generalized besicovitch spaces and applications to deterministic homogenization the purpose of the present work is to introduce a framework which enables us to study nonlinear homogenization problems the starting point is the theory of algebras with mean value very often in physics from very simple experimental data one gets complicated structure phenomena these phenomena are represented by functions which are permanent in mean but complicated in detail in addition the functions are subject to the verification of a functional equation which in general is nonlinear the problem is therefore to give an interpretation of these phenomena using functions having the following qualitative properties they are functions that represent a phenomenon on a large scale and which vary irregularly undergoing nonperiodic oscillations on a fine scale in this work we study the qualitative properties of spaces of such functions which we call generalized besicovitch spaces and we prove general compactness results related to these spaces we then apply these results in order to study some new homogenization problems one important achievement of this work is the resolution of the generalized weakly almost periodic homogenization problem for a nonlinear pseudo monotone parabolic type operator we also give the answer to the question raised by frid and silva in their paper 35 h frid j silva homogenization of nonlinear pde s in the fourier stieltjes algebras siam j math anal 41 4 2009 1589 1620 as regards whether there exist or do not exist ergodic algebras that are not subalgebras of the fourier stieltjes algebra

statistical properties of skew endomorphisms with bernoulli base abstract we analyse numerically the statistical properties of a class of mappings of the 2 torus t 2 onto itself whose investigation is suggested by some models of modulated diffusion these transformations can be written as a skew product of the endomorphism bp x px mod 1 p z 1 0 1 on the 1 torus t 1 0 1 and a translation on t 1 under suitable assumptions the skew product can be proven to be mixing w r t the lebesgue measure central limit cl and functional central limit fcl properties are numerically checked for analytic observables the result is remarkable because the mappings show no hyperbolic or quasi hyperbolic structure crucial for the proof of central limit theorem and donsker s invariance principle in all of the dynamical systems where these properties have been established up to now moreover cl and fcl behaviours seem to hold also in the case of purely ergodic endomorphisms and even for observables whose correlations do not decay at infinity

stochastic processes originating in deterministic microscopic dynamics we investigate the probability distribution of the scaled trajectory of a test particle moving in an equilibrium fluid according to the laws of classical mechanics i e ifq t is the displacement of the test particle we letqa t q at a and consider the distribution of the trajectory qa t in the limit a the randomness of the motion is due entirely to the randomness of the initial state of the fluid test particle or both and the process is generally non markovian nevertheless it can be proven in some cases and we expect it to be true in many more that qa t looks like brownian motion in the limit a some results for simple model systems are presented

microscopic chaos and gaussian diffusion processes abstract in this paper we construct and analyze a prototypical model of microscopic chaos in particular we extend the results of beck and shimizu to the case where the microscopic time scale is no longer small the upshot is that a non ornstein uhlenbeck deterministic process can generate a gaussian diffusion process

higher order correlations of tchebyscheff maps we determine the higher order correlation functions of iterates of tchebyscheff polynomials of arbitrary order n the non vanishing correlations can be described by suitable sets of graphs these graphs allow for a perturbative evaluation of invariant densities for langevin like systems that are driven by deterministic chaotic forces generated by tchebyscheff maps rather than gaussian white noise

ergodic properties of a kicked damped particle we investigate a class of nonlinear dynamical systems describing the movement of a particle in a viscous medium under the influence of a kick force these systems can be regarded as a generalization of the langevin approach to brownian motion in the sense that the fluctuating force on the particle is not gaussian white noise but an arbitrary non gaussian process generated by a nonlinear dynamical system we investigate how certain properties of the force periodicity ergodicity mixing property transfer to the velocity of the particle moreover the relaxation properties of the system are analysed

approach to gaussian stochastic behavior for systems driven by deterministic chaotic forces we consider skew product dynamical systems that describe the stroboscopic dynamics of a damped particle subjected to a chaotic kick force in a suitable scaling limit the dynamics converges to the ornstein uhlenbeck process we investigate the deterministic chaotic corrections in the vicinity of this gaussian limit case for various examples of chaotic forces we present numerical evidence that for certain classes of chaotic forces the deterministic chaotic corrections of the invariant density are universal we provide analytical results for forces generated by tchebyscheff maps and sketch a renormalization group theory in the space of probability densities

