nonuniformly expanding 1d maps this paper attempts to make accessible a body of ideas surrounding the following result typical families of possibly multi model 1 dimensional maps passing through misiurewicz points have invariant densities for positive measure sets of parameters

invariant measures exist under a summability condition for unimodal maps for unimodal maps with negative schwarzian derivative a sufficient condition for the existence of an invariant measure absolutely continuous with respect to lebesgue measure is given namely the derivatives of the iterations of the map in the unique critical value must be so large that the sum of some root of the inverses is finite

large derivatives backward contraction and invariant densities for interval maps in this paper we study the dynamics of a smooth multimodal interval map f with non flat critical points and all periodic points hyperbolic repelling assuming that dfn f c as n holds for all critical points c we show that f satisfies the so called backward contracting property with an arbitrarily large constant and that f has an invariant probability which is absolutely continuous with respect to lebesgue measure and the density of belongs to lp for all p lmax lmax 1 where lmax denotes the maximal critical order of f in the appendix we prove that various growth conditions on the derivatives along the critical orbits imply stronger backward contraction

hyperbolicity and invariant measures for general c 2 interval maps satisfying the misiurewicz condition in this paper we will show that piecewise c2 mappings on 0 1 or s1 satisfying the so called misiurewicz conditions are globally expanding in the sense defined below and have absolute continuous invariant probability measures of positive entropy we do not need assumptions on the schwarzian derivative of these maps instead we need the conditions that is piecewise c2 that all critical points of are non flat and that has no periodic attractors our proof gives an algorithm to verify this last condition our result implies the result of misiurewicz in mi where only maps with negative schwarzian derivatives are considered moreover as a byproduct the present paper implies and simplifies the proof of the results of mane in ma who considers general c2 maps without conditions on the schwarzian derivative and restricts attention to points whose forward orbit stay away from the critical points one of the main complications will be that in this paper we want to prove the existence of invariant measures and therefore have to consider points whose iterations come arbitrarily close to critical points misiurewicz deals with this problem using an assumption on the schwarzian derivative of the map this assumption implies very good control of the non linearity of even for high n in order to deal with this non linearity without an assumption on the schwarzian derivative we use the tools of m s it will turn out that the estimates we obtain are so precise that the existence of invariant measures can be proved in a very simple way in some sense much simpler than in mi the existence of these invariant measures under such general conditions was already conjectured a decade ago

non uniform hyperbolicity in complex dynamics we say that a rational function f satisfies the summability condition with exponent if for every critical point c which belongs to the julia set j there exists a positive integer nc so that sum n 1 infty f n prime f n c c alpha infty and f has no parabolic periodic cycles let max be the maximal multiplicity of the critical points

transformation of finite degree discrete volterra systems with cross products to finite input output forms in this paper the problem of finite input output representation is solved for a class of finite degree discrete volterra systems containing cross products among inputs and outputs via the so called spl delta spl epsiv operators a corresponding algorithm is supplied an application concerning the bilinear systems is also given

why computers like lebesgue measure let 0 1 0 1 be a transformation which has an absolutely continuous invariant measure let be a realistic deterministic model for we prove that if has long computer trajectories either periodic or non periodic then these computer trajectories have histograms which approach the density of for a large class of piecewise linear transformations we prove the existence of long periodic trajectories

regular or stochastic dynamics in real analytic families of unimodal maps in this paper we prove that in any non trivial real analytic family of quasiquadratic maps almost any map is either regular i e it has an attracting cycle or stochastic i e it has an absolutely continuous invariant measure to this end we show that the space of analytic maps is foliated by codimension one analytic submanifolds hybrid classes this allows us to transfer the regular or stochastic property of the quadratic family to any non trivial real analytic family

a generalized framework for nodal first derivative summation by parts operators a generalized framework is presented that extends the classical theory of finite difference summation by parts sbp operators to include a wide range of operators where the main extensions are i non repeating interior point operators ii nonuniform nodal distribution in the computational domain iii operators that do not include one or both boundary nodes necessary and sufficient conditions are proven for the existence of nodal approximations to the first derivative with the sbp property it is proven that the positive definite norm matrix of each sbp operator must be associated with a quadrature rule moreover given a quadrature rule there exists a corresponding sbp operator where for diagonal norm sbp operators the weights of the quadrature rule must be positive the generalized framework gives a straightforward means of posing many known approximations to the first derivative as sbp operators several are surveyed such as discontinuous galerkin discretizations based on the legendre gauss quadrature points and shown to be sbp operators moreover the new framework provides a method for constructing sbp operators by starting from quadrature rules this is illustrated by constructing novel sbp operators from known quadrature rules to demonstrate the utility of the generalization the legendre gauss and legendre gauss radau quadrature points are used to construct sbp operators that do not include one or both boundary nodes

