variable delay feedback control of unstable steady states in retarded time delayed systems we study the stability of unstable steady states in scalar retarded time delayed systems subjected to a variable delay feedback control the important aspect of such a control problem is that time delayed systems are already infinite dimensional before the delayed feedback control is turned on when the frequency of the modulation is large compared to the system s dynamics the analytic approach consists of relating the stability properties of the resulting variable delay system with those of an analogous distributed delay system otherwise the stability domains are obtained by a numerical integration of the linearized variable delay system the analysis shows that the control domains are significantly larger than those in the usual time delayed feedback control and that the complexity of the domain structure depends on the form and the frequency of the delay modulation

transition between kelvin s equilibria a transition between kelvin s equilibrium states is investigated using nonlinear theory we have shown that the transition of polygonal patterns of the hollow vortex core from mode n 2 through n 4 occurs in two steps quasiperiodicity and frequency locking we have also shown that this transition can be modeled by a one dimensional circle map we extrapolate the present result and hypothesize that the transition between kelvin s equilibria follows the same route and the ratios of locking frequencies form a farey sum and staircase function against the control parameter where the staircase corresponds to the rational frequency ratio n ensuremath 1 n

spectal dimension of fractal sets we consider an optimal partial covering of fractal sets in a two dimensional space using ellipses which become increasingly anisotropic as their size is reduced if the semi minor axis isand the semi major axis is w e set where 0 1 is an exponent characterizing the anisotropy of the covers the optimization involves varying the angle of the principal axis to maximize the measure covered by each ellipse for point set fractals in most cases we find that the number of points n which can be covered by an ellipse centred on any given point has expectation valuen where is a generalized dimension we term the spectal dimension because our covering strategy may be used to characterize specular light scattering from fractal sets we investigate the function numerically for various sets showing that it may be different for sets which have the same fractal dimension

long range order in the scaling behaviour of hyperbolic dynamical systems abstract detailed statistical analyses of chaotic signals have been so far limited to 2 d maps with one expanding and one contracting direction here by studying filtered signals we provide evidence that the existence of a further stable direction can transform self similarity into self affinity and generate phase transitions according to the thermodynamic formalism of dynamical systems in a hyperbolic case the dimension spectrum is computed exactly as a function of the filter cut off frequency moreover the validity of the kaplan yorke formula and of its generalization are rigorously proven finally the scaling function is shown to satisfy a stochastic recursive relation whose lyapunov exponent allows to control the extension of the range of interactions in the associated hamiltonian

the entropy function for characteristic exponents using a thermodynamic formalism we define an entropy function s a which measures large deviations of the liapunov characteristic exponents of certain hyperbolic dynamical systems the function s a is a legendre transform of a free energy or pressure associated with the dynamical system we show that s a is the noncompact topological entropy of the set of points a with characteristic exponent a that s a a is the hausdorff dimension of a and that a s a is the escape rate from a we explain how to use the formalism for cookie cutters to describe the distribution of scales in the universal period doubled attractor and critical golden circle mapping we relate s a to the renyi entropies prove a conjecture of kantz and grassberger relating the escape rate from hyperbolic repellors and saddles to the characteristic exponents and information dimension and study the fluctuations of escape rates we discuss the escape rate from f x 4 e x 1 x and the behaviour of the associated s a as e 0 finally we discuss how to apply these ideas to experimental time series and non hyperbolic attractors

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

simultaneous camera pose and correspondence estimation with motion coherence traditionally the camera pose recovery problem has been formulated as one of estimating the optimal camera pose given a set of point correspondences this critically depends on the accuracy of the point correspondences and would have problems in dealing with ambiguous features such as edge contours and high visual clutter joint estimation of camera pose and correspondence attempts to improve performance by explicitly acknowledging the chicken and egg nature of the pose and correspondence problem however such joint approaches for the two view problem are still few and even then they face problems when scenes contain largely edge cues with few corners due to the fact that epipolar geometry only provides a soft point to line constraint viewed from the perspective of point set registration the point matching process can be regarded as the registration of points while preserving their relative positions i e preserving scene coherence by demanding that the point set should be transformed coherently across views this framework leverages on higher level perceptual information such as the shape of the contour while thus potentially allowing registration of non unique edge points the registration framework in its traditional form is subject to substantial point localization error and is thus not suitable for estimating camera pose in this paper we introduce an algorithm which jointly estimates camera pose and correspondence within a point set registration framework based on motion coherence with the camera pose helping to localize the edge registration while the ambiguous edge information helps to guide camera pose computation the algorithm can compute camera pose over large displacements and by utilizing the non unique edge points can recover camera pose from what were previously regarded as feature impoverished sfm scenes our algorithm is also sufficiently flexible to incorporate high dimensional feature descriptors and works well on traditional sfm scenes with adequate numbers of unique corners

generalised composition law for symbolic itineraries a generalised composition law is proposed which includes the composition law as a special case the composition law preserves the order of the itineraries and their maximality as an application the symbolic dynamics of the intermittency is briefly sketched other possible generalisations of the composition law are mentioned

growth and integrability in the dynamics of mappings the growth of some numerical characteristics of the mappings under their iterations in the context of the general problem of integrability is discussed in the general case such characteristics as complexity by arnold or the number of the different images for the multiple valued mappings are growing exponentially it is shown that the integrability is closely related with thepolynomial growth the analogies with quantum integrable systems are discussed

dynamics of bimeromorphic maps of surfaces we classify bimeromorphic self maps f x inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i of compact kahler surfaces x in terms of their actions f h 1 1 x inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i on cohomology we observe that the growth rate of f n is invariant under bimeromorphic conjugacy and that by conjugating one can always arrange that f n f n we show that the sequence f n can be bounded grow linearly grow quadratically or grow exponentially in the first three cases we show that after conjugating f is an automorphism virtually isotopic to the identity f preserves a rational fibration or f preserves an elliptic fibration respectively in the last case we show that there is a unique up to scaling expanding eigenvector for f that is nef and that f is bimeromorphically conjugate to an automorphism if and only if 2 0 we go on in this case to construct a dynamically natural positive current representing and we study the growth rate of periodic orbits of f we conclude by illustrating our results with a particular family of examples

