lorenz map a simple extension of the theorems to the case of expansive maps of the interval with multiple discontinuities is described

prime and renormalisable kneading invariants and the dynamics of expanding lorenz maps abstract we study the dynamics and kneading theory of topologically expansive piecewise increasing maps of the interval with a single discontinuity lorenz maps we characterise those maps which are locally eventually onto locally onto and neither of the above in terms of the renormalisability or otherwise of the kneading invariant we also show how the decomposition of the non wandering set is related to the renormalisability of the kneading invariant we discuss our results in the context of other authors results on similar classes of 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

chaotic scattering and diffusion in the lorentz gas a chaotic scattering theory of diffusion in the lorentz gas is presented the scattering process is considered on disk scatterers of increasing sizes in this way chaotic and fractal properties of the scattering process are related to diffusion a formula is obtained that gives the diffusion coefficient in terms of the lyapunov exponent and the hausdorff codimension of the fractal repeller of orbits trapped in the scatterer numerical results are presented that support our theoretical results

dynamical crossover in deterministic diffusion we study diffusion in a one dimensional periodic array of scatterers modeled by a simple map the chaotic scattering process of the map can be changed by a control parameter and exhibits a dynamics analogous to a crisis in chaotic scattering we show that the associated strong backscattering induces a crossover between different asymptotic laws for the parameter dependent diffusion coefficient these laws are obtained from exact diffusion coefficient results and are supported by simple random walk models we conjecture that the main physical feature of this crossover is present in many other dynamical systems exhibiting nonequilibrium transport s1063 651x 97 50502 2 pacs number s 05 45 1b 05 60 1w

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

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

mean field games with common noise a theory of existence and uniqueness is developed for general stochastic differential mean field games with common noise the concepts of strong and weak solutions are introduced in analogy with the theory of stochastic differential equations and existence of weak solutions for mean field games is shown to hold under very general assumptions examples and counter examples are provided to enlighten the underpinnings of the existence theory finally an analog of the famous result of yamada and watanabe is derived and it is used to prove existence and uniqueness of a strong solution under additional assumptions

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

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

dispersion of particles in periodic media we discuss the long time properties of the dispersion of particles in periodic media using the random walk formalism exact asymptotic results are obtained for the average velocity and the diffusion coefficient expressed in terms of the green s function of the random walk inside the periodically repeated unit cell we explicitly calculate the transport coefficients for several specific cases of interest including a system with dead zones a simple model for field induced trapping and a one dimensional map leading to deterministic diffusion

state constrained stochastic optimal control problems via reachability approach this paper deals with a class of stochastic optimal control problems socps in the presence of state constraints it is well known that for such problems the value function is in general discontinuous and its characterization by a hamilton jacobi equation requires additional assumptions involving an interplay between the boundary of the set of constraints and the dynamics of the controlled system here we give a characterization of the epigraph of the value function without assuming the usual controllability assumptions to this end the socp is first translated into a state constrained stochastic target problem then a level set approach is used to describe the backward reachable sets of the new target problem it turns out that these backward reachable sets describe the value function the main advantage of our approach is that it allows us to easily handle the state constraints by an exact penalization however the target problem involves a new state variable and a new control variable that is u

matrices and eigenfunctions induced by markov maps abstract the matrix m induced by a markov map has 1 as the eigenvalue of maximum modulus if m is also irreducible then the algebraic and geometric multiplicities of the eigenvalue 1 are also 1 let g g 1 g 2 g n be an n row vector such that g 1 c and g i c i 2 then every such g is the unique upto constant multiples left eigenvector associated with the eigenvalue 1 of a matrix m induced by a markov map

