fractal basin boundaries abstract basin boundaries for dynamical systems can be either smooth or fractal this paper investigates fractal basin boundaries one practical consequence of such boundaries is that they can lead to great difficulty in predicting to which attractor a system eventually goes the structure of fractal basin boundaries can be classified as being either locally connected or locally disconnected examples and discussion of both types of structures are given and it appears that fractal basin boundaries should be common in typical dynamical systems lyapunov numbers and the dimension for the measure generated by inverse orbits are also discussed

an explicit description of the global attractor of the damped and driven sine gordon equation we prove that the size of the finite dimensional attractor of the damped and driven sine gordon equation stays small as the damping and driving amplitude become small a decomposition of finite dimensional attractors in banach space is found into a part that attracts all of phase space except sets whose finitedimensional projections have lebesgue measure zero and a partc that only attracts sets whose finite dimensional projections have lebesgue measure zero we describe the components of the attractor andc which is called the hyperbolic structure for the damped and driven sine gordon equation is low dimensional but the dimension ofc which is associated with transients is much larger we verify numerically that this is a complete description of the attractor for small enough damping and driving parameters and describe the bifurcations of the attractor in this small parameter region

varieties of periodic attractor in cellular automata we apply three alternate definitions of attractor to cellular automata examples are given to show that using the different definitions can give different answers to the question does this cellular automaton have a periodic attractor the three definitions are the topological notion of attractor as used by c conley a more measure theoretic version given by j milnor and a variant of milnor s deenition that is based on the concept of the center of attraction of an orbit restrictions on the types of periodic orbits that can be periodic attractors for cellular automata are described with any of these definitions a cellular automaton has at most one periodic attractor additionally if conley s definition is used then a periodic attractor must be a fixed point using milnor s definition each point on a periodic attractor must be fixed by all shifts so the number of symbols used is an upper bound on the period whether the actual upper bound is 1 is unknown with the third definition this restriction is removed and examples are given of onedimensional cellular automata on three symbols that have finite attractors of arbitrarily large size with the third definition a finite attractor is not necessarily a single periodic orbit the purpose of this paper is to describe the types of periodic orbits that can be attractors of a cellular automaton we will consider three different definitions of attractor one that is based upon topological dynamics a second that is based upon a mixture of topological and measure theoretic dynamics and a third that is even more measure theoretic in nature examples will be given to show that these three definitions can lead to different answers to the question of the existence of a periodic attractor for cellular automata we begin with an informal discussion of some of the results details and precise definitions are given in later sections a cellular automaton is a type of endomorphism of a certain function space let z x zm s where zm is the integer lattice in rm and s is a finite set called the symbol set a shift on z is a map ct z z of the form vtx n x n t for some t e zm a cellular automaton is a continuous map f z z that commutes with all of these shifts cellular automata have received much attention in recent years one reason for this attention is the fact that especially for m 1 or 2 cellular automata are readily accessible received by the editors july 20 1989 1980 mathematics subject classi fication 1985 revision primary 58f12 58f21 research supported in part by nsf grant dms 880758 d 1991 american mathematical society 0002 9947 91 1 00 25 per page

intertwined basins of attraction of dynamical systems in this paper we investigate a kind of intertwining phenomenon and give a new definition of intertwined attractors of the dynamical systems and get some related results

the liapunov dimension of strange attractors abstract many papers have been published recently on studies of dynamical processes in which the attracting sets appear quite strange in this paper the question of estimating the dimension of the attractor is addressed while more general conjectures are made here particular attention is paid to the idea that if the jacobian determinant of a map is greater than one and a ball is mapped into itself then generically the attractor will have positive two dimensional measure and most of this paper is devoted to presenting cases with such jacobians for which the attractors are proved to have non empty interior

rotational chaos and strange attractors on the 2 torus we construct a 2 torus homeomorphism h homotopic to the identity with an attracting r h bing s pseudocircle c such that the rotation set of h c is not a unique vector the only known examples of such an attractor were birkhoff s attractors arising for dissipative maps with a twist our construction relies heavily on the barge martin method for constructing attractors as inverse limits of graphs

comment on adaptive feedback control algorithm we point out a problem in the parametric estimation method based on the adaptive feedback algorithm proposed by d huang phys rev e 73 066204 2006 both illustratively and analytically furthermore under some hypotheses we prove that it is not the so called chaotic dynamical characteristic but the linear independence between the right hand functions and the estimated parameters of the intrinsic system in its attractor that counts in the availability of this algorithm in estimating the unknown parameters

