discovering multiple lyapunov functions for switched hybrid systems in this paper we analyze local asymptotic stability of switched hybrid systems whose subsystems have polynomial vector fields by discovering multiple lyapunov functions in quadratic forms we start with an algebraizable sufficient condition for the existence of quadratic multiple lyapunov functions then since different discrete modes are considered we apply real root classification together with a projection operator to underapproximate this sufficient condition step by step arriving at a set of semialgebraic sets which only involve the coefficients of the pre assumed multiple lyapunov function note that for each step we additionally use the information on the different discrete modes to optimize our intermediate computation results afterward we compute a sample point in the corresponding semialgebraic set for the coefficients resulting in a multiple lyapunov function finally we test our approach on some examples using a prototypical implementation and compare it with the generic quantifier e

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

maps of intervals with indifferent fixed points thermodynamic formalism and phase transitions we develop the thermodynamic formalism for a large class of maps of the interval with indifferent fixed points for such systems the formalism yields onedimensional systems with many body infinite range interactions for which the thermodynamics is well defined but gibbs states are not piecewise linear systems of this kind yield the soluble in a sense fisher models we prove that such systems exhibit phase transitions the order of which depends on the behavior at the indifferent fixed points we obtain the critical exponent describing the singularity of the pressure and analyze the decay of correlations of the equilibrium states at all temperatures our technique relies on establishing and exploiting a relation between the transfer operators of the original map and its suitable expanding induced version the technique allows one also to obtain a version of the bowen ruelle formula for the hausdorff dimension of repellers for maps with indifferent fixed points and to generalize fisher results to some nonsoluble models

providing a basin of attraction to a target region of polynomial systems by computation of lyapunov like functions in this paper we present a method for computing a basin of attraction to a target region for polynomial ordinary differential equations this basin of attraction is ensured by a lyapunov like polynomial function that we compute using an interval based branch and relax algorithm this algorithm relaxes the necessary conditions on the coefficients of the lyapunov like function to a system of linear interval inequalities that can then be solved exactly it iteratively refines these relaxations in order to ensure that whenever a nondegenerate solution exists it will eventually be found by the algorithm application of an implementation to a range of benchmark problems shows the usefulness of the approach

statistical properties of a nonuniformly hyperbolic map of the interval we prove a power law upper bound for the decay of the correlations for holder observables in the case of a nonuniformly hyperbolic map of the interval introduced by gaspard and wang as a piecewise linear approximation of the intermittent map of manneville pomeau the result is then applied to compute the central limit theorem for the same class of observables

subdivision direction selection in interval methods for global optimization the role of the interval subdivision selection rule is investigated in branch and bound algorithms for global optimization the class of rules that allows convergence for the model algorithm is characterized and it is shown that the four rules investigated satisfy the conditions of convergence a numerical study with a wide spectrum of test problems indicates that there are substantial differences between the rules in terms of the required cpu time the number of function and derivative evaluations and space complexity and two rules can provide substantial improvements in efficiency

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

ergodic and statistical properties of piecewise linear hyperbolic automorphisms of the 2 torus we study generic piecewise linear hyperbolic automorphisms of the 2 torus we explain why the resulting dynamical system is ergodic and mixing and prove the exponential decay of correlations

a method to estimate the parameters of an arma model a method of estimating the parameters of an arma model is proposed in the transient case we show that the formulas recently given by li and dickinson 7 for the ar parameters are easily obtained by considering the formulation of the one step predictor in the steady state case we can get separately the ar parameters and the ma ones

on the estimation of asymptotic stability regions state of the art and new proposals this paper deals with the problem of the estimation of regions of asymptotic stability for continuous autonomous nonlinear systems the first part of the work provides a comprehensive survey of the existing methods and of their applications in engineering fields in the second part certain topological considerations are first developed and the trajectory reversing method is then presented together with a theorem on which it is based in the final part several examples of application are reported showing the efficiency of the proposed technique for low order second and third systems

a comparison of two linear methods of estimating the parameters of arma models a finite order stationary and minimum phase arma autoregressive moving average p q model is equivalent to an infinite order ar autoregressive model two methods of estimating the parameters of the arma p q model by solving only linear equations are based on or closely related to this equivalence relation one method was derived directly from the equivalence relation by d graupe et al ibid vol ac 20 p 104 107 feb 1975 the other was derived by s li and b w dickinson ibid vol ac 31 p 275 278 mar 1986 and ieee trans acoust speech signal process vol assp 36 p 502 512 apr 1988 based on an iterated least squares regression approach the end results bear close resemblance to those of graupe et al the two methods are compared and ways to improve the parameter estimates are suggested

