measuring the strangeness of strange attractors we study the correlation exponent v introduced recently as a characteristic measure of strange attractors which allows one to distinguish between deterministic chaos and random noise the exponent v is closely related to the fractal dimension and the information dimension but its computation is considerably easier its usefulness in characterizing experimental data which stem from very high dimensional systems is stressed algorithms for extracting v from the time series of a single variable are proposed the relations between the various measures of strange attractors and between them and the lyapunov exponents are discussed it is shown that the conjecture of kaplan and yorke for the dimension gives an upper bound for v various examples of finite and infinite dimensional systems are treated both numerically and analytically

periodic shadowing sensitivity analysis of chaotic systems abstract the sensitivity of long time averages of a hyperbolic chaotic system to parameter perturbations can be determined using the shadowing direction the uniformly bounded in time solution of the sensitivity equations although its existence is formally guaranteed for certain systems methods to determine it are hardly available one practical approach is the least squares shadowing lss algorithm wang 2014 18 whereby the shadowing direction is approximated by the solution of the sensitivity equations with the least square average norm here we present an alternative potentially simpler shadowing based algorithm termed periodic shadowing the key idea is to obtain a bounded solution of the sensitivity equations by complementing it with periodic boundary conditions in time we show that this is not only justifiable when the reference trajectory is itself periodic but also possible and effective for chaotic trajectories our error analysis shows that periodic shadowing has the same convergence rates as lss when the time span t is increased the sensitivity error first decays as 1 t and then asymptotically as 1 t we demonstrate the approach on the lorenz equations and also show that as t tends to infinity periodic shadowing sensitivities converge to the same value obtained from long unstable periodic orbits lasagna 2018 14 for which there is no shadowing error finally finite difference approximations of the sensitivity are also examined and we show that subtle non hyperbolicity features of the lorenz system introduce a small yet systematic bias

linearly implicit imex runge kutta methods for a class of degenerate convection diffusion problems multispecies kinematic flow models with strongly degenerate diffusive corrections give rise to systems of nonlinear convection diffusion equations of arbitrary size applications of these systems include models of polydisperse sedimentation and multiclass traffic flow implicit explicit imex runge kutta rk methods are suitable for the solution of these convection diffusion problems since the stability restrictions coming from the explicitly treated convective part are much less severe than those that would be deduced from an explicit treatment of the diffusive term these schemes usually combine an explicit rk scheme for the time integration of the convective part with a diagonally implicit one for the diffusive part in r burger p mulet and l m villada siam j sci comput 35 2013 pp b751 b777 a scheme of this type is proposed where the nonlinear and nonsmooth systems of algebraic equations arising in the implicit treatment of the degenerate diffusive part are solved by smoothing o

flux explicit imex runge kutta schemes for hyperbolic to parabolic relaxation problems we consider the development of implicit explicit imex runge kutta r k schemes for hyperbolic systems of conservation laws with stiff diffusive relaxation the asymptotic behavior of such systems as the relaxation parameter vanishes is governed by a reduced parabolic system and it is desirable to have schemes that are able to capture the correct diffusive limit the hyperbolic part becomes stiff when the system relaxes towards the parabolic equation for this reason in previous works part of the hyperbolic terms are treated implicitly s jin l pareschi and g toscani siam j numer anal 35 1998 pp 2405 2439 s jin and l pareschi j comp phys 161 2000 pp 312 330 g naldi and l pareschi siam j numer anal 37 2000 pp 1246 1270 in particular in s boscarino l pareschi and g russo implicit explicit runge kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit submitted the scheme relaxes to an implicit method for the limit diffusi

numerical convergence of the block maxima approach to the generalized extreme value distribution in this paper we perform an analytical and numerical study of extreme value distributions in discrete dynamical systems in this setting recent works have shown how to get a statistics of extremes in agreement with the classical extreme value theory we pursue these investigations by giving analytical expressions of extreme value distribution parameters for maps that have an absolutely continuous invariant measure we compare these analytical results with numerical experiments in which we study the convergence to limiting distributions using the so called block maxima approach pointing out in which cases we obtain robust estimation of parameters in regular maps for which mixing properties do not hold we show that the fitting procedure to the classical extreme value distribution fails as expected however we obtain an empirical distribution that can be explained starting from a different observable function for which nicolis et al phys rev lett 97 21 210602 2006 have found analytical results

on spurious detection of linear response and misuse of the fluctuation dissipation theorem in finite time series abstract using a sensitive statistical test we determine whether or not one can detect the breakdown of linear response given observations of deterministic dynamical systems a goodness of fit statistics is developed for a linear statistical model of the observations based on results for central limit theorems for deterministic dynamical systems and used to detect linear response breakdown we apply the method to discrete maps which do not obey linear response and show that the successful detection of breakdown depends on the length of the time series the magnitude of the perturbation and on the choice of the observable we find that in order to reliably reject the assumption of linear response for typical observables sufficiently large data sets are needed even for simple systems such as the logistic map one needs of the order of 10 6 observations to reliably detect the breakdown with a confidence level of 95 if less observations are available one may be falsely led to conclude that linear response theory is valid the amount of data required is larger the smaller the applied perturbation for judiciously chosen observables the necessary amount of data can be drastically reduced but requires detailed a priori knowledge about the invariant measure which is typically not available for complex dynamical systems furthermore we explore the use of the fluctuation dissipation theorem fdt in cases with limited data length or coarse graining of observations the fdt if applied naively to a system without linear response is shown to be very sensitive to the details of the sampling method resulting in erroneous predictions of the response

high order asymptotic preserving methods for fully nonlinear relaxation problems we study solutions to nonlinear hyperbolic systems with fully nonlinear relaxation terms in the limit of both infinitely stiff relaxation and arbitrary late time in this limit the dynamics is governed by effective systems of parabolic type which may contain degenerate and or fully nonlinear diffusion terms for this class of problems we develop an implicit explicit method based on runge kutta discretization in time and we apply this method to the investigation of several examples of interest in fluid dynamics importantly we impose here a realistic stability condition on the time step and we demonstrate that solutions in the hyperbolic to parabolic regime can be computed numerically with high robustness and accuracy even in the presence of fully nonlinear relaxation