attractors for graph critical rational maps we call a rational map f graph critical if any critical point either belongs to an invariant finite graph g or has minimal limit set or is non recurrent and has limit set disjoint from g we prove that for any conformal measure either for almost every point of the julia set j f its limit set coincides with j f or for almost every point of j f its limit set coincides with the limit set of a critical point of f

crises sudden changes in chaotic attractors and transient chaos abstract the occurrence of sudden qualitative changes of chaotic dynamics as a parameter is varied is discussed and illustrated it is shown that such changes may result from the collision of an unstable periodic orbit and a coexisting chaotic attractor we call such collisions crises phenomena associated with crises include sudden changes in the size of chaotic attractors sudden appearances of chaotic attractors a possible route to chaos and sudden destructions of chaotic attractors and their basins this paper presents examples illustrating that crisis events are prevalent in many circumstances and systems and that just past a crisis certain characteristic statistical behavior whose type depends on the type of crisis occurs in particular the phenomenon of chaotic transients is investigated the examples discussed illustrate crises in progressively higher dimension and include the one dimensional quadratic map the two dimensional henon map systems of ordinary differential equations in three dimensions and a three dimensional map in the case of our study of the three dimensional map a new route to chaos is proposed which is possible only in invertible maps or flows of dimension at least three or four respectively based on the examples presented the following conjecture is proposed almost all sudden changes in the size of chaotic attractors and almost all sudden destruction or creations of chaotic attractors and their basins are due to crises

strange attractors of uniform flows on considere des attracteurs orbitalement stables de ces flots sur le tore solide ouvert d 2 s 1 qui ont une vitesse dans la direction s 1 on trouve qu un tel attracteur est la frontiere d une suite strictement emboitee de tores solides ouverts positivement invariants

ber performance of dqpsk in nakagami fading with selection diversity and maximal ratio combining a new expression for the average bit error rate ber of differential quadrature phase shift keying in slow frequency nonselective nakagami fading is derived for a space diversity receiver having a cascade arrangement of l groups of m branch selection combiners and an l branch maximal ratio combiner mrc this allows the use of a large number of antennas for performance improvement and a small number of m inputs to the combiners for low complexity the average ber performance of the cascade receiver is investigated for different fading severity conditions characterized by the nakagami m factor and compared with the conventional mrc receiver as the fading gets less severe m increases performance improvement over the conventional mrc scheme is only noticeable for the larger range of average signal to noise ratio

connection coefficients for laguerre sobolev orthogonal polynomials laguerre sobolev polynomials are orthogonal with respect to an inner product of the form p qs 0 p x q x x e x dx 0 p x q x d x w here 1 0 and p q p the linear space of polynomials with real coefficients if d x x e x dx the corresponding sequence of monic orthogonal polynomials q x has been studied by marcellan et al j com put appl math 71 1996 245 265 while if d x x dx the sequence of monic orthog onal polynomials l x was introduced by koekoek and meijer siam j math anal 24 1993 768 782 for each of these two families of laguerre sobolev polynomials here we give the explicit expression of the connection coefficients in their expansion as a series of standard la guerre polynomials the inverse connection problem of expanding laguerre polynomials in series of laguerre sobolev polynomials and the connection problem relating two families of laguerre sobolev polynomials with different parameters are also considered

on the unfolding of a blowout bifurcation abstract suppose a chaotic attractor a in an invariant subspace loses stability on varying a parameter at the point of loss of stability the most positive lyapunov exponent of the natural measure on a crosses zero at what has been called a blowout bifurcation we introduce the notion of an essential basin of an attractor a this is the set of points x such that accumulation points of the sequence of measures 1 n n 1 k 0 f k x are supported on a we characterise supercritical and subcritical scenarios according to whether the lebesgue measure of the essential basin of a is positive or zero we study a drift diffusion model and a model class of piecewise linear mappings of the plane in the supercritical case we find examples where a lyapunov exponent of the branch of attractors may be positive hyperchaos or negative depending purely on the dynamics far from the invariant subspace for the mappings we find asymptotically linear scaling of lyapunov exponents average distance from the subspace and basin size on varying a parameter we conjecture that these are general characteristics of blowout bifurcations

