chaotic functions with zero topological entropy on caracterise la classe m c i i des applications chaotiques en ce sens on montre que m contient certaines appliquations qui ont a la fois une entropie topologique nulle et des attracteurs infinis on montre que le complement de m consiste en applications qui ont seulement des trajectoires approchables par des cycles

building counterexamples to generalizations for rational functions of ritt s decomposition theorem abstract the classical ritt s theorems state several properties of univariate polynomial decomposition in this paper we present new counterexamples to the first ritt theorem which states the equality of length of decomposition chains of a polynomial in the case of rational functions namely we provide an explicit example of a rational function with coefficients in q and two decompositions of different length another aspect is the use of some techniques that could allow for other counterexamples namely relating groups and decompositions and using the fact that the alternating group a 4 has two subgroup chains of different lengths and we provide more information about the generalizations of another property of polynomial decomposition the stability of the base field we also present an algorithm for computing the fixing group of a rational function providing the complexity over the rational number field

scrambled sets of continuous maps of 1 dimensional polyhedra let k be a 1 dimensional simplicial complex in r3 without isolated vertexes x iki be the polyhedron of k with the metric dk induced by k and f x x be a continuous map in this paper we prove that if k is finite then the interior of every scrambled set of f in x is empty we also show that if k is an infinite complex then there exist continuous maps from x to itself having scrambled sets with nonempty interiors and if x r or r then there exist c maps of x with the whole space x being a scrambled set

hamiltonian mappings and circle packing phase spaces we introduce three area preserving maps with phase space structures which resemble circle packings each mapping is derived from a kicked hamiltonian system with one of the three different phase space geometries planar hyperbolic or spherical and exhibits an infinite number of coexisting stable periodic orbits which appear to pack the phase space with circular resonances

computing hopf bifurcations ii three examples from neurophysiology in guckenheimer myers and sturmfels siam j numer anal 34 1997 we present algorithms for detecting hopf bifurcations in two parameter families of vector fields based on classical algebraic constructions in addition to their utility as augmented systems for use with standard newton type continuation methods they are shown to be particularly well adapted for solution by computer algebra techniques for vector fields of small or moderate dimension the present study examines the performance of these methods on test problems selected from models of current research interest in neurophysiology implementation issues are examined and the numerical properties of the proposed methods are compared with several alternative algorithms for hopf pathfollowing that appear in the literature

convergence cones near bifurcation the performance of predictor solver continuation methods near bifurcation points is considered a general theory is developed for the convergence of two commonly used solvers namely newton s method and the chord method near singular points of a nonlinear equation this theory provides a uniform treatment of continuation for simple bifurcation multiple bifurcation and multiple limit point bifurcation problems for these types of bifurcation it is shown that there are conical domains of attraction for the above iterative methods centred on solution branches which correspond to isolated roots of the algebraic bifurcation equations

hopf bifurcation near a double singular point with z 2 symmetry and x 0 breaking this paper deals with nonlinear equations f x 0 and the corresponding odes xt f x satisfying f 0 0 and a z2 symmetry in particular we are interested in hopf points which indicate the bifurcation of periodic solutions of xt f x from steady state solutions of f x 0 it is shown that under suitable nondegeneracy conditions there bifurcate two paths of hopf points from a double singular point where x 0 and fx 0 has a double zero eigenvalue with one eigenvector symmetric and one anti symmetric this result gives a new example of finding hopf points through local singular points our main tools for analysis are some extended systems which also provide easily implemented algorithms for the numerical computation of the bifurcating hopf points a supporting numerical example for a brusselator model is also presented

skew and linearized reed solomon codes and maximum sum rank distance codes over any division ring abstract reed solomon codes and gabidulin codes have maximum hamming distance and maximum rank distance respectively a general construction using skew polynomials called skew reed solomon codes has already been introduced in the literature in this work we introduce a linearized version of such codes called linearized reed solomon codes we prove that they have maximum sum rank distance such distance is of interest in multishot network coding or in singleshot multi network coding to prove our result we introduce new metrics defined by skew polynomials which we call skew metrics we prove that skew reed solomon codes have maximum skew distance and then we translate this scenario to linearized reed solomon codes and the sum rank metric the theories of reed solomon codes and gabidulin codes are particular cases of our theory and the sum rank metric extends both the hamming and rank metrics we develop our theory over any division ring commutative or non commutative field we also consider non zero derivations which give new maximum rank distance codes over infinite fields not considered before