variational recursive joint estimation of dense scene structure and camera motion from monocular high speed traffic sequences we present an approach to jointly estimating camera motion and dense structure of a static scene in terms of depth maps from monocular image sequences in driver assistance scenarios at each instant of time only two consecutive frames are processed as input data of a joint estimator that fully exploits second order information of the corresponding optimization problem and effectively copes with the non convexity due to both the imaging geometry and the manifold of motion parameters additionally carefully designed gaussian approximations enable probabilistic inference based on locally varying confidence and globally varying sensitivity due to the epipolar geometry with respect to the high dimensional depth map estimation embedding the resulting joint estimator in an online recursive framework achieves a pronounced spatio temporal filtering effect and robustness we evaluate hundreds of images taken from a car moving at speed up to 100 km h and being part of a publicly available benchmark data set the results compare favorably with two alternative settings stereo based scene reconstruction and camera motion estimation in batch mode using multiple frames they however require a calibrated camera pair or storage for more than two frames which is less attractive from a technical viewpoint than the proposed monocular and recursive approach in addition to real data a synthetic sequence is considered which provides reliable ground truth

lattice valued convergence spaces and regularity we define a regularity axiom for lattice valued convergence spaces where the lattice is a complete heyting algebra to this end we generalize the characterization of regularity by a dual form of a diagonal condition we show that our axiom ensures that a regular t1 space is separated and that regularity is preserved under initial constructions further we present an extension theorem for a continuous mapping from a subspace to a regular space a characterization in the restricted case that the lattice is a complete boolean algebra in terms of the closure of an l filter is given

multifractality and the kauffman model the author simulates the spread of an initial damage in the kauffman cellular automata at its critical point on the square lattice he calculates various moments of the probability that a site has been damaged n times and checks for multifractality in the critical exponents for different ensembles specifically he finds no evidence for multifractality when the moments of the probabilities are evaluated with lattice size l but multifractal behaviour occurs when the moments are monitored as a function of time t

diagonal conditions for lattice valued uniform convergence spaces we define diagonal conditions for lattice valued uniform convergence spaces and show that these conditions are preserved under initial constructions further the forgetful functor from the category of stratified lattice valued uniform convergence spaces into the category of stratified lattice valued limit spaces maps these conditions to corresponding conditions we especially define uniform regularity and characterize this condition by certain closures of lattice valued filters as an application we generalize an extension theorem for uniformly continuous mappings

five turbulent problems abstract five questions which involve strange attractors in one way or another are discussed these are the choice of probability measures to describe turbulence the hausdorff dimension of julia sets for rational maps of the riemann sphere the existence of nonperiodic turbulent crystal structure the mechanism generating intermittency in developed hydrodynamic turbulence the distribution of characteristic exponents in a turbulent fluid

statistics of finite time lyapunov exponents in a random time dependent potential the sensitivity of trajectories over finite time intervals t to perturbations of the initial conditions can be associated with a finite time lyapunov exponent lambda obtained from the elements m ij of the stability matrix m for globally chaotic dynamics lambda tends to a unique value the usual lyapunov exponent lambda infinity for almost all trajectories as t is sent to infinity but for finite t it depends on the initial conditions of the trajectory and can be considered as a statistical quantity we compute for a particle moving in a randomly time dependent one dimensional potential how the distribution function p lambda t approaches the limiting distribution p lambda infinity delta lambda lambda infinity our method also applies to the tail of the distribution which determines the growth rates of moments of m ij the results are also applicable to the problem of wave function localization in a disordered one dimensional potential

discrete versions of some classical integrable systems and factorization of matrix polynomials discrete versions of several classical integrable systems are investigated such as a discrete analogue of the higher dimensional force free spinning top euler arnold equations the heisenberg chain with classical spins and a new discrete system on the stiefel manifold the integrability is shown with the help of a lax pair representation which is found via a factorization of certain matrix polynomials the complete description of the dynamics is given in terms of abelian functions the flow becomes linear on a prym variety corresponding to a spectral curve the approach is also applied to the billiard problem in the interior of ann dimensional ellipsoid

some relations between dimension and lyapounov exponents we consider differentiable maps and compact invariant sets we introduce dimensional quantities related to the ergodic invariant measures and prove some simple relations

nonlinear electronic circuit part i multiple routes to chaos abstract a nonlinear electronic oscillator suitable for synchronized chaotic communication is studied this circuit is capable of transmitting discrete chaotic signals although the chaotic modes of operation are controlled in an analog way in part i of this review paper the three routes to chaotic operation that appear namely the period doubling intermittency and crisis induced intermittency are thoroughly studied and discussed in all three routes to chaos the appropriate experimental distributions were calculated moreover the chaotic nature of the circuit operation was evaluated by using the grassberger procaccia method calculation of the corresponding minimum embedding dimension the kolmogorov sinai entropy as well as the maximal lyapunov exponent give useful information in order to fully characterize the circuit operation

the infinite number of generalized dimensions of fractals and strange attractors abstract we show that fractals in general and strange attractors in particular are characterized by an infinite number of generalized dimensions dq q 0 to this aim we develop a rescaling transformation group which yields analytic expressions for all the quantities dq we prove that lim q 0 dq fractal dimension d limq 1dq information dimension and dq 2 correlation exponent v dq with other integer q s correspond to exponents associated with ternary quaternary and higher correlation functions we prove that generally dq dq for any q q for homogeneous fractals dq dq a particularly interesting dimension is dq for two examples feigenbaum attractor generalized baker s transformation we calculate the generalized dimensions and find that d is a non trivial number all the other generalized dimensions are bounded between the fractal dimension and d