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

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

a supermartingale characterization of sets of stochastic integrals and applications a supermartingale characterization of sets of stochastic integrals is given along with its applications to control and diffusion approximation the characterization is convenient for passing to the limit under natural conditions it is proved that the set of distributions of controlled diffusion processes is convex and compact

generalized solutions in nonlinear stochastic control problems an optimal stochastic control problem is considered for systems with unbounded controls satisfying an integral constraint it is shown that there exists an optimal control within the class of generalized controls leading to impulse actions applying an approach of time transformation developed recently for deterministic systems the original control problem is shown to be equivalent to an optimal stopping problem moreover the description of generalized solutions is given in terms of stochastic differential equations governed by a measure

martingale approach to stochastic differential games of control and stopping we develop a martingale approach for studying continuous time stochastic differential games of control and stopping in a non markovian framework and with the control affecting only the drift term of the state process under appropriate conditions we show that the game has a value and construct a saddle pair of optimal control and stopping strategies crucial in this construction is a characterization of saddle pairs in terms of pathwise and martingale properties of suitable quantities

on the existence of optimal controls the optimal control problem where the state is governed by an ito stochastic differential equation possibly just an ordinary differential equation is formulated in martingale terms under a coercivity condition which is weaker than compactness of the control set a convexity condition and mild continuity hypotheses on the data it is shown by the direct method that optimal controls exist hard and soft constraints are allowed in the absence of soft constraints it is shown that there exists an optimal control that is a function only of the present time and state i e the synthesis problem has a solution the main tool here is krylov s markovian selection theorem

applicable stochastic control from theory to practice stochastic control has come of age it is currently applied in a wide variety of fields spanning cancer research and chemotherapy economics and finance engineering management and many other areas of scientific and social investigation both theoretical models formulated to elucidate some characteristics of a problem under study and practical applications which have sought to provide specific answers to well stated problems have been the subjects of intense research for example economists have simultaneously groped with the underlying theoretical constructs justifying the adoption of particular econometric models while at the same time attempting to devise algorithms and optimization approaches to provide economic policy guidelines based on these models in the same spirit t remendous interest in options management which are based on stochastic dynamic models has stimulated much research and numerical analyses not only in universities but also in the back rooms of trading firms the usefulness of stochastic control was recognized once it became clear that it can be used as a unifying framework for treating multi stage or multi period optimization problems subject to stochastic disturbances however just as it is convenient as a formalism to formulate such problems it is becoming increasingly clear that it is practically difficult to solve them there are in fact only few problems one can solve completely and obtain optimal solutions for this reason there is today an increased awareness that a broad range of techniques including stochastic approximations perturbation techniques markov decision processes etc are an essential part of stochastic control kits it is in such a spirit that computer software such as pandore of inria the adaptive control algorithms for econometric models developed by kendrick and many others has been devised providing thereby computer aided expertise for managing stochastic control problems this software is essential even after the great effort applied to solving such problems over the last three decades ever since bellman s principle of optimality there are very few non trivial problems we can solve theoretically these include the well known linear quadratic stochastic control scp problem some version of the multi armed bandit problem and some specific linear optimization problems for example even the adaptive control problems studied over the last few years by kendrick and his co workers and presumed to be convex are believed today to include some unspecified nonconvexities private communication by d kendrick and hans amman in this spirit the intent in this paper is to introduce salient aspects of stochastic control which are useful for applications in management a review of applications can be found in neck 1984 for example as well as in books such as tapiero 1977 1988 bensoussan hurst and naslund 1974 bensoussan kleindorfer and tapiero 1978 1980 sethi and thompson 1981 kamien and schwarz 1981 malliaris and brock 1982 and whittle 1982 further due to an awesome amount of research published in this area appearing under various connotations such a task has become unwieldy rather

the lagrange and the vanishing discount techniques to controlled diffusions with cost constraints abstract in this paper we introduce two useful methods to compute optimal control policies for either the discounted or the average payoff criterion with cost constraints when the dynamic system evolves as a n dimensional diffusion processes as for the attribute cost constraints we mean the coexistence of a given cost function that in general is dominated above by another function in particular by a constant playing the role of an extra constraint in the control model to deduce optimality results for the discounted case we employ the lagrange multipliers technique along with dynamic programming arguments then the vanishing discount method is applied to easily obtain average optimal policies we support our theory by providing an example on pollution accumulation problem

