synchronization time in a hyperbolic dynamical system with long range interactions abstract we show that the threshold of complete synchronization in a lattice of coupled non smooth chaotic maps is determined by linear stability along the directions transversal to the synchronization subspace we examine carefully the synchronization time and show that an inadequate observation of the system evolution leads to wrong results we present both careful numerical experiments and a rigorous mathematical explanation confirming this fact allowing for a generalization involving hyperbolic coupled map lattices

comparison results for parallel multisplitting methods with applications to aor methods abstract in this paper we present some comparison theorems between two general parallel multisplittings these comparison theorems can be applied to many classical splittings such as jacobi gauss seidel and aor splittings some significant improvements and generalizations of the existing comparison results are obtained

nonnegative splittings for rectangular matrices the extension of the nonnegative splitting for rectangular matrices called proper nonnegative splitting is proposed first different convergence and comparison theorems for the proper nonnegative splittings are established the notion of double nonnegative splitting is then generalized to rectangular matrices finally different convergence and comparison results are presented for this decomposition the case for singular square matrices is also studied

a class of preconditioners based on the i s α type preconditioning matrices for solving linear systems abstract the purpose of this paper is to present a class of preconditioners based on the i s type preconditioning matrices provided by evans et al d j evans m m martins m e trigo the aor iterative method for new preconditioned linear systems j comput appl math 132 2001 461 466 and zhang et al y zhang t z huang x p liu modified iterative methods for nonnegative matrices and m matrices linear systems comput math appl 50 2005 1587 1602 three mathematical models are presented to seek the optimal parameter for these preconditioners numerical results are presented which show the improvements on the convergence rate of the jacobi type and gauss seidel type preconditioned iterative methods

mechanisms for the hard bubbling transition in symmetrically coupled chaotic systems we investigate mechanisms for the hard bubbling transition in symmetrically coupled one dimensional maps a transition from strong to weak synchronization occurs via a first period doubling or pitchfork transverse bifurcation of a periodic saddle embedded in the synchronous chaotic attractor the consequence of such transverse bifurcations depends on the existence of an absorbing area controlling the global dynamics in the presence of an absorbing area a subcritical transverse bifurcation is found to give rise to abrupt appearance of transient intermittent bursts with large amplitude for this case any small parameter mismatch that is unavoidable in real systems induces a hard bubbling transition via an interior crisis through a detailed numerical analysis we present explicit examples for the subcritical period doubling and pitchfork transverse bifurcations leading to hard bubbling

block gauss elimination followed by a classical iterative method for the solution of linear systems in the last two decades many papers have appeared in which the application of an iterative method for the solution of a linear system is preceded by a step of the gauss elimination process in the hope that this will increase the rates of convergence of the iterative method this combination of methods has been proven successful especially when the matrix a of the system is an m matrix the purpose of this paper is to extend the idea of one to more gauss elimination steps consider other classes of matrices a e g p cyclic consistently ordered and generalize and improve the asymptotic convergence rates of some of the methods known so far

wada basin boundaries and basin cells in dynamical systems examples are common in which two or more attractors coexist and in such cases the basin boundary is nonempty we consider a two dimensional diffeomorphism f that is f is an invertible map and both f and its inverse are differentiable with continuous derivatives which has at least three basins fractal basin boundaries contain infinitely many periodic points generally only finitely many of these periodic points are outermost on the basin boundary that is accessible from a basin for many systems all accessible points lie on stable manifolds of periodic points a point x on the basic boundary is a wada point if every open neighborhood of x has a nonempty intersection with at least three different basins we call the boundary of a basin a wada basin boundary if all its points are wada points our main goal is to have definitions and hypotheses for wada basin boundaries that can be verified by computer the basic notion basin cell will play a fundamental role in our results for numerical verifications assuming each accessible point on the boundary of a basin b is on the stable manifold of some periodic orbit we show that partial derivative b over bar is a wada basin boundary if the unstable manifold of each of its accessible periodic orbits intersects at least three basins in addition we find conditions for basins b 1 b 2 b n n greater than or equal to 3 to satisfy partial derivative b over bar 1 partial derivative b over bar 2 partial derivative b over bar n our results provide numerically verifiable conditions guaranteeing that the boundary of a basin is a wada basin boundary our examples make use of an existing numerical procedure for finding the accessible periodic points on the basin boundary and another procedure for plotting stable and unstable manifolds to verify the existence of wada basin boundaries

new parallel sor method by domain partitioning in this paper we propose and analyze a new parallel sor method the psor method formulated by using domain partitioning and interprocessor data communication techniques we prove that the psor method has the same asymptotic rate of convergence as the red black r b sor method for the five point stencil on both strip and block partitions and as the four color r b g o sor method for the nine point stencil on strip partitions we also demonstrate the parallel performance of the psor method on four different mimd multiprocessors a ksr1 an intel delta a paragon and an ibm sp2 finally we compare the parallel performance of psor r b sor and r b g o sor numerical results on the paragon indicate that psor is more efficient than r b sor and r b g o sor in both computation and interprocessor data communication

chaos hyperchaos transition the chaos hyperchaos transition occurs when the second lyapunov exponent becomes positive we argue that this transition is mediated by changes in the stability of an infinite number of unstable periodic orbits embedded in the chaotic attractor bifurcations of unstable periodic orbits occur in the neighborhood of the chaos hyperchaos transition point where we observe unstable variable dimensionality we give evidence that the chaos hyperchaos transition is initiated by i the saddle repeller bifurcation of a particular unstable periodic orbit usually of low period ii the appearance of a repelling node in the saddle node bifurcation after which the chaotic attractor becomes riddled or iii the absorption of the repeller unstable node or focus originally located out of the attractor by the growing attractor