ω chaos and topological entropy we present a new concept of chaos chaos and prove some properties of chaos then we prove that chaos is equivalent to positive entropy on the interval we also prove that chaos is equivalent to the definition of chaos given by devaney on the interval

jordan holder theorem for imprimitivity systems and maximal decompositions of rational functions in this paper we prove several results about the lattice of imprimitivity systems of a permutation group containing a cyclic subgroup with at most two orbits as an application we generalize the first ritt theorem about functional decompositions of polynomials and some other related results besides we discuss examples of rational functions related to finite subgroups of aut cp 1 for which the first ritt theorem fails to be true

on the equation p f q g where p q are polynomials and f g are entire functions in 1922 ritt described polynomial solutions of the functional equation p f q g in this paper we describe solutions of the equation above in the case when p q are polynomials while f g are allowed to be arbitrary entire functions in fact we describe solutions of the more general functional equation s p f q g where s f g are entire functions and p q are arbitrary rational functions as an application we solve the problem of description of strong uniqueness polynomials for entire functions

computation of hopf branches bifurcating from a class of hopf steady state points abstract this paper is concerned with the computation of hopf branches bifurcating from a hopf steady state point of a two parameter non linear problem satisfying a subspace invariance condition we will present a new approach to the theoretical and computational analysis of the bifurcating branches of hopf points at this singular point utilizing the system used to calculate hopf points according to the analysis standard continuation and branch switching can be used to calculate these hopf branches in addition a more effective method based on the extended system of the singular point is proposed for the calculation of the branch of secondary hopf points the use of newton s method for solving the extended system will lead to a quadratically convergent scheme and its implementation with a suitable partitioning procedure is also discussed numerical examples are given

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

computing hopf bifurcations i this paper addresses the problems of detecting hopf bifurcations in systems of ordinary differential equations and following curves of hopf points in two parameter families of vector fields the established approach to this problem relies upon augmenting the equilibrium condition so that a hopf bifurcation occurs at an isolated regular point of the extended system we propose two new methods of this type based on classical algebraic results regarding the roots of polynomial equations and properties of kronecker products for matrices in addition to their utility as augmented systems for use with standard newton type continuation methods they are also particularly well adapted for solution by computer algebra techniques for vector fields of small or moderate dimension

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

a note on the calculation of paths of hopf bifurcations this paper is concerned with the analysis of a two parameter nonlinear problem whose linearisation has a double zero eigenvalue with only one eigenvector the aim of the paper is to present a new approach to the theoretical and computational analysis of the bifurcating branches at this singular point using a symmetry in the system used to calculate hopf bifurcations it is shown that standard branch switching can be used to jump on to the path of hopf bifurcations emanating from the singular point

computational methods for bifurcation problems with symmetries with special attention to steady state and hopf bifurcation points we show how group theoretical methods can be employed to utilize the symmetry of a bifurcation problem in numerical computations we extend the approach by werner 1988 by presenting methods for the detection of bifurcation points and the computation of multiple hopf points the essential numerical point is the utilization of certain reduced instead of full systems involving appropriate subgroups of the underlying symmetry group the group theoretical tool is an a priori knowledge of the interaction of certain subgroups 0 and of at possibly multiple steady state or hopf bifurcation points minimal 0 breaking bifurcation we introduce a bifurcation graph which shows graphically this a priori information its edges represent possible symmetry breaking bifurcations our analysis follows the lines of golubitsky stewart and schaeffer 1988 but it is aimed to numerical applications we have chosen a 4 box brusselator model in order to explain our notions and ideas and to discuss the numerical procedure

a direct method for the computation of hopf bifurcation points we present a method for the numerical computation of a hopf bifurcation point of an evolution equation an augmented time independent system of equations has to be solved for which the hopf bifurcation point is an isolated solution in the case of a system of ordinary differential equations we propose an efficient implementation of newton s method for the solution of the resulting system of algebraic equations if the method is applied directly to a partial differential system with one space variable a two point boundary value problem is obtained we also compare this technique with a related method and results are given for two systems of partial differential equations describing chemical reacting systems