stochastic one dimensional lorentz gas on a lattice we stud ya one dimensional stochastic lorentz gas wher ea light particle moves i na fixed array of nonidentical random scatterers arranged i na lattice each scatterer is characterized b ya random transmissionreflection coefficient we consider the case when the transmission coefficients of the scatterers are independent identically distributed random variables a symbolic program is presented which generates the exact velocity autocorrelation functio n vacf in terms of the moments of the transmission coefficients the vacf is found for different types of disorder for times up to 20 collision times we then conside ra specific type of disorder a two state lorentz gas in which two types of scatterers are arranged randomly i na lattice the na lattice point is occupied b ya scatterer whose transmission coefficient is with probability p or with probability 1 p a perturbation expansion with respect to is derived the 2 term in this expansion shows that the vacf oscillates with time the period of oscillation being twice the time of flight from one scatterer to its nearest neighbor the coarse grained vacf decays for long times like t 32 which is similar to the decay of the vacf of the random lorentz gas wit ha single type of scatterer the perturbation results and the exact one s found up to 20 collision times show good agreement

parameter estimation of unstable limit cycling systems using adaptive feedback linearization example of delta wing roll dynamics the problem of estimating the unknown parameters of a nonlinear plant model for an unstable limit cycling system is considered a feedback linearizing control is used that aims to cancel the nonlinear terms in the plant model and to alter the linear terms such that the closed loop plant model matches with a specified linear reference model the controller parameters are unknown and are evolved starting from zero by an adaptive law that aims to drive them towards their ideal values that would provide perfect model matching between the reference model and the closed loop plant model the converged controller parameters would then provide good estimates for the unknown plant parameters an external forcing signal is considered common to both the reference model and the plant and an adaptation law in the presence of this forcing function is derived using lyapunov methods significantly the same stability analysis is used both to derive a controller for the limit cycling system and also to provide a solution to the problem of parameter estimation simulations using an exponentially decaying sinusoidal forcing show that good estimates of the linear plant parameters are obtained for a wide range of values of the amplitude frequency and decay time of the forcing function and also in the presence of measurement noise in the plant output the problem of non convergence of the nonlinear controller parameters is examined in some detail the parameter estimation method is demonstrated in our paper on a nonlinear model for a rolling delta wing however it should be equally applicable to a wide class of limit cycling systems modeled by a set of nonlinear differential equations

limit cycles in highly non linear differential equations abstract this paper is concerned with both small amplitude and large amplitude limit cycle bifurcations of planar differential systems the analysis is not restricted to minimal models with few non linear terms in fact the novel approach adopted here is to consider differential equations containing highly non linear terms in both the damping and restoring coefficients the maximum number of limit cycles which may be bifurcated in a small region of the origin is given for certain classes of the more generalised mixed rayleigh lienard oscillator equations of the form f x h g x 0 certain mechanical systems are investigated

statistical properties of lorentz gas with periodic configuration of scatterers in our previous paper markov partitions for some classes of dispersed billiards were constructed using these partitions we estimate the decay of velocity auto correlation function and prove the central limit theorem of probability theory and donsker s invariance principle for lorentz gas with periodic configuration of scatterers

on infinite volume mixing in the context of the long standing issue of mixing in infinite ergodic theory we introduce the idea of mixing for observables possessing an infinite volume average the idea is borrowed from statistical mechanics and appears to be relevant at least for extended systems with a direct physical interpretation we discuss the pros and cons of a few mathematical definitions that can be devised testing them on a prototypical class of infinite measure preserving dynamical systems namely the random walks

strange attractors and asymptotic measures of discrete time dissipative systems we investigate asymptotic properties of certain discrete time dynamical systems in two and three dimensions with solenoidal attractor it is proved that the asymptotic measures relevant for the generalized version of the ergodic theorem all derive from one haar measure