symbolic fenchel conjugation of key importance in convex analysis and optimization is the notion of duality and in particular that of fenchel duality this work explores improvements to existing algorithms for the symbolic calculation of subdifferentials and fenchel conjugates of convex functions defined on the real line more importantly these algorithms are extended to enable the symbolic calculation of fenchel conjugates on a class of real valued functions defined on mathbb r n these algorithms are realized in the form of the maple package scat

convergence of parallel multisplitting iterative methods for m matrices abstract despite the major recent activity in parallel processing few effective new algorithms designed exclusively for multiprocessors have been put forth one such is the multisplitting iterative algorithm suggested by o leary and white although they have given some sufficient conditions for convergence a general convergence theory has not been developed even for the classical situation where the coefficient matrix a is an m matrix or is symmetric positive definite in this paper we study the m matrix case in detail the multisplitting process for a rn n is recast as an ordinary iterative process for a certain block matrix a r kn kn where k is the number of processors and standard convergence results are used to develop a convergence theory for multisplitting iterative methods where a is an m matrix comparison results between multisplitting methods are established in terms of monotonic norms and for the case where a is irreducible in terms of the asymptotic convergence rate a key observation here is that in certain cases the rate of global convergence of these parallel iterative methods is inherent in the splitting of a and is independent of the manner in which the work is distributed among the processors thus in general one can distribute the work for load balancing purposes without affecting the convergence rate

bd splittings of matrices abstract a new matrix decomposition of real square singular matrices called b d splitting is proposed by extending the notion of b splitting 16 for nonsingular matrices then different convergence and comparison theorems for these splittings are discussed another convergence theorem using a generalized regular splitting is also proved as a result a new characterization of monotone matrices using regular splitting is obtained as a corollary

more on modifications and improvements of classical iterative schemes for m matrices in the last four decades many articles have been devoted to the modifications and improvements of classes of preconditioners for linear systems whose matrix coefficient is an m matrix in order to improve on the convergence rates of the classical iterative schemes jacobi gauss seidel etc the present work is a contribution towards the generalization of the most common preconditioners used so far

letter to the editor comment on a comparison theorem of the sor iterative method in this work we point out that there are incorrect assertions in the article by li ying sun li ying sun a comparison theorem for the sor iterative method j comput appl math 181 2005 336 341 moreover we answer an open problem

a multi parameters preconditioned aor iterative method for linear systems abstract in order to improve the convergence rates of iterative method solving the linear system ax b a preconditioned aor iterative method with a multi parameters preconditioner i s is proposed some convergence and comparison results for i 0 1 are given when a is an l matrix when a is an h matrix the convergence of the corresponding methods is discussed for i 0 i where i 1 furthermore effectiveness of the new preconditioner method is shown by numerical experiments numerical experiments show that our methods are superior to the basic aor iterative methods

convergence of two stage and multisplitting methods using various types of splittings we first study the convergence of both two stage iterative method and multisplitting methods using various types of splittings for solving a linear system whose coefficient matrix is an m matrix or an h matrix we next study a parallel implementation of two stage multisplitting method associated with a special multisplitting and an application of the two stage multisplitting method to parallel preconditioner of krylov subspace methods lastly we provide parallel performance results of both two stage multisplitting method and krylov subspace method with a parallel preconditioner which is derived from two stage multisplitting method on the ibm p690 supercomputer

erratum to a note on the preconditioned gauss seidel method for m matrices j comput appl math 219 1 2008 59 71 in this note some errors in a recent article by niki et al h niki t kohno m morimoto the preconditioned gauss seidel method faster than the sor method j comput appl math 219 2008 59 71 are pointed out and a new proof for the corresponding result is presented

the convergence of the modified gauss seidel methods for consistent linear systems in this paper we present a convergence analysis for the modified gauss seidel methods given in gunawardena et al linear algebra appl 154 156 1991 125 and kohno et al linear algebra appl 267 1997 113 for consistent linear systems we prove that the modified gauss seidel method converges for some values of the parameters in the preconditioned matrix

optimal periodic orbits of chaotic systems invariant sets embedded in a chaotic attractor can generate time averages that differ from the average generated by typical orbits on the attractor motivated by two different topics namely controlling chaos and riddled basins of attraction we consider the question of which invariant set yields the largest optimal value of an average of a given smooth function of the system state we present numerical evidence and analysis which indicate that the optimal average is typically achieved by a low period unstable periodic orbit embedded in the chaotic attractor

pseudo riddling in chaotic systems abstract previous investigations of riddling have focused in the case where the dynamical invariant set in the symmetric invariant subspace of the system is a chaotic attractor a situation expected to arise commonly in dynamical systems however is that the dynamics in the invariant subspace is in a periodic window in such a case there are both a stable periodic attractor and a coexisting non attracting chaotic saddle in the invariant subspace we show that riddling can still occur in a generalized sense in particular we argue that the basin of the periodic attractor in the invariant subspace contains both an open set and a set of measure zero with riddled holes that belong to the basin of another attractor we call such basin pseudo riddled and we argue that pseudo riddling can be more pervasive than riddling of chaotic attractors because it can occur regardless of whether the chaotic saddle in the invariant subspace is transversely stable or unstable we construct an analyzable model derive a scaling law for pseudo riddling and provide numerical support we also show that the chaotic saddle in the invariant subspace can undergo an enlargement into the full phase space when it becomes transversely unstable

