chaotic systems counting the number of periods abstract characterization of chaotic motion may proceed both at an averaged macroscopic level using such notions as lyapunov exponents dimensions and entropies and at a microscopic level in the latter case the number of periodic orbits being a topological invariant plays an important role for various one dimensional mappings the counting problem itself has many interesting facets and may be solved more or less completely in different ways recent progress in this counting problem is summarized with the hope that the explicit results obtained may be useful for classification of higher dimensional dissipative chaotic systems

weighted max norms splittings and overlapping additive schwarz iterations summary weighted max norm bounds are obtained for algebraic additive schwarz iterations with overlapping blocks for the solution of ax b when the coefficient matrix a is an m matrix the case of inexact local solvers is also covered these bounds are analogous to those that exist using a norms when the matrix a is symmetric positive definite a new theorem concerningp regular splittings is presented which provides a useful tool for the a norm bounds furthermore a theory of splittings is developed to represent algebraic additive schwarz iterations this representation makes a connection with multisplitting methods with this representation and using a comparison theorem it is shown that a coarse grid correction improves the convergence of additive schwarz iterations when measured in weighted max norm

variational iteration method for autonomous ordinary differential systems in this paper a new iteration technique is proposed to solve autonomous ordinary differential systems in this method general lagrange multipliers are introduced to construct correction functionals for the systems the multipliers in the functionals can be identified by the variational theory the initial approximations can be freely chosen with possible unknown constants which can be determined by imposing the boundary initial conditions some examples are given the results reveal that the method is very effective and convenient

bifurcations in discrete dynamical systems with cubic maps a representative class of discrete dynamical systems with maps with two critical points is investigated numerically in the symmetrical and the asymmetrical case a new type of bifurcation which does not occur for maps with one critical point is described the corresponding reverse bifurcation is also found new formulas for reverse bifurcation points boundary curves and accumulation curves are given

error bound and convergence analysis of matrix splitting algorithms for the affine variational inequality problem consider the affine variational inequality problem it is shown that the distance to the solution set from a feasible point near the solution set can be bounded by the norm of a natural residual at that point this bound is then used to prove linear convergence of a matrix splitting algorithm for solving the symmetric case of the problem this latter result improves upon a recent result of luo and tseng that further assumes the problem to be monotone

symbolic dynamics of the cubic map abstract a symmetric cubic map on the interval describing isolated stable or bistable orbits and different kind of bifurcation was analyzed stable orbits were presented by itineraries permutation parentheses and matrices itineraries were used to order the cycles with respect to the parameter and also to order the coordinates of the cycle in the interval linear symbolic dynamics was introduced to study the periodic unstable points coexisting with stable orbits by means of a matrix of intervals the non zero entries of these matrices have an n shape corresponding to the n shape of the cubic the characteristic polynomial of the matrix of intervals was in one to one correspondence with the itinerary equations to calculate the number of different kind of orbits were explicitly written and used to calculate those numbers for a small number of periods

a domain decomposition method for a kind of optimization problems in this paper we are concerned with the monotone convergence of a multiplicative method for solving a kind of optimization problems we show that the iterate sequence produced by the method converges to the solution of the problem monotonically

what is the effective impact of the explosive orbital growth in discrete time one dimensional polynomial dynamical systems abstract we study the distribution of periodic orbits in one dimensional two parameter maps specifically we report an exact expression to quantify the growth of the number of periodic orbits for discrete time dynamical systems governed by polynomial equations of motion of arbitrary degree in addition we compute high resolution phase diagrams for quartic and for both normal forms of cubic dynamics and show that their stability phases emerge all distributed in a similar way preserving a characteristic invariant ordering such coincidences are remarkable since our exact expression shows the total number of orbits of these systems to differ dramatically by more than several millions even for quite low periods all this seems to indicate that surprisingly the total number and the distribution of stable phases is not significantly affected by the specific nature of the nonlinearity present in the equations of motion