limit sets of s unimodal maps with zero entropy one dimensional mappings at the limit of period doubling are studied in this paper without the use of the renormalization theory of feigenbaum and others the principal result is that the attracting part of the nonwandering set is a cantor set of measure zero under the additional assumption that the map has negative schwarzian derivative

computing invariant densities and metric entropy we present a method for accurately computing the metric entropy or equivalently the lyapunov exponent of the absolutely continuous invariant measure for a piecewise analytic expanding markov map t of the interval we construct atomic signed measures m supported on periodic orbits up to period m and prove that super exponentially fast we illustrate our method with several examples

ergodic properties of markov maps inr d for domainsi r d bounded or not the notion of a markov map fromi into itself is developed it is shown that under a condition of renyi type and the assumption that the map is markov any probability density tends inl1 norm to a unique invariant measure under the action of the perron frobenius operatorp the smoothness and ergodic properties of that invariant measure are studied the paper generalizes results of lasota and yorke from dimension one to higher dimension

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

the numerical viscosity of entropy stable schemes for systems of conservation laws i discrete approximations to hyperbolic systems of conservation laws are studied the amount of numerical viscosity present in such schemes is quantified and related to their entropy stability by means of comparison to this end conservative schemes which are also entropy conservative are constructed these entropy conservative schemes enjoy second order accuracy moreover they can be interpreted as piecewise linear finite element methods and hence can be formulated on various mesh configurations it is then shown that conservative schemes are entropy stable if and for three point schemes only they contain more viscosity than that present in the above mentioned entropy conservative ones

julia fatou sullivan theory for real one dimensional dynamics our aim is to show that the julia fatou sullivan structure theory for the dynamics of rational maps is also valid for smooth endomorphisms of the circle and of the interval under extremely mild smoothness and non flatness conditions in order to stress the similarity between real and complex one dimensional dynamics let us recall the main results from the fatou julia sullivan theory if f is a rational map then there is a dynamical decomposition of the riemann sphere into the disjoint union of two totally invariant i e both forward and backward invariant sets j f f f here f f is the domain of normality of the family of iterates of f and is called the fatou set its complement which is called the julia set off is a compact set which contains all the complications of the dynamics off the connected components of the open set f f are mapped onto each other by f hence the orbit of a component of f f is the union of some components of f f julia proved in the beginning of the century that if a component of f f is periodic and contains an attracting periodic point then the orbit of this component must contain a critical point sullivan in the remarkable paper su showed via quasi conformal deformations that the components are eventually periodic and fall into finitely many orbits let n be either the circle s j or a compact interval of the real line and f n n be a smooth endomorphism a critical point of f is a point where the derivative vanishes a critical point is non flat if some higher derivative is non zero a critical point is an inflection point if it has a neighbourhood where f is monotone otherwise it is called a turning point assume f is not a homeomorphism i f f is a homeomorphism then these maps correspond to degree 1 rational maps of the riemann sphere and the situation is

extended skew symmetric form for summation by parts operators and varying jacobians abstract a generalised analytical notion of summation by parts sbp methods is proposed extending the concept of sbp operators in the correction procedure via reconstruction cpr a framework of high order methods for conservation laws for the first time sbp operators with dense norms and not including boundary points are used to get an entropy stable split form of burgers equation moreover overcoming limitations of the finite difference framework stability for curvilinear grids and dense norms is obtained for sbp cpr methods by using a suitable way to compute the jacobian

technical communique magnitude bounds of generalized frequency response functions for nonlinear volterra systems described by narx model in order to reveal the relationship between system time domain model parameters and system frequency response functions new magnitude bounds of frequency response functions for nonlinear volterra systems described by narx model are established the magnitude bound of the nth order generalized frequency response function gfrf can be expressed as a simple n degree polynomial function of the magnitude of the first order gfrf whose coefficients are functions of the model parameters and frequency variables thus the system output spectrum can also be bounded by a polynomial function of the magnitude of the first order gfrf these results demonstrate explicitly the analytical relationship between model parameters and system frequency response functions and provide a significant insight into the magnitude based analysis and synthesis of nonlinear systems in the frequency domain

