stable sets and epsilon stable sets in positive entropy systems in this paper the chaoticity appearing in the stable and unstable sets of a dynamical system with positive entropy is investigated it is shown that in any positive entropy system there is a measure theoretically rather big set such that the closure of the stable or unstable set of any point from the set contains a weak mixing set moreover the bowen entropy of these weak mixing sets are also estimated at the same time it is proved that the topological entropy of any topological system can be calculated in terms of the dispersion of the pre images of epsilon stable sets which answers an open question posed by d fiebig u r fiebig and z h nitecki ergod th dynam sys 23 1785 1806 2003

complex dynamics of cellular automata rule 119 abstract in this paper the dynamical behaviors of cellular automata rule 119 are studied from the viewpoint of symbolic dynamics in the bi infinite symbolic sequence space 2 it is shown that there exists one bernoulli measure global attractor of rule 119 which is also the nonwandering set of the rule moreover it is demonstrated that rule 119 is topologically mixing on the global attractor and possesses the positive topological entropy therefore rule 119 is chaotic in the sense of both li yorke and devaney on the global attractor it is interesting that rule 119 a member of wolfram s class ii which was said to be simple as periodic before actually possesses a chaotic global attractor in 2 finally it is noted that the method presented in this work is also applicable to studying the dynamics of other rules especially the 112 bernoulli shift rules therein

distributional chaos revisited in their famous paper schweizer and smital introduced the definition of a distributionally chaotic pair and proved that the existence of such a pair implies positive topological entropy for continuous mappings of a compact interval further their approach was extended to the general compact metric space case in this article we provide an example which shows that the definition of distributional chaos and as a result li yorke chaos may be fulfilled by a dynamical system with intuitively regular dynamics embedded in 3 next we state strengthened versions of distributional chaos which as we show are present in systems commonly considered to have complex dynamics we also prove that any interval map with positive topological entropy contains two invariant subsets x y c i such that f x has positive topological entropy and f y displays distributional chaos of type 1 but not conversely

existence and roughness of exponential dichotomies of linear dynamic equations on time scales in this paper we define the exponential dichotomy of linear dynamic equations on time scales then we present perturbation theorems on the roughness of exponential dichotomy and develop several explicit sufficient criteria for linear dynamic equations to have an exponential dichotomy as applications of the criteria of exponential dichotomy we derive some new sufficient conditions for the existence of periodic solutions of semi linear dynamic equations and nonlinear dynamic equations on time scales

differentiability of multivalued functions on time scales and applications to multivalued dynamic equations abstract in this paper a class of new derivatives of multivalued functions on a time scale called the hukuhara hilger derivative suited to our study of multivalued dynamic equations is developed moreover some corresponding properties of the hukuhara hilger derivative are discussed the proposed approach forms the appropriate environment within which the study of multivalued dynamic inclusions on time scales can be developed a first example of multivalued dynamic equations is included

discrete chaos induced by heteroclinic cycles connecting repellers in banach spaces abstract this paper is concerned with chaos induced by heteroclinic cycles connecting repellers for maps in banach spaces several criteria of chaos are established in general banach spaces and finite dimensional spaces respectively by employing the coupled expansion theory all the maps presented in this paper are proved to be chaotic in the sense of both li yorke and devaney or in the sense of both li yorke and wiggins or in the sense of li yorke an illustrative example is provided with computer simulations

topological entropy of fuzzified dynamical systems a discrete dynamical system is given by a compact metric space x and any continuous self map defined on x this discrete dynamical system can be naturally extended to the space of fuzzy sets on x in this paper we study relations between the sizes of the topological entropies of the original dynamical system and of its fuzzy counterpart among other things we present a constructive proof of the fact that even very weak assumptions on the crisp discrete dynamical system ensure infinite topological entropy of the fuzzy system however we also show that there are subsystems of the fuzzy dynamical system with topological entropy equal to that of the crisp dynamical system

use of the repeated cross over designs in assessing bioequivalence we consider applications of the repeated 2 x 2 cross over design to evaluating bioequivalence between the two formulations the repeated 2 x 2 cross over design allows us not only to assess bioequivalence on average bioavailability and to examine the subject by formulation interaction but also to obtain independent unbiased estimates of intrasubject variability one consequence of unequal intrasubject variabilities is that the sum of squares of intersubject residuals and the sum of squares of subject by formulation residuals are not independent we also discuss the relative merits of this design as compared to the standard 2 x 2 cross over design without repeated measurements in terms of precision and sample size with respect to the ratio of the number of subjects to the repeated measurements per subject we investigate other uses of the 2 x 2 cross over in examining the bioequivalence between the two different dosing regimens possible applications of other repeated cross over designs to bioequivalence for more than two formulations are explored a numerical example illustrates the proposed procedure

