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

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

a revision of the lyapunov exponent in 1d quadratic maps abstract only misiurewicz points strictly have a lyapunov exponent positive measure in one dimensional 1d quadratic maps hence we represent here the lyapunov exponent of all the misiurewicz points which were found and inventoried by us in a former work as a result of this representation an infinity of alignments of misiurewicz points are obtained and two types of jumps in the values of are found

corrigendum to parameterized tractability of the maximum duo preservation string mapping problem theoret comput sci 646 2016 16 25 this is a corrigendum for our paper 1 as we have found that the first fpt algorithm for the maximum duo preservation string mapping problem we presented is incorrect however we show that by slightly modifying the color coding technique on which the algorithm is based we can fix the error thus giving a correct fpt algorithm for maximum duo preservation string mapping problem

collet eckmann and holder we prove that collet eckmann condition for rational functions which requires exponential expansion only along the critical orbits yields the holder regularity of fatou components this implies geometric regularity of julia sets with non hyperbolic and critically recurrent dynamics in particular polynomial collet eckmann julia sets are locally connected if connected and their hausdorff dimension is strictly less than 2 the same is true for rational collet eckmann julia sets with at least one non empty fully invariant fatou component

solving the maximum duo preservation string mapping problem with linear programming abstract in this paper we introduce the maximum duo preservation string mapping problem mpsm which is complementary to the minimum common string partition problem mcsp when each letter occurs at most k times in any input string the version of mpsm is called k mpsm in order to design approximation algorithms for mpsm we also introduce the constrained maximum induced subgraph problem cmis and the constrained minimum induced subgraph cnis problem we show that both cmis and cnis are np complete we also study the approximation algorithms for the restricted version of cmis which is called k cmis k 2 using linear programming method we propose an approximation algorithm for 2 cmis with approximation ratio 2 and an approximation algorithm for k cmis k 3 with approximation ratio k 2 based on approximation algorithms for k cmis we get approximation algorithms for k mpsm with the same approximation ratio

stochastic teams with nonclassical information revisited when is an affine law optimal in this note we consider a parameterized family of two stage stochastic control problems with nonclassical information patterns which includes the well known 1968 counterexample of witsenhausen we show that whenever the performance index does not contain a product term between the decision variables the optimal solution is linear in the observation variables the parameter space can be partitioned into two regions in one of which the optimal solution is linear whereas in the other it is inherently nonlinear extensive computations using two point piecewise constant policies and linear plus piecewise constant policies provide numerical evidence that nonlinear policies may indeed outperform linear policies when the product term is present

complexities of differentiable dynamical systems we define the notion of localizable property for a dynamical system then we survey three properties of complexity and relate how they are known to be typical among differentiable dynamical systems these notions are the fast growth of the number of periodic points the positive entropy and the high emergence we finally propose a dictionary between the previously explained theory on entropy and the ongoing one on emergence

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

non uniform hyperbolicity and universal bounds for s unimodal maps an s unimodal map f is said to satisfy the collet eckmann condition if the lower lyapunov exponent at the critical value is positive if the infimum of the lyapunov exponent over all periodic points is positive then f is said to have a uniform hyperbolic structure we prove that an s unimodal map satisfies the collet eckmann condition if and only if it has a uniform hyperbolic structure the equivalence of several non uniform hyperbolicity conditions follows one consequence is that some renormalization of an s unimodal map has an absolutely continuous invariant probability measure with exponential decay of correlations if and only if the collet eckmann condition is satisfied the proof uses new universal bounds that hold for any s unimodal map without periodic attractors