double precision errors in the logistic map statistical study and dynamical interpretation the nature of the round off errors that occur in the usual double precision computation of the logistic map is studied in detail different iterative regimes from the whole panoply of behaviors exhibited in the bifurcation diagram are examined histograms of errors in trajectories given and for the case of fully developed chaos an explicit formula is found it is shown that the statistics of the largest double precision error as a function of the map parameter is characterized by jumps whose location is determined by certain boundary crossings in the bifurcation diagram both jumps and locations seem to present geometric convergence characterized by the two first feigenbaum constants eventually a comparison with benford s law for the distribution of the leading digit of compilation of numbers is discussed

the lyapunov exponent of holomorphic maps we prove that for any polynomial map with a single critical point its lower lyapunov exponent at the critical value is negative if and only if the map has an attracting cycle similar statement holds for the exponential maps and some other complex dynamical systems we prove further that for the unicritical polynomials with positive area julia sets almost every point of the julia set has zero lyapunov exponent part of this statement generalizes as follows every point with positive upper lyapunov exponent in the julia set of an arbitrary polynomial is not a lebegue density point

a well balanced and entropy conservative discontinuous galerkin spectral element method for the shallow water equations in this work we design an arbitrary high order accurate nodal discontinuous galerkin spectral element type method for the one dimensional shallow water equations the novel method uses a skew symmetric formulation of the continuous problem we prove that this discretisation exactly preserves the local mass and momentum furthermore we show that combined with a special numerical interface flux function the method exactly preserves the entropy which is also the total energy for the shallow water equations finally we prove that the surface fluxes the skew symmetric volume integrals and the source term are well balanced numerical tests are performed to demonstrate the theoretical findings

approximation for the measures of ergodic transformations of the real line let p varphi l 1 r to l 1 r be the frobenius perron operator corresponding to a nonsingular point transformation of the real liner into itself and let for each natural numbern pn be the discrete analogue ofpf it is shown that under fairly weak restrictions on the equationf pf f has an unique solutionf0 such thatf0 0 a e f0 1 and that this solution can be approximated inl1 r in two different ways 1 by the sequence left 1 mathord left vphantom 1 n right kern nulldelimiterspace n sum limits k 0 n 1 p varphi k f right wheref 0 f 1 and 2 by the sequence s on of simple functions such thatsn pn s on

collapsing effects in numerical simulation of a class of chaotic dynamical systems and random mappings with a single attracting centre computer simulations of dynamical systems contain discretizations where finite machine arithmetic replaces continuum state spaces in some circumstances complicated theoretical behaviour has a tendency to collapse to trivial and degenerate behaviour as a result of discretizations characteristics of such collapsing effects often seem to depend on the corresponding discretization only in a random way we describe a procedure to construct a stochastic process with similar statistical characteristics results of computer modelling are discussed

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

dynamics of a map with a power law tail we analyze a one dimensional piecewise continuous discrete model proposed originally in studies on population ecology the map is composed of a linear part and a power law decreasing piece and has three parameters the system presents both regular and chaotic behavior we study numerically and in part analytically different bifurcation structures particularly interesting is the description of the abrupt order to chaos transition mediated by an attractor made of an infinite number of limit cycles with only a finite number of different periods it is shown that the power law piece in the map is at the origin of this type of bifurcation the system exhibits interior crises and crisis induced intermittency

on discretely entropy conservative and entropy stable discontinuous galerkin methods abstract high order methods based on diagonal norm summation by parts operators can be shown to satisfy a discrete conservation or dissipation of entropy for nonlinear systems of hyperbolic pdes 1 2 these methods can also be interpreted as nodal discontinuous galerkin methods with diagonal mass matrices 3 4 5 6 in this work we describe how use flux differencing quadrature based projections and sbp like operators to construct discretely entropy conservative schemes for dg methods under more arbitrary choices of volume and surface quadrature rules the resulting methods are semi discretely entropy conservative or entropy stable with respect to the volume quadrature rule used numerical experiments confirm the stability and high order accuracy of the proposed methods for the compressible euler equations in one and two dimensions

collapsing of chaos in one dimensional maps in their numerical investigation of the family of one dimensional maps f x d 1 2jxj where 2 diamond et al p diamond et al physica d 86 1999 559 571 have observed the surprising numerical phenomenon that a large fraction of initial conditions chosen at random eventually wind up at 1 a repelling fixed point this is a numerical artifact because the continuous maps are chaotic and almost every true trajectory can be shown to be dense in 1 1 the goal of this paper is to extend and resolve this obvious contradiction we model the numerical simulation with a randomly selected map while they used 27 bit precision in computing f we prove for our model that this numerical artifact persists for an arbitrary high numerical prevision the fraction of initial points eventually winding up at 1 remains bounded away from 0 for every numerical precision 2000 elsevier science b v all rights reserved