a model of modulated diffusion ii numerical results on statistical properties we investigate numerically the statistical properties of a model of modulated diffusion for which we have already computed analytically the diffusion coefficientd our model is constructed by adding a deterministic or random noise to the frequency of an integrable isochronous system we consider in particular the central limit theorem and the invariance principle and we show that they follow wheneverd is positive and for any magnitude of the noise we also investigate the asymptotic distribution in a case whend 0

application of secondary bifurcations to large amplitude limit cycles in mechanical systems large amplitude limit cycles have been described in terms of primary hopf secondary fold bifurcation pairs minimal models for two cases of large amplitude limit cycles have been developed for mechanical systems the mechanism responsible for large amplitude limit cycles in these models is non linear damping the theory is applied to the problems of surge oscillations in axial flow compressors and wing rock oscillations in aircraft flight dynamics to derive new and useful results

homogenization of degenerate porous medium type equations in ergodic algebras abstract we consider the homogenization problem for general porous medium type equations of the form u t f x x e u the pressure function f x y may be of two different types in the type 1 case f x y is a general strictly increasing function this is a mildly degenerate case in the type 2 case f x y has the form h x y f u s x y where f u is just a nondecreasing function this is a strongly degenerate case we address the initial boundary value problem for a general bounded or unbounded domain with null or more generally steady pressure condition on the boundary the homogenization is carried out in the general context of ergodic algebras as far as the authors know homogenization of such degenerate quasilinear parabolic equations is addressed here for the first time we also review the existence and stability theory for such equations and establish new results needed for the homogenization analysis further we include some new results on algebras with mean value specially a new criterion establishing the null measure of level sets of elements of the algebra which is useful in connection with the homogenization of porous medium type equations in the type 2 case

fractal structure and gaussian distribution in chaotic brownian motion in brownian motion driven by a chaotic sequence of iterates of a map f y x t x t f t where f t yn 1 for n t n 1 n 1 2 and y n 1 f yn the fractal structure and the dependence of the recurrence relation xn 1 xn where xn x t n are studied the recurrence relation is shown to change from a shape similar to f y into a diagonal strip which does not depend on the details of f y if is decreased consequently it is shown that the stationary distribution of xn changes from a similar shape as the invariant density of f y into a gaussian shape

from the perron frobenius equation to the fokker planck equation we show that for certain classes of deterministic dynamical systems the perron frobenius equation reduces to the fokker planck equation in an appropriate scaling limit by perturbative expansion in a small time scale parameter we also derive the equations that are obeyed by the first and second order correction terms to the fokker planck limit case in general these equations describe non gaussian corrections to a langevin dynamics due to an underlying deterministic chaotic dynamics for double symmetric maps the first order correction term turns out to satisfy a kind of inhomogeneous fokker planck equation with a source term for a special example we are able solve the first and second order equations explicitly

homogenization of linear and nonlinear transport equations we study the homogenization of the linear and nonlinear transport equations with oscillatory velocity fields two types of homogenized equations are derived for general n dimensional linear and nonlinear problems we derive homogenized equations by introducing additional independent variables to represent the small scales for the two dimensional linear transport equations we derive effective equations for the averaged quantities such equations take the form of either a degenerate non local diffusion equation with memory or a higher order hyperbolic equation to study the nonlinear transport equations we introduce the concept of two scale young measure and extend diperna s method to prove that it reduces to a family of dirac measures

multiscale young measures in homogenization of continuous stationary processes in compact spaces and applications abstract we introduce a framework for the study of nonlinear homogenization problems in the setting of stationary continuous processes in compact spaces the latter are functions f t r n q q with f t x f t x where q is a compact hausdorff topological space f c q and t x q q x r n is an n dimensional continuous dynamical system endowed with an invariant radon probability measure it can be easily shown that for almost all q the realization f t x belongs to an algebra with mean value that is an algebra of functions in buc r n containing all translates of its elements and such that each of its elements possesses a mean value this notion was introduced by zhikov and krivenko v v zhikov e v krivenko homogenization of singularly perturbed elliptic operators mat zametki 33 1983 571 582 english transl in math notes 33 1983 294 300 we then establish the existence of multiscale young measures in the setting of algebras with mean value where the compactifications of r n provided by such algebras plays an important role these parametrized measures are useful in connection with the existence of correctors in homogenization problems we apply this framework to the homogenization of a porous medium type equation in r n with a stationary continuous process as a stiff oscillatory external source this application seems to be new even in the classical context of periodic homogenization