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

decay of correlations in one and two dimensions an exposition of some methods of proving exponential stretched exponential decay of correlations is given one dimensional strictly hyperbolic and quadratic maps and two dimensional piecewise smooth uniformly hyperbolic maps are considered the emphasis is on the fundamental constructions of the markov sieve method due to bunimovich chernov sinai and those of liverani s hilbert metric method

the witsenhausen counterexample a hierarchical search approach for nonconvex optimization problems the witsenhausens counterexample 1968 is a difficult nonconvex functional optimization problem which has been outstanding for more than 30 years considerable amount of literature has been accumulated but optimal solutions remain elusive in this paper we develop a framework that allows one to gain additional new insights to the properties of a better solution for a benchmark instance through our approach we are able to zero in on a solution that is 13 better than the previously known best solution and more than 54 better than previous results obtained by other authors more importantly we demonstrate that our approach called hierarchical search can be useful in general optimization problems

asymptotic periodicity and banded chaos abstract treating one dimensional maps as dynamical systems we examine their evolution in terms of density flow in particular a type of density evolution known as asymptotic periodicity is studied unlike statistically stable or exact systems asymptotically periodic systems do not in general evolve to an invariant density even though they possess one consequently a nonequilibrium formalism using several metastable states rather than one invariant density is examined as an alternative to study the physical properties of asymptotically periodic systems asymptotic periodicity is demonstrated for the hat map and the quadratic map at the parameters where these maps generate banded chaos or quasiperiodicity using the ergodic properties of asymptotic periodicity a compact expression for the time correlation function is obtained that does not make any assumptions about decoupling the periodic and stochastic components of c t finally a generalization to the boltzmann gibbs entropy known as the conditional entropy is studied as an index characterizing the asymptotic density sequence that emerges in asymptotically periodic systems

dynamics of a two parameter family of maps of the interval on etudie la dynamique de la famille a 2 parametres des applications a une dimension x f x x 1 x x 0 1 ou les parametres et sont 0 et 1 on considere l effet de degres de liberte additionnels sur les varietes de comportement dynamique

brief paper optimal memoryless control in gaussian noise a simple counterexample in this paper we investigate control strategies for a scalar one step delay system in discrete time i e the state of the system is the input delayed by one time unit in contrast with classical approaches here the control action must be a memoryless function of the output of the plant which comprises the current state corrupted by measurement noise we adopt a first order state space representation for the delay system where the initial state is a gaussian random variable in addition we assume that the measurement noise is drawn from a white and gaussian process with zero mean and constant variance performance evaluation is carried out via a finite time quadratic cost that combines the second moment of the control signal and the second moment of the difference between the initial state and the state at the final time we show that if the time horizon is one or two then the optimal control is a linear function of the plant s output while for a sufficiently large horizon a control taking on only two values will outperform the optimal affine solution this paper complements the well known counterexample by hans witsenhausen which showed that the solution to a linear quadratic and gaussian optimal control paradigm might be nonlinear witsenhausen s counterexample considered an optimization horizon with two time steps two stage control in contrast with witsenhausen s work the solution to our counterexample is linear for one and two stages but it becomes nonlinear as the number of stages is increased the fact that our paradigm leads to nonlinear solutions in the multi stage case could not be predicted from prior results in contrast to prior work the validity of our counterexample is based on analytical proof methods our proof technique rests on a simple nonlinear strategy that is useful in its own right since it outperforms any affine solution

strange attractors in periodically kicked limit cycles and hopf bifurcations we prove the emergence of chaotic behavior in the form of horseshoes and strange attractors with srb measures when certain simple dynamical systems are kicked at periodic time intervals the settings considered include limit cycles and stationary points undergoing hopf bifurcations

slyrb measures natural invariant measures for chaotic systems in many applications it is useful to consider not only the set that constitutes an attractor but also if it exists the asymptotic distribution of a typical trajectory converging to the attractor indeed in the physics literature such a distribution is often assumed to exist when it exists it is called a natural invariant measure the results by lasota and yorke and by sinai ruelle and bowen represent two approaches both of which establish the existence of an invariant measure the goal of this paper is to relate the lasota yorke measure for chaotic attractors in one dimensional maps and the sinai ruelle bowen measure for chaotic attractors in higher dimensional dynamical systems we introduce the notion of slyrb measure we pronounce the term slyrb as a single word slurb the srb concept of measure can be motivated by asking how a trajectory from a typical initial point is distributed asymptotically similarly the slyrb concept of measure can be motivated by asking what the average distribution is for trajectories of a large collection of initial points in some region not necessarily restricted to a single basin the latter is analogous to ask where all the rain drops from a rain storm go and the former asks about where a single rain drop goes perhaps winding up distributed throughout a particular lake