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

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

hydrodynamic modes as singular eigenstates of the liouvillian dynamics deterministic diffusion hydrodynamic modes of diffusion and the corresponding nonequilibrium steady states are studied as an eigenvalue problem for the liouvillian dynamics of spatially extended suspension flows which are special continuous time dynamical systems including billiards defined on the basis of a mapping the infinite spatial extension is taken into account by spatial fourier transforms which decompose the observables and probability densities into sectors corresponding to the different values of the wave number the frobenius perron operator ruling the time evolution in each wave number sector is reduced to a frobenius perron operator associated with the mapping of the suspension flow in this theory the dispersion relation of diffusion is given as a pollicott ruelle resonance of the frobenius perron operator and the corresponding eigenstates are studied formulas are derived for the diffusion and the burnett coefficients in terms of the mapping of the suspension flow nonequilibrium steady states are constructed on the basis of the eigenstates and are given by mathematical distributions without density functions also referred to as singular measures the nonequilibrium steady states are shown to obey fick s law and to be related to zubarev s local integrals of motion the theory is applied to the regular lorentz gas with a finite horizon generalizations to the nonequilibrium steady states associated with the other transport processes are also obtained

optimal relaxed control of dissipative stochastic partial differential equations in banach spaces we study an optimal relaxed control problem for a class of semilinear stochastic pdes on banach spaces perturbed by multiplicative noise and driven by a cylindrical wiener process the state equation is controlled through the nonlinear part of the drift coefficient which satisfies a dissipative type condition with respect to the state variable the main tools of our study are the factorization method for stochastic convolutions in umd type 2 banach spaces and certain compactness properties of the factorization operator and of the class of young measures on suslin metrizable control sets

an ergodic control problem for constrained diffusion processes existence of optimal markov control an ergodic control problem for a class of constrained diffusion processes is considered the goal is the almost sure minimization of long term cost per unit time the main result of the paper is that there exists an optimal markov control for the considered problem it is shown that under the assumption of regularity of the skorohod map and the assumption that the drift vector field takes values in a certain cone of stability the class of controlled diffusion processes considered have strong uniform in control stability properties the role of the boundary is critical in obtaining the stability and ergodic control results for the class of controlled constrained diffusion processes considered in this work since the domains are unbounded and the corresponding unconstrained diffusions are typically transient these stability properties are key in obtaining appropriate tightness estimates once these estimates are available the remaining work lies in identifying weak limits of a certain family of occupation measures in this regard an extension to the echeverria weiss kurtz characterization of invariant measures of markov processes to the case of constrained controlled processes considered in this paper is proved this characterization result is also crucially used in proving the compactness of the family of invariant measures of markov processes corresponding to all possible markov controls

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

exact treatment of mode locking for a piecewise linear map a piecewise linear map with one discontinuity is studied by analytic means in the two dimensional parameter space when the slope of the map is less than unity periodic orbits are present and we give the precise symbolic dynamic classification of these the localization of the periodic domains in parameter space is given by closed expressions the winding number forms a devil s terrace a two dimensional function whose cross sections are complete devils s staircases in such a cross section the complementary set to the periodic intervals is a cantor set with dimensiond 0