from dynamical systems to the langevin equation abstract it is shown that some dynamical systems which in microscopic time are fully deterministic exhibit a long time stochastic behavior governed by the langevin equation the limit in which this becomes a precise statement is obtained by an appropriate scaling of time and force the results are exemplified by a computer experiment

on a comparative study between dependence scales determined by linear and non linear measures in this manuscript we present a comparative study about the determination of the relaxation i e independence time scales obtained from the correlation function the mutual information and a criterion based on the evaluation of a nonextensive generalisation of mutual entropy our results show that for systems with a small degree of complexity standard mutual information and the criterion based on its nonextensive generalisation provide the same scale whereas for systems with a higher complex dynamics the standard mutual information presents a time scale consistently smaller

driven chaotic motion in single and double well potentials abstract the stochastic motions in stable and bistable potentials are compared on the basis of a langevin like equation with a deterministic chaotic process replacing gaussian white noise ergodic and mixing properties are obtained under rather general circumstances in the case of the symmetric double well potential the system exhibits a behavior quite different from what one observes for stable potentials as the control parameters are varied the system undergoes a phase transition from an ergodic to a nonergodic phase lyapunov exponents and fractal dimensions are studied for more detailed insight

brownian motion as a problem of eliminating fast variables abstract the hamilton equations for a brownian particle involve its mass as a large parameter as a consequence its motion is relatively slow and the fast motion of the surrounding fluid molecules can be eliminated by the standard method of eliminating fast variables the result is equivalent to the known langevin equation plus additional higher orders all coefficients are expressed in terms of correlation functions of the microscopic force on the particle

free boundary convergence in the homogenization of the one phase stefan problem we consider the one phase stefan problem in a granular medium i e with nonconstant thermal diffusity and we study the asymptotic behaviour of the free boundary with respect to homogenization we prove the convergence of the coincidence set in measure and in the hausdorff metric we apply this result to the free boundary and we obtain the convergence in mean for the star shaped case and the uniform convergence for the one dimensional case respectively this gives an answer to a problem posed by j l lions in l 1 introduction we consider the exterior nonhomogeneous one phase stefan problem that is the melting of a frozen granular medium maintained at zero degrees in contact with a liquified region the unknown are the temperature and the free boundary which consists of the solid hquid interface consider a given bounded domain s0 c r the initial liquid region whose boundary consists of two connected manifolds t and t0 of class c1 with t lying in the interior of t0 and bounding a domain w let be a large ball with center 0 containing fi0 and set 0 is to consider the family of operators

fluctuation of the entropy production for the lorentz gas under small external forces in this paper we study the physical and statistical properties of the periodic lorentz gas with finite horizon driven to a non equilibrium steady state by the combination of non conservative external forces and deterministic thermostats a version of this model was introduced by chernov eyink lebowitz and sinai and subsequently generalized by chernov and the third author non equilibrium steady states for these models are srb measures and they are characterized by the positivity of the steady state entropy production rate our main result is to establish that the entropy production in this context equal to the phase space contraction satisfies the gallavotti cohen fluctuation relation the main tool needed in the proof is the family of anisotropic banach spaces introduced by the first and third authors to study the ergodic and statistical properties of billiards using transfer operator techniques

white noise limits for discrete dynamical systems driven by fast deterministic dynamics abstract we study a class of singularly perturbed dynamical systems that have fast and slow components e 1 being the fast to slow timescale ratio the fast components are governed by a strongly mixing discrete map which is iterated at time intervals e the slow components are governed by a first order finite difference equation that uses a time step e as e tends to zero the fast components may be eliminated giving rise to sdes for the slow components the emerging stochastic calculus is in the general case of neither ito nor stratonovich type but depends on the correlation time of the mixing process