on the capacity of free space optical intensity channels upper and lower bounds are derived on the capacity of the free space optical intensity channel this channel has a nonnegative input representing the transmitted optical intensity which is corrupted by additive white gaussian noise to preserve the battery and for safety reasons the input is constrained in both its average and its peak power for a fixed ratio of the allowed average power to the allowed peak power the difference between the upper and the lower bound tends to zero as the average power tends to infinity and their ratio tends to one as the average power tends to zero when only an average power constraint is imposed on the input the difference between the bounds tends to zero as the allowed average power tends to infinity and their ratio tends to a constant as the allowed average power tends to zero

evidence of dispersion relations for the nonlinear response of the lorenz 63 system along the lines of the nonlinear response theory developed by ruelle in a previous paper we have proved under rather general conditions that kramers kronig dispersion relations and sum rules apply for a class of susceptibilities describing at any order of perturbation the response of axiom a non equilibrium steady state systems to weak monochromatic forcings we present here the first evidence of the validity of these integral relations for the linear and the second harmonic response for the perturbed lorenz 63 system by showing that numerical simulations agree up to high degree of accuracy with the theoretical predictions some new theoretical results showing how to derive asymptotic behaviors and how to obtain recursively harmonic generation susceptibilities for general observables are also presented our findings confirm the conceptual validity of the nonlinear response theory suggest that the theory can be extended for more general non equilibrium steady state systems and shed new light on the applicability of very general tools based only upon the principle of causality for diagnosing the behavior of perturbed chaotic systems and reconstructing their output signals in situations where the fluctuation dissipation relation is not of great help

the extremal index hitting time statistics and periodicity abstract the extremal index appears as a parameter in extreme value laws for stochastic processes characterising the clustering of extreme events we apply this idea in a dynamical systems context to analyse the possible extreme value laws for the stochastic process generated by observations taken along dynamical orbits with respect to various measures we derive new easily checkable conditions which identify extreme value laws with particular extremal indices in the dynamical context we prove that the extremal index is associated with periodic behaviour the analogy of these laws in the context of hitting time statistics as studied in the authors previous works on this topic is explained and exploited extensively allowing us to prove for the first time the existence of hitting time statistics for balls around periodic points moreover for very well behaved systems uniformly expanding we completely characterise the extremal behaviour by proving that either we have an extremal index less than 1 at periodic points or equal to 1 at any other point this theory then also applies directly to general stochastic processes adding both useful tools to identify the extremal index and giving deeper insight into the periodic behaviour it suggests

stochastic climate dynamics random attractors and time dependent invariant measures a b s t r a c t this article attempts a unification of the two approaches that have dominated theoretical climate dynamics since its inception in the 1960s the nonlinear deterministic and the linear stochastic one this unification via the theory of random dynamical systems rds allows one to consider the detailed geometric structure of the random attractors associated with nonlinear stochastically perturbed systems we report on high resolution numerical studies of two idealized models of fundamental interest for climate dynamics the first of the two is a stochastically forced version of the classical lorenz model the second one is a low dimensional nonlinear stochastic model of the el nino southern oscillation enso these studies provide a good approximation of the two models global random attractors as well as of the time dependent invariant measures supported by these attractors the latter are shown to have an intuitive physical interpretation as random versions of sinai ruelle bowen srb measures

linearly implicit explicit schemes for the equilibrium dispersive model of chromatography abstract numerical schemes for the nonlinear equilibrium dispersive ed model for chromatographic processes with adsorption isotherms of langmuir type are proposed this model consists of a system of nonlinear convection dominated partial differential equations the nonlinear convection gives rise to sharp moving transitions between concentrations of different solute components this property calls for numerical methods with shock capturing capabilities based on results by donat guerrero and mulet appl numer math 123 2018 22 42 conservative shock capturing numerical schemes can be designed for this chromatography model since explicit schemes for diffusion problems can pose severe stability restrictions on the time step the novel schemes treat diffusion implicitly and convection explicitly to avoid the need to solve the nonlinear systems appearing in the implicit treatment of the nonlinear diffusion second order linearly implicit explicit runge kutta schemes limex rk schemes are employed numerical experiments demonstrate that the schemes produce accurate numerical solutions with the same stability restrictions as in the purely hyperbolic case

symmetry breaking mixing instability and low frequency variability in a minimal lorenz like system starting from the classical saltzman two dimensional convection equations we derive via a severe spectral truncation a minimal 10 ode system which includes the thermal effect of viscous dissipation neglecting this process leads to a dynamical system which includes a decoupled generalized lorenz system the consideration of this process breaks an important symmetry and couples the dynamics of fast and slow variables with the ensuing modifications to the structural properties of the attractor and of the spectral features when the relevant nondimensional number eckert number ec is different from zero an additional time scale of o text ec ensuremath 1 is introduced in the system as shown with standard multiscale analysis and made clear by several numerical evidences moreover the system is ergodic and hyperbolic the slow variables feature long term memory with 1 f 3 2 power spectra and the fast variables feature amplitude modulation increasing the strength of the thermal viscous feedback has a stabilizing effect as both the metric entropy and the kaplan yorke attractor dimension decrease monotonically with ec the analyzed system features very rich dynamics it overcomes some of the limitations of the lorenz system and might have prototypical value in relevant processes in complex systems dynamics such as the interaction between slow and fast variables the presence of long term memory and the associated extreme value statistics this analysis shows how neglecting the coupling of slow and fast variables only on the basis of scale analysis can be catastrophic in fact this leads to spurious invariances that affect essential dynamical properties ergodicity hyperbolicity and that cause the model losing ability in describing intrinsically multiscale processes

a high order semi implicit imex weno scheme for the all mach isentropic euler system abstract in this paper new high order schemes are constructed and analyzed for the numerical solution of euler equations of isentropic gas dynamics material waves are treated explicitly while acoustic waves are treated implicitly thus avoiding severe cfl restrictions for low mach flows high order accuracy in space is obtained by finite difference weno schemes while high order in time is obtained by imex methods with semi implicit linearization treatment the schemes are proven to be asymptotic preserving and asymptotic accurate as the mach number vanishes several tests in one and two space dimensions illustrate the effectiveness of the proposed schemes

strongly degenerate parabolic hyperbolic systems modeling polydisperse sedimentation with compression we show how existing models for the sedimentation of monodisperse flocculated suspensions and of polydisperse suspensions of rigid spheres differing in size can be combined to yield a new theory of the sedimentation processes of polydisperse suspensions forming compressible sediments sedimentation with compression or sedimentation consolidation process for n solid particle species this theory reduces in one space dimension to an n times n coupled system of quasi linear degenerate convection diffusion equations analyses of the characteristic polynomials of the jacobian of the convective flux vector and of the diffusion matrix show that this system is of strongly degenerate parabolic hyperbolic type for arbitrary n and particle size distributions bounds for the eigenvalues of both matrices are derived the mathematical model for n 3 is illustrated by a numerical simulation obtained by the kurganov tadmor central difference scheme for convection diffusion problems the numerical scheme exploits