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

phase transition like phenomenon in a piecewise linear map abstract we introduce a one hump fully developed chaotic piecewise linear map with a control parameter which tunes the shape and therefore the value of the lyapounov exponent of the map the graph of the lyapounov exponent as a function of this parameter shows several discontinuities which are explained both qualitatively and quantitatively

experimental evaluation of local wave speed in the presence of reflected waves wave speed also called pulse wave velocity is the speed by which disturbance travels along the medium and it depends on the mechanical and geometrical properties of the vessel and on the density of the blood wave speed is a parameter of clinical relevance because it is an indicator of arterial stiffness and cardiovascular diseases the aim of this work is to compare different methods for the determination of local wave speed in bench experiments and investigate their relative accuracy when reflections are present pressure p flow q and diameter d were measured along a flexible tube far and close to three positive and three negative reflection sites wave speed was calculated using pu loop lnd u loop qaloop d 2 p loop sum of squares and characteristic impedance methods results were compared to the foot to foot method we found that far from the reflections almost all methods give uniform results close to positive reflections the methods that rely on p and q or u overestimate the wave speed value while techniques based on d or a and q or u underestimate it on the contrary close to negative reflections the methods that rely on p and q or u underestimate the wave speed value while techniques based on d or a and q or u overestimate it the d 2 p loop does not seem to be affected by positive or negative

equivalence of nonequilibrium ensembles for simple maps abstract in this paper we discuss the equivalence of different approaches to the same transport problems with particular attention to recent works by breyman et al these suggest a singular equivalence between the stationary states of models from gaussian dynamics and those from chaotic scattering theory although the mentioned equivalence may concern but a limited set of the properties of those systems it is very important from a theoretical standpoint we argue that a special form of dissipativity in the evolution equations of the phase space distributions is to be invoked to understand those facts we discuss one application of periodic orbit theory to illustrate how the stationary states of thermostatted systems can be investigated and some qualitative features obtained

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

survey paper a partial history of the early development of continuous time nonlinear stochastic systems theory this article is a survey of the early development of selected areas in nonlinear continuous time stochastic control key developments in optimal control and the dynamic programming principle existence of optimal controls under complete and partial observations nonlinear filtering stochastic stability the stochastic maximum principle and ergodic control are discussed issues concerning wide bandwidth noise for stability modeling filtering and ergodic control are dealt with the focus is on the earlier work but many important topics are omitted for lack of space

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

a converse lyapunov theorem for almost sure stabilizability we prove a converse lyapunov theorem for almost sure stabilizability and almost sure asymptotic stabilizability of controlled diffusions given a stochastic system a s stochastic open loop stabilizable at the origin we construct a lower semicontinuous positive definite function whose level sets form a local basis of viable neighborhoods of the equilibrium this result provides with the direct lyapunov theorems proved in a companion paper a complete lyapunov like characterization of the a s stabilizability

almost sure stabilizability of controlled degenerate diffusions we develop a direct lyapunov method for the almost sure open loop stabilizability and asymptotic stabilizability of controlled degenerate diffusion processes the infinitesimal decrease condition for a lyapunov function is a new form of hamilton jacobi bellman partial differential inequality of second order we give local and global versions of the first and second lyapunov theorems assuming the existence of a lower semicontinuous lyapunov function satisfying such an inequality in the viscosity sense an explicit formula for a stabilizing feedback is provided for affine systems with smooth lyapunov function several examples illustrate the theory

a general characterization of the mean field limit for stochastic differential games the mean field limit of large population symmetric stochastic differential games is derived in a general setting with and without common noise on a finite time horizon minimal assumptions are imposed on equilibrium strategies which may be asymmetric and based on full information it is shown that approximate nash equilibria in the n player games admit certain weak limits as n tends to infinity and every limit is a weak solution of the mean field game mfg conversely every weak mfg solution can be obtained as the limit of a sequence of approximate nash equilibria in the n player games thus the mfg precisely characterizes the possible limiting equilibrium behavior of the n player games even in the setting without common noise the empirical state distributions may admit stochastic limits which cannot be described by the usual notion of mfg solution