pseudo riddling in chaotic systems abstract previous investigations of riddling have focused in the case where the dynamical invariant set in the symmetric invariant subspace of the system is a chaotic attractor a situation expected to arise commonly in dynamical systems however is that the dynamics in the invariant subspace is in a periodic window in such a case there are both a stable periodic attractor and a coexisting non attracting chaotic saddle in the invariant subspace we show that riddling can still occur in a generalized sense in particular we argue that the basin of the periodic attractor in the invariant subspace contains both an open set and a set of measure zero with riddled holes that belong to the basin of another attractor we call such basin pseudo riddled and we argue that pseudo riddling can be more pervasive than riddling of chaotic attractors because it can occur regardless of whether the chaotic saddle in the invariant subspace is transversely stable or unstable we construct an analyzable model derive a scaling law for pseudo riddling and provide numerical support we also show that the chaotic saddle in the invariant subspace can undergo an enlargement into the full phase space when it becomes transversely unstable

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

dynamics of homoclinic tangles in periodically perturbed second order equations we obtain a comprehensive description on the overall geometrical and dynamical structures of homoclinic tangles in periodically perturbed second order ordinary differential equations with dissipation let be the size of perturbation and be the entire homoclinic tangle we prove in particular that i for infinitely many disjoint open sets of contains nothing else but a horseshoe of infinitely many branches ii for infinitely many disjoint open sets of contains nothing else but one sink and one horseshoe of infinitely many branches and iii there are positive measure sets of so that admits strange attractors with sinai ruelle bowen measure we also use the equation d2qdt2 q2 dqdt q q2 q2sin t to illustrate how to apply our theory to the analysis and to the numerical simulations of a given equation

observations of order and chaos in nonlinear systems experiments on nonlinear electrical oscillators the belousov zhabotinskii reaction rayleigh benard convection and couette taylor flow have revealed several common routes to chaos that have also been found in numerical studies of models with a few degrees of freedom experimental results are presented illustrating the following transition sequences period doubling and the u sequence intermittency the periodic quasiperiodic chaotic sequence frequency locking and an alternating periodic chaotic sequence

parameterized tractability of the maximum duo preservation string mapping problem in this paper we investigate the parameterized complexity of the maximum duo preservation string mapping problem the complementary of the minimum common string partition problem we show that this problem is fixed parameter tractable when parameterized by the number k of conserved duos by first giving a parameterized algorithm based on the color coding technique and then presenting a reduction to a kernel of size o k 6

intractable problems in control theory this paper is an attempt to understand the apparent intractability of problems in decentralized decision making using the concepts and methods of computational complexity we first establish that the discrete version of an important paradigm for this area proposed by witsenhausen is np complete thus explaining the failures reported in the literature to attack it computationally in the rest of the paper we show that the computational intractability of the discrete version of a control problem the team decision problem in our particular example can imply that there is no satisfactory continuous algorithm for the continuous version to this end we develop a theory of continuous algorithms and their complexity and a quite general proof technique which can prove interesting by themselves

the collet eckmann condition for rational functions on the riemann sphere we show that the set of collet eckmann maps has positive lebesgue measure in the space of rational maps on the riemann sphere for any fixed degree d 2

generalizations of srb measures to nonautonomous random and infinite dimensional systems we review some developments that are direct outgrowths of or closely related to the idea of srb measures as introduced by sinai ruelle and bowen in the 1970s these new directions of research include the emergence of strange attractors in periodically forced dynamical systems random attractors in systems defined by stochastic differential equations srb measures for infinite dimensional systems including those defined by large classes of dissipative pdes quasi static distributions for slowly varying time dependent systems and surviving distributions in leaky dynamical systems

entropy chaos and weak horseshoe for infinite dimensional random dynamical systems in this paper we study the complicated dynamics of infinite dimensional random dynamical systems that include deterministic dynamical systems as their special cases in a polish space without assuming any hyperbolicity we prove if a continuous random map has a positive topological entropy then it contains a topological horseshoe we also show that the positive topological entropy implies the chaos in the sense of li yorke the complicated behavior exhibited here is induced by the positive entropy but not the randomness of the system 2017 wiley periodicals inc

