the attractor of an equation of tricomi s type consider the attractor a of an equation of pendulum type with friction driven by a constant torque the results of m levi and also obtained independently by q min s xian and z jinyan show that if the friction coefficient is larger than a certain bound then a is homeomorphic to the circle we shall study the bifurcation diagram of a particular class of equations of pendulum type and show that the bounds obtained before are optimal

existence of dynamical hull functions with two variables for the ac driven frenkel kontorova model abstract in this paper we are concerned with the dynamics of incommensurate structures in the overdamped limit case of the frenkel kontorova model with periodic driving force we show in the infinite dimensional phase space there exists an ordered circle on which both the poincare map and the space shift map induce orientation preserving circle homeomorphisms with rotation numbers and respectively the latter characterizing the mean spacing of particles furthermore we prove that the average velocity for the particle chain with irrational exists and t where t denotes the period of the driving force and it is continuous and nondecreasing with respect to f the average of the external driving force if the shift map on the invariant ordered circle is ergodic we demonstrate the existence of two variable dynamical hull functions and that there are no mode locking behaviors

a hybrid gradient and feasible direction pivotal solution algorithm for general linear programs the simplex solution algorithm for linear programs lp can be considered as a sub gradient direction method therefore a full gradient solution algorithm might be more efficient we have developed a full gradient method which consists of three phases the initialization phase provides the initial tableau which may not have a full set of basis the push phase uses a full gradient vector of the objective function to obtain a feasible vertex this is then followed by a series of pivotal steps using the sub gradient which leads to an optimal solution if exists in the final iteration phase at each of these iterations the sub gradient provides the desired direction of motion within the feasible region the algorithm hits and or moves on the constraint hyper planes and their intersections to reach an optimal vertex if exists the algorithm works in the original decision variables space therefore there is no need to introduce any new extra variables such as artificial variables the proposed algorithm is practical easy for the user to understand and provides useful information for the decision makers

applications of the parametric programming procedure abstract many classes of mathematical programming problems can be formulated as a linear program with a parametric objective function gass and saaty developed in the early 1950 s a parametric programming procedure for solving the latter problem this procedure is relevant to solution strategies for many problem types but is underutilized often because the relationship of the procedure to the relevant problem class is not recognized in this paper five recent applications of the parametric programming procedure are presented

one dimensional random attractor and rotation number of the stochastic damped sine gordon equation abstract this paper is devoted to the study of the asymptotic dynamics of the stochastic damped sine gordon equation with homogeneous neumann boundary condition it is shown that for any positive damping and diffusion coefficients the equation possesses a random attractor and when the damping and diffusion coefficients are sufficiently large the random attractor is a one dimensional random horizontal curve regardless of the strength of noise hence its dynamics is not chaotic it is also shown that the equation has a rotation number provided that the damping and diffusion coefficients are sufficiently large which implies that the solutions tend to oscillate with the same frequency eventually and the so called frequency locking is successful

fractional iterates for n dimensional maps this paper constitutes an extension of results of the paper by narayaninsamy int j bifurcations chaos 6 1 1996 55 67 we study here the solutions of the problem of the fractional iteration for n dimensional maps

rotation number of the overdamped frenkel kontorova model with ac driving abstract the dynamical behavior of the overdamped and overdamped limit cases of the frenkel kontorova model with ac driving is studied the relationship is established between the average velocity of the particles and the rotation number of the orientation preserving homeomorphism on the circle induced from the poincare map the dynamical behavior for both cases is determined completely by the rotation number numerical results are also provided for the second order system to support the theoretical conclusions

a mathematical programming approach to clusterwise regression model and its extensions abstract the clusterwise regression model is used to perform cluster analysis within a regression framework while the traditional regression model assumes the regression coefficient to be identical for all subjects in the sample the clusterwise regression model allows to vary with subjects of different clusters since the cluster membership is unknown the estimation of the clusterwise regression is a tough combinatorial optimization problem in this research we propose a generalized clusterwise regression model which is formulated as a mathematical programming mp problem a nonlinear programming procedure with linear constraints is proposed to solve the combinatorial problem and to estimate the cluster membership and simultaneously moreover by integrating the cluster analysis with the discriminant analysis a clusterwise discriminant model is developed to incorporate parameter heterogeneity into the traditional discriminant analysis the cluster membership and discriminant parameters are estimated simultaneously by another nonlinear programming model