additive schwarz algorithm for the nonlinear complementarity problem with m function in this paper an additive schwarz algorithm is considered for solving the finite dimensional nonlinear complementarity problem with m function the monotone convergence of the algorithm is obtained with special choices of initial values moreover the weighted max norm bound is obtained for the iterative errors

counting the number of periods in one dimensional maps with multiple critical points abstract the problem of counting the number of different periodic orbits in continuous maps of an interval is solved a map with m monotone pieces laps and m 1 critical points may have at most m 1 independent kneading sequences which provide the most convenient parameters for the map when one or more kneading sequences are kept constant or bound to vary simultaneously various degenerated cases of the map arise the number of period n orbits of the general m lap map as various degenerated cases is given by combinations of nink n with k m where nm n is the number of period n orbits of a particular kind of degenerated m lap map with only one kneading sequence the quantity nm n may be calculated or enumerated by many different methods which are all discussed in this paper we also give tabulated values for nm n for m 2 to 7

homotopy perturbation method a new nonlinear analytical technique in this paper a new perturbation method is proposed in contrast to the traditional perturbation methods this technique does not require a small parameter in an equation in this method according to the homotopy technique a homotopy with an imbedding parameter email protected 0 1 is constructed and the imbedding parameter is considered as a small parameter so the method is called the homotopy perturbation method which can take the full advantages of the traditional perturbation methods and homotopy techniques to illustrate its effectiveness and its convenience a duffing equation with high order of nonlinearity is used the result reveals that its first order of approximation obtained by the proposed method is valid uniformly even for very large parameter and is more accurate than the perturbation solutions

some flesh on the skeleton the bifurcation structure of bimodal maps abstract one of the main tools in the numerical study of two parameter families of one dimensional maps is the drawing of curves in parameter space corresponding to the existence of superstable periodic orbits we use kneading theory to describe the structure of these sets of curves for the case of maps with at most two turning points then we explain how the bifurcation structure hangs on this skeleton

chaotic iterative methods for the linear complementarity problems a class of parallel multisplitting chaotic relaxation methods is established for the large sparse linear complementarity problems and the global and monotone convergence is proved for the h matrix and the l matrix classes respectively moreover comparison theorem is given which describes the influences of the parameters and the multiple splittings upon the monotone convergence rates of the new methods

weighted max norm estimate of additive schwarz iteration scheme for solving linear complementarity problems in this paper we consider an algebraic additive schwarz iteration scheme for solving the finite dimensional linear complementarity problem that involves an m matrix the scheme contains some existing algorithms as special cases we establish monotone convergence of the iteration scheme under appropriate conditions moreover using the concept of weak regular splitting we estimate weighted max norm bounds for iteration errors thereby we show that the sequence generated by the iteration scheme converges to the unique solution of the problem without any restriction on the initial point

the exp function method and generalized solitary solutions the exp function method is applied to a system of nonlinear partial differential equations and generalized solitary solutions with free parameters are obtained the solution procedure is simple with the help of symbolic computation

on monotone and geometric convergence of schwarz methods for two sided obstacle problems convergence of schwarz methods is discussed for certain discretizations of two sided obstacle problems monotone and especially geometrical convergence are established if the stiffness matrix is a kind of m matrix and an h independent convergence rate is proved for uniformly overlapping decomposition

symbolic dynamics and skeletons of circle maps a circle map maps the circumference of a circle into itself such maps are of physical and biological interest because they arise naturally in many circumstances as for example the periodic forcing of limit cycle oscillators the canonical example of circle maps is x tx a sin2 x mod 1 where a and are two constants t is an integer that gives the topological degree of cycles that contain extremal points of are stable and are called superstable cycles the locus of superstable cycles in a parameter space is called the skeleton the critical points of x 0 and the discontinuous points of x 1 break into monotonic branches in this paper we propose a novel notation for symbolically representing the branches of x and discuss the symbolic sequences for the skeleton simple rules based on continuity of periodic orbits as parameters change allow us to construct the skeleton associated with all periodic orbits for the circle maps with t 0 0 a 1 0 and t 1 0 a 1 0 recursion formulae to compute the numbers of periodic orbits for t 0 and t 1 at 0 are given