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

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

existence of optimal stochastic controls and global solutions of forward backward stochastic differential equations we consider an optimal stochastic control problem assuming lipschitz conditions and allowing degeneracy of the diffusion coefficient under some structural constraint on the state equation we formulate the problem in the strong form i e we fix the probability space we relate the value function and the feedback law to a forward backward stochastic differential system we prove existence and uniqueness of a global solution to the latter and deduce existence and in some cases uniqueness of an optimal control to solve the coupled forward backward system we use a priori estimates which follow from its control theoretic interpretation

dissipative control system for the stochastic nonlinear h problems abstract we characterize functions satisfying a dissipative inequality associated with a stochastic control problem such a characterization is provided in terms of an upper generalized gaussian solution or a viscosity supersolution to a partial differential equation called hamilton jacobi equation h j links between upper generalized gaussian solutions and viscosity supersolutions to hamilton jacobi equation are studied finally it shows that generalized gaussian solutions is identical to viscosity solutions to hamilton jacobi equation

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

singular stochastic control problems in this paper we study an optimal singular stochastic control problem by using a time transformation this problem is shown to be equivalent to an auxiliary control problem defined as a combination of an optimal stopping problem and a classical control problem for this auxiliary control problem the controller must choose a stopping time optimal stopping and the new control variables belong to a compact set this equivalence is obtained by showing that the discontinuous state process governed by a singular control is given by a time transformation of an auxiliary state process governed by a classical bounded control it is proved that the value functions for these two problems are equal for a general form of the cost the existence of an optimal singular control is established under certain technical hypotheses moreover the problem of approximating singular optimal control by absolutely continuous controls is discussed in the same class of admissible controls

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

hamiltonian transport on unstable periodic orbits transport in hamiltonian systems is approached on the basis of the unstable periodic orbits upos embedded in the chaotic region correlation functions and diffusion rates are defined for upo ensembles of two dimensional periodic maps global diffusion in these maps is interpreted in terms of the ensembles ut containing all the upos of period t as t the relative populations of closed orbits and accelerator modes in ut approach a gaussian distribution characterized by the diffusion coefficient d evidence for the validity of this mechanism is provided for the cat and sawtooth maps prime lattice ensembles of the cat maps appear to be optimal in reproducing the global diffusion these ensembles are generically small subsets of ut for some t it is shown however that the corresponding diffusion rates assume for t odd precisely the known value of d in the entire time interval t t for t even all orbits are closed and d is reproduced only in the interval t 12 t the diffusion rates associated with ensembles ut of the sawtooth maps appear to approximate well the value of d if t is of the order of the diffusion time the case of local transport rates associated with ensembles that are localized in phase space is illustrated

intrinsic irreversibility and the validity of the kinetic description of chaotic systems irreversibility for a class of chaotic systems is seen to be an exact consequence of the dynamics through the use of a generalized spectral representation of the time evolution operator of probability densities the generalized representation is valid for one dimensional systems when the initial probability density satisfies certain physical conditions of smoothness the formalism is first applied to the one dimensional multi bernoulli map which is a simple map displaying deterministic diffusion the two dimensional invertible baker and multibaker transformations are then studied and the physical conditions determining which discrete spectral values are realized are seen to depend on the smoothness of both the density as well as the observable considered the generalized representation is constructed using a resolvent formalism the eigenstates of the diffusive systems are seen to be of a fractal nature

deterministic diffusion in flower shaped billiards we propose a flower shaped billiard in order to study the irregular parameter dependence of chaotic normal diffusion our model is an open system consisting of periodically distributed obstacles in the shape of a flower and it is strongly chaotic for almost all parameter values we compute the parameter dependent diffusion coefficient of this model from computer simulations and analyze its functional form using different schemes all generalizing the simple random walk approximation of machta and zwanzig the improved methods we use are based either on heuristic higher order corrections to the simple random walk model on lattice gas simulation methods or they start from a suitable green kubo formula for diffusion we show that dynamical correlations or memory effects are of crucial importance in reproducing the precise parameter dependence of the diffusion coefficent