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

numerical solutions to the witsenhausen counterexample by approximating networks approximate solutions to the witsenhausen counterexample 1968 are derived by constraining the unknown control functions to take on fixed structures containing free parameters to be optimized such structures are given by nonlinear approximating networks i e linear combinations of parametrized basis functions that benefit by density properties in normed linear spaces this reduces the original functional problem to a nonlinear programming one which is solved via stochastic approximation the method yields lower values of the costs than the ones achieved so far in the literature and most of all provides rather a complete overview of the shapes of the optimal control functions when the two parameters that characterize the witsenhausen counterexample vary one hidden layer neural networks are chosen as approximating networks

brief an ordinal optimization approach to optimal control problems we introduce an ordinal optimization approach to the study of optimal control law design as illustration of the methodology we find the optimal feedback control law for a simple lqg problem without the benefit of theory for the famous unsolved witsenhausen problem 1968 siam j control 6 1 a solution that is 50 better than the witsenhausen solution is found

fixed points of feigenbaum s type for the equationf p λx λf x existence and hyperbolicity of fixed points for the mapn p f x 1f p x withf p p fold iteration and f p 0 are given forp large these fixed points come close to being quadratic functions and our proof consists in controlling perturbation theory about quadratic functions

on the continuity of the srb entropy for endomorphisms we consider classes of dynamical systems admitting markov induced maps under general assumptions which in particular guarantee the existence of srb measures we prove that the entropy of the srb measure varies continuously with the dynamics we apply our result to a vast class of non uniformly expanding maps of a compact manifold and prove the continuity of the entropy of the srb measure in particular we show that the srb entropy of viana maps varies continuously with the map

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

quadratic maps without asymptotic measure an interval map is said to have an asymptotic measure if the time averages of the iterates of lebesgue measure converge weakly we construct quadratic maps which have no asymptotic measure we also find examples of quadratic maps which have an asymptotic measure with very unexpected properties e g a map with the point mass on an unstable fix point as asymptotic measure the key to our construction is a new characterization of kneading sequences

on continuity of bowen ruelle sinai measures in families of one dimensional maps let us consider a family of mapsq a x ax 1 x from the unit interval 0 1 to itself wherea 0 4 is the parameter we show that for any 2 there exists a subsete 4 in 0 4 with the properties

optimization formulation and monotonic solution method for the witsenhausen problem the witsenhausen counterexample is examined the problem is reduced to an optimization problem over the space of quantile functions calculus of variation methods are applied and necessary conditions are generated aspects of the structure of the problem and the solution are discussed a numerical method generating a sequence of solution approximations with monotonically decreasing cost is constructed based on the necessary conditions the limit of the method satisfies the necessary conditions

from invariant curves to strange attractors we prove that simple mechanical systems when subjected to external periodic forcing can exhibit a surprisingly rich array of dynamical behaviors as parameters are varied in particular the existence of global strange attractors with fully stochastic properties is proved for a class of second order odes

dynamics and convergence rate of ordinal comparison of stochastic discrete event systems this paper addresses ordinal comparison in the simulation of discrete event systems it examines dynamic behaviors of ordinal comparison in a fairly general framework it proves that for regenerative systems the probability of obtaining a desired solution using ordinal comparison approaches converges at exponential rate while the variances of the performance measures converge at best at rate o 1 t sup 2 where t is the simulation time heuristic arguments are provided to explain that exponential convergence holds for general systems