the boundary constraint method for constructing vortex surface fields we develop a boundary constraint method for constructing the vortex surface field vsf in a three dimensional fluid velocityvorticity field the isosurface of the vsf is a vortex surface consisting of vortex lines which can be used to characterize the evolution of vortical structures in a lagrangian sense the evolution equation with pseudo time is solved under the vsf boundary constraint to obtain a numerical solution of the vsf compared with the existing two time method the boundary constraint method constructs the vsf from a single velocity dataset at a given time instead of a time series of velocity fields starting from a simple condition this improvement significantly increases the applicability of the vsf method and reduces the demanding computational cost and required velocity data size using the boundary constraint method we construct the vsfs in taylorgreen flow and transitional channel flow in addition the uniqueness of the vsf solution is discussed and the convergence of the error in the calculation of vsfs is analyzed

on the global structure of invariant regions of flows with asymptotically stable attractors several authors have pointed out the usefulness of shape theory which was introduced by k borsuk in 1968 as a tool in the study of dynamical systems in particular bogatyi and gutsu 4 garay 10 gunther and segal 11 hastings 12 robbin and salamon 16 rogers 17 and tezer 24 have obtained interesting results in dynamical systems using shape theoretical techniques one of the authors of the present article has given another connection between shape and topological dynamics 20 based on the approach to shape developed in 18 and 19 see also 21 for applications of shape theory to the study of uniform attractors most of the mentioned papers study properties of flows although 16 and 24 are devoted to discrete dynamical systems in particular a shape conley index is constructed in 16 for discrete dynamical systems and some shape theoretical ideas are used in 24 to study properties of topological conjugacy of shifts and homotopical shift equivalence for maps of polyhedra in the papers 4 10 11 12 20 21 some topological properties of the attractors of flows are studied for instance hastings established in 12 a higher dimensional poincare bendixson theorem proving the existence of attractors of certain flows in the interior of submanifolds of the euclidean space and comparing the shape of the attractor with that of the manifold other properties studied in the above papers are related to strong cellularity of the global attractors 10 and polyhedral shape of asymptotically stable compacta 4 11 20 a remarkable fact proved by

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

a unified approach to computing error probabilities of diversity combining schemes over correlated fading channels the average bit error rate performance of one stage and two stage diversity combining schemes operating over correlated fading channels is investigated two channel models that can significantly simplify the performance analysis are considered in particular a linear correlation channel model having equal branch variances can be decorrelated at the receiver so that the branches become independent it is shown that in general employing diversity combining schemes for decorrelated or orthogonalized branches can recover some of the diversity gain lost due to the branch correlations this is observed for example for the case of hybrid selection maximum ratio combining operating over decorrelated and orthogonalized non zero mean gaussian fading channels furthermore a fading amplitude channel model is proposed assuming vector norm superposition of the impinging plane waves this channel model is well suited for the performance analysis of maximum ratio and equal gain combining schemes operating over correlated fading channels finally the average bit error rates of several diversity combining schemes are evaluated analytically using the prony approximation method as well as using computer simulation

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

intertwined basins of attraction generated by the stable manifold of a saddle point this article deals with the intertwined basins of attraction we give a necessary and sufficient condition for the existence of intertwined basins of attraction

the dimension of chaotic attractors dimension is perhaps the most basic property of an attractor in this paper we discuss a variety of different definitions of dimension compute their values for a typical example and review previous work on the dimension of chaotic attractors the relevant definitions of dimension are of two general types those that depend only on metric properties and those that depend on the frequency with which a typical trajectory visits different regions of the attractor both our example and the previous work that we review support the conclusion that all of the frequency dependent dimensions take on the same value which we call the dimension of the natural measure and all of the metric dimensions take on a common value which we call the fractal dimension furthermore the dimension of the natural measure is typically equal to the lyapunov dimension which is defined in terms of lyapunov numbers and thus is usually far easier to calculate than any other definition because it is computable and more physically relevant we feel that the dimension of the natural measure is more important than the fractal dimension

on the intertwined basins of attraction for planar flows in this paper we discuss the intertwined basins of attraction one of the simplest cases of obstruction to predictability some sufficient conditions to guarantee the existence of intertwined basins of attraction are given