properties of border states of transient chaos transient chaos with absolutely continuous conditionally invariant measure is studied and a general solution of the frobenius perron equation is presented for complete one dimensional maps in particular properties of borderline situations exhibiting phase transitions within the framework of thermodynamical formalism are investigated

bifurcation and chaotic motion of a soliton in a long josephson junction oscillator the fluxon dynamics of a long josephson junction ljj is investigated numerically the effect of the surface resistance of the superconductor on the bifurcation phenomenon is examined in detail we show the occurrence of a period doubling route to chaos and the reappearance of low order periodic motions in the ljj by varying the magnitude of the external force the chaotic attractors of the ljj are characterized using maximal lyapunov exponents and a fractal correlation dimension we show the existence of noise induced oscillations in a force free ljj system finally the influence of noise on the periodic and chaotic dynamics is also presented

rationality for isolated expansive sets abstract an expansive isolated set of a diffeomorphism with nondegenerate periodic points has a rational zeta function

on the complexity of some birational transformations using three different approaches we analyse the complexity of various birational maps constructed from simple operations inversions on square matrices of arbitrary size the first approach comprises the study of the images of lines and relies mainly on univariate polynomial algebra the second approach is a singularity analysis and the third method is more numerical using integer arithmetics these three methods have their own domain of application but they give corroborating results and lead us to a conjecture on the algebraic entropy of a class of maps constructed from matrix inversions

the category of stratified l filter spaces abstract in this paper the concepts of stratified l filter space complete stratified l filter space and symmetric stratified l kent convergence space are introduced it is shown that 1 the category of stratified l filter spaces with cauchy continuous maps is a strong topological universe 2 the category of complete stratified l filter spaces as a bicoreflective subcategory of the category of stratified l filter spaces is isomorphic to the category of symmetric stratified l kent convergence spaces 3 the category of complete stratified l filter spaces as an isomorphism closed full subcategory of the category of stratified l filter spaces is strongly cartesian closed

real arnold complexity versus real topological entropy for birational transformations we consider a family of birational measure preserving transformations of two variables depending on one parameter for which simple rational expressions with integer coefficients for the exact expression of the dynamical zeta function have been conjectured together with an equality between the multiplicative rate of growth of the arnold complexity and the exponential of the topological entropy this identification takes place for the birational mapping seen as a mapping bearing on two complex variables acting in a complex projective space we revisit this identification between these two quite universal complexities by considering now the mapping as a mapping bearing on two real variables the definitions of the two previous topological complexities arnold complexity and topological entropy are modified according to this real variables point of view most of the universality is lost however the results presented here are again in agreement with an identification between the multiplicative rate of growth of some real arnold complexity and the exponential of some real topological entropy a detailed analysis of this real arnold complexity as a function of the parameter of this family of birational transformations of two variables is given one can also slightly modify the definition of the dynamical zeta function introducing a real dynamical zeta function associated with the counting of the real cycles only similarly one can also introduce some real arnold complexity generating functions we show that several of these two real generating functions seem to have the same singularities furthermore we actually conjecture several simple rational expressions for them yielding again algebraic values for the exponential of the real topological entropy in particular when the parameter of our family of birational transformations becomes large we obtain two interesting compatible nontrivial rational expressions these rational results for real mappings cannot be understood in terms of any obvious markov s partition or symbolic dynamics hyperbolic systems interpretation the birational transformation is far from being hyperbolic it is measure preserving it would be useful to know whether this relation between the arnold complexity and the topological entropy as well as the rationality of the degree generating functions and dynamical zeta functions are a consequence of the measure preserving property of the mapping

real arnold complexity versus real topological entropy for a one parameter dependent two dimensional birational transformation we consider a family of birational transformations of two variables depending on one parameter for which simple rational expressions with integer coefficients for the exact expression of the dynamical zeta function have been conjectured moreover an equality between the asymptotic of the arnold complexity and the exponential of the topological entropy has been conjectured this identification takes place for the birational mapping seen as a mapping bearing on two complex variables acting in a complex projective space we revisit this identification between these two quite universal complexities by considering now the mapping as a mapping bearing on two real variables the definitions of the two previous topological complexities arnold complexity and topological entropy are modified according to this real variables point of view most of the universality is lost however the results presented here are again in agreement with an identification between the asymptotic of the real arnold complexity and the exponential of the real topological entropy a detailed analysis of the real arnold complexity as a function of the parameter of this family of birational transformations of two variables is given

dimensions and entropies of strange attractors from a fluctuating dynamics approach abstract it is shown that the fluctuations in the divergence of near by trajectories on strictly deterministic strange attractors can be modelled by stochastic concepts in particular we propose kramers moyal type equations for correlation functions between points on the attractor the drift terms are the lyapunov exponents the diffusion terms depend on the above fluctuations from this we obtain bounds on generalized dimensions and entropies numerical results show that in nearly all studied cases henon map zaslavskii map mackey glass eq the attractors are fractal measures in the sense of farmer information dimension hausdorff dimension metric entropy topological entropy

completion of stratified l m filter tower spaces the notion of stratified l m filter tower spaces is introduced and the resulting category is shown to be a strong topological universe completions of stratified l m filter tower spaces are considered and a sufficient and necessary condition for a stratified l m filter tower space to have a completion is also given it is proved that the reflective modification of completion for a stratified l m cauchy tower space considered as a stratified l m filter tower space is still a completion for this stratified l m cauchy tower space