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

coupling induced statistical cycling in two diffusively coupled maps abstract this paper discusses the effects of diffusively coupling two identical one dimensional maps attention is focused on situations where the local isolated maps are statistically stable but where the coupled system is not a biologically motivated map and the quadratic map are numerically shown to display this behavior the piecewise linear tent map is then investigated analytically and we give a phase diagram of this system which displays the location of nonequilibrium phase transitions it is conjectured that the diffusive coupling of two chaotic but statistically stable maps i e with asymptotically stable perron frobenius operators can yield a two dimensional system which is not statistically stable whose associated perron frobenius operator is asymptotically periodic

revisiting the parameterized complexity of maximum duo preservation string mapping abstract in the maximum duo preservation string mapping max duo psm problem the input consists of two related strings a and b of length n and a nonnegative integer k the objective is to determine whether there exists a mapping m from the set of positions of a to the set of positions of b that maps only to positions with the same character and preserves at least k duos which are pairs of adjacent positions we develop a randomized algorithm that solves max duo psm in 4 k n o 1 time and a deterministic algorithm that solves this problem in 6 855 k n o 1 time the previous best known deterministic algorithm for this problem has 8 e 2 k o k n o 1 running time beretta et al 2016 1 2 we also show that max duo psm admits a problem kernel of size o k 3 improving upon the previous best known problem kernel of size o k 6

regularity and other properties of absolutely continuous invariant measures for the quadratic family in the current paper we study in more detail some properties of the absolutely continuous invariant measures constructed in the course of the proof of jakobson s theorem in particular we show that the density of the invariant measure is continuous at misiurewicz points from this we deduce that the lyapunov exponent is also continuous at these points our considerations apply just to the parameters constructed in the proof of jakobson s theorem other properties like the positivity of the lyapunov exponent uniqueness of the absolutely continuous invariant measure and exactness of the corresponding dynamical system are also proved

positive lyapunov exponents in families of one dimensional dynamical systems we give a generalization of jakobson s theorem ja and a precise estimate for the density of parameters corresponding to systems with absolutely continuous invariant measures

determinant sums for undirected hamiltonicity we present a monte carlo algorithm for hamiltonicity detection in an n vertex undirected graph running in o 1 657 n time to the best of our knowledge this is the first superpolynomial improvement on the worst case runtime for the problem since the o 2 n bound established for the traveling salesman problem tsp over 50 years ago r bellman j assoc comput mach 9 1962 pp 61 63 m held and r m karp j soc indust appl math 10 1962 pp 196 210 o f n suppresses polylogarithmic functions in f n it answers in part the first open problem in woeginger s 2003 survey on exact algorithms for np hard problems for bipartite graphs we improve the bound to o sqrt 2 n subset o 1 415 n time both the bipartite and the general algorithm can be implemented to use space polynomial in n we combine several recently resurrected ideas to get the results our main technical contribution is a new algebraic characterization of hamiltonian graphs we introduce an ex

sensitive dependence on parameters for a family of dynamical systems we define sensitive dependence on parameters in a way resembling guckenheimer s definition of sensitive dependence on initial conditions while sensitive dependence on initial conditions tells us that if we know the initial condition only approximately then we cannot make deterministic predictions sensitive dependence on parameters tells us that if we know the parameter value only approximately then we cannot make statistical predictions we show that the family of logistic maps has sensitive dependence on parameters moreover the long time response of the system to the change of the parameter is practically independent of the immediate response

the hausdorff dimension of the boundary of the mandelbrot set and julia sets it is shown that the boundary of the mandelbrot set m has hausdorff dimension two and that for a generic c e am the julia set of z i z2 c also has hausdorff dimension two the proof is based on the study of the bifurcation of parabolic periodic points

large deviation principle in one dimensional dynamics we study the dynamics of smooth interval maps with non flat critical points for every such a map that is topologically exact we establish the full level 2 large deviation principle for empirical means in particular the large deviation principle holds for every non renormalizable quadratic map this includes the maps without physical measure found by hofbauer and keller and challenges the widely shared view of the large deviation principle as a refinement of laws of large numbers

dissipative homoclinic loops of two dimensional maps and strange attractors with one direction of instability we prove that when subjected to periodic forcing of the form p t d h x y c sin t certain two dimensional vector fields with dissipative homoclinic loops generate strange attractors with sinai ruelle bowen measures for a set of forcing parameters of positive lebesgue measure the proof extends ideas of afraimovich and shilnikov and applies the recent theory of rank 1 maps developed by wang and young we prove a general theorem and then apply this theorem to an explicit model a forced duffing equation of the form d 2 q