spectral galerkin methods for transfer operators in uniformly expanding dynamics markov expanding maps a class of simple chaotic systems are commonly used as models for chaotic dynamics but existing numerical methods to study long time statistical properties such as invariant measures have a poor trade off between computational effort and accuracy we develop a spectral galerkin method for these maps transfer operators estimating statistical quantities using finite submatrices of the transfer operators infinite fourier or chebyshev basis coefficient matrices rates of convergence of these estimates are obtained via quantitative bounds on the full transfer operator matrix entries we find the method furnishes up to exponentially accurate estimates of statistical properties in only a polynomially large computational time to implement these results we suggest and demonstrate two algorithms a rigorously validated algorithm and a fast more convenient adaptive algorithm using the first algorithm we prove rigorous bounds on some exemplar quantities that are substantially more accurate than previous we show that the adaptive algorithm can produce double floating point accuracy estimates in a fraction of a second on a personal computer

differentiation of srb states correction and complements taking into account criticism by d dolgopyat and m jiang we present here an improved derivation of the formula for the derivative of an srb measure with respect to parameters

dimension and product structure of hyperbolic measures we prove that every hyperbolic measure invariant under a c 1 fi difieomorphism of a smooth riemannian manifold possesses asymptotically almost local product structure i e its density can be approximated by the product of the densities on stable and unstable manifolds up to small exponentials this has not been known even for measures supported on locally maximal hyperbolic sets using this property of hyperbolic measures we prove the long standing eckmann ruelle conjecture in dimension theory of smooth dynamical systems the pointwise dimension of every hyperbolic measure invariant under a c 1 fi difieomorphism exists almost everywhere this implies the crucial fact that virtually all the characteristics of dimension type of the measure including the hausdorfi dimension box dimension and information dimension coincide this provides the rigorous mathematical justiflcation of the concept of fractal dimension for hyperbolic measures

renormalization group theory for finite size scaling in extreme statistics we present a renormalization group rg approach to explain universal features of extreme statistics applied here to independent identically distributed variables the outlines of the theory have been described in a previous paper the main result being that finite size shape corrections to the limit distribution can be obtained from a linearization of the rg transformation near a fixed point leading to the computation of stable perturbations as eigenfunctions here we show details of the rg theory which exhibit remarkable similarities to the rg known in statistical physics besides the fixed points explaining universality and the least stable eigendirections accounting for convergence rates and shape corrections the similarities include marginally stable perturbations which turn out to be generic for the fisher tippett gumbel class distribution functions containing unstable perturbations are also considered we find that after a transitory divergence they return to the universal fixed line at the same or at a different point depending on the type of perturbation

universal behaviour of extreme value statistics for selected observables of dynamical systems the main results of the extreme value theory developed for the investigation of the observables of dynamical systems rely up to now on the block maxima approach in this framework extremes are identified with the block maxima of the time series of the chosen observable in the limit of infinitely long blocks it has been proved that assuming suitable mixing conditions for the underlying dynamical systems the extremes of a specific class of observables are distributed according to the so called generalised extreme value gev distribution direct calculations show that in the case of quasi periodic dynamics the block maxima are not distributed according to the gev distribution in this paper we show that considering the exceedances over a given threshold instead of the block maxima approach it is possible to obtain a generalised pareto distribution also for extremes computed in systems which do not satisfy mixing conditions requiring that the invariant measure locally scales with a well defined exponent the local dimension we show that the limiting distribution for the exceedances of the observables previously studied with the block maxima approach is a generalised pareto distribution where the parameters depend only on the local dimensions and the values of the threshold but not on the number of observations considered we also provide connections with the results obtained with the block maxima approach in order to provide further support to our findings we present the results of numerical experiments carried out considering the well known chirikov standard map

response theory for equilibrium and non equilibrium statistical mechanics causality and generalized kramers kronig relations we consider the general response theory recently proposed by ruelle for describing the impact of small perturbations to the non equilibrium steady states resulting from axiom a dynamical systems we show that the causality of the response functions entails the possibility of writing a set of kramers kronig k k relations for the corresponding susceptibilities at all orders of nonlinearity nonetheless only a special class of directly observable susceptibilities obey k k relations specific results are provided for the case of arbitrary order harmonic response which allows for a very comprehensive k k analysis and the establishment of sum rules connecting the asymptotic behavior of the harmonic generation susceptibility to the short time response of the perturbed system these results set in a more general theoretical framework previous findings obtained for optical systems and simple mechanical models and shed light on the very general impact of considering the principle of causality for testing self consistency the described dispersion relations constitute unavoidable benchmarks that any experimental and model generated dataset must obey the theory exposed in the present paper is dual to the time dependent theory of perturbations to equilibrium states and to non equilibrium steady states and has in principle similar range of applicability and limitations in order to connect the equilibrium and the non equilibrium steady state case we show how to rewrite the classical response theory by kubo so that response functions formally identical to those proposed by ruelle apart from the measure involved in the phase space integration are obtained these results taking into account the chaotic hypothesis by gallavotti and cohen might be relevant in several fields including climate research in particular whereas the fluctuation dissipation theorem does not work for non equilibrium systems because of the non equivalence between internal and external fluctuations k k relations might be robust tools for the definition of a self consistent theory of climate change

stochastic perturbations to dynamical systems a response theory approach using the formalism of the ruelle response theory we study how the invariant measure of an axiom a dynamical system changes as a result of adding noise and describe how the stochastic perturbation can be used to explore the properties of the underlying deterministic dynamics we first find the expression for the change in the expectation value of a general observable when a white noise forcing is introduced in the system both in the additive and in the multiplicative case we also show that the difference between the expectation value of the power spectrum of an observable in the stochastically perturbed case and of the same observable in the unperturbed case is equal to the variance of the noise times the square of the modulus of the linear susceptibility describing the frequency dependent response of the system to perturbations with the same spatial patterns as the considered stochastic forcing this provides a conceptual bridge between the change in the fluctuation properties of the system due to the presence of noise and the response of the unperturbed system to deterministic forcings using kramers kronig theory it is then possible to derive the real and imaginary part of the susceptibility and thus deduce the green function of the system for any desired observable we then extend our results to rather general patterns of random forcing from the case of several white noise forcings to noise terms with memory up to the case of a space time random field explicit formulas are provided for each relevant case analysed as a general result we find using an argument of positive definiteness that the power spectrum of the stochastically perturbed system is larger at all frequencies than the power spectrum of the unperturbed system we provide an example of application of our results by considering the spatially extended chaotic lorenz 96 model these results clarify the property of stochastic stability of srb measures in axiom a flows provide tools for analysing stochastic parameterisations and related closure ansatz to be implemented in modelling studies and introduce new ways to study the response of a system to external perturbations taking into account the chaotic hypothesis we expect that our results have practical relevance for a more general class of system than those belonging to axiom a