stereo processing by semiglobal matching and mutual information this paper describes the semiglobal matching sgm stereo method it uses a pixelwise mutual information ml based matching cost for compensating radiometric differences of input images pixelwise matching is supported by a smoothness constraint that is usually expressed as a global cost function sgm performs a fast approximation by pathwise optimizations from all directions the discussion also addresses occlusion detection subpixel refinement and multibaseline matching additionally postprocessing steps for removing outliers recovering from specific problems of structured environments and the interpolation of gaps are presented finally strategies for processing almost arbitrarily large images and fusion of disparity images using orthographic projection are proposed a comparison on standard stereo images shows that sgm is among the currently top ranked algorithms and is best if subpixel accuracy is considered the complexity is linear to the number of pixels and disparity range which results in a runtime of just 1 2 seconds on typical test images an in depth evaluation of the ml based matching cost demonstrates a tolerance against a wide range of radiometric transformations finally examples of reconstructions from huge aerial frame and pushbroom images demonstrate that the presented ideas are working well on practical problems

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

difficulties in detecting chaos in a complex system abstract the sequences given by a 7d map are analysed by means of the methods widely used to detect chaos in the real world in order to test their sensitivity to chaotic features of a non linear system determined by comparatively high number of parameters the same diagnostic approaches are applied to the 3d lorenz map for comparison the results show that for some of the sequences yielded from the 7d map the adopted methods are not able to give a straightforward answer to the question if the system is chaotic as in the 3d case since the sequences subject to the analysis are not contaminated by noise and are sufficiently long it could be assumed that such difficulties arise likely due to specific internal features of the more complex system it is found also that an increase from 0 01 to 0 5 of the sampling step determining the sequences obtained from the 7d map masks the chaos in some of them

singular value decomposition in attractor reconstruction pitfalls and precautions applicability of singular value decomposition for reconstructing the strange attractor from one dimensional chaotic time series proposed by broomhead and king is extensively tested and discussed previously published doubts about its reliability are confirmed singular value decomposition by nature a linear method can bring distorted and misleading results when nonlinear structures are studied

symmetry breaking bifurcation for coupled chaotic attractors the authors consider transitions from synchronous to asynchronous chaotic motion in two identical dissipatively coupled one dimensional mappings they show that the probability density of the asymmetric component satisfies a scaling law the exponent in this scaling law varies continuously with the distance from the bifurcation point and is determined by the spectrum of local lyapunov exponents of the uncoupled map finally they show that the topology of the invariant set is rather unusual though the attractor for supercritical coupling is a line it is surrounded by a strange invariant set which is dense in a two dimensional neighbourhood of the attractor

singularity complexity and quasi integrability of rational mappings we investigate global properties of the mappings entering the description of symmetries of integrable spin and vertex models by exploiting their nature of birational transformations of projective spaces we give an algorithmic analysis of the structure of invariants of such mappings we discuss some characteristic conditions for their quasi integrability and in particular its links with their singularities in the 2 plane finally we describe some of their propertiesqua dynamical systems making contact with arnol d s notion of complexity and exemplify remarkable behaviours

essential matrix estimation using gauss newton iterations on a manifold a novel approach for essential matrix estimation is presented this being a key task in stereo vision processing we estimate the essential matrix from point correspondences between a stereo image pair assuming that the internal camera parameters are known the set of essential matrices forms a smooth manifold and a suitable cost function can be defined on this manifold such that its minimum is the desired essential matrix we seek a computationally efficient optimization scheme towards meeting the demands of on line processing of video images our work extends and improves the earlier research by ma et al who proposed an intrinsic riemannian newton method for essential matrix computations in contrast to ma et al we propose three gauss newton type algorithms that have improved convergence properties and reduced computational cost the first one is based on a novel intrinsic newton method using the normal riemannian metric on the manifold consisting of all essential matrices the other two methods are newton like methods that are more efficient from a numerical point of view local quadratic convergence of the algorithms is shown based on a careful analysis of the underlying geometry of the problem

symmetry complexity and multicritical point of the two dimensional spin glass we analyse models of spin glasses on the two dimensional square lattice by exploiting symmetry arguments the replicated partition functions of the ising and related spin glasses are shown to have many remarkable symmetry properties as functions of the edge boltzmann factors it is shown that the applications of homogeneous and hadamard inverses to the edge boltzmann matrix indicate reduced complexities when the elements of the matrix satisfy certain conditions suggesting that the system has special simplicities under such conditions using these duality and symmetry arguments we present a conjecture on the exact location of the multicritical point in the phase diagram

post critical set and non existence of preserved meromorphic 2 forms we present a family of birational transformations in cp2 depending on two or three parameters which do not generically preserve meromorphic 2 forms with the introduction of the orbit of the critical set vanishing condition of the jacobian also called the post critical set we get some new structures some non analytic 2 form which reduce to meromorphic 2 forms for particular subvarieties in the parameter space on these subvarieties the iterates of the critical set have a polynomial growth in the degrees of the parameters while one has an exponential growth out of these subspaces the analysis of our birational transformation in cp2 is first carried out using the diller favre criterion in order to find the complexity reduction of the mapping the integrable cases are found the identification between the complexity growth and the topological entropy is once again verified we perform plots of the post critical set as well as calculations of lyapunov exponents for many orbits confirming that generically no meromorphic 2 form can be preserved for this mapping these birational transformations in cp2 which generically do not preserve any meromorphic 2 form are extremely similar to other birational transformations we previously studied which do preserve meromorphic 2 forms we note that these two sets of birational transformations exhibit totally similar results as far as topological complexity is concerned but drastically different results as far as a more probabilistic approach of dynamical systems is concerned lyapunov exponents with these examples we see that the existence of a preserved meromorphic 2 form explains most of the numerical discrepancies between the topological and probabilistic approaches of dynamical systems