letter to the editor a note on the preconditioned gauss seidel gs method for linear systems wen li j comput appl math 182 2005 81 90 asserted that there are some errors in article by hiroshi niki kyouji harada munenori morimoto and michio sakakihara j comput appl math 164 165 2004 587 600 and li presented a new proof for the corresponding results in h niki et al in this paper we point out some errors in li s assertion moreover we show that a new proof presented by li is imperfect

comparison theorems for splittings of matrices this paper investigates the comparisons of asymptotic rates of convergence of two iteration matrices on the basis of nonnegative matrix theory comparisons between two nonnegative splittings and between two parallel multisplitting methods are derived when the coefficient matrix a is hermitian positive semi definite comparison theorems about two p regular splittings and two parallel multisplitting methods are proved

characterization of blowout bifurcation by unstable periodic orbits blowout bifurcation in chaotic dynamical systems occurs when a chaotic attractor lying in some invariant subspace becomes transversely unstable we establish quantitative characterization of the blowout bifurcation by unstable periodic orbits embedded in the chaotic attractor we argue that the bifurcation is mediated by changes in the transverse stability of an infinite number of unstable periodic orbits there are two distinct groups of periodic orbits one transversely stable and another transversely unstable the bifurcation occurs when some properly weighted transverse eigenvalues of these two groups are balanced s1063 651x 97 50902 0 pacs number s 05 45 1b recently a novel type of bifurcation has been discovered in chaotic dynamical systems 1 2 this is the so called blowout bifurcation that occurs in dynamical systems with a symmetric invariant subspace let s be the invariant subspace in which there is a chaotic attractor since s is invariant initial conditions in s result in trajectories that remain in s forever whether the chaotic attractor in s is also an attractor in the full phase space depends on the sign of the largest lyapunov exponent l computed for trajectories in s with respect to perturbations in the subspace t which is transverse to s when l is negative s attracts trajectories transversely in the vicinity of s and hence the chaotic attractor in s is an attractor in the full phase space if l is positive trajectories in the neighborhood of s are repelled away from it and consequently the attractor in s is transversely unstable and it is hence not an attractor in the full phase space blowout bifurcation occurs when l changes from negative to positive values there are distinct physical phenomena associated with the blowout bifurcation for example near the bifurcation point where l is negative if there are other attractors in the phase space then typically the basin of the chaotic attractor in s is riddled 3 when l is slightly positive if there are no other attractors in the phase space the dynamics in the transverse subspace t exhibits an extreme type of temporally intermittent bursting behavior the on off intermittency 4 5 recent study has also revealed that blowout bifurcation can lead to symmetry breaking in chaotic systems 6 in the study of chaos theory it is important to be able to understand a bifurcation in terms of unstable periodic orbits of the system because the knowledge of periodic orbits usually yields a great deal of information about the dynamics 7 9 periodic orbits are known to be responsible for many different types of bifurcations in chaotic systems for example the period doubling bifurcation 10 and the saddlenode bifurcation are bifurcations of periodic orbits catastrophic events in chaotic systems such as crises 11 and basin boundary metamorphoses 12 are triggered by collision of periodic orbits usually of low period embedded in different dynamical invariant sets the birth of wada basin boundaries meaning common boundaries of more than two basins of attraction is caused by a saddle node bifurcation on the basin boundary 13 more recent study indicates that the riddling bifurcation bifurcation that gives birth to a riddled basin is triggered by the loss of transverse stability of some periodic orbit of low period embedded in the chaotic attractor in s 14 in view of the role of periodic orbits played in these major bifurcations it is desirable to study the blowout bifurcation by periodic orbits in this regard ashwin buescu and stewart have noticed that as a system parameter changes towards the blowout bifurcation point more and more atypical invariant measures become transversely unstable 2 at the bifurcation the natural measure of the chaotic attractor in s becomes unstable in this paper we establish a quantitative characterization of the blowout bifurcation by unstable periodic orbits embedded in the chaotic attractor in the invariant subspace s i n particular we argue that near the bifurcation there exist two groups of periodic orbits s s and s u each having an infinite number of members one transversely stable and another transversely unstable respectively the sign of the largest transverse lyapunov exponent l is determined by the relative weights of s s and s u l is negative positive when s s s u weighs over s u s s a precise definition of the weights will be described in the sequel at the bifurcation the weights of s s and s u are balanced in contrast to most known bifurcations in chaotic systems that usually involve only one or a few periodic orbits 10 14 blowout bifurcation is induced by a change in the transverse stability of an infinite number of unstable periodic orbits the num

two new modified gauss seidel methods for linear system with m matrices in 2002 h kotakemori et al proposed the modified gauss seidel mgs method for solving the linear system with the preconditioner p i s m a x h kotakemori k harada m morimoto h niki a comparison theorem for the iterative method with the preconditioner i s m a x j comput appl math 145 2002 373 378 since this preconditioner is constructed by only the largest element on each row of the upper triangular part of the coefficient matrix the preconditioning effect is not observed on the nth row in the present paper to deal with this drawback we propose two new preconditioners the convergence and comparison theorems of the modified gauss seidel methods with these two preconditioners for solving the linear system are established the convergence rates of the new proposed preconditioned methods are compared in addition numerical experiments are used to show the effectiveness of the new mgs methods