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

on scrambled sets for chaotic functions some recent research has raised questions concerning the possible sizes of scrambled sets for chaotic functions we answer these questions by showing that a scrambled set can have full measure but cannot be residual although a scrambled set can be second category in every interval we also indicate relationships that exist between chaotic functions and transitive functions

numerical computation of periodic orbits that bifurcate from stationary solutions of ordinary differential equations the computation of periodic orbits of autonomous ordinary differential equations is considered a new method especially suited for computing limit cycles that bifurcate from stationary solutions is established this novel approach is very easy to implement the method requiring the programming of the right hand side only another topic of this paper is the demonstration of how a direct method should be applied in the computation of hopf bifurcation points these methods solve boundary value problems by means of standard software only the methods are tested on four examples arising in different application areas including chemistry and biology

numerical computation of hopf bifurcation points for parabolic diffusion reaction differential equations a numerical method for the direct computation of hopf bifurcation points for a system of parabolic diffusion reaction partial differential equations is described a finite difference discretization in space is used to approximate these equations by a system of first order ordinary differential equations the hopf bifurcation of the latter system is computed by using kubicek s method the results are improved by richardson extrapolation for approximation of the space operator a standard three point difference formula as well as the stormer numerov technique are used the method proposed is illustrated on an example describing heat and mass transfer in a porous catalyst particle

quadratically convergent methods for the computation of unfolded singularities an underdetermined system of equations for the determination of the lyapunov schmidt reduction is presented this system can easily be extended to an augmented system for the computation of nontrivial singularities by adding suitable equations which characterize the singularity necessary and sufficient conditions for these additional equations are given so that the corresponding gauss newton method converges locally and quadratically for several types of singularities possible equations are derived from invariance conditions numerical examples illustrating the results are included

a direct method for computation of simple bifurcations in the present study we focus on the computational analysis of partial differential equations with emphasis on the stability of the equilibrium states and on their bifurcations in practical applications it is not sufficient to obtain an equilibrium solution at a point in the parameter space the equilibrium solution branches their stability characteristics and particularly the critical points of transition from one state to another e g bifurcation points are required for understanding the physics of the problem in principle the linear stability of an equilibrium state can be investigated by solving an eigenvalue problem and consequently the points of bifurcations can be detected we review alternative techniques for detecting bifurcation points which are direct and numerically efficient and therefore more practical starting with a large dimension dynamical system which represents a projection of a set of coupled partial differential equations onto a basis function we discuss the relative effectiveness of the time evolution approach the test function approach and the direct method we will then extend the direct method for a more practical and efficient implementation with this technique we compute the sequence of transitions from steady state to chaotic flow in a two dimensional lid driven cavity of aspect ratio 0 8 1 0 and 1 5 we demonstrate the effectiveness of this technique by computing interesting new dynamics in this relatively simple hydrodynamic system in particular we show that depending on the aspect ratio the first transition from steady state could be through a supercritical or a subcritical hopf bifurcation leading the system to a time periodic state we construct the destabilizing disturbance structure and conclude that the first bifurcation of the primary steady state is due to the centrifugal instability of the primary eddy the mechanism of transition to chaos is low dimensional the transition to chaos occurs after a secondary hopf bifurcation

hopf bifurcation near a double singular point abstract the nonlinear equation f x 0 f x r2 x where x is a banach space and f satisfies a z2 symmetry relation is considered interest centres on a certain type of double singular point where the solution x is symmetric and fx has a double zero eigenvalue with one eigenvector symmetric and one antisymmetric we show that under certain nondegeneracy conditions which are stated both algebraically and geometrically there exists a path of hopf bifurcations or imaginary hopf bifurcations passing through the double singular point and for which x is not symmetric except at the double singular point an easy geometrical test is found to decide which type of phenomenon occurs a biproduct of the analysis is that explicit expressions are obtained for quantities which help to provide a reliable numerical method to compute these paths a pseudo spectral method was used to obtain numerical results for the brusselator equations to illustrate the theory