extracting qualitative dynamics from experimental data abstract we consider the notion of qualitative information and the practicalities of extracting it from experimental data our approach based on a theorem of takens draws on ideas from the generalized theory of information known as singular system analysis due to bertero pike and co workers we illustrate our technique with numerical data from the chaotic regime of the lorenz model

integrability versus non integrability hard hexagons and hard squares compared in this paper we compare the integrable hard hexagon model with the nonintegrable hard squares model by means of partition function roots and transfer matrix eigenvalues we consider partition functions for toroidal cylindrical and free free boundary conditions up to sizes 40 40 and transfer matrices up to 30 sites for all boundary conditions the hard squares roots are seen to lie in a bounded area of the complex fugacity plane along with the universal hard core line segment on the negative real fugacity axis the density of roots on this line segment matches the derivative of the phase difference between the eigenvalues of largest and equal moduli and exhibits much greater structure than the corresponding density of hard hexagons we also study the special point z 1 of hard squares where all eigenvalues have unit modulus and we give several conjectures for the value at z 1 of the partition functions

real and complex dynamics of a family of birational maps of the plane the golden mean subshift we describe the real dynamics of a family of birational mappings of the plane by combining complex intersection theory and techniques from smooth dynamical systems we are able to give an essentially complete account of the behavior of both wandering and nonwandering orbits in particular the golden mean subshift provides a topological model for the dynamics on the nonwandering set while the mappings are not hyperbolic they are shown to possess many of the structures associated with hyperbolicity

largest and smallest t2 compactifications of lattice valued convergence spaces we give necessary and sufficient conditions for a lattice valued convergence space to possess a largest or a smallest t2 compactification

on diagonal completion of lattice valued diagonal cauchy spaces we define and study diagonal axioms for lattice valued cauchy spaces a completion of a weakly diagonal lattice valued cauchy space is constructed which is weakly diagonal and the coarsest among such completions it is at the same time at least as fine as any weakly regular completion

integrable mappings and polynomial growth abstract we describe birational representations of discrete groups generated by involutions having their origin in the theory of exactly solvable vertex models in lattice statistical mechanics these involutions correspond respectively to two kinds of transformations on q q matrices the inversion of the q q matrix and an involutive permutation of the entries of the matrix we concentrate on the case where these permutations are elementary transpositions of two entries in this case the birational transformations fall into six different classes for each class we analyze the factorization properties of the iteration of these transformations these factorization properties enable to define some canonical homogeneous polynomials associated with these factorization properties some mappings yield a polynomial growth of the complexity of the iterations for three classes the successive iterates for q 4 actually lie on elliptic curves this analysis also provides examples of integrable mappings in arbitrary dimension even infinite moreover for two classes the homogeneous polynomials are shown to satisfy non trivial non linear recurrences the relations between factorizations of the iterations the existence of recurrences on one or several variables as well as the integrability of the mappings are analyzed

rational dynamical zeta functions for birational transformations we propose a conjecture for the exact expression of the unweighted dynamical zeta function for a family of birational transformations of two variables depending on two parameters this conjectured function is a simple rational expression with integer coecients this yields an algebraic value for the topological entropy furthermore the generating function for the arnold complexity is also conjectured to be a rational expression with integer coecients with the same singularities as for the dynamical zeta function this leads at least in this example to an equality between the asymptotic of the arnold complexity and the exponential of the topological entropy we also give a semi numerical method to eectively compute the arnold complexity c 1999 published by elsevier science b v all rights reserved

a possible scheme for classifying phase transitions in the generalized entropy spectrum of a nonhyperbolic system abstract in this contribution the generalized thermodynamic formalism is applied to a nonhyperbolic dynamical system the system is used to discuss the occurence of combination of phase transitions among different entropy like scaling functions for the discussion the associated generalized entropy function is calculated from an approximation of finite level and for the asymptotic case from this unifying description of the system the information contained in the particular spectra as well as the relation among the different spectra can be derived and made visible in an easy way it is shown how the phase transition like behavior leads to the manifestation of a critical line in the surface described by the generalized entropy function which in turn determines the occurence of phase transitions for the particular spectra furthermore it is indicated how apparent phase transitions are encountered in approximations of a finite level which do not persist in the asymptotic limit

lucas kanade meets horn schunck combining local and global optic flow methods differential methods belong to the most widely used techniques for optic flow computation in image sequences they can be classified into local methods such as the lucas kanade technique or bigun s structure tensor method and into global methods such as the horn schunck approach and its extensions often local methods are more robust under noise while global techniques yield dense flow fields the goal of this paper is to contribute to a better understanding and the design of novel differential methods in four ways i we juxtapose the role of smoothing regularisation processes that are required in local and global differential methods for optic flow computation ii this discussion motivates us to describe and evaluate a novel method that combines important advantages of local and global approaches it yields dense flow fields that are robust against noise iii spatiotemporal and nonlinear extensions as well as multiresolution frameworks are presented for this hybrid method iv we propose a simple confidence measure for optic flow methods that minimise energy functionals it allows to sparsify a dense flow field gradually depending on the reliability required for the resulting flow comparisons with experiments from the literature demonstrate the favourable performance of the proposed methods and the confidence measure

on stratified l convergence spaces pretopological axioms and diagonal axioms this paper focuses on pretopological and diagonal axioms for stratified l convergence spaces recently defined by the authors in 2011 this notion is slightly stronger than jager s stratified l generalized convergence spaces in particular one pretopological axiom is stated without resorting to the notion of neighborhood l filters some generalizations of kowalsky s and fischer s diagonal conditions are presented the relationship between these axioms are investigated