inclusion functions and global optimization 11 inclusion functions combined with special subdivision strategies are an effective means of solving the global unconstrained optimization problem although these techniques were determined and numerically tested about ten years ago they are nearly unknown and scarcely used in order to make the role of inclusion functions and subdivision strategies more widespread and transparent we will discuss a related simplified basic algorithm it computes approximations of the global minimum and at the same time bounds the absolute approximation error we will show that the algorithm works and converges under more general assumptions than it has been known hitherto that is only appropriate inclusion functions are expected to exist the number of minimal points finite or infinite is not of importance lipschitz conditions or continuity are not assumed as shown in the appendix the required inclusion functions can be constructed and programmed without difficulty in a natural way using interval analysis

ergodic properties of invariant measures for piecewise monotonic transformations during the last decade a lot of research has been done on onedimensional dynamics different kinds of invariant measures for certain classes of piecewise monotonic transformations have been considered in most of these cases the perron frobenius operator plays an important role in this paper we try to unify these different examples which are discussed in detail below first we give a discription of the results proved in this paper

safe bounds in linear and mixed integer linear programming current mixed integer linear programming solvers are based on linear programming routines that use floating point arithmetic occasionally this leads to wrong solutions even for problems where all coefficients and all solution components are small integers an example is given where many state of the art milp solvers fail it is then shown how using directed rounding and interval arithmetic cheap pre and postprocessing of the linear programs arising in a branch and cut framework can guarantee that no solution is lost at least for mixed integer programs in which all variables can be bounded rigorously by bounds of reasonable size

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

a new technique for arma system identification and rational approximation the fundamental problem of identifying a linear time invariant system from measured samples of its output response to a known input is addressed utilizing a new and simple deterministic theory founded on well established passive network concepts the analysis together with documented numerical results demonstrates that the proposed method achieves two goals stable rational minimum phase transfer functions can be identified with a priori knowledge of either numerator or denominator degrees and stable rational minimum phase pade like approximations appear to be generated automatically in the nonrational case to substantiate these claims a detailed theoretical exposition of the basic ideas an extensive discussion of numerical results and a summary of related results are given

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

revised cpa method to compute lyapunov functions for nonlinear systems the cpa method uses linear programming to compute continuous and piecewise affine lyapunov function for nonlinear systems with asymptotically stable equilibria in it was shown that the method always succeeds in computing a cpa lyapunov function for such a system the size of the domain of the computed cpa lyapunov function is only limited by the equilibrium s basin of attraction however for some systems an arbitrary small neighborhood of the equilibrium had to be excluded from the domain a priori this is necessary if the equilibrium is not exponentially stable because the existence of a cpa lyapunov function in a neighborhood of the equilibrium is equivalent to its exponential stability as shown in however if the equilibrium is exponentially stable then this was an artifact of the method in this paper we overcome this artifact by developing a revised cpa method we show that this revised method is always able to compute a cpa lyapunov function for a system with an exponentially stable equilibrium the only conditions on the system are that it is c and autonomous the domain of the cpa lyapunov function can be any a priori given compact neighborhood of the equilibrium which is contained in its basin of attraction whereas in a previous paper we have shown these results for planar systems in this paper we cover general n dimensional systems

underapproximating backward reachable sets by semialgebraic sets underapproximations uas of backward reachable sets play an important role in controller synthesis and trajectory analysis for constrained nonlinear dynamical systems but there are few methods available to compute them given a nonlinear system a target region of simply connected compact type and a time duration we present a method using boundary analysis to compute an ua of the backward reachable set the ua is represented as a semialgebraic set formed by what we term polynomial mathtt level set functions the polynomial mathtt level set function is a semidefinite positive function with one real root such that the interior and closure of a semialgebraic set formed by it are both simply connected and have the same boundary the function can be computed by solving a convex program which is constructed based on sum of squares decomposition and linear interval inequalities we test our method on several examples and compare them with existing methods the results show that our method can obtain better estimations more efficiently in terms of time for these special examples

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

sporadicity between periodic and chaotic dynamical behaviors abstract we define the class of sporadic dynamical systems as the systems where the algorithmic complexity of kolmogorov kolmogorov a n 1983 russ math surv 38 29 40 and chaitin chaitin g j 1987 algorithmic information theory cambridge univ press cambridge u k as well as the logarithm of separation of initially nearby trajectories grow as nv0 log n v1 with 0 v0 1 or v0 1 and v1 0 as time n these systems present a behavior intermediate between the multiperiodic v0 0 v1 1 and the chaotic ones v0 1 v1 0 we show that intermittent systems of manneville manneville p 1980 j phys paris 41 1235 1243 as well as some countable markov chains may be sporadic and furthermore that the dynamical fluctuations of these systems may be of levy s type rather than gaussian