existence and uniqueness of splittings for stationary iterative methods with applications to alternating methods given a nonsingular matrix a and a matrix t of the same order under certain very mild conditions there is a unique splitting a b c such that t b 1 c moreover all properties of the splitting are derived directly from the iteration matrix t these results do not hold when the matrix a is singular in this case given a matrix t and a splitting a b c such that t b 1 c there are infinitely many other splittings corresponding to the same matrices a and t and different splittings can have different properties for instance when t is nonnegative some of these splittings can be regular splittings while others can be only weak splittings analogous results hold in the symmetric positive semidefinite case given a singular matrix a not for all iteration matrices t there is a splitting corresponding to them necessary and sufficient conditions for the existence of such splittings are examined as an illustration of the theory developed the convergence of certain alternating iterations is analyzed different cases where the matrix is monotone singular and positive semi definite are studied

comparisons of regular splittings of matrices in this article new comparison theorems for regular splittings of matrices are derived in so doing the initial results of varga in 1960 on regular splittings of matrices and the subsequent unpublished results of wo nicki in 1973 on regular splittings of matrices will be seen to be special cases of these new comparison theorems

comparison theorems using general cones for norms of iteration matrices we prove comparison theorems for norms of iteration matrices in splittings of matrices in the setting of proper cones in a finite dimensional real space by considering cone linear absolute norms and cone max norms subject to mild additional hypotheses we show that these comparison theorems can hold only for such norms within the class of cone absolute norms finally in a banach algebra setting we prove a comparison theorem for spectral radii without appealing to perron frobenius theory

comparison results for solving preconditioned linear systems in this paper we present some comparison theorems between two different modified gauss seidel mgs methods the second preconditioning based on the first preconditioning is also discussed in this paper

a comparison theorem of the improving gauss seidel method for h matrix and its comparison matrix in 1997 kohno et al have reported that the improving modified gauss seidel method which was referred to as the imgs method was superior to the classical iterative methods h matrix is a widely used matrix in this paper we apply this method to h matrix and its comparison matrix with a more general splitting we present the corresponding convergence results and the comparison results of the spectral radius the convergence rates are as well as compared some recent results are improved

gauss type preconditioning techniques for linear systems abstract many researchers have considered preconditioners chosen to eliminate the off diagonal elements of the coefficient matrix of a linear system in this work we generalize the left gauss type preconditioners y zhang t z huang x p liu modified iterative methods for nonnegative matrices and m matrices linear systems comput math appl 50 2005 1587 1602 which eliminate the strictly lower triangular elements right gauss type preconditioners that eliminate strictly upper triangular elements are proposed in this paper these gauss type preconditioners are partly derived from the lu factorization method theoretic analysis on spectral radii of the two kinds of gauss type preconditioners is given numerical experiments are used to show the performance of the improved inbuilt left and right gauss type preconditioning algorithms associated with jacobi type and gauss seidel type iterative methods

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

an extended gs method for dense linear systems davey and rosindale k davey i rosindale an iterative solution scheme for systems of boundary element equations internat j numer methods engrg 37 1994 1399 1411 derived the gsor method which uses an upper triangular matrix w in order to solve dense linear systems by applying functional analysis the authors presented an expression for the optimum w moreover davey and bounds k davey s bounds a generalized sor method for dense linear systems of boundary element equations siam j comput 19 1998 953 967 also introduced further interesting results in this note we employ a matrix analysis approach to investigate these schemes and derive theorems that compare these schemes with existing preconditioners for dense linear systems we show that the convergence rate of the gauss seidel method with preconditioner p g is superior to that of the gsor method moreover we define some splittings associated with the iterative schemes some numerical examples are reported to confirm the theoretical analysis we show that the egs method with preconditioner p g c o p t produces an extremely small spectral radius in comparison with the other schemes considered

a comparison theorem for the sor iterative method in 1997 kohno et al have reported numerically that the improving modified gauss seidel method which was referred to as the imgs method is superior to the sor iterative method in this paper we prove that the spectral radius of the imgs method is smaller than that of the sor method and gauss seidel method if the relaxation parameter w 0 1 as a result we prove theoretically that this method is succeeded in improving the convergence of some classical iterative methods some recent results are improved

on algebraic multi level methods for non symmetric systems comparison results abstract we establish theoretical comparison results for algebraic multi level methods applied to non singular non symmetric m matrices we consider two types of multi level approximate block factorizations or amg methods the amli and the mamli method we compare the spectral radii of the iteration matrices of these methods this comparison shows that the spectral radius of the mamli method is less than or equal to the spectral radius of the amli method moreover we establish how the quality of the approximations in the block factorization effects the spectral radii of the iteration matrices we prove comparisons results for different approximations of the fine grid block as well as for the used schur complement we also establish a theoretical comparison between the amg methods and the classical block jacobi and block gauss seidel methods