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

compound windows of the henon map abstract for the two parameter second order henon map the shapes and locations of the periodic windows continua of parameter values for which solutions x 0 x 1 can be stably periodic embedded in larger regions where chaotic solutions or solutions of other periods prevail are found by a random searching procedure and displayed graphically many windows have a typical shape consisting of a central body from which four narrow antennae extend such windows to be called compound windows are often arranged in bands to be called window streets that are made up largely of small detected but poorly resolved compound windows for each fundamental subwindow the portion of a window where a fundamental period prevails a stability measure u is introduced where the solution is stable u 1 curves of constant u are found by numerical integration along one line in parameter space the henon map reduces to the one parameter first order logistic map and two antennae from each compound window intersect this line the curves where u 1 and u 1 that bound either antenna are close together within these intersections but as either curve with u 1 leaves the line it diverges from the curve where u 1 crosses the other curve where u 1 and nears the other curve where u 1 forming another antenna the region bounded by the numerically determined curves coincides with the subwindow as found by random searching a fourth degree equation for an idealized curve of constant u is established points in parameter space producing periodic solutions where x 0 x m 0 for given values of m are found to lie on cantor sets of curves that closely fit the window streets points producing solutions where x 0 x m 0 and satisfying a third condition approximating the condition that x n be bounded as n lie on curves to be called street curves of order m that approximate individual members of the cantor set and individual window streets compound windows of period m m tend to occur near the intersections of street curves of orders m and m some exceptions to what appear to be fairly general results are noted the exceptions render it difficult to establish general theorems

strange attractors that are not chaotic abstract it is shown that in certain types of dynamical systems it is possible to have attractors which are strange but not chaotic here we use the word strange to refer to the geometry or shape of the attracting set while the word chaotic refers to the dynamics of orbits on the attractor in particular the exponential divergence of nearby trajectories we first give examples for which it can be demonstrated that there is a strange nonchaotic attractor these examples apply to a class of maps which model nonlinear oscillators continuous time which are externally driven at two incommensurate frequencies it is then shown that such attractore are persistent under perturbations which preserve the original system type i e there are two incommensurate external driving frequencies this suggests that for systems of the typw which we have considered nonchaotic strange attractors may be expected to occur for a finite interval of parameter values on the other hand when small perturbations which do not preserve the system type are numerically introduced the strange nonchaotic attractor is observed to be converted to a periodic or chaotic orbit thus we conjecture that in general continuous time systems flows which are not externally driven at two incommensurate frequencies should not be expected to have strange nonchaotic attractors except possibly on a set of measure zero in the parameter space

binary tree approach to scaling in unimodal maps ge rusjan and zweifel introduced a binary tree which represents all the periodic windows in the chaotic regime of iterated one dimensional unimodal maps we consider the scaling behavior in a modified tree which takes into account the self similarity of the window structure a nonuniversal geometric convergence of the associated superstable parameter values towards a misiurewicz point is observed for almost all binary sequences with periodic tails for these sequences the window period grows arithmetically down the binary tree there are an infinite number of exceptional sequences however for which the growth of the window period is faster numerical studies with a quadratic maximum suggest more rapid than geometric scaling of the superstable parameter values for such sequences