a mechanical model of brownian motion we consider a dynamical system consisting of one large massive particle and an infinite number of light point particles we prove that the motion of the massive particle is in a suitable limit described by the ornstein uhlenbeck process this extends to three dimensions previous results by holley in one dimension

from ruelle s transfer operator to the schro dinger operator abstract we show that ruelle s transfer operator defined for an appropriate class of strongly chaotic mappings reduces to the schrodinger operator with imaginary time in an appropriate scaling limit we also derive the equations for the next order corrections to the schrodinger behavior if the scaling limit is not completely performed in the simplest case the first order corrections turns out to obey a schrodinger equation with a source term this means a possible chaotic dynamics underlying the feynman kac formula on a very small time scale e g the planc scale leads generalized quantum mechanics with an additional source term

the number of limit cycle bifurcation diagrams for the generalized mixed rayleigh lienard oscillator abstract this paper investigates the generalized mixed rayleigh lienard oscillator with highly nonlinear terms not restrict to the number of limit cycles this analysis considers mainly the number of limit cycle bifurcation diagrams of the system first the singularity theory approach is applied to the first order averaged approximation of the system with lower order nonlinear terms to reveal all possible bifurcation diagrams by summarizing the generating rule and structural distinction of different bifurcation diagrams a numerical procedure is then developed calculation suggests that the number of bifurcation diagrams increase very fast as the order of nonlinear terms lastly numerical simulations are adopted to approve the analytical results

multiscale young measures in almost periodic homogenization and applications we prove the existence of multiscale young measures associated with almost periodic homogenization we give applications of this tool in the homogenization of nonlinear partial differential equations with an almost periodic structure such as scalar conservation laws nonlinear transport equations hamilton jacobi equations and fully nonlinear elliptic equations motivated by the application in nonlinear transport equations we also prove basic results on flows generated by lipschitz almost periodic vector fields which are of interest in their own in our analysis an important role is played by the so called bohr compactification mathbb g n of mathbb r n this is a natural parameter space for the young measures our homogenization results provide also the asymptotic behavior for the whole set of mathbb g n translates of the solutions which is in the spirit of recent studies on the homogenization of stationary ergodic processes

brownian motion from deterministic dynamics abstract certain deterministic dynamical systems exhibit a transition from non gaussian chaotic to brownian motion behavior when a suitable scaling limit is applied for a simple hyperbolic model system we analyze the structure of the corresponding strange attractor determine the invariant measure and elucidate the transition scenario leading to a gaussian stochastic process

limit behaviour and scaling relations of two kinds of noisy logistic map in the vicinity of chaos threshold and their robustness abstract in this paper we numerically investigate the distribution of the sums of the iterates of the logistic map and the relationships among the important properties of the nonlinear dynamics in the vicinity of the chaos threshold by adding two kinds of contributions with different densities the first one is the well known white noise whereas the second is a newly defined one named as quartic term which makes contributions from the own structure of the map as the chaos threshold is approached the iterates of the standard logistic map i e noise free have strong correlations and the standard central limit theorem is not valid anymore in a recent work tirnakli 2009 it has been shown that the limit distribution seems to converge to q gaussian distribution which maximizes the nonadditive entropy s q 1 i p i q q 1 under appropriate conditions in this work we investigate the effect of these contributions i e white noise and quartic term on the limit distribution and on the range of the obtained q gaussian distribution as a result of these findings under the existence of white noise and also the quartic term we analyse the validity of the scaling relations among correlation fractality the lyapunov divergence and q gaussian distributions which have recently been observed in afsar 2014 the results obtained here strengthen the argument that the central limit behaviour is given by a q gaussian as the chaos threshold is approached and indicate that the scaling relations obtained for the standard logistic map among the range of the q gaussian the correlation dimension the correlation length the lyapunov exponent fractality and the distance from the chaos threshold are robust under the existence of white noise and the quartic term

the transition from deterministic chaos to a stochastic process abstract deterministic dynamical systems of langevin type recently introduced by beck and roepsdorf physica a 145 1987 1 generate langevin dynamics in a suitable limit we discuss the transition from chaos to randomness in these systems under the point of view of the attractor dimension and dynamical entropy and show that the limit of a stochastic process is reached via a shift of the deterministic properties towards the infinitesimal length scales