factorization properties of birational mappings abstract we analyse birational mappings generated by transformations on q q matrices which correspond respectively to two kinds of transformations the matrix inversion and a permutation of the entries of the q q matrix remarkable factorization properties emerge for quite general involutive permutations it is shown that factorization properties do exist even for birational transformations associated with noninvolutive permutations of entries of q q matrices and even for more general transformation which are rational transformations but no longer birational the existence of factorization relations independent of q the size of the matrices is underlined the relations between the polynomial growth of the complexity of the iterations the existence of recursions in a single variable and the integrability of the mappings are sketched for the permutations yielding these properties all these results show that permutations of the entries of the matrix yielding factorization properties are not so rare in contrast the occurrence of recursions in a single variable or of the polynomial growth of the complexity are of course less frequent but not completely exceptional

discrete symmetry groups of vertex models in statistical mechanics we analyze discrete symmetry groups of vertex models in lattice statistical mechanics represented as groups of birational transformations they can be seen as generated by involutions corresponding respectively to two kinds of transformations onq q matrices the inversion of theq q matrix and an involutive permutation of the entries of the matrix we show that the analysis of the factorizations of the iterations of these transformations is a precious tool in the study of lattice models in statistical mechanics this approach enables one to analyze two dimensionalq4 state vertex models as simply as three dimensional vertex models or higher dimensional vertex models various examples of birational symmetries of vertex models are analyzed a particular emphasis is devoted to a three dimensional vertex model the 64 state cubic vertex model which exhibits a polynomial growth of the complexity of the calculations a subcase of this general model is seen to yield integrable recursion relations we also concentrate on a specific two dimensional vertex model to see how the generic exponential growth of the calculations reduces to a polynomial growth when the model becomes yang baxter integrable it is also underlined that a polynomial growth of the complexity of these iterations can occur even for transformations yielding algebraic surfaces or higher dimensional algebraic varieties

chaos in segments from korean traditional singing and western singing the time series of the segments from a korean traditional song gwansanyungma and a western song la mamma morta were investigated using chaotic analysis techniques it is found that the phase portrait in the reconstructed state space of the time series of the segment from the korean traditional song has a more complex structure in comparison with the segment from the western songs the segment from the korean traditional song has the correlation dimension 4 4 and two positive lyapunov exponents which show that the dynamic related to the korean traditional song is a high dimensional hyperchaotic process on the other hand the segment from the western song with only one positive lyapunov exponent and the correlation dimension 2 5 exhibits low dimensional chaotic behavior

subcategories of lattice valued convergence spaces we study several subcategories of heyting algebra valued convergence spaces these categories generalize the well known categories of kent convergence spaces of limit spaces of pretopological spaces of pseudo topological spaces and of topological limit spaces to the many valued setting moreover a functorial mechanism is described for changing the basis lattice

singularity confinement and chaos in discrete systems we present a number of second order maps which pass the singularity confinement test commonly used to identify integrable discrete systems but which nevertheless are non integrable as a more sensitive integrability test we propose the analysis of the complexity algebraic entropy of the map using the growth of the degree of its iterates integrability is associated with polynomial growth while the generic growth is exponential for chaotic systems

gahler s neighborhood condition for lattice valued convergence spaces we generalize a neighborhood condition from the category of convergence spaces to the category of lattice valued convergence spaces for a space in the category of convergence spaces this condition is equivalent to being a topological space it turns out that there are two meaningful generalizations of this condition to the category of lattice valued convergence spaces the first one guarantees that a space in a certain subcategory is a lattice valued topological space the other one is a levelwise condition which is equivalent to a generalization of fischer s diagonal condition the latter generalization is preserved under the embedding of the categories of convergence approach spaces and of probabilistic limit spaces into the category of lattice valued convergence spaces

a comparative study of energy minimization methods for markov random fields with smoothness based priors among the most exciting advances in early vision has been the development of efficient energy minimization algorithms for pixel labeling tasks such as depth or texture computation it has been known for decades that such problems can be elegantly expressed as markov random fields yet the resulting energy minimization problems have been widely viewed as intractable algorithms such as graph cuts and loopy belief propagation lbp have proven to be very powerful for example such methods form the basis for almost all the top performing stereo methods however the trade offs among different energy minimization algorithms are still not well understood in this paper we describe a set of energy minimization benchmarks and use them to compare the solution quality and runtime of several common energy minimization algorithms we investigate three promising methods graph cuts lbp and tree reweighted message passing in addition to the well known older iterated conditional mode icm algorithm our benchmark problems are drawn from published energy functions used for stereo image stitching interactive segmentation and denoising we also provide a general purpose software interface that allows vision researchers to easily switch between optimization methods the benchmarks code images and results are available at http vision middlebury edu mrf

accurate approximations for posterior moments and marginal densities abstract this article describes approximations to the posterior means and variances of positive functions of a real or vector valued parameter and to the marginal posterior densities of arbitrary i e not necessarily positive parameters these approximations can also be used to compute approximate predictive densities to apply the proposed method one only needs to be able to maximize slightly modified likelihood functions and to evaluate the observed information at the maxima nevertheless the resulting approximations are generally as accurate and in some cases more accurate than approximations based on third order expansions of the likelihood and requiring the evaluation of third derivatives the approximate marginal posterior densities behave very much like saddle point approximations for sampling distributions the principal regularity condition required is that the likelihood times prior be unimodal

let s baxterise we recall the concept of baxterisation of an r matrix or of a monodromy matrix which corresponds to build from one point in the r matrix parameter space the algebraic variety where the spectral parameter s live we show that the baxterisation which amounts to studying the iteration of a birational transformation is a win win strategy it enables to discard efficiently the non integrable situations focusing directly on the two interesting cases where the algebraic varieties are of the so called general type finite order iteration or are abelian varieties infinite order iteration we emphasize the heuristic example of the sixteen vertex model and provide a complete description of the finite order iterations situations for the baxter model we show that the baxterisation procedure can be introduced in much larger frameworks where the existence of some underlying yang baxter structure is not used we baxterise l operators local quantum lax matrices and quantum hamiltonians