an illustrative case study on application of learning based ordinal optimization approach to complex deterministic problem in this paper we consider complex deterministic problems where there are two models that can be used to predict the performance for a given design one of the models can give a precise estimation but is complex and time consuming the other model is simple and fast but can only give a very crude estimation we have proposed a learning based ordinal optimization approach to tackle this problem in this approach we first run a simple model for all the designs and a complex model for a few designs and then through regression analysis we estimate the noise trend and this noise trend together with the crude estimates from the simple model will be used to screen the designs the proposed approach is applied to solve an integrally bladed rotor ibr manufacturing problem where the production sequence and the production parameters need to be determined in order to minimize the overall manufacturing cost while satisfying the manufacturing constraints the results indicate that by using a very crude and simple model we are able to identify good designs with a high degree of confidence

a new approach to discrete stochastic optimization problems solving a discrete stochastic optimization problem involves two efforts one is to sample and search the design space and the other is to estimate the performance values of the sampled designs when the amount of computational resources is limited we need to know how to balance these two efforts in order to obtain the best result in this paper two performance measures which quantify the marginal contribution of the searching process as well as the performance evaluation process are proposed using these two measures we recommend a framework that can dynamically allocate the computational resources to the search process and the performance evaluation process two algorithms based on the proposed framework are tested on several scenarios and the results produced are very promising

knotted periodic orbits in suspensions of smale s horseshoe torus knots and bifurcation sequences we construct a suspension of smale s horseshoe diffeomorphism of the two dimensional disc as a flow in an orientable three manifold such a suspension is natural in the sense that it occurs frequently in periodically forced nonlinear oscillators such as the duffing equation from this suspension we construct a knot holder or template a branched two manifold with a semiflow in such a way that the periodic orbits are isotopic to those in the full three dimensional flow we discuss some of the families of knotted periodic orbits carried by this template in particular we obtain theorems of existence uniqueness and non existence for families of torus knots we relate these families to resonant hamiltonian bifurcations which occur as horseshoes are created in a one parameter family of area preserving maps and we also relate them to bifurcations of families of one dimensional quadratic like maps which can be studied by kneading theory thus using knot theory kneading theory and hamiltonian bifurcation theory we are able to connect a countable subsequence of one dimensional bifurcations with a subsequence of area preserving bifurcations in a two parameter family of suspensions in which horseshoes are created as the parameters vary one implication is that infinitely many bifurcation sequences are reversed as one passes from the one dimensional to the area preserving family there are no universal routes to chaos

hierarchy of chaotic maps with an invariant measure and their coupling abstract hierarchy of one parameter families of chaotic maps with an invariant measure have been introduced where their appropriate coupling has led to the generation of some coupled chaotic maps with an invariant measure it is shown that these chaotic maps also the coupled maps do not undergo any period doubling or period n tupling cascade bifurcation to chaos but they have either single fixed point attractor at certain values of the parameters or they are ergodic in the complementary region using the invariant measure or sinai ruelle bowen measure the kolmogrov sinai entropy of the chaotic maps coupled maps have been calculated analytically where the numerical simulations support the results

continuous measures in one dimension families of unimodal maps satisfying 1 t 1 1 1 1 witht 1 1 and t 1 1 2 t x isc2 inx2 and and symmetric inx 3 t0 0 0 t1 0 1 with frac d d lambda t 0 0 are considered the results of guckenheimer 1982 are extended to show that there is a positive measure of for whicht has a finite absolutely continuous invariant measure

rigorous bounds on the power spectrum of arbitrary prime eta order renormalisation group equations the authors give rigorous bound inequalities for the universal scaling factors with their numerical values n 3 4 5 of the power spectrum for a renormalisation group equation of arbitrary prime order eta by using a generalisation of the method of collet eckmann thomas

frontiers in complex dynamics rational maps on the riemann sphere occupy a distinguished niche in the general theory of smooth dynamical systems first rational maps are complexanalytic so a broad spectrum of techniques can contribute to their study quasiconformal mappings potential theory algebraic geometry etc the rational maps of a given degree form a finite dimensional manifold so exploration of this parameter space is especially tractable finally some of the conjectures once proposed for smooth dynamical systems and now known to be false seem to have a definite chance of holding in the arena of rational maps in this article we survey a small constellation of such conjectures centering around the density of hyperbolic rational maps those which are dynamically the best behaved we discuss some of the evidence and logic underlying these conjectures and sketch recent progress towards their resolution