response operators for markov processes in a finite state space radius of convergence and link to the response theory for axiom a systems using straightforward linear algebra we derive response operators describing the impact of small perturbations to finite state markov processes the results can be used for studying empirically constructed e g from observations or through coarse graining of model simulations finite state approximation of statistical mechanical systems recent results concerning the convergence of the statistical properties of finite state markov approximation of the full asymptotic dynamics on the srb measure in the limit of finer and finer partitions of the phase space are suggestive of some degree of robustness of the obtained results in the case of axiom a system our findings give closed formulas for the linear and nonlinear response theory at all orders of perturbation and provide matrix expressions that can be directly implemented in any coding language plus providing bounds on the radius of convergence of the perturbative theory in particular we relate the convergence of the response theory to the rate of mixing of the unperturbed system one can use the formulas derived for finite state markov processes to recover previous findings obtained on the response of continuous time axiom a dynamical systems to perturbations by considering the generator of time evolution for the measure and for the observables a very basic low tech and computationally cheap analysis of the response of the lorenz 63 model to perturbations provides rather encouraging results regarding the possibility of using the approximate representation given by finite state markov processes to compute the system s response

almost every real quadratic map is either regular or stochastic in this paper we complete a program to study measurable dynamics in the real quadratic family our goal was to prove that almost any real quadratic map pc z t x2 c c c 2 1 4 has either an attracting cycle or an absolutely continuous invariant measure the final step completed here is to prove that the set of infinitely renormalizable parametric values c c 2 1 4 has zero lebesgue measure we derive this from a renormalization theorem which asserts uniform hyperbolicity of the full renormalization operator this theorem gives the most general real version of the feigenbaum coullet tresser universality simultanuously for all combinatorial types

on differentiability of srb states for partially hyperbolic systems consider a one parameter family of diffeomorphisms f e such that f 0 is an anosov element in a standard abelian anosov action having sufficiently strong mixing properties let e be any u gibbs state for f e we prove theorem 1 that for any c function a the map e e a is differentiable at e 0 this implies corollary 2 2 that the difference of birkhoff averages of the perturbed and unperturbed systems is proportional to e we apply this result corollary 3 3 to show that a generic perturbation of the time one map of geodesic flow on the unit tangent bundle over a surface of negative curvature has a unique srb measure with good statistical properties

spectral theory zeta functions and the distribution of periodic points for collet eckmann maps we study unimodal interval mapst with negative schwarzian derivative satisfying the collet eckmann condition dtn tc k cn for some constantsk 0 and c 1 c is the critical point oft we prove exponential mixing properties of the unique invariant probability density oft describe the long term behaviour of typical in the sense of lebesgue measure trajectories by central limit and large deviations theorems for partial sum processes of the form s n sigma i 0 n 1 f t i x and study the distribution of typical periodic orbits also in the sense of a central limit theorem and a large deviations theorem

deterministic homogenization for fast slow systems with chaotic noise consider a fast slow system of ordinary differential equations of the form x a x y e 1b x y y e 2g y where it is assumed that b averages to zero under the fast flow generated by g we give conditions under which solutions x to the slow equations converge weakly to an it o diffusion x as e 0 the drift and diffusion coefficients of the limiting stochastic differential equation satisfied by x are given explicitly our theory applies when the fast flow is anosov or axiom a as well as to a large class of nonuniformly hyperbolic fast flows including the one defined by the well known lorenz equations and our main results do not require any mixing assumptions on the fast flow

on the computation of the extremal index for time series the extremal index is a quantity introduced in extreme value theory to measure the presence of clusters of exceedances in the dynamical systems framework it provides important information about the dynamics of the underlying systems in this paper we provide a review of the meaning of the extremal index in dynamical systems depending on the observables used this quantity can inform on local properties of attractors such as periodicity stability and persistence in phase space or on global properties such as the lyapunov exponents we also introduce a new estimator of the extremal index and show its relation with those previously introduced in the statistical literature we reserve a particular focus to the systems perturbed with noise as they are a good paradigm of many natural phenomena different kind of noises are investigated in the annealed and quenched situations applications to climate data are also presented

physical layer security for indoor visible light communications secrecy capacity analysis this paper investigates the physical layer security for an indoor visible light communication network consisting of a transmitter a legitimate receiver and an eavesdropper both the main channel and the wiretapping channel have non negative inputs which are corrupted by additive white gaussian noises considering the illumination requirement and the physical characteristics of lighting source the input is also constrained in both its average and peak optical intensities two scenarios are investigated one is only with an average optical intensity constraint and the other is with both average and peak optical intensity constraints based on the information theory closed form expressions of the upper and lower bounds on secrecy capacity for the two scenarios are derived numerical results show that the upper and lower bounds on secrecy capacity are tight which validates the derived closed form expressions moreover the asymptotic behaviors in the high signal to noise ratio snr regime are analyzed from the theoretical aspects at high snr when only considering the average optical intensity constraint a small performance gap exists between the asymptotic upper and lower bounds on secrecy capacity when considering both average and peak optical intensity constraints the asymptotic upper and lower bounds on secrecy capacity coincide with each other these conclusions are also confirmed by numerical results

multi level dynamical systems connecting the ruelle response theory and the mori zwanzig approach we consider the problem of deriving approximate autonomous dynamics for a number of variables of a dynamical system which are weakly coupled to the remaining variables in a previous paper we have used the ruelle response theory on such a weakly coupled system to construct a surrogate dynamics such that the expectation value of any observable agrees up to second order in the coupling strength to its expectation evaluated on the full dynamics we show here that such surrogate dynamics agree up to second order to an expansion of the mori zwanzig projected dynamics this implies that the parametrizations of unresolved processes suited for prediction and for the representation of long term statistical properties are closely related if one takes into account in addition to the widely adopted stochastic forcing the often neglected memory effects