on the sum of kappa stochastic variates and applications to equal gain combining in this paper we study the statistics of the sum of not necessarily identically distributed kappa that is k random variables rv s assuming half integer values for the shaping parameters novel closed form expressions for the probability density function pdf of the sum of independent k rvs are obtained while for arbitrary values of the shaping parameters a corresponding pdf expression is derived in terms of fast converging infinite series furthermore an infinite series representation for the pdf of the sum of two arbitrarily correlated k rvs is derived the proposed analysis is employed to the performance analysis of equal gain combining egc receivers operating over composite fading shadowing channels modeled by the k distribution more specifically the outage and the average bit error probabilities as well as the average channel capacity of egc receivers operating over such composite environment are studied considering different channel fading shadowing conditions and correlation effects various numerical performance evaluation results are presented these results are complemented by equivalent computer simulated ones that validate the accuracy of the proposed analysis

performance analysis of mimo free space optical systems in gamma gamma fading atmospheric turbulence induced fading is one of the main impairments affecting free space optics fso communications in recent years gamma gamma fading has become the dominant fading model for fso links because of its excellent agreement with measurement data for a wide range of turbulence conditions however in contrast to rf communications the analysis techniques for fso are not well developed and prior work has mostly resorted to simulations and numerical integration for performance evaluation in gamma gamma fading in this paper we express the pairwise error probabilities of single input single output siso and multiple input multiple output mimo fso systems with intensity modulation and direct detection im dd as generalized infinite power series with respect to the signal to noise ratio for numerical evaluation these power series are truncated to a finite number of terms and an upper bound for the associated approximation error is provided the resulting finite power series enables fast and accurate numerical evaluation of the bit error rate of im dd fso with on off keying and pulse position modulation in siso and mimo gamma gamma fading channels furthermore we extend the well known rf concepts of diversity and combining gain to fso and gamma gamma fading in particular we provide simple closed form expressions for the diversity gain and the combining gain of mimo fso with repetition coding across lasers at the transmitter and equal gain combining or maximal ratio combining at the receiver

role of invariant and minimal absorbing areas in chaos synchronization in this paper the method of critical curves a tool for the study of the global dynamical properties of two dimensional noninvertible maps is applied to the study of chaos synchronization and related phenomena of riddling blowout and on off intermittency a general procedure is suggested in order to obtain the boundary of a particular two dimensional compact trapping region called absorbing area containing the onedimensional chaotic set on which synchronized dynamics occur the main purpose of the paper is to show that only invariant and minimal absorbing areas are useful to characterize the global dynamical behavior of the dynamical system when a milnor attractor with locally riddled basin or a chaotic saddle exists and may strongly influence the effects of riddling and blowout bifurcations some examples are given for a system of two coupled logistic maps and some practical methods and numerical tricks are suggested in order to ascertain the properties of invariance and minimality of an absorbing area some general considerations are given concerning the transition from locally riddled to globally riddled basins and the role of the absorbing area in the occurrence of such transition is discussed s1063 651x 98 11611 2

prey predator systems with intertwined basins of attraction in this article we show the existence of intertwined basins of attraction for a class of prey predator systems which improves the extant results by deleting some crucial and hard testing conditions

small random perturbations of dynamical systems and the definition of attractors the strange attractors plotted by computers and seen in physical experiments do not necessarily have an open basin of attraction in view of this we study a new definition of attractors based on ideas of conley we argue that the attractors observed in the presence of small random perturbations correspond to this new definition

basins of wada abstract we describe situations in which there are several regions more than two with the wada property namely that each point that is on the boundary of one region is on the boundary of all we argue that such situations arise even in studies of the forced damped pendulum where it is possible to have three attractor regions coexisting and the three basins of attraction have the wada property

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

some properties of the sobolev laguerre polynomials abstract in this paper we study the case 0 of the sobolev laguerre polynomials we determine a generating function for the polynomials and an expansion formula

pollaczek polynomials and pade approximants some closed form expressions recently some work has been devoted to the task of obtaining closed form expressions for important classes of orthogonal polynomials this has been done successfully for the associated jacobi polynomials and their special cases including the associated laguerre and hermite polynomials such expressions allow one to write closed form expressions for the n 1 n pade approximants to important transcendental functions for instance expressions for the associated jacobi polynomials provide in closed form the truncates of gauss continued fraction in this paper we continue this work by developing closed form expressions for the pollaczek polynomials pn x a b c these expressions involve a finite number of terms each algebraic in character this is to be contr with the traditional representation which is a ratio of cross products of gaussian hypergeometric functions an expression not algebraic in the variable and not identifiably a polynomial we examine some special cases of this formula including the associated ultraspherical polynomials and the chihara ismail polynomials finally we determine the n 1 n pade approximants to a certain ratio of gaussian hypergeometric functions