diffusion coefficient and drift velocity in periodic media abstract we present a novel method of computing the asymptotic drift velocity v and the diffusion coefficient d of a particle diffusing in an arbitrary periodic medium we focus on a particular case of 1d systems with the nearest neighbor transition rates and explain in detail how our method works in this case we show that the problem of finding v and d can be reduced to a simple graph counting problem using this method and a computer algebra system we were able to prove fisher and kolomeisky s m e fisher and a b kolomeisky physica a 274 1999 241 conjecture about the maximal force exerted by a molecular motor in the linear response regime for lattice periods l 16

tight binding approach to overdamped brownian motion on a multidimensional tilted periodic potential we present a theoretical treatment of overdamped brownian motion on a multidimensional tilted periodic potential that is analogous to the tight binding model of quantum mechanics in our approach we expand the continuous smoluchowski equation in the localized wannier states of the periodic potential to derive a discrete master equation this master equation can be interpreted in terms of hopping within and between bloch bands and for weak tilting and long times we show that a single band description is valid in the limit of deep potential wells we derive a simple functional dependence of the hopping rates and the lowest band eigenvalues on the tilt we also derive formal expressions for the drift and diffusion in terms of the lowest band eigenvalues

interconnection between set and fuzzy differential equations abstract the original formulation of fuzzy differential equations suffers from the disadvantage since the solutions increases as time increases in this paper employing the results of set differential equations obtained in v laksmikantham s leela a s vatsala setvalued hybrid differential equations and stability in terms of two measures j hybrid systems 2 2 2002 169 188 a new approach is suggested in order to capture the vagueness and the rich properties of solutions without fuzziness

sharkovsky s program for the classification of triangular maps is almost completed abstract for a continuous map of the interval there are more than 50 conditions characterizing zero topological entropy some are applicable to the class of triangular maps x y f x g x y of the square but only a few of them are equivalent in this more general setting in 1989 a n sharkovsky posed the problem of proving or disproving all possible implications between them during last 20 years 32 conditions were considered and most of the work was done only 45 relations out of 992 remained not clear in this paper we give a survey of known results provide two new examples disproving another 26 possible implications and spell out the remaining 19 open problems all but one concern distributional chaos

a note on distributional chaos with respect to a sequence abstract the aim of this note is to use methods developed by kuratowski and mycielski to prove that some more common notions in topological dynamics imply distributional chaos with respect to a sequence in particular we show that the notion of distributional chaos with respect to a sequence is only slightly stronger than the definition of chaos due to li and yorke namely positive topological entropy and weak mixing both imply distributional chaos with respect to a sequence which is not the case for distributional chaos as introduced by schweizer and smital

on the fuzzy initial value problem abstract the initial value problem x t f t x t x 0 x 0 with fuzzy initial value and with deterministic or fuzzy function f is considered two different approaches viz the extension principle and the use of extremal solutions of deterministic initial value problems are applied generalizations to fuzzy integral equations and fuzzy functional differential equations are indicated

almost periodicity of set valued functions and set dynamic equations on time scales in this paper we first propose the concept of the almost periodicity of set valued functions on almost periodic time scales then based on the exponential dichotomy we discuss the existence and uniqueness of almost periodic solutions of set dynamic equations on time scales as an application several examples are presented to exhibit that one can obtain the existence of almost periodic solutions to interval valued dynamic equations or fuzzy dynamic equations and their discrete counterparts in a united means

dynamics of a class of odes more general than almost periodic abstract a temporally global solution if it exists of a nonautonomous ordinary differential equation need not be periodic almost periodic or almost automorphic when the forcing term is periodic almost periodic or almost automorphic respectively an alternative class of functions extending periodic and almost periodic functions which has the property that a bounded temporally global solution solution of a nonautonomous ordinary differential equation belongs to this class when the forcing term does is introduced here specifically the class of functions consists of uniformly continuous functions defined on the real line and taking values in a banach space which have pre compact ranges besides periodic and almost periodic functions this class also includes many nonrecurrent functions assuming a hyperbolic structure for the unperturbed linear equation and certain properties for the linear and nonlinear parts the existence of a special bounded entire solution as well the existence of stable and unstable manifolds of this solution are established moreover it is shown that this solution and these manifolds inherit the temporal behaviour of the vector field equation in the stable case it is shown that this special solution is the pullback attractor of the system a class of infinite dimensional examples involving a linear operator consisting of a time independent part which generates a c 0 semigroup plus a small time dependent part is presented and applied to systems of coupled heat and beam equations

chaos induced by regular snap back repellers abstract this paper is concerned with chaos induced by regular snap back repellers one new criterion of chaos induced by strictly coupled expanding maps in compact sets of metric spaces is established by employing this criterion the nondegenerateness assumption in the marotto theorem established in 1978 is weakened in addition it is proved that a regular snap back repeller and a regular homoclinic orbit to a regular expanding fixed point in finite dimensional spaces imply chaos in the sense of li yorke an illustrative example is provided with computer simulations