application of hyperbolic dynamics to physics some problems and conjectures the long time behavior of smooth dynamical systems is in good cases given by an srb measure the measure is expected often to have very discontinuous dependence on parameters of the dynamical system this is a very unsatisfactory situation for physical applications where one would like to differentiate with respect to parameters here we propose a solution to this difficulty based on analytic continuation in a frequency variable rene thom was a philosopher and among other things but most successfully a lover of that part of philosophy which we call mathematics i use here the term philosophy with its old meaning which some contemporary philosophers and mathematicians might not like inside and outside mathematics thom s interest was turned to forms and ideas his passion was to understand geometrically the nature of things and for this he used mathematical proofs but he was not a formalist and for him proofs remained secondary to the conceptual landscape that they revealed one area where he invested a lot of effort was the study of differentiable dynamical systems this smooth dynamics is where geometric forms come closest to life i attended the regular seminar that he ran on the subject under the influence of thom the ihes was an important center for the development of differentiable dynamical systems especially hyperbolic dynamics in the golden years of the 1960 s and 1970 s steve smale and his friends were frequent visitors and the russians came when they could the big question in smooth dynamics is what do general smooth dynamical systems look like what one wants is an understanding of a large set of dynamical systems this is also a significant question for physical applications more specifically one would like to understand the long time behavior for typical time evolutions and how this changes when the parameters controlling the time evolution are changed the uniformly hyperbolic systems studied by anosov smale thom and many others form a large set and smale had conjectured that they might be dense among all dynamical systems but this is not the case and it may be too much to hope for a very large set of dynamical systems that are understood in great detail like the uniformly hyperbolic systems from a practical point of view the numerical study of dynamical systems often reveals extremely long characteristic times which received by the editors october 1 2003 2000 mathematics subject classification primary 37c40 secondary 37d45

towards a general theory of extremes for observables of chaotic dynamical systems in this paper we provide a connection between the geometrical properties of the attractor of a chaotic dynamical system and the distribution of extreme values we show that the extremes of so called physical observables are distributed according to the classical generalised pareto distribution and derive explicit expressions for the scaling and the shape parameter in particular we derive that the shape parameter does not depend on the chosen observables but only on the partial dimensions of the invariant measure on the stable unstable and neutral manifolds the shape parameter is negative and is close to zero when high dimensional systems are considered this result agrees with what was derived recently using the generalized extreme value approach combining the results obtained using such physical observables and the properties of the extremes of distance observables it is possible to derive estimates of the partial dimensions of the attractor along the stable and the unstable directions of the flow moreover by writing the shape parameter in terms of moments of the extremes of the considered observable and by using linear response theory we relate the sensitivity to perturbations of the shape parameter to the sensitivity of the moments of the partial dimensions and of the kaplan yorke dimension of the attractor preliminary numerical investigations provide encouraging results on the applicability of the theory presented here the results presented here do not apply for all combinations of axiom a systems and observables but the breakdown seems to be related to very special geometrical configurations

whitney holder continuity of the srb measure for transversal families of smooth unimodal maps we consider c 2 families t mapsto f t of c 4 nondegenerate unimodal maps we study the absolutely continuous invariant probability srb measure mu t of f t as a function of t on the set of collet eckmann ce parameters upper bounds assuming existence of a transversal ce parameter we find a positive measure set of ce parameters delta and for each t 0 in delta a set delta 0 subset delta of polynomially recurrent parameters containing t 0 as a lebesgue density point and constants c ge 1 gamma 4 so that for every 1 2 holder function a begin aligned big int a d mu t int a d mu t 0 big le c vert a vert c 1 2 t t 0 1 2 log t t 0 gamma forall t in delta 0 end aligned in addition for all t in delta 0 the renormalisation period p t of f t satisfies p t le p t 0 and there are uniform bounds on the rates of mixing of f t p t for all t with p t p t 0 if f t x tx 1 x the set delta contains almost all ce parameters lower bounds assuming existence of a transversal mixing misiurewicz thurston parameter t 0 we find a set of ce parameters delta mt accumulating at t 0 a constant c ge 1 and a c infty function a 0 so that begin aligned c t t 0 1 2 ge big int a 0 d mu t int a 0 d mu t 0 big ge c 1 t t 0 1 2 forall t in delta mt end aligned

high order approximation of the frobenius perron operator we formulate a general convergence theory for the finite dimensional projection approximation of the fixed point of a class of linear operators in particular we explore the relation with the standard framework of stability and consistency of the approximation imply convergence applications include the frobenius perron operator p s corresponding to a nonsingular measurable transformation from 0 1 to itself numerical experiments are also given to support the theoretical analysis

hitting time statistics and extreme value theory we consider discrete time dynamical systems and show the link between hitting time statistics the distribution of the first time points land in asymptotically small sets and extreme value theory distribution properties of the partial maximum of stochastic processes this relation allows to study hitting time statistics with tools from extreme value theory and vice versa we apply these results to non uniformly hyperbolic systems and prove that a multimodal map with an absolutely continuous invariant measure must satisfy the classical extreme value laws with no extra condition on the speed of mixing for example we also give applications of our theory to higher dimensional examples for which we also obtain classical extreme value laws and exponential hitting time statistics for balls we extend these ideas to the subsequent returns to asymptotically small sets linking the poisson statistics of both processes

response and sensitivity using markov chains dynamical systems are often subject to forcing or changes in their governing parameters and it is of interest to study how this affects their statistical properties a prominent real life example of this class of problems is the investigation of climate response to perturbations in this respect it is crucial to determine what the linear response of a system is as a quantification of sensitivity alongside previous work here we use the transfer operator formalism to study the response and sensitivity of a dynamical system undergoing perturbations by projecting the transfer operator onto a suitable finite dimensional vector space one is able to obtain matrix representations which determine finite markov processes further using perturbation theory for markov matrices it is possible to determine the linear and nonlinear response of the system given a prescribed forcing here we suggest a methodology which puts the scope on the evolution law of densities the liouville fokker planck equation allowing to effectively calculate the sensitivity and response of two representative dynamical systems

on iterations of 1 ax 2 on 1 1 on etudie des iterations de f x a 1 ax u 2 1 x 1 ou a est un parametre dans l intervalle 0 2 on etablit la non existence de cycles attractifs on definit une partition on donne une estimation uniforme de la a derivee