imperfection sensitivity of interacting hopf and steady state bifurcations and their classification abstract the bifurcation problem of interacting time periodic and stationary solutions of nonlinear evolution equations with double degeneracy is discussed in terms of singularity and imperfect bifurcation theory a complete classification up to symmetry covariant contact equivalence and codimension three of generic perturbations of interacting hopf and steady state bifurcations is presented the sensitivity of the bifurcation diagrams to imperfections is analyzed normal forms describing sequences of secondary and tertiary bifurcations leading to motions on tori are determined a variety of phenomena such as gaps in hopf branches periodic motions not stably connected to steady states and the formation of islands is discovered which one can expect to find in perturbed evolution equations on pure geometric grounds implications for physical systems are discussed

new algorithms for the evaluation of complex bifurcation points in ordinary differential equations a comparative numerical study four iterative algorithms two of them new for the evaluation of complex hopf bifurcation points in ordinary differential equations are compared a comparison of effectiveness of the proposed algorithms is made for two examples taken from chemical reaction engineering the applicability of the algorithms for the solution of various types of problems is also discussed

distributionally scrambled invariant sets in a compact metric space abstract the paper solves a question posed by oprocha on the existence of invariant distributionally chaotic scrambled sets we show among other things that a continuous map f acting on compact metric space x d with a weak specification property fixed point and infinitely many mutually distinct periods has a dense mycielski i e c dense set of type f invariant distributionally scrambled set as a consequence we describe a class of maps with a distributionally scrambled invariant set of full lebesgue measure in the case when x is a k dimensional cube

detection of hopf points by counting sectors in the complex plane let a alpha alpha real be a family of real n by n matrices a value alpha 0 of alpha is called a hopf value if a alpha 0 has a conjugate pair of purely imaginary eigenvalues pm rm i omega 0 omega 0 0 we describe a technique for detecting hopf values based on the evolution of the schur complement of a zi in a bordered extension of a zi where z varies along the positive imaginary axis of the complex plane we compare the efficiency of this method with more obvious methods such as the use of the qr algorithm and of the determinant function of a as well as with recent work on the cayley transform in particular we show the advantages of the schur complement method in the case of large sparse matrices arising in dynamical problems by discretizing boundary value problems the hopf values of the jacobian matrices are important in this setting because they are related to the hopf bifurcation phenomenon where steady state solutions bifurcate into periodic solutions

discretizing dynamical systems with generalized hopf bifurcations we consider the discretizations of parameter dependent continuous time dynamical systems we show that the general one step methods shift a generalized hopf bifurcation and turn it into a generalized neimark sacker point we analyze the effect of discretization methods on the emanating hopf curve in particular we obtain estimates for the eigenvalues of the discretized system along this curve a detailed analysis of the discretized first lyapunov coefficient is also given the results are illustrated by a numerical example dynamical consequences are discussed

computation of hopf bifurcation with bordered matrices hopf bifurcation for dynamical systems dot x f x lambda is characterized by rank deficiency two of a 2 omega 2 i where a f x x lambda in mathbb r n times n is the jacobian at the hopf point and pm i omega are the hopf eigenvalues replacing w 2 by nu and allowing nu leq 0 we introduce real and imaginary hopf matrices a with hopf numbers nu defining a codimension 2 manifold mathcal m h nu subset mathbb r n times n times mathbb r which can be described by two scalar equations alpha a nu 0 beta a nu 0 given by the solution of certain linear bordered systems these two scalar equations form a suitable augmentation of f x lambda 0 for the numerical computation of branches of hopf points in two parameter systems in contrast to chu govaerts and spence siam j numer anal 31 1994 pp 524 539 we suggest an a dependent bordering having certain advantages we construct a scalar test function for hopf points in the