almost periodically forced circle flows abstract we study general dynamical and topological behaviors of minimal sets in skew product circle flows in both continuous and discrete settings with particular attentions paying to almost periodically forced circle flows when a circle flow is either discrete in time and unforced i e a circle map or continuous in time but periodically forced behaviors of minimal sets are completely characterized by classical theory the general case involving almost periodic forcing is much more complicated due to the presence of multiple forcing frequencies the topological complexity of the forcing space and the possible loss of mean motion property on one hand we will show that to some extent behaviors of minimal sets in an almost periodically forced circle flow resemble those of denjoy sets of circle maps in the sense that they can be almost automorphic cantorian and everywhere non locally connected but on the other hand we will show that almost periodic forcing can lead to significant topological and dynamical complexities on minimal sets which exceed the contents of denjoy theory for instance an almost periodically forced circle flow can be positively transitive and its minimal sets can be li yorke chaotic and non almost automorphic as an application of our results we will give a complete classification of minimal sets for the projective bundle flow of an almost periodic sl 2 r valued continuous or discrete cocycle continuous almost periodically forced circle flows are among the simplest non monotone multi frequency dynamical systems they can be generated from almost periodically forced nonlinear oscillators through integral manifolds reduction in the damped cases and through mather theory in the damping free cases they also naturally arise in 2d almost periodic floquet theory as well as in climate models discrete almost periodically forced circle flows arise in the discretization of nonlinear oscillators and discrete counterparts of linear schrodinger equations with almost periodic potentials they have been widely used as models for studying strange non chaotic attractors and intermittency phenomena during the transition from order to chaos hence the study of these flows is of fundamental importance to the understanding of multi frequency driven dynamical irregularities and complexities in non monotone dynamical systems

chaotic sets of shift and weighted shift maps abstract in this paper li yorke chaotic sets generated by shift and weighted shift maps are studied the characterization of li yorke chaotic sets by p scrambled sets maximal scrambled sets and orbit invariants are proved for the general shift maps for weighted shift maps on infinite dimensional spaces the necessary and sufficient conditions for li yorke chaotic are proved both in the abstract sequence setting and in the eigenfunction setting besides a constructive proof is provided for the devaney chaos of weighted shift maps on the schwartz space

calculus of fuzzy vector valued functions and almost periodic fuzzy vector valued functions on time scales abstract in this paper we establish an embedding theorem for the fuzzy multidimensional space and introduce the new types of multiplications of fuzzy vectors determined by the determinant algorithm then some fundamental results of calculus of fuzzy vector valued functions on time scales are established based on these results we introduce the concept of shift almost periodic fuzzy vector valued functions and develop a theory of almost periodic fuzzy functions on time scales this function theory can be used to study almost periodic fuzzy dynamic equations on time scales including a new type of fuzzy dynamic equations called fuzzy q dynamic equation i e quantum fuzzy dynamic equation

dynamic parrondo s paradox abstract this paper is focused on studying parrondo s paradox in non linear dynamics specifically how the periodic combination of the individual maps f and g can give rise to chaos or order we construct dynamical systems exhibiting the paradox for several notions of chaos derived from topological dynamics the effect of altering the order of the combination i e considering f g or g f is analyzed as well as the robustness of the parrondo effect under small perturbations conditions for avoiding parrondian dynamics are also obtained placing special emphasis on the notion of chaos given by turbulence

chaos induced by heteroclinic cycles connecting repellers and saddles in locally compact metric spaces abstract this paper is concerned with chaos induced by heteroclinic cycles connecting repellers and saddles for maps in locally compact metric spaces one criterion of chaos for such types of maps is established in locally compact metric spaces by employing the coupled expansion theory where the map is only continuous in some domains of interest in particular one criterion of chaos in the euclidean space r n is also established the maps in the criteria are proved to have a positive topological entropy and to be chaotic in the sense of li yorke an illustrative example is provided with computer simulations

independence in topological and c dynamics we develop a systematic approach to the study of independence in topological dynamics with an emphasis on combinatorial methods one of our principal aims is to combinatorialize the local analysis of topological entropy and related mixing properties we also reframe our theory of dynamical independence in terms of tensor products and thereby expand its scope to c dynamics

positive topological entropy and δ weakly mixing sets abstract the notion of weakly mixing set is introduced and studied it is proved that weakly mixing sets share many properties with weakly mixing sets in particular if a dynamical system has positive topological entropy then the collection of weakly mixing sets is residual in the closure of the collection of entropy sets in the hyperspace the existence of weakly mixing sets in a topological dynamical system admitting an ergodic invariant measure which is not measurable distal is obtained these results generalize several well known results and also answer several open questions

chaotic sets of continuous and discontinuous maps abstract this paper discusses li yorke chaotic sets of continuous and discontinuous maps with particular emphasis to shift and subshift maps scrambled sets and maximal scrambled sets are introduced to characterize li yorke chaotic sets the orbit invariant for a scrambled set is discussed some properties about maximality equivalence and uniqueness of maximal scrambled sets are also discussed it is shown that for shift maps the set of all scrambled pairs has full measure and chaotic sets of some discontinuous maps such as the gauss map interval exchange transformations and a class of planar piecewise isometries are studied finally some open problems on scrambled sets are listed and remarked