decay of correlations for certain quadratic maps we prove exponential decay of correlations for f wheref belongs in a positive measure set of quadratic maps of the interval and is its absolutely continuous invariant measure these results generalize to other interval maps

performance analysis of estimation algorithms of nonstationary arma processes the correlation analysis based methods are not suitable for identifying parameters of nonstationary autoregressive ar moving average ma and arma systems by using estimation residuals in place of unmeasurable noise terms in information vector or matrix we develop a least squares based and gradient based algorithms and establish the consistency of the proposed algorithms without assuming noise stationarity ergodicity or existence of higher order moments furthermore we derive the conditions for convergence of the parameter estimation the simulation results validate the convergence theorems proposed

weak and strong solvability of interval linear systems of equations and inequalities abstract we consider weak and strong solvability of general interval linear systems consisting of mixed equations and inequalities with mixed free and sign restricted variables we generalize the well known weak solvability characterizations by oettli prager for equations and gerlach for inequalities to a unified framework in the same manner we extend strong solvability theorems to general interval linear systems next we propose a sufficient condition for checking strong solvability we give an application in linear programming with interval data by means of weak and strong solvability we determine limits of the optimal values for any form of the problem setting

bracketing backward reach sets of a dynamical system in this paper we present a new method for bracketing i e characterising from inside and from outside backward reach set of the target region t of a continuous time dynamical system the princip

renewal sequences and intermittency in this paper we examine the generating function z of a renewal sequence arising from the distribution of return times in the turbulent region for a class of piecewise affine interval maps introduced by gaspard and wang and studied by several authors we prove that it admits a meromorphic continuation to the entire complex z plane with a branch cut along the ray 1 moreover we compute the asymptotic behavior of the coefficients of its taylor expansion at z 0 from this we obtain the exact polynomial asympotics for the rate of mixing when the invariant measure is finite and of the scaling rate when it is infinite

rigorous lower and upper bounds in linear programming we consider the computation of rigorous lower and upper error bounds for the optimal value of linear programming problems the input data of the lp problem may be exactly given or may vary between given lower and upper bounds the results are then verified for the family of lp problems with input data inside these bounds in many cases only a small computational effort is required for problems with finite simple bounds the rigorous lower bound of the optimal value can be computed with o n2 operations the error bounds can be used as well to perform a sensitivity analysis provided the width of the uncertainties is not too large some numerical examples are presented

estimating the robust domain of attraction for non smooth systems using an interval lyapunov equation the lyapunov equation allows finding a quadratic lyapunov function for an asymptotically stable fixed point of a linear system applying this equation to the linearization of a nonlinear system can also prove the exponential stability of its fixed points this paper proposes an interval version of the lyapunov equation which allows investigating a given lyapunov candidate function for non smooth nonlinear systems inside an explicitly given neighborhood leading to rigorous estimates of the domain of attraction eda of exponentially stable fixed points these results are developed in the context of uncertain systems experiments are presented which show the interest of the approach including with respect to usual approaches based on sum of squares for the computation of eda

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

an efficient method to compute consistent estimates of the ar parameters of an arma model a computationally efficient method for consistent estimation of the ar parameters of an arma model is obtained based on an iterated least squares approach proposed by tsay and tiao the method is derived from the lattice filter structure which generates an orthogonal basis for the hilbert space of interest it is shown that for the arma p q model we can get consistent estimates of the ar parameters by using a least squares lattice estimation algorithm and solving a system of q linear equations

computation of lyapunov functions for smooth nonlinear systems using convex optimization linearly parameterized non quadratic lyapunov functions for smooth nonlinear systems are computed numerically using large scale linear or quadratic programming

towards a complete determination of the spectrum of a transfer operator associated with intermittency the physical phenomenon of intermittency can be investigated via the spectral analysis of a transfer operator associated with the dynamics of an interval map with indifferent fixed point for an example of such an intermittent map the farey map we give a simple proof that the transfer operator is self adjoint on a suitably defined hilbert space and characterize its spectrum using a suitable first return map we present a highly efficient numerical method for the determination of all the eigenvalues including those embedded in the continuous spectrum

limit theorems and markov approximations for chaotic dynamical systems we prove the central limit theorem and weak invariance principle for abstract dynamical systems based on bounds on their mixing coefficients we also develop techniques of markov approximations for dynamical systems we apply our results to expanding interval maps axiom a diffeomorphisms chaotic billiards and hyperbolic attractors

discrete maximum principle for higher order finite elements in 1d we formulate a sufficient condition on the mesh under which we prove the discrete maximum principle dmp for the one dimensional poisson equation with dirichlet boundary conditions discretized by the hp fem the dmp holds if a relative length of every element k in the mesh is bounded by a value h rel p 0 9 1 where p 1 is the polynomial degree of the element k the values h rel p are calculated for 1 p 100