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

the generalized hss method for solving singular linear systems for the singular non hermitian and positive semidefinite linear systems we propose an alternating direction iterative method with two parameters based on the hermitian and skew hermitian splitting the semi convergence analysis and the quasi optimal parameters of the proposed method are discussed moreover the corresponding preconditioner based on the splitting is given to improve the semi convergence rate of the gmres method numerical examples are given to illustrate the theoretical results and the efficiency of the generalized hss method either as a solver or a preconditioner for gmres

characterization of the natural measure by unstable periodic orbits in chaotic attractors the natural measure of a chaotic set in a phase space region can be related to the dynamical properties of the unstable periodic orbits embedded in that set this result has been proven to be valid for hyperbolic chaotic systems we test the goodness of such a periodic orbit characterization of the natural measure for nonhyperbolic chaotic systems by comparing the natural measure of a typical chaotic trajectory with that computed from unstable periodic orbits our results suggest that the unstable periodic orbit formulation of the natural measure is typically valid for nonhyperbolic chaotic systems s0031 9007 97 03650 8 pacs numbers 05 45 b in studying chaotic systems one is often interested in long term statistics such as averages lyapunov exponents dimensions and other invariants of the probability density or the measure both these statistical quantities are physically meaningful only when the measure being considered is the one generated by a typical trajectory in phase space this measure is called the natural measure 1 and it is invariant under the evolution of the dynamics therefore it is of paramount physical importance to be able to understand and to be able to characterize the natural measure 2 in terms of fundamental dynamical quantities and there is nothing more fundamental than to express the natural measure in terms of the periodic orbits embedded in a chaotic attractor a key contribution along these lines was made in ref 3 in which the authors obtained an expression for the invariant natural measure in terms of the magnitude of the eigenvalues of the unstable periodic orbits embedded in the chaotic attractor they proved 3 the correctness of their expression but only for the special case of a hyperbolic dynamics 4 the validity of their results for physical which are typically nonhyperbolic situations remained however only a conjecture the purpose of this letter is to provide strong evidence for the applicability of the results of grebogi et al 3 to nonhyperbolic chaotic systems and hence validating their conjecture the long time probability density or the natural measure generated by typical trajectories of chaotic dynamical systems is generally highly singular a trajectory originated from a random initial condition in the basin of attraction of a chaotic attractor visits different parts of the attractor with drastically different probabilities call regions with high probabilities hot spots and regions with low probabilities cold spots such hot and cold spots in the attractor can be interwoven on arbitrarily fine scales in this sense chaotic attractors are said to possess a multifractal structure which is a property of the natural measure to obtain the natural measure one covers the chaotic attractor with a grid of cubes and examines the frequency with which a typical trajectory visits these cubes in the limit that both the length of the trajectory goes to infinity and the size of the grid goes to zero 5 except for an initial condition set of lebesgue measure zero in the basin of attraction of the chaotic attractor these frequencies in the cubes are the same for different choices of the initial condition x0 and they are called the natural measure specifically let fsx0 t eid be the amount of time that a trajectory from a random initial condition x0 in the basin of attraction spends in the ith covering cube ci of edge length ei in a time t the natural measure of the attractor in the cube ci is

on upper triangular block weak regular splittings of a singular m matrix abstract let a be in the frobenius normal form and let a m n be a weak regular splitting we call this weak regular splitting an upper triangular block weak regular splitting if m and n are upper triangular block matrices partitioned in the same way as a first we study weak regular splittings of a singular irreducible m matrix and obtain some interesting results we also investigate the index of the spectral radius of the iteration matrix m 1 n when the splitting a m n is assumed to be upper triangular weak regular and we obtain ind 1 m 1 n ind 0 a by this result we improve the main results obtained by h schneider

how to transform one convex function continuously into another the proximal average operator provides a parametric family of convex functions that continuously transform one convex function into another even when the domains of the two functions do not intersect we prove that the proximal average operator is a homotopy with respect to the epi topology study its properties and present several explicit formulas for specific classes of functions the parametric family inherits desirable properties such as differentiability and strict convexity from the given functions the results illustrate the powerful tools available in convex and variational analysis from both a theoretical and a computational point of view

periodic orbit theory of the blowout bifurcation this paper presents a theory for characterization of the blowout bifurcation by periodic orbits blowout bifurcation in chaotic systems occurs when a chaotic attractor lying in some symmetric invariant subspace becomes transversely unstable we present an analysis and numerical results that indicate that the bifurcation is mediated by changes in the transverse stability of an infinite number of unstable periodic orbits embedded in the chaotic attractor there are two distinct groups of periodic orbits one transversely stable and another transversely unstable the bifurcation occurs when some properly weighted transverse eigenvalues of these two groups are balanced our results thus categorize the blowout bifurcation as a unique type of bifurcation that involves an infinite number of periodic orbits in contrast to most previously known bifurcations that are mediated by only a finite number of periodic orbits

applications of convex analysis within mathematics in this paper we study convex analysis and its theoretical applications we first apply important tools of convex analysis to optimization and to analysis we then show various deep applications of convex analysis and especially infimal convolution in monotone operator theory among other things we recapture the minty surjectivity theorem in hilbert space and present a new proof of the sum theorem in reflexive spaces more technically we also discuss autoconjugate representers for maximally monotone operators finally we consider various other applications in mathematical analysis

improving the modified gauss seidel method for z matrices abstract in 1991 a d gunawardena et al reported that the convergence rate of the gauss seidel method with a preconditioning matrix i s is superior to that of the basic iterative method in this paper we use the preconditioning matrix i s if a coefficient matrix a is an irreducibly diagonally dominant z matrix then i s a is also a strictly diagonally dominant z matrix it is shown that the proposed method is also superior to other iterative methods

conditions for strict inequality in comparisons of spectral radii of splittings of different matrices we present new comparison theorems for the spectral radii of matrices arising from splittings of different matrices under nonnegativity assumptions our focus is on establishing strict inequalities of the spectral radii without imposing strict inequalities of the matrices but we also obtain new results for nonstrict inequalities of the spectral radii we emphasize two different approaches one combinatorial and the other analytic and discuss their merits in the light of the results obtained we try to get fairly general results and indicate by counter examples that some of our hypotheses cannot be relaxed in certain directions