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

generalizations of the differentiability of fuzzy number valued functions with applications to fuzzy differential equations the usual concept of differentiability of fuzzy number valued functions has the following shortcoming if c is a fuzzy number and g a b r is an usual real valued function differentiable on x 0 a b with g x 0 0 then f x c g x is not differentiable on x 0 in this paper we introduce and study generalized concepts of differentiability of any order n n which solves this shortcoming newton leibnitz type formula is obtained and existence of the solutions of fuzzy differential equations involving generalized differentiability is studied also some concrete applications to partial and ordinary fuzzy differential equations with fuzzy input data of the form c g x are given

cauchy matrix for linear almost periodic systems and some consequences abstract a novel method to construct the fundamental matrix for a linear almost periodic system is proposed provided that the diagonal terms satisfy an average separation condition and the off diagonal coefficients are l small the idea is to transform the system in a set of riccati type equations and use exponential dichotomy and its consequences it is shown that the method yields easy computation procedures with simple and direct conditions depending on the coefficients finally our result enables us to obtain i explicit almost periodic matrices q t q 1 t and q t which diagonalize the original system and ii sufficient conditions for the stability two illustrative examples are shown

periodic solutions of functional dynamic equations with infinite delay abstract in this paper sufficient criteria are established for the existence of periodic solutions of some functional dynamic equations with infinite delays on time scales which generalize and incorporate as special cases many known results for differential equations and for difference equations when the time scale is the set of the real numbers or the integers respectively the approach is mainly based on the krasnosel ski fixed point theorem which has been extensively applied in studying existence problems in differential equations and difference equations but rarely applied in studying dynamic equations on time scales this study shows that one can unify such existence studies in the sense of dynamic equations on general time scales

on li yorke pairs the li yorke definition of chaos proved its value for interval maps in this paper it is considered in the setting of general topological dynamics we adopt two opposite points of view on the one hand sufficient conditions for li yorke chaos in a topological dynamical system are given we solve a long standing open question by proving that positive entropy implies li yorke chaos on the other hand properties of dynamical systems without li yorke pairs are investigated in addition to having entropy 0 they are minimal when transitive and the property is stable under factor maps arbitrary products and inverse limits finally it is proved that minimal systems without li yorke pairs are disjoint from scattering systems

stability criteria for set dynamic equations on time scales very recently a new theory known as set dynamic equations on time scales has been built in this paper notions of stability for the solutions of set dynamic equations on time scales using lyapunov like functions are considered criteria for the equistability equiasymptotic stability uniform and uniform asymptotic stability are developed

fieller s confidence intervals for the ratio of two means in the assessment of average bioequivalence from crossover data the two period crossover design is the most commonly used study design for bioequivalence of one test formulation to be assessed in comparison to one reference formulation consequently in this paper all derivation is based on this particular design it is assumed that for the underlying statistical model the usual assumptions of normality and additivity are satisfied on the original scale of measurement and that it is wanted to base the assessment of average bioavailability on the ratio of the unknown population means for the test and reference formulation the purpose of this paper is to illustrate that it is reasonable to assume a uniform covariance structure for the two period crossover design because the demand of equal variability in bioavailabilities in addition to equal average bioavailabilities for the reference and test formulation makes the assumption of uniform covariance structure very realistic and also because the properties of a decision rule based upon a fieller s confidence interval under a uniform covariance structure are competitive with those of the corresponding rule based on a general covariance structure

invariant manifolds for stochastic partial differential equations invariant manifolds provide the geometric structures for describing and understanding dynamics of nonlinear systems the theory of invariant manifolds for both finite and infinite dimensional autonomous deterministic systems and for stochastic ordinary differential equations is relatively mature in this paper we present a unified theory of invariant manifolds for infinite dimensional random dynamical systems generated by stochastic partial differential equations we first introduce a random graph transform and a fixed point theorem for nonautonomous systems then we show the existence of generalized fixed points which give the desired invariant manifolds

almost periodic sequence solution of a discrete hassell varley predator prey system with feedback control in this paper we investigate a discrete hassell varley response function predator prey system with feedback control first sufficient conditions are established for the permanence of the system then assuming that the coefficients of the system is almost periodic sequences we obtain conditions for the existence and unique of the almost periodic solution moreover the almost periodic solution we obtained is uniformly asymptotically stable

stable periodic solution of a discrete periodic lotka volterra competition system with a feedback control this paper discusses a discrete lotka volterra competition system with feedback controls we first obtain the persistence of the system assuming that the coefficients in the system are periodic we obtain the existence of a periodic solution moreover under some additional conditions this periodic solution is globally stable