measures of chaos and a spectral decomposition of dynamical systems on the interval let f 0 1 0 1 be continuous for x y e 0 1 the upper and lower distance distribution functions fx y and fxy are defined for any t 0 as the lim sup and lim inf as n oc of the average number of times that the distance ifi x fi y i between the trajectories of x and y is less than t during the first n iterations the spectrum of f is the system f of lower distribution functions which is characterized by the following properties 1 the elements of f are mutually incomparable 2 for any f e f there is a perfect set pf 8 0 such that fuv f and fuv 1 for any distinct u v e pf 3 if s is a scrambled set for f then there are f g in f and a decomposition s sf u sg sg may be empty such that fuv f if u v e sf and fuv g if u v e sg our principal results are 1 if f has positive topological entropy then f is nonempty and finite and any f e f is zero on an interval 0 e where e 0 and hence any pf is a scrambled set in the sense of li and yorke 2 if f has zero topological entropy then f f where f 1 it follows that the spectrum of f provides a measure of the degree of chaos of f in addition a useful numerical measure is the largest of the numbers fo 1 f t dt where f e f

an algorithm for the computation of hopf bifurcation points in comparison with other methods we present an algorithm for the computation of a hopf bifurcation point based on a direct method i e an augmented time independent system is solved the algorithm is designed to deal efficiently with problems representing a finite difference discretization of a partial differential system in this case the bandstructure of the jacobian matrix is exploited we also compare this technique with some other methods in regard to its efficiency and reliability

approximation of hopf bifurcation we make several assumptions on a nonlinear evolution problem ensuring the existence of a hopf bifurcation under a fairly general approximation condition we define a discrete problem which retains the bifurcation property and we prove an error estimate between the branches of exact and approximate periodic solutions

on algebraic curves a x b y 0 of genus zero using a geometric approach involving riemann surface orbifolds we provide lower bounds for the genus of an irreducible algebraic curve of the form mathcal e a b a x b y 0 where a b in mathbb c z we also investigate series of curves mathcal e a b of genus zero where by a series we mean a family with the same a we show that for a given rational function a a sequence of rational functions b i such that deg b i rightarrow infty and all the curves a x b i y 0 are irreducible and have genus zero exists if and only if the galois closure of the field extension mathbb c z mathbb c a has genus zero or one

the approximation of generalized turning points newline by krylov subspace methods certain types of singular solutions of nonlinear parameter dependent operator equations were characterized by griewank and reddien 5 6 as regular solutions of suitable augmented systems for their numerical approximation an approach based on the use of krylov subspaces is here presented the application to boundary value problems is illustrated by numerical examples

operators with dense invariant cyclic vector manifolds abstract we study a class of banach space operators patterned after the weighted backward shifts on hilbert space and show that any non scalar operator in the commutant of one of these generalized backward shifts has a dense invariant linear manifold whose non zero members are cyclic vectors under appropriate extra hypotheses on the commuting operator stronger forms of cyclicity are possible the most extreme being hypercyclicity density of an orbit motivated by these results we examine the cyclic behavior of two seemingly unrelated classes of operators adjoint multiplications on hilbert spaces of holomorphic functions and differential operators on the frechet space of entire functions we show that each of these operators other than the scalar multiples of the identity possesses a dense invariant linear submanifold each of whose non zero elements is hypercyclic finally we explore some connections with dynamics many of the hypercyclic operators discussed here are in at least one of the commonly accepted senses of the word chaotic

a new perspective of cyclicity in convolutional codes in this paper we propose a new way of providing cyclic structures to convolutional codes we define the skew cyclic convolutional codes as left ideals of a quotient ring of a suitable non commutative polynomial ring in contrast to the previous approaches to cyclicity for convolutional codes we use ore polynomials with coefficients in a field the rational function field over a finite field so their arithmetic is very well known and we may proceed similarly to cyclic block codes in particular we show how to obtain easily skew cyclic convolutional codes of a given dimension and we compute an idempotent generator of the code and its dual