homogenization of a generalized stefan problem in the context of ergodic algebras abstract we address the deterministic homogenization in the general context of ergodic algebras of a doubly nonlinear problem which generalizes the well known stefan model and includes the classical porous medium equation it may be represented by the differential inclusion for a real valued function u x t t u x e x u x x e x t u u f x e x t u on a bounded domain r n t 0 t together with initial boundary conditions where z x is strictly convex and z x t u is a c 1 convex function both with quadratic growth satisfying some additional technical hypotheses as functions of the oscillatory variable x u x t u and f x t u belong to the generalized besicovitch space b 2 associated with an arbitrary ergodic algebra a the periodic case was addressed by visintin 2007 based on the two scale convergence technique visintin s analysis for the periodic case relies heavily on the possibility of reducing two scale convergence to the usual l 2 convergence in the cartesian product r n where is the periodic cell this reduction is no longer possible in the case of a general ergodic algebra to overcome this difficulty we make essential use of the concept of two scale young measures for algebras with mean value associated with bounded sequences in l 2

fast chaos versus white noise entropy analysis and a fokker planck model for the slow dynamics abstract we show in detail how methods of time series analysis such as dimension and entropy estimates support the idea that fast low dimensional chaos can be modelled properly by noise motivated by this observation we derive a formalism by which fast chaotic degrees of freedom in systems with time scale separation can be replaced by a suitable stochastic process a fokker planck equation for the phase space density of the slow degrees of freedom is derived from the full deterministic system by a projection operator technique together with a perturbation expansion we compare two different projection strategies and illustrate the resulting equations by specifying them for a simple model system

coarse grained dynamical entropies investigation of high entropic dynamical systems abstract coarse grained dynamical entropies characterize in a quantitative way the precision dependent predictability of a system this gives a much more complete description than solely the kolmogorov sinai entropy the concept is applied to two classes of high entropic systems to one class which simulates stochastic processes in some suitable limit and another class of extensively chaotic spatially extended systems in both cases the coarse grained entropies yield new insight into the length scale dependent complexity of the systems

a derivation theory for generalized besicovitch spaces and its application for partial differential equations we present an extension of besicovitch spaces which has its origin in the works of zhikov et al and kozlov and oleinik we show that these spaces have a similar behaviour to the l p spaces and we give a notion of derivative that allows us to treat partial differential problems in this frame to the best of our knowledge this is new even for the usual besicovitch spaces

dynamical systems of langevin type abstract dynamical systems of langevin type are deterministic mappings that arise if the gaussian white noise of a langevin equation is replaced by a deterministic chaotic dynamics we describe the various types of mappings that can be obtained in this way and investigate typical transition scenarios from complicated non gaussian to langevin like behavior that occur if a time scale parameter is changed we define a new characteristic quantify fir ergodic mappings called effective langevin radius which is related to the critical time scale where the marginal invariant density of the dynamical system of langevin type loses its differentiability the invariant density is shown to contain useful information on the entire characteristic functional of the chaotic driving force some general symmetry properties of the invariant densities are discussed

the two scale convergence method applied to generalized besicovitch spaces the twoscale convergence method has proved to be a very useful tool for dealing with periodic homogenization problems in the present paper we develop this theory to generalized besicovitch spaces

on intrinsic randomness of dynamical systems we discuss the problem of nonunitary equivalence via positivity preserving similarity transformations between the unitary groups associated with deterministic dynamical evolution and semigroups associated with stochastic processes dynamical systems admitting such nonunitary equivalence with stochastic markov processes are said to beintrinsically random in a previous work it was found that the so called bernoulli systems discrete time are intrinsically random in this sense this result is extended here by showing that a more general class of dynamical systems the so calledk systems andk flows are intrinsically random the connection of intrinsic randomness with local instability of motion is briefly discussed we also show that markov processes associated through nonunitary equivalence tononisomorphic k flows are necessarily non isomorphic