stable sets and mean li yorke chaos in positive entropy systems abstract it is shown that in a topological dynamical system with positive entropy there is a measure theoretically rather big set such that a multivariant version of mean li yorke chaos happens on the closure of the stable or unstable set of any point from the set it is also proved that the intersections of the sets of asymptotic tuples and mean li yorke tuples with the set of topological entropy tuples are dense in the set of topological entropy tuples respectively

graph theoretic structure of maps of the cantor space abstract in this paper we develop unifying graph theoretic techniques to study the dynamics and the structure of spaces h 0 1 n and c 0 1 n the space of homeomorphisms and the space of self maps of the cantor space respectively using our methods we give characterizations which determine when two homeomorphisms of the cantor space are conjugate to each other we also give a new characterization of the comeager conjugacy class of the space h 0 1 n the existence of this class was established by kechris and rosendal and a specific element of this class was described concretely by akin glasner and weiss our characterization readily implies many old and new dynamical properties of elements of this class for example we show that no element of this class has a li yorke pair implying the well known glasner weiss result that there is a comeager subset of h 0 1 n each element of which has topological entropy zero our analogous investigation in c 0 1 n yields a surprising result there is a comeager subset of c 0 1 n such that any two elements of this set are conjugate to each other by an element of h 0 1 n our description of this class also yields many old and new results concerning dynamics of a comeager subset of c 0 1 n

syndetically proximal pairs abstract for continuous self maps of compact metric spaces we study the syndetically proximal relation and in particular we identify certain sufficient conditions for the syndetically proximal cell of each point to be small we show that any interval map f with positive topological entropy has a syndetically scrambled cantor set and an uncountable syndetically scrambled set invariant under some power of f in the process of proving this we improve a classical result about interval maps and establish that if f is an interval map with positive topological entropy and m 2 then there is n n such that the one sided full shift on m symbols is topologically conjugate to a subsystem of f 2 n the classical result gives only semi conjugacy

asymptotic pairs stable sets and chaos in positive entropy systems abstract we consider positive entropy g systems for certain countable discrete infinite left orderable amenable groups g we will use local analysis to study the connection between the existence of asymptotic pairs chaotic sets and stable sets examples are given for the case of integer lattice groups the heisenberg group and the groups of integral unipotent upper triangular matrices

chaotic and topological properties of β transformations article history in this paper we prove that the transformations are chaotic in the sense of both li yorke and devaney the topological and metric properties of the sets of points with dense or non dense orbits are investigated we also prove the result that the set of points with non dense orbits under the transformation is of full hausdorff dimension for any 1

on fuzzy entropy and topological entropy of fuzzy extensions of dynamical systems the goal of this paper is to establish connections between two entropy concepts namely the fuzzy entropy of a fuzzy set which appears in fuzzy set theory and the topological entropy of zadeh s extensions of continuous maps defined on metric compact spaces which is a topological dynamics concept we develop several concepts that allow us to simultaneously address the two entropy concepts except for stating several basic properties we are partially capable of specifying in which classes of fuzzy sets dynamically complex behavior can appear by using these instruments such instruments are necessary for a proper understanding of the behavior of fuzzy systems generated from crisp discrete dynamical systems

fluctuation theorem and large deviation function for a solvable model of a molecular motor we study a discrete stochastic model of a molecular motor this discrete model can be viewed as a minimal ratchet model we extend our previous work on this model by further investigating the constraints imposed by the fluctuation theorem on the operation of a molecular motor far from equilibrium in this work we show the connections between different formulations of the fluctuation theorem one formulation concerns the generating function of the currents while another one concerns the corresponding large deviation function which we have calculated exactly for this model a third formulation concerns the ratio of the probability of observing a velocity v to the same probability of observing a velocity ensuremath v finally we show that all the formulations of the fluctuation theorem can be understood from the notion of entropy production

variation of constant formula for first order fuzzy differential equations in this paper we study first order linear fuzzy differential equations by using the generalized differentiability concept and we present the general form of their solutions we also correct and complete some previous results finally some examples are given to illustrate our results

exponential stability for set dynamic equations on time scales abstract a new theory known as set dynamic equations on time scales has been built the criteria for the equistability equiasymptotic stability uniform and uniformly asymptotic stability were developed in hong 2010 1 in this paper we consider the exponential stability exponentially asymptotic stability uniform and uniformly exponentially asymptotic stability for the trivial solution of set dynamic equations on time scales by using lyapunov like functions

on the distal and asymptotic sets for β transformations abstract in this paper we study the distal and asymptotic sets of a given point in the transformation dynamical system and prove that the hausdorff dimension of the distal set of any given point is one and the hausdorff dimension of the asymptotic set of any given point is zero

stochastic set differential equations in this paper we present the existence and uniqueness of solutions to the stochastic set differential equations driven by a wiener process in the paper we also study their continuous dependence on initial conditions and a stability property

a simplified ratchet model of molecular motors a limiting case of one of the ratchet models of ajdari prost et al is analyzed an explicit solution is obtained for the probability distribution as a function of the time for any initial distribution with all the transients included in the long time limit the drift velocity and diffusion coefficient are obtained in terms of the microscopic transition rates that are the parameters in the model in spite of its extreme simplicity with realistic values of its kinetic parameters the model yields values of the drift velocity and effective force that are of the right magnitude for a molecular motor the model proves to be a simple special case of derrida s periodic one dimensional hopping model for which he found a solution in the long time limit