modified iterative methods for consistent linear systems abstract in order to solve a linear system ax b certain elementary row operations are performed on a before applying the gauss seidel or jacobi iterative methods it is shown that when a is a nonsingular m matrix or a singular tridiagonal m matrix the modified method yields considerable improvement in the rate of convergence for the iterative method it is also shown that in some cases this method is superior to certain other modified iterative methods the performance of this modified method on some matrices other than m matrices is also investigated

the preconditioned gauss seidel method faster than the sor method in recent years a number of preconditioners have been applied to linear systems a d gunawardena s k jain l snyder modified iterative methods for consistent linear systems linear algebra appl 154 156 1991 123 143 t kohno h kotakemori h niki m usui improving modified gauss seidel method for z matrices linear algebra appl 267 1997 113 123 h kotakemori k harada m morimoto h niki a comparison theorem for the iterative method with the preconditioner i s m a x j comput appl math 145 2002 373 378 h kotakemori h niki n okamoto accelerated iteration method for z matrices j comput appl math 75 1996 87 97 m usui h niki t kohno adaptive gauss seidel method for linear systems internat j comput math 51 1994 119 125 10 since these preconditioners are constructed from the elements of the upper triangular part of the coefficient matrix the preconditioning effect is not observed on the nth row of matrix a in the present paper in order to deal with this drawback we propose a new preconditioner in addition the convergence and comparison theorems of the proposed method are established simple numerical examples are also given and we show that the convergence rate of the proposed method is better than that of the optimum sor

the transition to chaotic attractors with riddled basins abstract recently it has been shown that there are chaotic attractors whose basins are such that any point in the basin has pieces of another attractor basin arbitrarily nearby the basin is riddled with holes here we consider the dynamics near the transition to this situation as a parameter is varied using a simple analyzable model we obtain the characteristic behaviors near this transition numerical tests on a more typical system are consistent with the conjecture that these results are universal for the class of systems considered

a new iterative method for solving linear systems abstract a new iterative method for solving linear systems is derived it can be considered as a modification of the gauss seidel method the modified method can be two times faster than the original one

preconditioned aor iterative methods for m matrices linear systems with m matrices often appear in a wide variety of areas in this paper we give general preconditioners for solving the systems with nonsingular m matrix we show that our preconditioners increase the convergence rate of aor iterative methods numerical results for corresponding preconditioned gmres methods are also given

a note on comparison theorems for nonnegative matrices in a recent paper 4 csordas and varga have unified and extended earlier theorems of varga in 10 and wo nicki in 11 on the comparison of the asymptotic rates of convergence of two iteration matrices induced by two regular splittings the main purpose of this note is to show a connection between the csordas varga paper and a paper by beauwens 1 in which a comparison theorem is developed for the asymptotic rate of convergence of two nonnegative iteration matrices induced by two splittings which are not necessarily regular monotonic norms already used in 1 play an important role in our work here

periodic orbit analysis and scaling laws of intermingled basins of attraction in an ecological dynamical system chaotic dynamical systems with two or more attractors lying on invariant subspaces may provided certain mathematical conditions are fulfilled exhibit intermingled basins of attraction each basin is riddled with holes belonging to basins of the other attractors in order to investigate the occurrence of such phenomenon in dynamical systems of ecological interest two species competition with extinction we have characterized quantitatively the intermingled basins using periodic orbit theory and scaling laws the latter results agree with a theoretical prediction from a stochastic model and also with an exact result for the scaling exponent we derived for the specific class of models investigated we discuss the consequences of the scaling laws in terms of the predictability of a final state extinction of either species in an ecological experiment

collective behavior in coupled chaotic map lattices with random perturbations abstract numerical simulations of coupled map lattices with non local interactions i e the coupling of a given map occurs with all lattice sites often involve a large computer time if the lattice size is too large in order to study dynamical effects which depend on the lattice size we considered the use of small truncated lattices with random inputs at their boundaries chosen from a uniform probability distribution this emulates a thermal bath where deterministic degrees of freedom exhibiting chaotic behavior are replaced by random perturbations of finite amplitude we demonstrate the usefulness of this idea to investigate the occurrence of completely synchronized chaotic states as the coupling parameters are varied we considered one dimensional lattices of chaotic logistic maps at outer crisis x 4 x 1 x

computing the conjugate of convex piecewise linear quadratic bivariate functions we present a new algorithm to compute the legendre fenchel conjugate of convex piecewise linear quadratic plq bivariate functions the algorithm stores a function using a primal planar arrangement it then computes the dual arrangement associated with the conjugate by looping through vertices edges and faces in the primal arrangement and building associated dual vertices edges and faces using optimal computational geometry data structures the algorithm has a linear time worst case complexity we present the algorithm and illustrate it with numerical examples we proceed to build a toolbox for convex bivariate plq functions by implementing the addition and scalar multiplication operations finally we compose these operators to compute classical convex analysis operators such as the moreau envelope and the proximal average

comparisons of weak regular splittings and multisplitting methods comparison results for weak regular splittings of monotone matrices are derived as an application we get upper and lower bounds for the convergence rate of iterative procedures based on multisplittings this yields a very simple proof of results of neumann plemmons on upper bounds and establishes lower bounds which has in special cases been conjectured by these authors

