integrability analyticity isochrony equilibria small oscillations and diophantine relations results from the stationary korteweg de vries hierarchy the isochronous variant is exhibited of the dynamical system corresponding to the mth ordinary differential equation of the stationary korteweg de vries kdv hierarchy new diophantine relations are thereby obtained in the guise of matrices of arbitrary order having integer eigenvalues or equivalently of polynomials of arbitrary degree having integer zeros generalizations of these formulas to relations among rational functions are also obtained the basic idea to arrive at such relations is not new but the specific application reported in this paper is new and it is likely to open the way to several analogous new findings

properties of the zeros of generalized basic hypergeometric polynomials we define the generalized basic hypergeometric polynomial of degree n in terms of the generalized basic hypergeometric function by choosing one of its parameters to allow the termination of the series after a finite number of summands in this paper we obtain a set of nonlinear algebraic equations satisfied by the n zeros of the polynomial moreover we obtain an n n matrix m defined in terms of the zeros of the polynomial which in turn depend on the parameters of the polynomial the eigenvalues of this remarkable matrix m are given by neat expressions that depend only on some of the parameters of the polynomial that is the matrix m is isospectral moreover in case the parameters that appear in the expressions for the eigenvalues of m are rational the matrix m has rational eigenvalues a diophantine property

correspondence comments on identification of hammerstein nonlinear armax systems this note points out some errors in the proof of the main result of ding and chen 2005 identification of hammerstein nonlinear armax systems automatica 41 9 1479 1489 and mentions the well known results that have solved the problem discussed in theorem 1 of the above paper by ding and chen

connection between dispersive transport and statistics of extreme events a length dependence of the effective mobility in the form of a power law b l1 1 is observed in dispersive transport in amorphous substances with 0 1 we deduce this behavior as a simple consequence of the statistical theory of extreme events we derive various quantities related to the largest value in samples of n trials for the exponential and power law probability densities of the individual events

chaos order statistics and unstable periodic orbits we present an interesting connection between order statistics and unstable periodic orbits of chaotic maps this can be used to locate all the unstable periodic points of a given order of one dimensional chaotic maps with continuous invariant density the densities of the ordered variates of the iterates are discontinuous exactly at unstable periodic points of the map this is illustrated using the logistic map where densities corresponding to a small number of iterates have been obtained in closed form this scheme can also be applied to a class of continuous time systems where the successive maxima of the time series behave as if they were generated from a unimodal map we demonstrate this by using the lorenz model

extreme value statistics of 2d chaotic systems we study the extreme value statistics of 2d chaotic systems we consider the extreme value of n phase space points of a 2d chaotic trajectory under a suitably defined norm to order the points and calculate analytically its density we find that the extreme value density is non differentiable on a set of points and the number of such singular points increases with n however for identically distributed independent 2d random variables the number of singular points is independent of n

a technique to identify solvable dynamical systems and a solvable generalization of the goldfish many body problem a simple approach is discussed which associates to solvable matrix equations solvable dynamical systems generally interpretable as interesting many body problems possibly involving auxiliary dependent variables in addition to those identifying the positions of the moving particles we then focus on cases in which the auxiliary variables can be altogether eliminated reobtaining thereby via this unified approach well known solvable many body problems and moreover a solvable extension of the goldfish model

integrability analyticity isochrony equilibria small oscillations and diophantine relations results from the stationary burgers hierarchy an isochronous system is introduced by modifying the nth ode of the stationary burgers hierarchy and then by investigating its behaviour near its equilibria neat diophantine relations are identified involving well known polynomials of arbitrary degree having integer zeros or equivalently matrices the determinants of which yield such polynomials the basic idea to arrive at such relations is not new but the specific application reported in this paper is new and it is likely to open the way to several analogous new findings

invariant measurement of strange sets in terms of cycles we argue that extraction of unstable cycles and their eigenvalues is not only experimentally feasible but is also a theoretically optimal measurement of the invariant properties of a dynamical system

a linear second order ode with only polynomial solutions abstract a linear second order ode featuring an arbitrary number of free parameters is identified all solutions of which are polynomials

can the general solution of the second order ode characterizing jacobi polynomials be polynomial the answer to the question posed in the title is affirmative indeed while the second order ode 1 x2 p 2 x p n n 1 p 0 for generic values of the two parameters and and n a non negative integer identifies uniquely up to an overall multiplicative constant characterizing its normalization the jacobi polynomial p n x as its polynomial solution if instead n m and n n with m and n being two non negative integers satisfying the restriction m n n then the general solution of this ode is a polynomial of degree n depending in addition of course to the three parameters n m and n on two arbitrary parameters i e on an additional one besides the overall multiplicative parameter characterizing its normalization