δ weakly mixing subset in positive entropy actions of a nilpotent group abstract the notion of weakly mixing subsets is introduced for countable torsion free discrete group actions it is shown that for a finitely generated torsion free discrete nilpotent group action positive topological entropy implies the existence of weakly mixing subsets and while there exists a finitely generated torsion free discrete solvable group action which has positive topological entropy but without any weakly mixing subsets

invariant manifolds global attractors and almost periodic solutions of nonautonomous difference equations the article is devoted to the study of quasi linear nonautonomous difference equations invariant manifolds compact global attractors almost periodic and recurrent solutions and chaotic sets first we prove that such equations admit an invariant continuous section an invariant manifold then we obtain the conditions for the existence of a compact global attractor and characterize its structure third we derive a criterion for the existence of almost periodic and recurrent solutions of the quasi linear nonautonomous difference equations finally we prove that quasi linear maps with chaotic base admit a chaotic compact invariant set the obtained results are applied while studying triangular maps invariant manifolds compact global attractors almost periodic and recurrent solutions and chaotic sets

on distributionally chaotic and null systems abstract by a topological dynamical system we mean a pair x f where x is a compactum and f is a continuous self map on x a system is said to be null if its topological sequence entropies are zero along all strictly increasing sequences of natural numbers we show that there exists a null system which is distributionally chaotic this system admits open distributionally scrambled sets and its collection of all maximal distributionally scrambled sets has the same cardinality as the collection of all subsets of the phase space finally such system can even exist on continua

phase spaces and periodic solutions of set functional dynamic equations with infinite delay abstract very recently a new theory known as set dynamic equations on time scales has been built in this paper a phase space is built for set functional dynamic equations with infinite delay on time scales and sufficient criteria are established for the existence of periodic solutions of such equations which generalize and incorporate as special cases some known results for set differential equations and for set difference equations when the time scale is the real number set or the integer set respectively moreover for differential inclusions and difference inclusions if the variable under consideration is a single valued mapping our results show that one can unify the study of some continuous or discrete problems in the sense of set dynamic equations on general time scales

on proximality with banach density one abstract let x t be a topological dynamical system a pair of points x y x 2 is called banach proximal if for any e 0 the set n z d t n x t n y e has banach density one we study the structure of the banach proximal relation a useful tool is the notion of the support of a topological dynamical system we show that a dynamical system is strongly proximal if and only if every pair in x 2 is banach proximal a subset s of x is banach scrambled if every two distinct points in s form a banach proximal pair but not asymptotic we construct a dynamical system with the whole space being a banach scrambled set even though the banach proximal relation of the full shift is of first category it has a dense mycielski invariant banach scrambled set we also show that for an interval map it is li yorke chaotic if and only if it has a cantor banach scrambled set

dynamical compactness and sensitivity abstract to link the auslander point dynamics property with topological transitivity in this paper we introduce dynamically compact systems as a new concept of a chaotic dynamical system x t given by a compact metric space x and a continuous surjective self map t x x observe that each weakly mixing system is transitive compact and we show that any transitive compact m system is weakly mixing then we discuss the relationships between it and other several stronger forms of sensitivity we prove that any transitive compact system is li yorke sensitive and furthermore multi sensitive if it is not proximal and that any multi sensitive system has positive topological sequence entropy moreover we show that multi sensitivity is equivalent to both thick sensitivity and thickly syndetic sensitivity for m systems we also give a quantitative analysis for multi sensitivity of a dynamical system

on open problems concerning distributional chaos for triangular maps abstract we show that in the class t of the triangular maps x y f x g x y of the square there is a map of type 2 with non minimal recurrent points which is not dc3 we also show that every dc1 continuous map of a compact metric space has a trajectory which cannot be weakly approximated by trajectories of compact periodic sets these two results make possible to answer some open questions concerning classification of maps in t with zero topological entropy and contribute to an old problem formulated by a n sharkovsky

uniqueness and attractivity of the carrying simplex for discrete time competitive dynamical systems abstract for the discrete time dissipative competitive dynamical system on r n it is proved that generically there is a unique n 1 dim invariant carrying simplex attracting all nontrivial orbits which solved the conjecture raised by h l smith in 1986