comparison theorems for weak splittings in respect to a proper cone of nonsingular matrices the asymptotic rates of convergence of two iteration matrices induced by two splittings of the same nonsingular matrix have been compared in the works of many authors in this paper new comparison theorems which extend song s results y song numer math 65 1993 245 252 for weak splittings of the same or different types are presented these new comparison theorems are derived by using nonnegativity in respect to a proper cone k n ew converses which partially solve an open question asked by v a miller m neumann numer math 47 1985 427 434 are also presented 1999 elsevier science inc all rights reserved

comparison theorems for the convergence factor of iterative methods for singular matrices abstract iterative methods for the solution of consistent singular systems of linear equations are governed by the convergence factor of the iteration matrix t i e by the quantity t max t 1 where t is the spectrum of t theorems are presented comparing the convergence factor of two iterative methods the comparison is based on the relationship between the matrices of the splittings a cone other than the usual nonnegative hyperoctant is used to define the order used in this comparison although this cone is based on the unknown projection onto the null space of a matrix the characterization provided in the paper allows in specific instances the cone to be readily computable

optimization of the spectral radius of a product for nonnegative matrices abstract let a be an n n irreducible nonnegative matrix we show that over the set n of all n n doubly stochastic matrices s the multiplicative spectral radius sa attains a minimum and a maximum at a permutation matrix for the case when a is a symmetric nonnegative matrix a by product of our technique of proof yields a result allowing us to show that s 1 a s 2 a when s 1 and s 2 are two symmetric matrices such that both s 1 a and s 2 a are nonnegative matrices and s 1 s 2 is a positive semidefinite matrix this result has several corollaries one corollary is that s 1 a s 2 a when s 1 1 n j and s 2 1 n 1 j i where j is the matrix of all 1 s a second corollary is a comparison theorem for weak regular splittings of two monotone matrices

symmetry increasing bifurcation as a predictor of a chaos hyperchaos transition in coupled systems in weakly coupled systems it is possible to observe the coexistence of the chaotic attractors which are located out of the invariant manifold and are not symmetrical in relation to this manifold when the control parameter is changed these attractors can undergo a chaos hyperchaos transition we give numerical evidence that before this transition the coexisting attractors merge together creating an attractor symmetrical with respect to the invariant manifold we argue that the attractors that are not located at the invariant manifold can exhibit dynamical behavior similar to bubbling and on off intermittency previously observed for the attractors located at the invariant manifold and we describe the mechanism of these phenomena

a note on the improving modified gauss seidel imgs method abstract in this paper we first provide another proof for a theorem given by sun l sun a comparison theorem for the sor iterative method j comput appl math 181 2005 336 341 by showing that spectral radius of the gauss seidel method is not greater than that of the sor method when 0 1 next convergence results of the imgs method for both an h matrix and a symmetric positive definite matrix are provided

theorems on m splittings of a singular m matrix which depend on graph structure abstract let a m ne r n n be a splitting we investigate the spectral properties of the iteration matrix m 1n by considering the relationships of the graphs of a m n and m 1n we call a splitting an m splitting if m is a nonsingular m matrix and n 0 for an m splitting of an irreducible z matrix a we prove that the circuit index of m 1n is the greatest common divisor of certain sets of integers associated with the circuits of a for m splittings of a reducible singular m matrix we show that the spectral radius of the iteration matrix is 1 and that its multiplicity and index are independent of the splitting these results hold under somewhat weaker assumptions

a general preconditioner for linear complementarity problem with an m matrix abstract in this paper we first present a general preconditioner p for solving linear complementarity problem lcp associated with an m matrix a and a vector f and prove that the lcp a f is equivalent to the lcp p a p f then based on this general preconditioner p two preconditioned ssor methods for solving the linear complementarity problems are proposed we show that this general preconditioner p accelerates the convergence of two ssor methods under the assumption that p a is a z matrix in addition we also give a practically concrete choice for the preconditioner p satisfying aforementioned assumption numerical examples are used to illustrate the theoretical results obtained

comparison results of the preconditioned aor methods for l matrices abstract in this paper we first provide comparison results of several types of the preconditioned aor paor methods for solving a linear system whose coefficient matrix is an l matrix satisfying some weaker conditions than those used in the recent literature next we propose an application of paor method to a preconditioner of krylov subspace method lastly numerical results are provided to show that krylov subspace method with the paor preconditioner performs quite well as compared with the ilu 0 preconditioner

on optimal improvements of classical iterative schemes for z matrices many researchers have considered preconditioners applied to linear systems whose matrix coefficient is a z or an m matrix that make the associated jacobi and gauss seidel methods converge asymptotically faster than the unpreconditioned ones such preconditioners are chosen so that they eliminate the off diagonal elements of the same column or the elements of the first upper diagonal milaszewicz laa 93 1987 161 170 gunawardena et al laa 154 156 1991 123 143 in this work we generalize the previous preconditioners to obtain optimal methods good jacobi and gauss seidel algorithms are given and preconditioners that eliminate more than one entry per row are also proposed and analyzed moreover the behavior of the above preconditioners to the krylov subspace methods is studied

modified iterative methods for nonnegative matrices and m matrices linear systems the purpose of this paper is to present new preconditioning techniques for solving nonnegative matrices linear system and m matrices linear system ax b based on the i s a type preconditioning matrices provided by hadjidimos et al 1 and evans et al 2 convergence analysis of the proposed methods are given numerical results are presented which show the improvements on the convergence rate of the jacobi type and gauss seidel type preconditioned iterative methods