absolutely continuous invariant measures for one parameter families of one dimensional maps given a one parameter familyf x of maps of the interval 0 1 we consider the set of parameter values for whichf has an invariant measure absolutely continuous with respect to lebesgue measure we show that this set has positive measure for two classes of maps i f x f x where 0 4 andf x is a functionc3 near the quadratic mapx 1 x and ii f x f x mod 1 wheref isc3 f 0 f 1 0 andf has a unique nondegenerate critical point in 0 1

linear response function for coupled hyperbolic attractors the srb measures of a hyperbolic system are widely accepted as the measures that are physically relevant it has been shown by ruelle that they depend smoothly on the system furthermore ruelle showed by a separate argument that the first derivative i e the linear response function admits a geometric interpretation in this paper we consider thermodynamic limits of srb measures in lattices of coupled hyperbolic attractors in a previous paper using markov partitions and thermodynamic formalism we had established the smooth dependence of thermodynamic limits of srb measures here we establish that the linear response function admits a geometric interpretation the formula is analogous to the one found by ruelle for finite dimensional systems if one term is reinterpreted appropriately we show that the limiting derivative is the thermodynamic limit of the derivatives in finite volume we also obtain similar results for the derivatives of the entropy

on the susceptibility function of piecewise expanding interval maps we study the susceptibility function psi z sum n 0 infty int z n x y rho 0 y frac partial partial y varphi f n y dy associated to the perturbation f t f tx circ f of a piecewise expanding interval map f and to an observable 1 is the formal derivative at t 0 of the average mathcal r t int varphi rho t dx of with respect to the srb measure of ft our analysis is based on a spectral description of transfer operators it gives in particular sufficient conditions on f x and which guarantee that z is holomorphic in a disc of larger than one or which ensure that a number may be associated to the possibly divergent series 1 we present examples of f x and so that mathcal r t is not lipschitz at 0 and we propose a new version of ruelle s conjecture

on a class of uniformly accurate imex runge kutta schemes and applications to hyperbolic systems with relaxation in this paper we consider hyperbolic systems with relaxation in which the relaxation time varepsilon may vary from values of order one to very small values when varepsilon is very small the relaxation term becomes very strong and highly stiff and underresolved numerical schemes may produce spurious results in such cases it is important to have schemes that work uniformly with respect to varepsilon implicit explicit imex runge kutta r k schemes have been widely used for the time evolution of hyperbolic partial differential equations but the schemes existing in literature do not exhibit uniform accuracy with respect to the relaxation time we develop new imex r k schemes for hyperbolic systems with relaxation that present better uniform accuracy than the ones existing in the literature and in particular produce good behavior with high order accuracy in the asymptotic limit i e when varepsilon is very small these schemes are obtained by imposing new additional order conditions to guarantee better accuracy over a wide range of the relaxation time we propose the construction of new third order imex r k schemes of type ck s boscarino siam j numer anal 45 2008 pp 1600 1621 in several test problems these schemes with a fixed spatial discretization exhibit for all range of the relaxation time an almost uniform third order accuracy

numerical estimates of local dimension by waiting time and quantitative recurrence abstract waiting time indicators are defined by measuring the time needed for a point x to approach a given point y our aim is to test the use of waiting time indicators and quantitative recurrence ones to numerically estimate the local dimension of attractors in dynamical systems as suggested by some rigorous results we test this approach on a collection of examples whose dimensions are already known rigorously or with a large amount of precision the main outcome is that the use of waiting time compared to recurrence increases largely the speed of convergence of the estimators

linear response of the lorenz system the present numerical study provides strong evidence that at standard parameters the response of the lorenz system to small perturbations of the control parameter r is linear this evidence is obtained not only directly by determining the response in the observable a x z but also indirectly by validating various implications of the assumption of a linear response like a quadratic response at twice the perturbation frequency a vanishing response in a x x the kramers kronig relations and relations between different response functions since the lorenz system is nonhyperbolic the present results indicate that in contrast to a recent speculation the large system limit thermodynamic limit need not be invoked to obtain a linear response for chaotic systems of this type

sensitivity of long periodic orbits of chaotic systems the properties of long numerically determined periodic orbits of two low dimensional chaotic systems the lorenz equations and the kuramoto sivashinsky system in a minimal domain configuration are examined the primary question is to establish whether the sensitivity of period averaged quantities with respect to parameter perturbations computed over long orbits can be used as a sufficiently good proxy for the response of the chaotic state to finite amplitude parameter perturbations to address this question an inventory of thousands of orbits at least two orders of magnitude longer than the shortest admissible cycles is constructed the expectation of period averages floquet exponents and sensitivities over such set is then obtained it is shown that all these quantities converge to a limiting value as the orbit period is increased however while period averages and floquet exponents appear to converge to analogous quantities computed from chaotic trajectories the limiting value of the sensitivity is not necessarily consistent with the response of the chaotic state similar to observations made with other shadowing algorithms

linear response susceptibility and resonances in chaotic toy models abstract we consider simple examples illustrating some new features of the linear response theory developed by ruelle for dissipative and chaotic systems d ruelle smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics j stat phys 95 1999 393 468 in this theory the concepts of linear response susceptibility and resonance which are familiar to physicists have been revisited due to the dynamical contraction of the whole phase space onto attractors in particular the standard framework of the fluctuation dissipation theorem breaks down and new resonances can show up outside the power spectrum in previous papers we proposed and used new numerical methods to demonstrate the presence of the new resonances predicted by ruelle in a model of chaotic neural networks in this article we deal with simpler models which can be worked out analytically in order to gain more insights into the genesis of the stable resonances and their consequences on the linear response of the system we consider a class of two dimensional time discrete maps describing simple rotator models with a contracting radial dynamics onto the unit circle and a chaotic angular dynamics t 1 2 t mod 2 a generalisation of this system to a network of interconnected rotators is also analysed and related with our previous studies b cessac j a sepulchre stable resonances and signal propagation in a chaotic network of coupled units phys rev e 70 2004 056111 b cessac j a sepulchre transmitting a signal by amplitude modulation in a chaotic network chaos 16 2006 013104 1 13104 12 these models permit us to classify the different types of resonances in the susceptibility and to discuss in particular the relation between the relaxation time of the system to equilibrium with the mixing time given by the decay of the correlation functions also it enables one to propose some general mechanisms responsible for the creation of stable resonances with arbitrary frequencies widths and dependence on the pair of perturbed observed variables