extracting unstable periodic orbits from chaotic time series data a general nonlinear method to extract unstable periodic orbits from chaotic time series is proposed by utilizing the estimated local dynamics along a trajectory we devise a transformation of the time series data such that the transformed data are concentrated on the periodic orbits thus one can extract unstable periodic orbits from a chaotic time series by simply looking for peaks in a finite grid approximation of the distribution function of the transformed data our method is demonstrated using data from both numerical and experimental examples including neuronal ensemble data from mammalian brain slices the statistical significance of the results in the presence of noise is assessed using surrogate data s1063 651x 97 15205 9

parametric identification of hammerstein systems with consistency results using stochastic inputs the extended least squares algorithm is applied to identify the hammerstein system where the nonlinear static function f is expressed as a linear combination of basic functions with unknown coefficients strong consistency of the estimates is established and their convergent rates are obtained as well

extreme value distributions in chaotic dynamics a theory of extremes is developed for chaotic dynamical systems and illustrated on representative models of fully developed chaos and intermitent chaos the cumulative distribution and its associated density for the largest value occurring in a data set for monotonically increasing or decreasing sequences and for local maxima are evaluated both analytically and numerically substantial differences from the classical statistical theory of extremes are found arising from the deterministic origin of the underlying dynamics

diophantine properties of the zeros of certain laguerre and para jacobi polynomials matrices and of arbitrary rank n given by simple expressions in terms of the n zeros of certain laguerre or para jacobi polynomials of degree n feature a diophantine property in the laguerre case this property states that the 2n zeros of the polynomial are all integers indeed we conjecture that the results in the para jacobi case are somewhat analogous they refer to the zeros of the general solution of the ode generally characterizing jacobi polynomials in the special case in which this solution is a para jacobi polynomial featuring an additional arbitrary parameter

properties of the zeros of the polynomials belonging to the q askey scheme in this paper we provide properties which are to the best of our knowledge new of the zeros of the polynomials belonging to the q askey scheme these findings include diophantine relations satisfied by these zeros when the parameters characterizing these polynomials are appropriately restricted

external theory for stochastic processes abstract the purpose of this paper is to provide an overview of the asymptotic distributional theory of extreme values for a wide class of dependent stochastic sequences and continuous parameter processes the theory contains the standard classical extreme value results for maxima and extreme order statistics as special cases but is richer on account of the diverse behavior possible under dependence in both discrete and continuous time contexts emphasis is placed on stationary cases but other important classes e g mark of sequences are included significant ideas and methods are described rather than details and in particular the nature and role of important underlying point processes such as exceedances and upcrossings are emphasized applications are given to particular classes of process e g normal moving average and connections with related theory such as convergence of sums are indicated

finding periodic points from short time series we present an algorithm for finding low order periodic points of chaotic maps from possibly very short time series no information about the map other than the time series is used the method finds all the periodic points of a piecewise linear approximation of the map we present examples showing the effectiveness of the method for the henon and ikeda maps and a chaotic electronic circuit including a cycle expansion calculation of the hausdorff dimension for the henon map

finite dimensional representations of difference operators and the identification of remarkable matrices two square matrices of arbitrary order n are introduced they are defined in terms of n arbitrary numbers zn and of an arbitrary additional parameter a respectively q and provide finite dimensional representations of the two operators acting on a function f z as follows f z a f z a respectively f qz f z q 1 z these representations are exact in a sense explained in the paper when the function f z is a polynomial in z of degree less than n this formalism allows to transform difference equations valid in the space of polynomials of degree less than n into corresponding matrix vector equations as an application of this technique several remarkable square matrices of order n are identified which feature explicitly n arbitrary numbers zn or the n zeros of polynomials belonging to the askey and q askey schemes several of these findings have a diophantine character

an isochronous variant of the ruijsenaars toda model equilibrium configurations behavior in their neighborhood diophantine relations an isochronous variant of the ruijsenaars toda integrable many body problem is introduced an equilibrium configuration of this dynamical system is identified and by investigating the motions in its neighborhood diophantine relations are obtained

topological entropy of one dimensional maps approximations and bounds we present a method for computing the topological entropy of one dimensional maps as an approximation scheme the algorithm converges rapidly and provides both upper and lower bounds

novel solvable variants of the goldfish many body model a recent technique to identify solvable many body problems in two dimensional space yields via a new twist new many body problems of goldfish type some of these models are isochronous namely their generic solutions are completely periodic with a fixed period independent of the initial data the investigation of the behavior of some of these isochronous systems in the vicinity of their equilibrium configurations yields some amusing diophantine relations

identification of hammerstein nonlinear armax systems two identification algorithms an iterative least squares and a recursive least squares are developed for hammerstein nonlinear systems with memoryless nonlinear blocks and linear dynamical blocks described by armax carma models the basic idea is to replace unmeasurable noise terms in the information vectors by their estimates and to compute the noise estimates based on the obtained parameter estimates convergence properties of the recursive algorithm in the stochastic framework show that the parameter estimation error consistently converges to zero under the generalized persistent excitation condition the simulation results validate the algorithms proposed