what shape is your conjugate a survey of computational convex analysis and its applications computational convex analysis algorithms have been rediscovered several times in the past by researchers from different fields to further communications between practitioners we review the field of computational convex analysis which focuses on the numerical computation of fundamental transforms arising from convex analysis current models use symbolic numeric and hybrid symbolic numeric algorithms our objective is to disseminate widely the most efficient numerical algorithms useful for applications in image processing computing the distance transform the generalized distance transform and mathematical morphology operators partial differential equations solving hamilton jacobi equations and using differential equations numerical schemes to compute the convex envelope max plus algebra computing the equivalent of the fast fourier transform multifractal analysis etc the fields of applications include among others computer vision robot navigation thermodynamics electrical networks medical imaging and network communication

a preconditioned two step modulus based matrix splitting iteration method for linear complementarity problem abstract in this paper a preconditioned two step modulus based matrix splitting iteration method for linear complementarity problems associated with an m matrix is proposed the convergence analysis of the presented method is given in particular we provide a comparison theorem between preconditioned two step modulus based gauss seidel ptmgs iteration method and two step modulus based gauss seidel tmgs iteration method which shows that ptmgs method improves the convergence rate of original tmgs method for linear complementarity problem numerical tested examples are used to illustrate the theoretical analysis

letter to the editor a note on sor ω splitting of an m matrix recently wang and huang j comput appl math 135 2001 325 corollary 4 7 assumed that sor w splitting with w 1 is a weak regular splitting of an m matrix in this note we point out that sor w splitting with w 1 can never be a weak regular splitting of an m matrix this shows that chang s counterexample j comput appl math 167 2004 251 example 1 is not an appropriate one since the sor w splitting with w 1 given there is not a weak regular splitting

convergence analysis of the preconditioned gauss seidel method for h matrices in 1997 kohno et al toshiyuki kohno hisashi kotakemori hiroshi niki improving the modified gauss seidel method for z matrices linear algebra appl 267 1997 113 123 proved that the convergence rate of the preconditioned gauss seidel method for irreducibly diagonally dominant z matrices with a preconditioner i s a is superior to that of the basic iterative method in this paper we present a new preconditioner i k b which is different from the preconditioner given by kohno et al toshiyuki kohno hisashi kotakemori hiroshi niki improving the modified gauss seidel method for z matrices linear algebra appl 267 1997 113 123 and prove the convergence theory about two preconditioned iterative methods when the coefficient matrix is an h matrix meanwhile two novel sufficient conditions for guaranteeing the convergence of the preconditioned iterative methods are given

symmetry breaking bifurcation with on off intermittency in chaotic dynamical systems when a dynamical system possesses certain symmetry there can be an invariant subspace in the phase space in the invariant subspace there can be a chaotic attractor as a parameter changes through a critical value the chaotic attractor can lose stability with respect to perturbations transverse to the invariant subspace we show that the loss of the transverse stability can lead to a symmetry breaking bifurcation characterized by lack of the system symmetry in the asymptotic attractor an accompanying physical phenomenon is an extreme type of temporally intermittent bursting behavior the mechanism for this type of symmetry breaking bifurcation is elucidated s1063 651x 96 51505 9

a note on the preconditioned gauss seidal gs method for linear systems in this note recent comparison results for preconditioned gauss seidel gs methods are discussed a new strict comparison result between two different preconditioned gs methods is given some errors in a recent article by niki et al j comput appl math 164 165 2004 587 are pointed out and a new proof for the corresponding results in niki et al is presented

on stein rosenberg type theorems for nonnegative and perron frobenius splittings abstract the stein rosenberg theorem is extended and generalized to the class of nonnegative splittings a m 1 n 1 m 2 n 2 as well as to the most generalized class of perron frobenius splittings two types of the stein rosenberg theorem are stated and proved for both classes these theorems allow us to obtain comparison results for the rate of convergence of the associated iterative methods specific assumptions are given under which the inequalities of the spectral radii become equalities or strict inequalities the theoretical results are confirmed by numerical examples

modified gauss seidel type methods and jacobi type methods for z matrices abstract in this paper we present the convergence analysis for some modified gauss seidel and jacobi type iterative methods and provide a comparison of spectral radius among the gauss seidel iterative method and these modified methods some recent results are improved

on off intermittency power spectrum and fractal properties of time series abstract some dynamical systems possess invariant submanifolds such that the dynamics restricted to the invariant submanifold is chaotic this situation arises in systems with a spatial symmetry or in the synchronization of chaotic oscillators the invariant submanifold could become unstable to perturbations in the transverse directions when a parameter of the system is changed through a critical blow out value this could result in an extreme form of temporally intermittent bursting called on off intermittency we propose a model that incorporates the universal features of systems that display on off intermittency we study this model both with and without additive noise and we derive scaling results for the power spectral density of the on off intermittent process and for the box counting dimension for the set of time intervals when the process takes on values above a given threshold we then present numerical simulations realizing these results

the survey of preconditioners used for accelerating the rate of convergence in the gauss seidel method several preconditioned iterative methods reported in the literature have been used for improving the convergence rate of the gauss seidel method in this article on the basis of nonnegative matrix comparisons between some splittings for such preconditioned matrices are derived simple numerical examples are also given

convergence analysis of the two preconditioned iterative methods for m matrix linear systems in this paper we present the two preconditioners i s and i s r for solving m matrix linear systems and discuss the convergence of the two preconditioned iterative methods meanwhile we obtain comparison theorems between the two preconditioned iterative methods and consider the solution of m matrix linear systems by preconditioned krylov subspace methods numerical experiments are given to validate the performance of the preconditioners