on the average number of steps of the simplex method of linear programming the goal is to give some theoretical explanation for the efficiency of the simplex method of george dantzig fixing the number of constraints and using dantzig s self dual parametric algorithm we show that the number of pivots required to solve a linear programming problem grows in proportion to the number of variables on the average

a lagrangian treatment of certain nonlinear clustering problems abstract a clustering problem is described in which the objective function involving both within and between subset distances in order to induce separation between clusters and yield a realistic measure is highly nonlinear and difficult to analyze by fixing the number of within group distances however a linear objective function is achieved and the problem becomes of the more common set partitioning kind with the further assumption of collinear objects yielding an important subclass of problem the spp matrix is unimodular and the simplex method suffices to solve the integer programming problem a lagrange relaxation on the constraint for the number of within group distances allows then the nonlinear clustering problem to be solved exactly up to the occurrence of gaps as a sequence of lp s an implementation for small problems using the mpsx of ibm is presented as are the results from various examples for the non collinear case the mapping of data into a single dimension is discussed as are strategies for solving larger problems the behavior of the lagrangian generated function values is examined and found to be exploitable useful approximations in the event of gaps are found to be easily computed

global behavior in the dynamical equation of j j type on considere l equation de la forme 1 ecos sin asinwt on etudie le cas e 0 2 on montre que chaque trajectoire est asymptotiquement periodique ou quasi periodique

dynamical behavior in coupled systems of j j type abstract in this paper we analyze the structure of the global attractor of the coupled josephson junction system system and prove that there exists a unique invariant restricted horizontal curve when the damping is not small and the coupling is not very weak this invariant curve plays a very important role in analyzing the dynamical behavior of the system when the damping is large enough the unique invariant curve is globally attracting so that the system behaves exactly as a onedimensional system

on fractional iterates of a brouwer homeomorphism embeddable in a flow we present a method for finding continuous and consequently homeomorphic orientation preserving iterative roots of a brouwer homeomorphism which is embeddable in a flow to obtain the roots we use a countable family of maximal parallelizable regions of the flow which is a cover of the plane the maximal parallelizable regions are unions of equivalence classes of an appropriate equivalence relation we show that if an equivalence class is invariant under the nth iterate of a brouwer homeomorphism g then it is invariant under g we use this fact to prove that each maximal parallelizable region of the flow must be invariant under all homeomorphic orientation preserving iterative roots of the given brouwer homeomorphism

classroom note initialization of the simplex algorithm an artificial free approach the simplex algorithm requires artificial variables for solving linear programs which lack primal feasibility at the origin point we present a new general purpose solution algorithm which obviates the use of artificial variables the algorithm searches for a feasible segment of a boundary hyperplane a face of feasible region or an intersection of several faces by using rules similar to the ordinary simplex

periodic solutions of the forced pendulum exchange of stability and bifurcations abstract we study the t periodic solutions of the forced pendulum equation u cu a sin u f t where f satisfies f t t 2 f t we prove that this equation always has at least two geometrically distinct t periodic solutions u0 and u1 we then investigate the dependence of the set of t periodic solutions on the forcing strength we prove that under some restriction on the parameters a c the periodic solutions found before can be smoothly parameterized by and that there are some intervals for which u0 u1 are the only t periodic solutions up to geometrical equivalence but there are other intervals in which additional t periodic solutions bifurcate off the branches u0 u1 we characterize the global structure of the bifurcating branches related to this bifurcation phenomenon is the phenomenon of exchange of stability in some intervals u0 is dynamically stable and u1 is unstable while in other intervals the reverse is true a phenomenon which has important consequences for the dynamics of the forced pendulum as we show by both theoretical analysis and numerical simulation

structure of the global attractor for a second order strongly damped lattice system in this paper we consider the dynamical behavior of a second order strongly damped lattice system where the coupled operator is nonnegative definite symmetric firstly we prove the existence of a global attractor and give an upper bound of hausdorff dimension of the global attractor which keeps bounded for large strongly damping then we show that when the damping term is linear and the damping is suitable large the system has an unbounded one dimensional global attractor which is a restricted horizontal curve

dynamics in coupled oscillators with recurrent random forcing a pde approach the current paper is devoted to the study of coupled oscillators with recurrent random forcing special attention is given to the solutions having the same recurrence randomness as that of the forcing recurrent random solutions for short by embedding coupled oscillators into coupled parabolic equations it establishes a general theorem on the existence of recurrent random solutions it also finds conditions under which such solutions are unique when the recurrent forcing is actually quasi periodic or almost periodic recurrent solutions are refereed to as quasi periodic or almost periodic solutions in a weak sense and they are quasi periodic or almost periodic in the classical sense under the uniqueness conditions in addition applications of the general theory to coupled duffing type oscillators and josephson junctions are considered and the results obtained extend several existing ones for quasi periodic duffing oscillators