on the metric properties of the feigenbaum attractor the two standard literature definitions of the function associated with the feigenbaum attractor are not equivalent the method due to vulet al and feigenbaum is used to calculate the haussdorff dimension of the feigenbaum attractor using as input the trajectory scaling functions the two calculations yield the same hausdorff dimensiond 0 5380451435 to within the accuracy of the computation

dense versus sparse approaches for estimating the fundamental matrix there are two main strategies for solving correspondence problems in computer vision sparse local feature based approaches and dense global energy based methods while sparse feature based methods are often used for estimating the fundamental matrix by matching a small set of sophistically optimised interest points dense energy based methods mark the state of the art in optical flow computation the goal of our paper is to show that this separation into different application domains is unnecessary and can be bridged in a natural way as a first contribution we present a new application of dense optical flow for estimating the fundamental matrix comparing our results with those obtained by feature based techniques we identify cases in which dense methods have advantages over sparse approaches motivated by these promising results we propose as a second contribution a new variational model that recovers the fundamental matrix and the optical flow simultaneously as the minimisers of a single energy functional in experiments we show that our coupled approach is able to further improve the estimates of both the fundamental matrix and the optical flow our results prove that dense variational methods can be a serious alternative even in classical application domains of sparse feature based approaches

growth complexity spectrum of some discrete dynamical systems we first study the iteration of birational mappings generated by the composition of the matrix inversion and of a permutation of the entries of 3 3 matrices and consider the degree d n of the numerators or denominators of the corresponding successive rational expressions for the nth iterate the growth of this degree is generically exponential with n d n n is called the growth complexity we introduce a semi numerical analysis which enables to compute these growth complexities for all the 9 possible birational transformations these growth complexities correspond to a spectrum of eighteen algebraic values we then drastically generalize these results replacing permutations of the entries by homogeneous polynomial transformations of the entries possibly depending on many parameters again it is shown that the associated birational or even rational transformations yield algebraic values for their growth complexities 1999 elsevier science b v all rights reserved

a note on regularity an alternative characterization of gahler s definition of regularity of a lattice valued convergence space is given in particular a space is regular exactly whenever convergence of filters remains invariant with respect to taking closures verification employs a slight modification of the work by jager

on the control of complex dynamic systems a method is described for the limited control of the dynamics of systems which generally have several dynamic attractors associated either with maps or first order ordinary differential equations ode in rn the control is based on the existence of convergent regions cc k 1 2 in the phase space of such systems where there is local convergence of all nearby orbits the character of the control involves the entrainment and subsequent possible migration of the experimental system from one attractor to another entrainment means that limt x t g t 0 where x t rn is the system s controlled dynamics and the goal dynamics g t gk has any topological form but is limited dynamically and to regions of phase space gk contained in some ck gk ck the control process is initiated only when the system enters a basin of entrainment bek gk associated with the goal region gk aside from this macroscopic initial state information about the system no further feedback of dynamic information concerning the response of the system is required the experimental reliability of the control requires that the regions bek be convex regions in the phase space which can apparently be assured if gk ck simple illustrations of these concepts are given using a general linear and a piecewise linear ode in rn in addition to these entrainment goals migration goal dynamics is introduced which intersects two convergent regions g cj g cj i j and permits transferring the dynamics of a system from one attractor to another or from one convergent region to another in the present study these concepts are illustrated with various one dimensional maps involving one or more attractors and convergent regions several theorems concerning entrainment are derived for very general continuous one dimensional maps sufficient conditions are also established which ensure near entrainment for a system when the dynamic model of the system is not exactly known the applications of these concepts to higher dimensional maps and flows will be presented in subsequent studies

evaluation of probabilistic and dynamical invariants from finite symbolic substrings comparison between two approaches abstract while evaluating scaling functions of fractal dimensions and lyapunov exponents from time series in the traditional way assumptions are made concerning the grammar of the underlying dynamical system it is implicitly assumed that the length of the substrings considered is sufficient to capture the grammatical properties of the system in this contribution we show where this assumption becomes relevant we give an example of a simple grammatical rule which leads to badly behaving convergence properties of the associated scaling functions as another consequence of our investigations we conclude that whenever a finite grammar is encountered the cycle expansion approach of cvitanovic using periodic orbits should preferably be used

lyapunov analysis from dynamical systems theory to applications the study of deterministic laws of evolution has characterized the development of science since newton s times chaos namely the manifestation of irregular and unpredictable dynamics not random but look random 1 entered the debate on determinism at the end of the 19th century with the discovery of sensitivity to initial conditions meaning that small infinitesimal differences in the initial state might lead to dramatic differences at later times poincare 2 3 was the first to realize that solutions of the three body problem are generically highly sensitive to initial conditions at about the same time this property was recognized in geodesic flows with negative curvature by hadamard 4 one of the first experimental observations of chaos as understood much later was when irregular noise was heard by van der pol in 1927 5 while studying a periodically forced nonlinear oscillator nevertheless it was only with the advent of digital computing that chaos started to attract the interest of the wider scientific community after the pioneering investigation of ergodicity in a chain of nonlinear oscillators by fermi pasta and ulam in 1955 6 it was in the early 1960s that the numerical studies of lorenz 7 and henon and heiles 8 revealed that irregular and unpredictable motions are a generic feature of low dimensional nonlinear deterministic systems the existence and onset of chaos was then rigorously analyzed in several systems while an exhaustive list of such mathematical proofs is beyond the scope of this preface one should mention the contributions of kolmogorov 9 10 chirikov 11 smale 12 ruelle and takens 13 li and yorke 14 and feigenbaum 15 the characteristic lyapunov exponents introduced by oseledets in 1968 16 are the fundamental quantities for measuring the sensitivity to initial conditions oseledets work generalized the concept of lyapunov stability to irregular trajectories building upon earlier studies of birkhoff 17 von neumann 18 krylov 19 3 and asonov and sinai 20 on ergodic theory lyapunov exponents quantify exponential sensitivity to initial conditions and provide direct access to the entropy production in ergodic systems via the pesin theory 21