nonlinear dynamics of networks the groupoid formalism a formal theory of symmetries of networks of coupled dynamical systems stated in terms of the group of permutations of the nodes that preserve the network topology has existed for some time global network symmetries impose strong constraints on the corresponding dynamical systems which affect equilibria periodic states heteroclinic cycles and even chaotic states in particular the symmetries of the network can lead to synchrony phase relations resonances and synchronous or cycling chaos symmetry is a rather restrictive assumption and a general theory of networks should be more flexible a recent generalization of the group theoretic notion of symmetry replaces global symmetries by bijections between certain subsets of the directed edges of the network the input sets now the symmetry group becomes a groupoid which is an algebraic structure that resembles a group except that the product of two elements may not be defined the groupoid formalism makes it possible to extend group theoretic methods to more general networks and in particular it leads to a complete classification of robust patterns of synchrony in terms of the combinatorial structure of the network many phenomena that would be nongeneric in an arbitrary dynamical system can become generic when constrained by a particular network topology a network of dynamical systems is not just a dynamical system with a high dimensional phase space it is also equipped with a canonical set of observables the states of the individual nodes of the network moreover the form of the underlying ode is constrained by the network topology which variables occur in which component equations and how those equations relate to each other the result is a rich and new range of phenomena only a few of which are yet properly understood

applying hankel s expansion for performance analysis in double nakagami generalized k fading channels the double scattering fading models have found applications in many transmission scenarios in this paper approximate probability density functions pdfs are proposed for double nakagami fading and gamma gamma fading based on hankel s expansion of the modified bessel function of the second kind the proposed pdf is not only consistent with the exact pdf when the difference between the two shape parameters is equal to an integer plus one half but also accurately approximates the distribution when the shape parameters are both greater than or equal to 2 the proposed approximate pdf has been compared with the power series expansion and the mixture gamma model it is found that the proposed pdf is in simple form and no parameters need to be tabulated moreover the proposed pdf does not require an increasing number of terms to maintain precision with increasing diversity order of the reception scheme closed form outage probability ergodic capacity and symbol error rate expressions have been derived using the proposed approximate pdfs for equal gain combining selection combining and two stage reception schemes the proposed approximations are also leveraged for performance analysis of space time block codes and free space optical links the precision of the proposed approximations is validated by comparing with the simulation results

laguerre sobolev orthogonal polynomials abstract in this paper polynomials that are orthogonal with respect to the inner product f g s 0 f x g x x e x dx 0 f x g x x e x dx where 1 and 0 are studied for these nonstandard orthogonal polynomials algebraic and differential properties as well as the relation with the classical laguerre polynomials are obtained finally some properties concerning the localization and separation of the zeros of these polynomials are deduced

n ast nakagami a novel stochastic model for cascaded fading channels a generic and novel distribution referred to as nakagami constructed as the product of n statistically independent but not necessarily identically distributed nakagami m random variables rvs is introduced and analyzed the proposed distribution turns out to be a very convenient tool for modelling cascaded nakagami m fading channels and analyzing the performance of digital communications systems operating over such channels the moments generating probability density cumulative distribution and moments functions of the n nakagami distribution are developed in closed form using the meijer s g function using these formulas generic closed form expressions for the outage probability amount of fading and average error probabilities for several binary and multilevel modulation signals of digital communication systems operating over the n nakagami fading and the additive white gaussian noise channel are presented complementary numerical and computer simulation performance evaluation results verify the correctness of the proposed formulation the suitability of the n nakagami fading distribution to approximate the lognormal distribution is also being investigated using kolmogorov smirnov tests the rate of convergence of the central limit theorem as pertaining to the multiplication of nakagami m rvs is quantified