on the eigenvalue problem for toeplitz band matrices abstract a formula is given for the characteristic polynomial of an n th order toeplitz band matrix with bandwidth k n in terms of the zeros of a k th degree polynomial with coefficients independent of n the complexity of the formula depends on the bandwidth k and not on the order n also given is a formula for eigenvectors in terms of the same zeros and k coefficients which can be obtained by solving a k k homogeneous system

optimization of observations a stochastic control approach we study a stochastic control problem for the optimization of observations in a partially observable stochastic system using a method of discontinuous time transformation we associate with the original problem with unbounded controls a problem that has bounded controls this latter problem allows us to construct nearly optimal nonanticipative lipschitz markov controls with finite observation power for the original problem since the controlled observation equation may degenerate we also derive a corresponding filtering result and show a separation property of the optimal controls

resolving the time lag between pressure and flow for the determination of local wave speed in elastic tubes and arteries it is well established that wave speed can be determined using the initial linear part of the pressure velocity loop pu loop however the frequency response of most flow measuring devices is usually slower than that of solid state pressure transducers making flow waveforms lagging in time behind pressure waveforms if this lag which is traditionally determined by eye is not corrected prior to the analysis the pu loop method may provide inaccurate wave speeds the main aim of this work is therefore to introduce an objective technique to establish the value of this lag the new technique relies on the linearity between pressure and velocity in the absence of reflections and determines the highest correlation factor between pressure and velocity in the range of minimum pressure to maximum velocity we shifted the flow waveform backwards in time steps equal to the sampling interval and the time shift associated with the highest correlation indicates the correct time lag of the flow waveform we first tested the new technique in vitro using a uniform latex tube and compared the results to those established using the traditional by eye method whilst varying the filter setting of the flowmeter then we applied the new technique to pressure and flow measured in the ascending aorta of anaesthetised open chested dogs we found the time lag between pressure and velocity calculated by the new technique in good agreement with that determined by eye in vitro and that increasing the filtering power generated greater delay between the measured pressure and flow the results obtained in vivo using the new technique were also in good agreement with those determined by eye we therefore conclude that the new technique provides a convenient and objective way of correcting the lag and can reliably align pressure and flow

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

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

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

topological entropy of one dimensional maps approximations and bounds we present a method for computing the topological entropy of one dimensional maps as an approximation scheme the algorithm converges rapidly and provides both upper and lower bounds

stability ordering of cycle expansions may be made to only a short time of order l 21 1 in the future where l1 is the largest lyapunov exponent despite this unpredictability much can be said about the average behavior of the system in many cases using only a small number of unstable periodic orbits or cycles this can be achieved by using cycle expansions 1 2 of ruelle s dynamical zeta function 3 cycle expansions have proved very useful in both classical and quantum chaos giving accurate estimates of the escape rate of open billiard systems 4 and the energy levels of helium 5 using a surprisingly small number of classical cycles the main idea of this approach is that a long generic trajectory may be approximated by various periodic orbits at different times and that longer periodic orbits may often be shadowed by shorter fundamental cycles closely following the shorter cycles along different sections of its length thus averages are calculated using fundamental cycles with small corrections due to longer cycles periodic orbits which are exact repetitions of smaller cycles are explicitly summed so that all expressions are written in terms of the remaining prime cycles the expansions work best when the symbolic dynamics is well understood and long periodic orbits are well shadowed by shorter ones in this paper we investigate a system in which neither of these conditions holds the strong field lorentz gas in spite of these difficulties reasonable results may be obtained by ordering the expansion in terms of stability rather than the length of periodic orbits this approach should be valid wherever cycle expansions can be applied including flows for which a natural topological length is difficult to define the origin of cycle expansions is a well developed theory of trace formulas and dynamical zeta functions 3 6 here we will need only the expression for the classical time average of some quantity a in a closed system 2

maximum principle for singular stochastic control problems in this paper an optimal singular stochastic control problem is considered for this model a general stochastic maximum principle is obtained by using a time transformation this is the first version of the stochastic maximum principle that covers nonlinear cases