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

additive semi implicit runge kutta methods for computing high speed nonequilibrium reactive flows this paper is concerned with time stepping numerical methods for computing stiff semi discrete systems of ordinary differential equations for transient hypersonic flows with thermo chemical nonequilibrium the stiffness of the equations is mainly caused by the viscous flux terms across the boundary layers and by the source terms modeling finite rate thermo chemical processes implicit methods are needed to treat the stiff terms while more efficient explicit methods can still be used for the nonstiff terms in the equations this paper studies three different semi implicit runge kutta methods for additively split differential equations in the form ofu f u g u wherefis treated by explicit runge kutta methods andgis simultaneously treated by three implicit runge kutta methods a diagonally implicit runge kutta method and two linearized implicit runge kutta methods the coefficients of up to third order accurate additive semi implicit runge kutta methods have been derived such that the methods are both high order accurate and strongly a stable for the implicit terms the results of two numerical tests on the stability and accuracy properties of these methods are also presented in the paper

unconditionally stable compact theta schemes for solving the linear and semi linear fourth order diffusion equations abstract this paper is concerned with numerical methods for solving a class of fourth order diffusion equations combining the compact difference operator in space discretization and the linear method in time the compact theta scheme for the linear problem is first proposed by virtue of the fourier method the suggested scheme is shown to be unconditionally stable and convergent in the discrete l2 norm for any 1 2 1 and then this idea is generalized to the semi linear case the corresponding compact theta scheme is constructed and analyzed in detail numerical experiments corresponding to the linear and semi linear situations are carried out to support our theoretical statements

structure and f dependence of the a c i m for a unimodal map f of misiurewicz type by using a suitable banach space on which we let the transfer operator act we make a detailed study of the ergodic theory of a unimodal map f of the interval in the misiurewicz case we show in particular that the absolutely continuous invariant measure can be written as the sum of 1 square root spikes along the critical orbit plus a continuous background we conclude by a discussion of the sense in which the map f mapsto rho may be differentiable

differentiation of srb states abstract let f be a diffeomorphism of a manifold m and f a generalized srb state for f if supp f is a hyperbolic compact set we show that the map is differentiable in a suitable functional setup and we compute the derivative when supp f is an attractor the derivative is given by where x is the vector field this formula can be extended to time dependent situations and also at least formally to nonuniformly hyperbolic situations the above results will find their use in the study of the onsager reciprocity relations and the fluctuation dissipation formula of nonequilibrium statistical mechanics

bandlimited optical intensity modulation under average and peak power constraints bandlimited optical intensity channels arising in applications such as visible light communications require that all signals satisfy a bandwidth constraint as well as average peak and non negativity amplitude constraints in this paper a 2 d signal space for bandlimited optical intensity channels is presented a novel feature of this model is that the non negativity and peak constraints are relaxed and the signal space parameterizes the probability of the negative or peak amplitude excursions in the output although the intensity channel only supports non negative amplitudes the impact of clipping on system performance is shown to be negligible if the likelihood of the negative excursion is small a tractable approximation using finite series is used to accurately compute the probability of clipping under average and peak optical power constraints the optical power and spectral efficiencies using hexagonal lattice constellations are computed schemes designed in this paper have higher average and peak optical power gains than m ary pulse amplitude modulation pam using previously established techniques for spectral efficiencies greater than 2 5 and 3 5 bits s hz respectively for high spectral efficiency e g greater than 6 bits s hz the proposed scheme attains a more than 2 db average and peak optical power gain over 16 pam using the previous approaches

statistical properties of unimodal maps the quadratic family we prove that almost every nonregular real quadratic map is colleteckmann and has polynomial recurrence of the critical orbit proving a conjecture by sinai it follows that typical quadratic maps have excellent ergodic properties as exponential decay of correlations keller and nowicki young and stochastic stability in the strong sense baladi and viana this is an important step in achieving the same results for more general families of unimodal maps

linear response for intermittent maps we consider the one parameter family alpha mapsto t alpha alpha in 0 1 of pomeau manneville type interval maps t alpha x x 1 2 alpha x alpha for x in 0 1 2 and t alpha x 2x 1 for x in 1 2 1 with the associated absolutely continuous invariant probability measure mu alpha for alpha in 0 1 sarig and gouezel proved that the system mixes only polynomially with rate n 1 1 alpha in particular there is no spectral gap we show that for any psi in l q the map alpha to int 0 1 psi d mu alpha is differentiable on 0 1 1 q and we give a linear response formula for the value of the derivative this is the first time that a linear response formula for the srb measure is obtained in the setting of slowly mixing dynamics our argument shows how cone techniques can be used in this context for alpha ge 1 2 we need the n 1 alpha decorrelation obtained by gouezel under additional conditions

on the validity of linear response theory in high dimensional deterministic dynamical systems this theoretical work considers the following conundrum linear response theory is successfully used by scientists in numerous fields but mathematicians have shown that typical low dimensional dynamical systems violate the theory s assumptions here we provide a proof of concept for the validity of linear response theory in high dimensional deterministic systems for large scale observables we introduce an exemplary model in which observables of resolved degrees of freedom are weakly coupled to a large inhomogeneous collection of unresolved chaotic degrees of freedom by employing statistical limit laws we give conditions under which such systems obey linear response theory even if all the degrees of freedom individually violate linear response we corroborate our result with numerical simulations

homogenization for deterministic maps and multiplicative noise a recent paper of melbourne stuart a note on diffusion limits of chaotic skew product flows nonlinearity 24 2011 1361 1367 gives a rigorous proof of convergence of a fast slow deterministic system to a stochastic differential equation with additive noise in contrast to other approaches the assumptions on the fast flow are very mild in this paper we extend this result from continuous time to discrete time moreover we show how to deal with one dimensional multiplicative noise this raises the issue of how to interpret certain stochastic integrals it is proved that the integrals are of stratonovich type for continuous time and neither

regularized nonlinear solvers for imex methods applied to diffusively corrected multispecies kinematic flow models implicit explicit imex methods are suitable for the solution of nonlinear convection diffusion equations since the stability restrictions coming from the explicitly treated convective part are much less severe than those that would be deduced from an explicit treatment of the diffusive term these schemes usually combine an explicit runge kutta scheme for the time integration of the convective part with a diagonally implicit one for the diffusive part the application of these schemes to multispecies kinematic flow models with strongly degenerate diffusive corrections requires the solution of highly nonlinear and nonsmooth systems of algebraic equations since the efficient solution of these systems by the newton raphson method requires some degree of smoothness it is proposed to regularize the diffusion coefficients in the model and to apply suitable techniques to solve these nonlinear systems in an efficient way numerical examples arising from models of polydisperse sedimentation and multiclass