nonunitary transformations and hydrodynamic limit in the one dimensional hard point gas abstract the ergodic theoretical formalism of nonunitary transformations is compared with the derivation of kinetic equations by the hydrodynamic scaling limit the one dimensional hard point system is used as an illustration both methods rest on the k mixing property of the physical systems considered we show that the specific form of the operator is important only on the microscopic time scale on the macroscopic scale it acts like a projector with broken time reversal symmetry we also discuss the connection between the kolmogorov sinai k s entropy of the model dynamical system the relative entropy production rate in the nonunitary formalism and the macroscopic entropy production rate

estimation of the parameters of an arma signal model based on an orthogonal search an iterative procedure for estimating the parameters of an arma signal model input not accessible is empirically compared to the results of several other algorithms the arma identifier reported includes inherent order estimation the empirical results indicate unbiased estimates with less variance than those obtained by the other methods

application of the lattice filter to robust estimation of ar and arma models a method for estimating the parameters of an autoregressive moving average arma model is proposed a p q stage lattice whitening filter is used to obtained consistent estimates of the ar parameters of an arma p p model it is shown that a set of ma parameter estimates can also be obtained using this approach with little added computation the method is used for estimation of the ar parameters of an arma process in additive white noise it is shown that the lattice filter is also useful in robust estimation of the ar parameter on an arma process with additive outliers

an interval approach to compute invariant sets this paper proposes an original interval based method to compute an outer approximation of all invariant sets such as limit cycles of a continuous time nonlinear dynamic system which are included inside a prior set of the state space contrary to all other existing approaches our method has the following properties first it is guaranteed a solution cannot be lost second it is applicable to a large class of systems without any specific assumption such as the knowledge of a lyapunov function or any partial linearity and third there is no need to integrate the system

the statistics of the trajectory of a certain billiard in a flat two torus we consider a billiard in the punctured torus obtained by removing a small disk of radius 0 from the flat torus 2 with trajectory starting from the center of the puncture in this case the phase space is given by the range of the velocity only let and respectively r denote the first exit time length of the trajectory and respectively the number of collisions with the side cushions when 2 is being identified with 0 1 2 we prove that the probability measures on 0 associated with the random variables and r are weakly convergent as varepsilon rightarrow 0 and explicitly compute the densities of the limits

recurrence properties of a special type of heavy tailed random walk in the proof of the invariance principle for locally perturbed periodic lorentz process with finite horizon a lot of delicate results were needed concerning the recurrence properties of its unperturbed version these were analogous to the similar properties of simple symmetric random walk however in the case of lorentz process with infinite horizon the analogous results for the corresponding random walk are not known either in this paper these properties are ascertained for the appropriate random walk this happens to be in the non normal domain of attraction of the normal law as a tool an estimation of the remainder term in the local limit theorem for the corresponding random walk is computed

on the distribution of free path lengths for the periodic lorentz gas consider the domain and let the free path length be defined as the distribution of values of is studied in the limit as for all it is shown that the value is critical for this problem in other words the limiting behavior of depends only on whether is larger or smaller than

survey paper set invariance in control the properties of positively invariant sets are involved in many different problems in control theory such as constrained control robustness analysis synthesis and optimization in this paper we provide an overview of the literature concerning positively invariant sets and their application to the analysis and synthesis of control systems

flow functions control flow functions and the reach control problem the paper studies the reach control problem rcp to make trajectories of an affine system defined on a polytopic state space reach and exit a prescribed facet of the polytope in finite time without first leaving the polytope we introduce the notion of a flow function which provides the analog of a lyapunov function for the equilibrium stability problem a flow function comprises a scalar function that decreases along closed loop trajectories and its existence is a necessary and sufficient condition for closed loop trajectories to exit the polytope it provides an analysis tool for determining if a specific instance of rcp is solved without the need for calculating the state trajectories of the closed loop system results include a variant of the lasalle principle tailored to rcp an open problem is to identify suitable classes of flow functions we explore functions of the form v x max v i x and we give evidence that these functions arise naturally when rcp is solved using continuous piecewise affine feedbacks next we introduce the notion of a control flow function it is shown that the artstein sontag theorem of control lyapunov functions has direct analogies to rcp via control flow functions

cluster approximation for the farey fraction spin chain we consider the farey fraction spin chain in an external field h utilising ideas from dynamical systems the free energy of the model is derived by means of an effective cluster energy approximation this approximation is valid for divergent cluster sizes and hence appropriate for the discussion of the magnetizing transition we calculate the phase boundaries and the scaling of the free energy at h 0 we reproduce the rigorously known asymptotic temperature dependence of the free energy for h 0 our results are largely consistent with those found previously using mean field theory and renormalization group arguments