global and cocycle attractors for non autonomous reaction diffusion equations the case of null upper lyapunov exponent abstract in this paper we obtain a detailed description of the global and cocycle attractors for the skew product semiflows induced by the mild solutions of a family of scalar linear dissipative parabolic problems over a minimal and uniquely ergodic flow we consider the case of null upper lyapunov exponent for the linear part of the problem then two different types of attractors can appear depending on whether the linear equations have a bounded or an unbounded associated real cocycle in the first case e g in periodic equations the structure of the attractor is simple whereas in the second case which occurs in aperiodic equations the attractor is a pinched set with a complicated structure we describe situations when the attractor is chaotic in measure in the sense of li yorke besides we obtain a non autonomous discontinuous pitchfork bifurcation scenario for concave equations applicable for instance to a linear dissipative version of the chafee infante equation

generalized hukuhara differentiability of interval valued functions and interval differential equations abstract in the present paper we use a recently proposed generalization of the hukuhara difference for compact convex sets to introduce and study a generalization of the hukuhara differentiability to the case of interval valued functions we also consider several possible definitions for the derivative of an interval valued function recently proposed in the literature and we study connections between them and their properties using these concepts we study interval differential equations local existence and uniqueness of two solutions is obtained together with characterizations of the solutions of an interval differential equation by ode systems several examples are also shown

estimation of direct formulation effect under log normal distribution in bioavailability bioequivalence studies this paper considers estimation of a measure of relative bioavailability of a test formulation to a reference formulation on the original scale under the assumption of a log normal distribution for the data from a 2 2 crossover bioavailability bioequivalence study we propose the minimum variance unbiased estimator which is equal to the maximum likelihood estimator adjusted for bias by a correction factor we also derive the variance of the minimum variance unbiased estimator and its unbiased estimator we derive the biases and mean square errors of the maximum likelihood estimator the ratio of the least squares means and the least squares mean of individual subject ratios with respect to this measure and compare them in a simulation study both theoretical and empirical results strongly suggest that one always consider the minimum variance unbiased estimator a numerical example illustrates the proposed estimation procedure

mean li yorke chaos for random dynamical systems abstract in this paper we studied the complexity of some random dynamical systems with positive topological entropy the random dynamical systems are usually generated by stochastic partial differential equations spdes and contain randomness in many ways we proved that there exists mean li yorke chaotic phenomenon in some random dynamical systems with positive entropy

p attracting and p invariant sets for a class of impulsive stochastic functional differential equations in this paper a nonlinear and nonautonomous impulsive stochastic functional differential equation is considered by establishing an l operator differential inequality and stochastic analysis technique we obtain the p attracting set and p invariant set of the impulsive stochastic functional differential equation an example is also discussed to illustrate the efficiency of the obtained results

procedures for comparing samples with multiple endpoints summary five procedures are considered for the comparison of two or more multivariate samples these procedures include a newly proposed nonparametric rank sum test and a generalized least squares test also considered are the following tests ordinary least squares hotelling s p and a bonferroni per experiment error rate approach applications are envisaged in which each variable represents a qualitatively different measure of response to treatment the null hypothesis of no treatment difference is tested with power directed towards alternatives in which at least one treatment is uniformly better than the others in all simulations the nonparametric procedure provided relatively good power and accurate control over the size of the test and is recommended for general use alternatively the generalized least squares procedure may also be useful with normally distributed data in moderate or large samples a convenient expression for this procedure is obtained and its asymptotic relative efficiency with respect to the ordinary least squares test is evaluated clinical trials are often conducted for the purpose of evaluating the relative efficacy of two or more modes of therapy when the therapies are administered to independent groups and efficacy is measured on the basis of a single response variable appropriate parametric and nonparametric one way analysis of variance anova procedures and their properties are well known often however efficacy is measured by more than one variable although univariate methods for assessing each characteristic individually are useful in this setting there is often the additional need for a single overall objective probability statement that addresses the question of whether or not the experimental therapy is efficacious this need is particularly acute when the medical question is controversial and the sample size is small although a large literature exists on the comparison of multivariate samples the procedures tend to be infrequently used in practice as pointed out by meier 1975 p 523 the standard analysis for the comparison of two multivariate samples which is based on hotelling s t2 statistic addresses somewhat the wrong question and consequently has very poor power for the alternatives of primary interest specifically the t2 procedure addresses the question are one or more of the treatments different making no distinction between variables that change favorably and variables that change unfavorably a second approach would be to assign per experiment error rates to each of the univariate tests by using bonferroni s inequality i e by multiplying each univariate p value by the number of variables studied the desired overall probability statement is then given by the minimum of the per experiment univariate p values although this approach may be useful in some settings it may lack power for alternatives in which most or all measures of efficacy are improved this will be of particular concern when the number k of endpoints studied is large relative to sample size in fact statistical significance will not be possible in some instances this problem will be exacerbated when the measures of efficacy are highly