a note on invariant measures for markov maps of an interval we present two examples of markov maps which satisfy the expanding condition renyi s condition and do not admit any absolutely continuous invariant measures these examples are counterexamples to the theorem formulated in 1 p 1

discretely conservative finite difference formulations for nonlinear conservation laws in split form theory and boundary conditions the lax wendroff theorem stipulates that a discretely conservative operator is necessary to accurately capture discontinuities the discrete operator however need not be derived from the divergence form of the continuous equations indeed conservation law equations that are split into linear combinations of the divergence and product rule form and then discretized using any diagonal norm skew symmetric summation by parts sbp spatial operator yield discrete operators that are conservative furthermore split form discretely conservation operators can be derived for periodic or finite domain sbp spatial operators of any order examples are presented of a fourth order sbp finite difference operator with second order boundary closures sixth and eighth order constructions are derived and are supplied in an accompanying text file

noisy integer maps abstract the combined effects of discretization and noise in iterated maps are studied with simulations and a transfer matrix technique the resulting changes in a typical map s invariant distribution are illustrated

feedback bounded stabilisation of certain discrete volterra systems abstractthroughout this paper we present a method for designing feedback laws which stabilise nonlinear discrete volterra systems our method is based on a factorisation algorithm which decomposes the original system as a composition of a polynomial and a linear series

finite input output representative of a class of volterra polynomial systems n this paper the problem of finite input output representation of a special class of nonlinear volterra polynomial systems is studied via the notion of linear factorization of d series this is an algebraic method based mainly on the star product operation and on a related euclidean type algorithm

multidimensional summation by parts operators general theory and application to simplex elements summation by parts sbp finite difference discretizations share many attractive properties with galerkin finite element methods fems including time stability and superconvergent functionals however unlike fems sbp operators are not completely determined by a basis so the potential exists to tailor sbp operators to meet different objectives to date application of high order sbp discretizations to multiple dimensions has been limited to tensor product domains this paper presents a definition for multidimensional sbp finite difference operators that is a natural extension of one dimensional sbp operators theoretical implications of the definition are investigated for the special case of a diagonal norm mass matrix in particular a diagonal norm sbp operator exists on a given domain if and only if there is a cubature rule with positive weights on that domain and the polynomial basis matrix has full rank when evaluated at the cubature nodes appropriate simultaneous approximation terms are devel

statistical properties of discretizations of a class of chaotic dynamical systems computer simulations of dynamical systems contain discretizations where finite machine arithmetic replaces continuum state spaces in some circumstances complicated theoretical behavior has a tendency to collapse to trivial and degenerate behavior as a result of discretizations various statistical estimators associated with such collapsing effects often seem to depend on the corresponding discretization in a random way behavior of some statistical properties of collapse is discussed results of computer modeling of mappings x 1 1 2x l x 0 1 l 2 are presented

generalised summation by parts operators and variable coefficients abstract high order methods for conservation laws can be highly efficient if their stability is ensured a suitable means mimicking estimates of the continuous level is provided by summation by parts sbp operators and the weak enforcement of boundary conditions recently there has been an increasing interest in generalised sbp operators both in the finite difference and the discontinuous galerkin spectral element framework however if generalised sbp operators are used the treatment of the boundaries becomes more difficult since some properties of the continuous level are no longer mimicked discretely interpolating the product of two functions will in general result in a value different from the product of the interpolations thus desired properties such as conservation and stability are more difficult to obtain here new formulations are proposed allowing the creation of discretisations using general sbp operators that are both conservative and stable thus several shortcomings that might be attributed to generalised sbp operators are overcome cf nordstrom and ruggiu 2017 38 and manzanero et al 2017 39

invariant measures exist without a growth condition given a non flat s unimodal interval map f we show that there exists c which only depends on the order of the critical point c such that if dfn f c c for all n sufficiently large then f admits an absolutely continuous invariant probability measure acip as part of the proof we show that if the quotients of successive intervals of the principal nest of f are sufficiently small then f admits an acip as a special case any s unimodal map with critical order l 2 having no central returns possesses an acip these results imply that the summability assumptions in the theorems of nowicki van strien 21 and martens nowicki 17 can be weakened considerably