analysis of non overlapping domain decomposition algorithms with inexact solves in this paper we construct and analyze new non overlapping domain decomposition preconditioners for the solution of second order elliptic and parabolic boundary value problems the preconditioners are developed using uniform preconditioners on the subdomains instead of exact solves they exhibit the same asymptotic condition number growth as the corresponding preconditioners with exact subdomain solves and are much more efficient computationally moreover this asymptotic condition number growth is bounded independently of jumps in the operator coefficients across subdomain boundaries we also show that our preconditioners fit into the additive schwarz framework with appropriately chosen subspace decompositions condition numbers associated with the new algorithms are computed numerically in several cases and compared with those of the corresponding algorithms in which exact subdomain solves are used

algebraic theory of multiplicative schwarz methods the convergence of multiplicative schwarz type methods for solving linear systems when the coefficient matrix is either a nonsingular m matrix or a symmetric positive definite matrix is studied using classical and new results from the theory of splittings the effect on convergence of algorithmic parameters such as the number of subdomains the amount of overlap the result of inexact local solves and of coarse grid corrections global coarse solves is analyzed in an algebraic setting results on algebraic additive schwarz are also included

convergence theory of restricted multiplicative schwarz methods convergence results for the restricted multiplicative schwarz rms method the multiplicative version of the restricted additive schwarz ras method for the solution of linear systems of the form ax b are provided an algebraic approach is used to prove convergence results for nonsymmetric m matrices several comparison theorems are also established these theorems compare the asymptotic rate of convergence with respect to the amount of overlap the exactness of the subdomain solver and the number of domains moreover comparison theorems are given between the rms and ras methods as well as between the rms and the classical multiplicative schwarz method

multigrid algorithms for variational inequalities we present an iterative scheme for obstacle problems described by variational inequalities at each step of the iteration the method leads to a reduced linear resp nonlinear algebraic system whi

asymptotology by homotopy perturbation method in the paper an heuristical example is given to illustrate the basic idea of the homotopy perturbation method so that homotopy perturbation method has made all that is necessary simple and all that is complex unnecessary

investigation on non linear vibration in arched beam for bridges construction via agm method analyzing and modeling the vibrational behavior of arched bridges during earthquake in order to decrease the exerted damages to the structure is a very hard task to do in the present paper this item has been performed analytically for the first time due to the importance of arched bridges building as a great structure in the human being civilization and its specifications such as transferring vertical loads to its arcs and the lack of bending moments and shearing forces this case study is devoted to this special issue therefore the nonlinear vibration of arched bridges has been modeled and simulated by an arched beam with harmonic vertical loads and its behavior has been investigated by analyzing a nonlinear partial differential equation governing on the system it is notable that the procedure has been done analytically by agm akbari ganji s method it has found that angular frequency is increased significantly by escalating the amount of the center outlet furthermore comparisons have been made between the obtained results by numerical method rkf 45 and agm in order to assess the scientific validity obtained results indicate the precision of the achieved answers by agm method

exact solutions of the invariant density for piecewise linear approximation to cubic maps a two parameter piecewise linear approximation to cubic maps has been proposed and analytically investigated the notion of n k sequences has been introduced to find the exact solutions of the invariant density of the model via the frobenius perron operator for a one parameter version of the present map the authors obtain the exact invariant density and the corresponding parameter equation their results can be regarded as a generalisation to the consequences obtained by derrida et al 1978 for the tent map

solitary solutions periodic solutions and compacton like solutions using the exp function method in this paper the exp function method is used for seeking solitary solutions periodic solutions and compacton like solutions of nonlinear differential equations the combined kdv mkdv equation and the liouville equation are chosen to illustrate the effectiveness and convenience of the proposed method