the dynamical systems approach to differential equations bulletin new series of the american mathematical society volume 11 number 1 july 1984 the dynamical systems approach to differential equations by morris w hirsch 1 this harmony that human intelligence believes it discovers in nature does it exist apart from that intelligence no without doubt a reality completely independent of the spirit which conceives it sees it or feels it is an impossibility a world so exterior as that even if it existed would be forever inaccessible to us but what we call objective reality is in the last analysis that which is common to several thinking beings and could be common to all this common part we will see can be nothing but the harmony expressed by mathematical laws h poincare la valeur de la science p 9 ignorance of the roots of the subject has its price no one denies that modern formulations are clear elegant and precise it s just that it s impossible to comprehend how any one ever thought of them m spivak a comprehensive introduction to differential geometry introduction the mathematical subject we call dynamical systems was fathered by poin care developed sturdily under birkhoff and has enjoyed a vigorous new growth for the last twenty years as i try to show in chapter i it is interesting to look at this mathematical development as the natural outcome of a much broader theme which is as old as science itself the universe is a sytem which changes in time every mathematician has a particular way of thinking about mathematics but rarely makes it explicit yet such perspectives in any particular field can be of great value to nonexperts who must apply the results of the field or who want to learn them in some fields the points of view of its practitioners cluster around several comparatively clearly defined concepts and insights dynamical systems theory is one of them the basic concepts go back to poincare the main thrust of chapters i and ii is to explain rather discursively some of received by the editors december 14 1983 1980 mathematics subject classification primary 00a25 00a99 01a45 01a55 01a60 34a40 34c35 34d10 34d30 35b05 35b35 35b40 35b50 35b30 35k55 54h20 58f25 58f40 58f12 58f10 58d25 58d07 46a40 46e10 47h07 47h20 92a15 92a17 secondary 06f30 35b65 58f13 35j60 46e05 92 03 90 03 90a16 supported in part by national science foundation and institute for advanced study 1984 american mathematical society 0273 0979 84 1 00 25 per page

closed form statistics for the sum of squared nakagami m variates and its applications we present closed form expressions for the probability density function pdf and the cumulative distribution function cdf of the sum of non identical squared nakagami m random variables rvs with integer order fading parameters as it is shown they can be written as a weighted sum of erlang pdfs and cdfs respectively while the analysis includes both independent and correlated sums of rvs the proposed formulation significantly improves previously published results which are either in the form of infinite sums or higher order derivatives of the fading parameter m the obtained formulas can be applied to the performance analysis of diversity combining receivers operating over nakagami m fading channels

some remarks on a paper by l carlitz we study a family of orthogonal polynomials which generalizes a sequence of polynomials considered by l carlitz we show that they are a special case of the sheffer polynomials and point out some interesting connections with certain sobolev orthogonal polynomials

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

basin boundary metamorphoses changes in accessible boundary orbits abstract basin boundaries sometimes undergo sudden metamophoses these metamorphoses can lead to the conversion of a smooth basin boundary to one which is fractal or else can cause a fractal or else can basin boundary to suddenly jump in size and change its character although remaining fractal for an invertible map in the plane there may be an infinite number of saddle periodic orbits in a basin boundary that is fractal nonetheless we have found that typically only one of them can be reached or accessed directly from a given basin the other periodic orbits are buried beneath infinitely many layers of the fractal structure of the boundary the boundary metamorphoses which we investigate are characterized by a sudden replacement of the basin boundary s accessible orbit

renormalization in the henon family ii the heteroclinic web we study highly dissipative henon maps f c b x y mapsto c x 2 by x with zero entropy they form a region in the parameter plane bounded on the left by the curve w of infinitely renormalizable maps we prove that morse smale maps are dense in but there exist infinitely many different topological types of such maps even away from w we also prove that in the infinitely renormalizable case the average jacobian bf on the attracting cantor set mathcal o f is a topological invariant these results come from the analysis of the heteroclinic web of the saddle periodic points based on the renormalization theory along these lines we show that the unstable manifolds of the periodic points form a lamination outside mathcal o f if and only if there are no heteroclinic tangencies

adaptive classification of temporal signals in fixed weight recurrent neural networks an existence proof recurrent neural networks with fixed weights have been shown in practice to successfully classify adaptively signals that vary as a function of time in the presence of additive noise and parametric perturbations we address the question can this ability be explained theoretically we provide a mathematical proof that these networks have this ability even when parametric perturbations enter the signals nonlinearly the restrictions that we impose on the signals to be classified are that they satisfy an assumption of nondegeneracy and that noise amplitude is sufficiently small further we demonstrate that the recurrent neural networks may not only classify uncertain signals adaptively but also can recover the values of uncertain parameters of the signals up to their equivalence classes