multi objective ordinal optimization for simulation optimization problems ordinal optimization oo has been successfully applied to accelerate the simulation optimization process with single objective by quickly narrowing down the search space in this paper we extend the oo techniques to address multi objective simulation optimization problems by using the concept of pareto optimality we call this technique the multi objective oo moo to define the good enough set and the selected set we introduce two performance indices based on the non dominance relationship among the designs then we derive several lower bounds for the alignment probability under various scenarios by using a bayesian approach numerical experiments show that the lower bounds of the alignment probability are valid when they are used to estimate the size of the selected set as well as the expected alignment level though the lower bounds are conservative they have great practical value in terms of narrowing down the search space

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

perturbing misiurewicz parameters in the exponential family in one dimensional real and complex dynamics a map whose post singular or post critical set is bounded and uniformly repelling is often called a misiurewicz map in results hitherto perturbing a misiurewicz map is likely to give a non hyperbolic map as per jakobson s theorem for unimodal interval maps this is despite genericity of hyperbolic parameters at least in the interval setting we show the contrary holds in the complex exponential family z mapsto lambda exp z misiurewicz maps are lebesgue density points for hyperbolic parameters as a by product we also show that lyapunov exponents almost never exist for exponential misiurewicz maps the lower lyapunov exponent is almost everywhere the upper lyapunov exponent is non negative and depends on the choice of metric

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

constraint ordinal optimization ordinal optimization has been developed as the solution to single performance measurement problem however in reality multiple performance measures are required to solve optimization problems constraint optimization was developed so that single measure could be optimized while the remaining performance measures were used as constraints in this paper we will build a new framework and quantification methods for dealing with the constraint optimization problem we call the framework constraint ordinal optimization

non uniformly hyperbolic diffeomorphisms derived from the standard map we prove that the system resulting from coupling the standard map with a fast hyperbolic system is robustly non uniformly hyperbolic

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

statistical stability and continuity of srb entropy for systems with gibbs markov structures we present conditions on families of diffeomorphisms that guarantee statistical stability and srb entropy continuity they rely on the existence of horseshoe like sets with infinitely many branches and variable return times as an application we consider the family of henon maps within the set of benedicks carleson parameters

regularity of the metric entropy for expanding maps the main result of the current paper is an estimate of the radius of the nonperipheral part of the spectrum of the perron frobenius operator for expanding mappings as a consequence we are able to show that the metric entropy of an expanding map has modulus of continuity x log 1 x on the space of c2 expandings we also give an explicit estimate of the rate of mixing for cl functions in terms of natural constants it seems that the method we present can be generalized to other classes of dynamical systems which have a distinguished invariant measure like axiom a diffeomorphisms it also can be adopted to show that the entropy of the quadratic family f x 1 ux2 computed with respect to the absolutely continuous invariant measure found in jakobson s theorem varies continuously the last result is going to appear somewhere else

onset of stochasticity for some one dimensional systems we study the appearance of stochastic dynamics in two situations 1 i for one parameter families of maps of the interval f undergoing an infinite sequence of period doubling bifurcations we consider 0 0 e beyond feigenbaum s parameter value 0 2 ii for a two parameter family of circle maps fb x x b2pisin2 x we consider b 1 1 e 0 1 in both cases we estimate the relative measure of parameter values corresponding to stochastic behavior

on the concept of attractor this note proposes a definition for the concept of attractor based on the probable asymptotic behavior of orbits the definition is sufficiently broad so that every smooth compact dynamical system has at least one attractor

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

transition to chaos through a converging sequence of maps the author considers an infinite sequence or one parameter family of 1d maps fm x which converges uniformly to a target map f x the parameter m can be continuous or discrete fm x have non linear dependence on m the target map f x generates fully chaotic dynamics the author analyses the dynamics generated by the sequence fm as the target map f x is approached through this sequence the first results of computer experiments regarding two special cases of the above situation are presented