asynchronous domain decomposition methods for continuous casting problem two asynchronous domain decomposition methods which appear to be a two stage schwarz alternating algorithms to solve the finite difference schemes approximating dynamic continuous casting problem are theoretically and numerically studied fully implicit and semi implicit implicit for the diffusion operator while explicit for the nonlinear convective term finite difference schemes are considered unique solvability of the finite difference schemes as well as a monotone dependence of the solution on the right hand side the so called comparison theorem are proved geometric rate of convergence for the iterative methods is investigated the comparison theorem being the main tool of this study numerical results are included and analyzed

a multiplicative schwarz iteration scheme for solving the linear complementarity problem with an h matrix in this paper we consider an algebraic multiplicative schwarz iteration scheme for solving the linear complementarity problem that involves an h matrix we show that the sequence generated by the multiplicative schwarz iteration scheme converges to the unique solution of the problem without any restriction on the initial point for different overlapping sizes the convergence rate of the proposed method is analyzed in an algebraic setting moreover we establish monotone convergence of the proposed method under appropriate conditions numerical results show that efficiency can be achieved by the multiplicative schwarz iteration scheme

overlapping additive and multiplicative schwarz iterations for h matrices abstract in recent years an algebraic framework was introduced for the analysis of convergence of schwarz methods for the solution of linear systems of the form ax b within this framework additive and multiplicative schwarz were shown to converge when the coefficient matrix a is a nonsingular m matrix or a symmetric positive definite matrix in this paper many of these results are extended to the case of a being an h matrix the case of inexact local solves is also considered in addition the two level scheme is studied i e when a coarse grid correction is used in conjunction with the additive or the multiplicative schwarz iterations

on the schwarz alternating method with more than two subdomains for nonlinear monotone problems in a hilbert space an iterative process is defined which generalizes the schwarz alternating method for nonlinear monotone problems to more than two subdomains and the convergence of this process is proved in the case of the problems with the solution in a convex of h 0 1 omega it is proved that the method converges geometrically numerical results are given concerning the problem of elastic plastic torsion of prismatic bars and the problem of seepage of fluids through porous media

a multisplitting method for symmetric linear complementarity problems abstract over the years many methods for solving the linear complementarity problem lcp have been developed most of these methods have their origin in solving a system of linear equations in particular much attention has recently been paid on the class of iterative methods called the splitting method which is an extension of the matrix splitting method for solving a system of linear equations such as jacobi gauss seidel and sor methods furthermore as a method for solving a system of linear equations o leary and white have proposed a parallel iterative method called the multisplitting method this method makes use of a set of different splittings of the coefficient matrix which may be dealt with independently of each other the results obtained from those splitting iterations are combined to define the multisplitting iterates thus the method may be effectively implemented on multiprocessors in this paper we extend the idea of the multisplitting to the symmetric lcp in particular we establish some convergence results for the multisplitting method which generalize the corresponding convergence results for the splitting method for lcp we also report some computational results with the proposed method

comparison of homotopy perturbation method and homotopy analysis method comparison of homotopy perturbation method hpm and homotopy analysis method is made revealing that the former is more powerful than the later furthermore the hpm is further developed in this paper by applying the modern perturbation methods

an additive schwarz method for variational inequalities this paper proposes an additive schwarz method for variational inequalities and their approximations by finite element methods the schwarz domain decomposition method is proved to converge with a geometric rate depending on the decomposition of the domain the result is based on an framework of convergence analysis established for general variational inequalities in hilbert spaces

finding general and explicit solutions of high nonlinear equations by the exp function method in this paper davey stewartson equation and coupled klein gordon schrodinger kgs equations and homogeneous nonlinear convection diffusion problem are solved using the exp function method the capabilities and wide range applications of the exp function method are illustrated this method can be used as an alternative to obtain analytical and approximate solutions of different types of differential equations in engineering mathematics