central schemes for balance laws of relaxation type several models in mathematical physics are described by quasi linear hyperbolic systems with source term and in several cases the production term can become stiff here suitable central numerical schemes for such problems are developed and applications to the broadwell model and extended thermodynamics are presented the numerical methods are a generalization of the nessyahu tadmor scheme to the nonhomogeneous case by including the cell averages of the production terms in the discrete balance equations a second order scheme uniformly accurate in the relaxation parameter is derived and its properties analyzed numerical tests confirm the accuracy and robustness of the scheme

equivalence of non equilibrium ensembles and representation of friction in turbulent flows the lorenz 96 model we construct different equivalent non equilibrium statistical ensembles in a simple yet instructive n degrees of freedom model of atmospheric turbulence introduced by lorenz in 1996 the vector field can be decomposed into an energy conserving time reversible part plus a non time reversible part including forcing and dissipation we construct a modified version of the model where viscosity varies with time in such a way that energy is conserved and the resulting dynamics is fully time reversible for each value of the forcing the statistical properties of the irreversible and reversible model are in excellent agreement if in the latter the energy is kept constant at a value equal to the time average realized with the irreversible model in particular the average contraction rate of the phase space of the time reversible model agrees with that of the irreversible model where instead it is constant by construction we also show that the phase space contraction rate obeys the fluctuation relation and we relate its finite time corrections to the characteristic time scales of the system a local version of the fluctuation relation is explored and successfully checked the equivalence between the two non equilibrium ensembles extends to dynamical properties such as the lyapunov exponents which are shown to obey to a good degree of approximation a pairing rule these results have relevance in motivating the importance of the chaotic hypothesis in explaining that we have the freedom to model non equilibrium systems using different but equivalent approaches and in particular that using a model of a fluid where viscosity is kept constant is just one option and not necessarily the only option for describing accurately its statistical and dynamical properties

implicit explicit runge kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit we consider implicit explicit imex runge kutta r k schemes for hyperbolic systems with stiff relaxation in the so called diffusion limit in such a regime the system relaxes towards a convection diffusion equation the first objective of this paper is to show that traditional partitioned imex r k schemes will relax to an explicit scheme for the limit equation with no need of modification of the original system of course the explicit scheme obtained in the limit suffers from the classical parabolic stability restriction on the time step the main goal of this paper is to present an approach based on imex r k schemes that in the diffusion limit relaxes to an imex r k scheme for the convection diffusion equation in which the diffusion is treated implicitly this is achieved by a novel reformulation of the problem and subsequent application of imex r k schemes to it an analysis of such schemes to the reformulated problem shows that the schemes reduce to imex r k schemes for the limit equation unde

differentiating the absolutely continuous invariant measure of an interval map f with respect to f let the map f 1 1 1 1 have a c i m absolutely continuous f invariant measure with respect to lebesgue let be the change of corresponding to a perturbation x f f 1 of f formally we have for differentiable a

parametric smoothness and self scaling of the statistical properties of a minimal climate model what beyond the mean field theories abstract a quasi geostrophic intermediate complexity model of the mid latitude atmospheric circulation is considered featuring simplified baroclinic conversion and barotropic convergence processes the model undergoes baroclinic forcing towards a given latitudinal temperature profile controlled by the forced equator to pole temperature difference t e as t e increases a transition takes place from a stationary regime hadley equilibrium to a periodic regime and eventually to a chaotic regime where evolution takes place on a strange attractor the attractor dimension metric entropy and bounding box volume in phase space have a smooth dependence on t e which results in power law scaling properties power law scalings with respect to t e are detected also for the statistical properties of global physical observables the total energy of the system and the averaged zonal wind the scaling laws which constitute the main novel result of the present work can be thought to result from the presence of a statistical process of baroclinic adjustment which tends to decrease the equator to pole temperature difference and determines the properties of the attractor of the system the self similarity could be of great help in setting up a theory for the overall statistical properties of the general circulation of the atmosphere and in guiding on a heuristic basis both data analysis and realistic simulations going beyond the unsatisfactory mean field theories and brute force approaches a leading example for this would be the possibility of estimating the sensitivity of the output of the system with respect to changes in the parameters

scaling laws for invariant measures on hyperbolic and nonhyperbolic atractors the analysis of dynamical systems in terms of spectra of singularities is extended to higher dimensions and to nonhyperbolic systems prominent roles in our approach are played by the generalized partial dimensions of the invariant measure and by the distribution of effective liapunov exponents for hyperbolic attractors the latter determines the metric entropies and provides one constraint on the partial dimensions for nonhyperbolic attractors there are important modifications we discuss them for the examples of the logistic and henon map we show in particular that the generalized dimensions have singularities with noncontinuous derivative similar to first order phase transitions in statistical mechanics

chaotic hypothesis onsager reciprocity and fluctuation dissipation theorem it is shown that the chaoticity hypothesis recently introduced in statistical mechanics which is analogous to ruelle s principle for turbulence implies the onsager reciprocity and the fluctuation dissipation theorem in various reversible models for coexisting transport phenomena

response formulae for n point correlations in statistical mechanical systems and application to a problem of coarse graining predicting the response of a system to perturbations is a key challenge in mathematical and natural sciences under suitable conditions on the nature of the system of the perturbation and of the observables of interest response theories allow to construct operators describing the smooth change of the invariant measure of the system of interest as a function of the small parameter controlling the intensity of the perturbation in particular response theories can be developed both for stochastic and chaotic deterministic dynamical systems where in the latter case stricter conditions imposing some degree of structural stability are required in this paper we extend previous findings and derive general response formulae describing how n point correlations are affected by perturbations to the vector flow we also show how to compute the response of the spectral properties of the system to perturbations we then apply our results to the seemingly unrelated problem of coarse graining in multiscale systems we find explicit formulae describing the change in the terms describing parameterisation of the neglected degrees of freedom resulting from applying perturbations to the full system all the terms envisioned by the mori zwanzig theory the deterministic stochastic and non markovian terms are affected at 1st order in the perturbation the obtained results provide a more comprehesive understanding of the response of statistical mechanical systems to perturbations and contribute to the goal of constructing accurate and robust parameterisations and are of potential relevance for fields like molecular dynamics condensed matter and geophysical fluid dynamics we envision possible applications of our general results to the study of the response of climate variability to anthropogenic and natural forcing and to the study of the equivalence of thermostatted statistical mechanical systems