lucas kanade 20 years on a unifying framework since the lucas kanade algorithm was proposed in 1981 image alignment has become one of the most widely used techniques in computer vision applications range from optical flow and tracking to layered motion mosaic construction and face coding numerous algorithms have been proposed and a wide variety of extensions have been made to the original formulation we present an overview of image alignment describing most of the algorithms and their extensions in a consistent framework we concentrate on the inverse compositional algorithm an efficient algorithm that we recently proposed we examine which of the extensions to lucas kanade can be used with the inverse compositional algorithm without any significant loss of efficiency and which cannot in this paper part 1 in a series of papers we cover the quantity approximated the warp update rule and the gradient descent approximation in future papers we will cover the choice of the error function how to allow linear appearance variation and how to impose priors on the parameters

regularity lattice valued cauchy spaces a category of lattice valued cauchy spaces is defined and its properties are investigated the relationship between this category and the recent work by jager is presented the notion of regularity in the cauchy space context is the primary emphasis here in particular the category consisting of the regular cauchy spaces is shown to be bireflective in the category of all lattice valued cauchy spaces having cauchy continuous maps as its morphisms moreover completions are also investigated

new integrable cases of a cremona transformation a finite order orbits analysis we analyse the properties of a particular birational mapping of two variables cremona transformation depending on two free parameters e and associated with the action of a discrete group of non linear birational transformations on the entries of a q q matrix this mapping originates from the analysis of birational transformations obtained from very simple algebraic calculations namely taking the inverse of q q matrices and permuting some of the entries of these matrices it has been seen to yield weak chaos and integrability we have found new integrable cases of this cremona transformation corresponding to the values of 0 when e 12 13 1 besides the already known values e 0 and e 1 and also arbitrary when e 0 for these cases one has a foliation of the parameter space in elliptic curves we give the equations of these elliptic curves based on this very example we show how one can find these integrability cases of the cremona transformation and actually integrate it using a method based on the systematic study of the finite order conditions of the cremona transformation the method is shown to be efficient and straightforward the various integrability cases are revisited using many different representations of this very mapping birational transformations recursion in one variable

characterization of strange attractors a new measure of strange attractors is introduced which offers a practical algorithm to determine their character from the time series of a single observable the relation of this new measure to fractal dimension and information theoretic entropy is discussed

using lyapunov exponents to predict the onset of chaos in nonlinear oscillators an analytic technique for predicting the emergence of chaotic instability in nonlinear nonautonomous dissipative oscillators is proposed the method is based on the lyapunov type stability analysis of an arbitrary phase trajectory and the standard procedure of calculating the lyapunov characteristic exponents the concept of temporally local lyapunov exponents is then utilized for specifying the area in the phase space where any trajectory is asymptotically stable and therefore the existence of chaotic attractors is impossible the procedure of linear coordinate transform optimizing the linear part of the vector field is developed for the purpose of maximizing the stability area in the vicinity of a stable fixed point by considering the inverse conditions of asymptotic stability this approach allows formulating a necessary condition for chaotic motion in a broad class of nonlinear oscillatory systems including many cases of practical interest the examples of externally excited one and two well duffing oscillators and a planar pendulum demonstrate efficiency of the proposed method as well as a good agreement of the theoretical predictions with the results of numerical experiments the comparison of the proposed method with melnikov s criterion shows a potential advantage of using the former one at high values of dissipation parameter and or multifrequency type of excitation in dynamical systems

real topological entropy versus metric entropy for birational measure preserving transformations abstract we consider a family of birational measure preserving transformations of two complex variables depending on one parameter for which simple rational expressions for the dynamical zeta function have been conjectured together with an equality between the topological entropy and the logarithm of the arnold complexity divided by the number of iterations similar results have been obtained for the adaptation of these two concepts to dynamical systems of real variables yielding to introduce a real topological entropy and a real arnold complexity we try to compare here the kolmogorov sinai metric entropy and this real arnold complexity or real topological entropy on this particular example of a one parameter dependent birational transformation of two variables more precisely we analyze using an infinite precision calculation the lyapunov characteristic exponents for various values of the parameter of the birational transformation in order to compare these results with the ones for the real arnold complexity we find a quite surprising result for this very birational example and in fact for a large set of birational measure preserving mappings generated by involutions the lyapunov characteristic exponents seem to be equal to zero or at least extremely small for all the orbits we have considered and for all values of the parameter birational measure preserving transformations generated by involutions could thus allow to better understand the difference between the topological description and the probabilistic description of discrete dynamical systems many birational measure preserving transformations generated by involutions seem to provide examples of discrete dynamical systems which can be topologically chaotic while they are metrically almost quasi periodic heuristically this can be understood as a consequence of the fact that their orbits seem to form some kind of transcendental foliation of the two dimensional space of variables

a birational mapping with a strange attractor post critical set and covariant curves we consider some two dimensional birational transformations one of them is a birational deformation of the henon map for some of these birational mappings the post critical set i e the iterates of the critical set is infinite and we show that this gives straightforwardly the algebraic covariant curves of the transformation when they exist these covariant curves are used to build the preserved meromorphic 2 form one may also have an infinite post critical set yielding a covariant curve which is not algebraic transcendental for two of the birational mappings considered the post critical set is finite and we claim that there is no algebraic covariant curve and no preserved meromorphic 2 form for these two mappings with finite post critical sets attracting sets occur and we show that they pass the usual tests lyapunov exponents and the fractal dimension for being strange attractors the strange attractor of one of these two mappings is unbounded