a structure theorem in one dimensional dynamics on considere la classe 7b a des applications c f 0 1 0 1 telles que f 0 f 1 0 et f a un unique point critique c 0 1 si le point critique de f 7b a est non plat alors f n a pas d intervalle errant

summation by parts operators for correction procedure via reconstruction the correction procedure via reconstruction cpr formerly known as flux reconstruction is a framework of high order methods for conservation laws unifying some discontinuous galerkin spectral difference and spectral volume methods linearly stable schemes were presented by vincent et al 2011 2015 but proofs of non linear entropy stability in this framework have not been published yet to the knowledge of the authors we reformulate cpr methods using summation by parts sbp operators with simultaneous approximation terms sats a framework popular for finite difference methods extending the results obtained by gassner 2013 for a special discontinuous galerkin spectral element method this reformulation leads to proofs of conservation and stability in discrete norms associated with the method recovering the linearly stable cpr schemes of vincent et al 2011 2015 additionally extending the skew symmetric formulation of conservation laws by additional correction terms entropy stability for burgers equation is proved for general sbp cpr methods not including boundary nodes

summation by parts operators and high order quadrature summation by parts sbp operators are finite difference operators that mimic integration by parts the sbp operator definition includes a weight matrix that is used formally for discrete integration however the accuracy of the weight matrix as a quadrature rule is not explicitly part of the sbp definition we show that sbp weight matrices are related to trapezoid rules with end corrections whose accuracy matches the corresponding difference operator at internal nodes for diagonal weight matrices the accuracy of sbp quadrature extends to curvilinear domains provided the jacobian is approximated with the same sbp operator used for the quadrature this quadrature has significant implications for sbp based discretizations in particular the diagonal norm accurately approximates the l 2 norm for functions and multi dimensional sbp discretizations accurately approximate the divergence theorem

limit cycles in a quadratic discrete iteration i study the truncated logistic equation as a map of d integers the number of limit cycles and the size of longest cycle are averaged exhaustively over many values of d for parameter values a beyond the first accumulation point fits of these quantities are compared with estimates obtained from random maps the results suggest some form of self organization i also present some analytical results on the existence and position of fixed points and a fast matrix method to enumerate limit cycles of arbitrary length for given a and d

absolutely continuous invariant measures forc2 unimodal maps satisfying the collet eckmann conditions in this paper we show that unimodal mappingsf 0 1 0 1 have absolutely continuous measures of positive entropy if these maps arec2 and satisfy the so called collet eckmann conditions no conditions on the schwarzian derivative off are assumed

singular measures without restrictive intervals for the transformationt 0 1 0 1 defined byt x x 1 x with 0 4 a is shown to exist for whicht has no restrictive intervals hence is sensitive to initial conditions but for which no finite absolutely continuous invariant measure exists fort

the model matching problem for a certain class of nonlinear systems sufficient conditions are described for the solution of the model matching problem for a specific class of nonlinear systems represented by the so called polynomials and linear desired models an algorithm yielding causal controllers and a bibo closed loop connection is supplied

periodicity versus chaos in one dimensional dynamics we survey recent results in one dimensional dynamics and as an application we derive rigorous basic dynamical facts for two standard models in population dynamics the ricker and the hassell families we also informally discuss the concept of chaos in the context of one dimensional discrete time models first we use the model case of the quadratic family for an informal exposition we then review precise generic results before turning to the population models our focus is on typical asymptotic behavior seen for most initial conditions and for large sets of maps parameter sets corresponding to different types of attractors are described in particular it is shown that maps with strong chaotic properties appear with positive frequency in parameter space in our population models natural measures asymptotic distributions and their stability properties are considered

a skew symmetric discontinuous galerkin spectral element discretization and its relation to sbp sat finite difference methods this paper shows that the discontinuous galerkin collocation spectral element method with gauss lobatto points dgsem gl satisfies the discrete summation by parts sbp property and can thus be classified as an sbp sat simultaneous approximation term scheme with a diagonal norm operator in the same way sbp sat finite difference schemes can be interpreted as discontinuous galerkin type methods with a corresponding weak formulation based on an inner product formulation common in the finite element community this relation allows the use of matrix vector notation common in the sbp sat finite difference community to show discrete conservation for the split operator formulation of scalar nonlinear conservation laws for dgsem gl and diagonal norm sbp sat based on this result a skew symmetric energy stable discretely conservative dgsem gl formulation applicable to general diagonal norm sbp sat schemes for the nonlinear burgers equation is constructed