evaluating the efficiency of the δ distribution mean estimator la distribution delta a ete proposee pour modeliser la distribution des donnees de capture par trait de chalut issues de sondages concernant les poissons et les planctons la formulation exacte de la variance de k estimateur de la moyenne de la distribution delta est extraite et etudiee dans le cas d echantillons de petite taille

chaos among self maps of the cantor space glasner and weiss have shown that a generic homeomorphism of the cantor space has zero topological entropy hochman has shown that a generic transitive homeomorphism of the cantor space has the property that it is topologically conjugate to the universal odometer and hence far from being chaotic in any sense we show that a generic self map of the cantor space has zero topological entropy moreover we show that a generic self map of the cantor space has no periodic points and hence is not devaney chaotic nor devaney chaotic on any subsystem however we exhibit a dense subset of the space of all self maps of the cantor space each element of which has infinite topological entropy and is devaney chaotic on some subsystem

homoclinic groups ie groups and expansive algebraic actions we give algebraic characterizations for expansiveness of algebraic actions of countable groups the notion of p expansiveness is introduced for algebraic actions and we show that for countable amenable groups a finitely presented algebraic action is 1 expansive exactly when it has finite entropy we also study the local entropy theory for actions of countable amenable groups on compact groups by automorphisms and show that the ie group determines the pinsker factor for such actions for an expansive algebraic action of a polycyclic by finite group on x it is shown that the entropy of the action is equal to the entropy of the induced action on the pontryagin dual of the homoclinic group the homoclinic group is a dense subgroup of the ie group the homoclinic group is nontrivial exactly when the action has positive entropy and the homoclinic group is dense in x exactly when the action has completely positive entropy

existence and stability of almost periodic solutions in impulsive neural network models abstract the existence and global exponential stability of an almost periodic solution of an impulsive neural network model with distributed delays is considered in a matrix setting the approach transforms the original network into a matrix analysis problem where a set of sufficient conditions based on spectral radius is presented a concrete hopfield model shows the advantages in comparison with a classical norm approach

gallavotti cohen type symmetry related to cycle decompositions for markov chains and biochemical applications we slightly extend the fluctuation theorem obtained in lebowitz and spohn in j stat phys 95 333 365 1999 for sums of generators considering continuous time markov chains on a finite state space whose underlying graph has multiple edges and no loop this extended frame is suited when analyzing chemical systems as simple corollary we derive by a different method the fluctuation theorem of d andrieux and p gaspard for the fluxes along the chords associated to a fundamental set of oriented cycles andrieux and gaspard in j stat phys 127 107 131 2007

generalized specification property and distributionally scrambled sets abstract suppose the dynamical system x f has generalized specification property in this paper we research the existence of the invariant mycielski distributionally scrambled sets for dynamical system x f moreover some properties about measure center are discussed and a dichotomy theorem is proved

on pairwise sensitivity we define the concept of pairwise sensitivity with respect to initial conditions for the endomorphisms on lebesgue metric spaces the idea is that the orbits of almost every pair of nearby initial points for the product of the invariant measure of a pairwise sensitive map may diverge from a positive quantity independent of the initial points we study the relationships between pairwise sensitivity and weak mixing pairwise sensitivity and positiveness of metric entropy and we compute the largest sensitivity constant

stability of set differential equations and applications abstract the paper introduces some results on stability of sheaf solutions exponential stability of solutions and sheaf solutions of set differential equations and their applications to set control differential equations

almost periodic fuzzy number valued functions in this paper we develop a theory of almost periodic fuzzy functions i e of the almost periodic functions of real variable and with values fuzzy real numbers although the class of fuzzy real numbers does not form a linear normed space the majority of main properties of almost periodic functions with values in banach spaces are extended to this case applications to fuzzy differential equations and to fuzzy dynamical systems are given

conditional entropy and fiber entropy for amenable group actions abstract in this paper we introduce the notions of topological conditional entropy and fiber entropy for a given factor map between two amenable group actions and prove three variational principles for conditional entropy and fiber entropy moreover as an application of our variational principle we prove that the countable to one extension and the distal extension have zero conditional topological entropy in particular the topological entropy of a distal system is zero

generalized derivative and π derivative for set valued functions in this paper we study the generalized derivative and the p derivative for interval valued functions we show the connections between these derivatives some illustrative examples and applications to interval differential equations and fuzzy functions are presented

set differential equations versus fuzzy differential equations the study of fuzzy differential equations fdes forms a suitable setting for a mathematical modeling of real world problems in which uncertainties or vagueness pervades in recent years the theory of fdes has been investigated extensively in the original formulation as well as in an alternative framework which leads to ordinary multivalued differential inclusions it has recently been realized that initiating the study of set differential equations in a metric space has several advantages in addition to providing a natural setting for considering fdes in this paper we present some interesting results in this direction with the necessary background material