a practical anti cycling procedure for linearly constrained optimization a procedure is described for preventing cycling in active set methods for linearly constrained optimization including the simplex method the key ideas are a limited acceptance of infeasibilities in all variables and maintenance of a working feasibility tolerance that increases over a long sequence of iterations the additional work per iteration is nominal and stalling cannot occur with exact arithmetic the method appears to be reliable based on computational results for the first 53 linear programming problems in thenetlib set

coupling schemes for cluster synchronization in coupled josephson equations abstract we propose an approach to constructing different coupling schemes to stabilize selected cluster synchronization patterns for coupled josephson equations in particular we select a coupling scheme to create synchronization with frequency ratio m 1 m 2 m n which can be arbitrarily chosen and is independent of the frequencies of the uncoupled oscillators we also discuss coupled discrete systems

a recursive lagrangian method for clustering problems abstract the problem of partitioning a finite set of objects into subsets or clusters so as to optimize an objective function is treated it is shown that for the particular case where all the objects lie upon a line the clustering problem may be solved exactly by an algorithm utilizing only lagrangian relaxation and dynamic programming by exploiting the approximately unimodal shape of the envelope of optimal objective values a search procedure in the space is devised and this allows the solution in reasonable computation time of the combinatorially large problem several examples are presented and the method is applied to one set of noncollinear data which is first transformed so as to become collinear further promising directions for computational enhancement are noted the important aspects of computer implementation are given

the effect of inversely unstable solutions on the attractor of the forced pendulum equation with friction abstract consider the attractor a of a periodically forced equation of pendulum type with linear friction in the cylinder levi and independently min xian and jinyan show that if the friction coefficient is larger than a certain bound then a is homeomorphic to the circle we shall give a topological version of the definition of inversely unstable solution of n levinson and show that the appearance of such solutions imply that a is not homeomorphic to the circle as an application we shall show that the bounds on the friction coefficient obtained before are optimal

some global bifurcations of two dimensional endomorphisms by use of critical lines many studies have been devoted to discrete dynamical systems represented by homeomorphisms and diffeomorphisms since the works by smale l and gavrilov and shilnikov 2 31 but comparatively few works have been devoted to endomorphisms that is to maps with a nonunique inverse however the mathematical models derived from the description of evolutive phenomena in several applicative fields are often represented by discrete dynamical systems or maps with a nonunique inverse this is particularly true in the socio economic field 4 61 but examples can also be found in ecology 7 81 biology 9 l l and control systems 121 this work concerns the study of the global dynamics of recurrences represented by twodimensional endomorphisms of applicative interest in the economic context useful notions for the local and global analysis of nonlinear endomorphisms have been introduced by gumowski and mira ll 13 141 for two dimensional endomorphisms in particular this is the notion of critical lines or critical curves representing the twodimensional extension of the critical points in one dimensional maps and the notions of absorbing area and chaotic area bounded by elements of critical lines which determine phase plane domains where chaotic solutions are located these notions which will be recalled in the next section together with some bifurcations involving the chaotic areas are the basic concepts and properties that we shall use in the analysis of the subsequent sections the aim of this paper is to illustrate by an example how powerful the critical lines may be in studying the global dynamics and bifurcations of plane nonlinear endomorphisms particularly when they are used in combination with more traditional techniques it is known that in diffeomorphisms global bifurcations which qualitatively change the structure of a chaotic attractor called interior crises or catastrophes by some authors 15 181 are characterized by the stable manifold of a saddle approaching and touching the chaotic attractor we shall find that this phenomenology also occurs in the endomorphisms of our model with a particular role being played by the critical lines in section 3 we shall describe our mathematical model a two dimensional piecewise linear map which we call map f together with some of its elementary properties and the definition of three classes of endomorphisms into which the analysis can be conveniently subdivided endomorphisms of the first class have been the object of a previous work 191 here we shall show in section 4 how several bifurcations involving chaotic areas can be explained by use of critical lines the main result is the analytical determination of the bifurcation which discriminates between the global attractivity of an absorbing area and the nonglobal attractivity

making progress during a stall in the simplex algorithm abstract all the standard methods for avoiding cycling in the simplex algorithm use row selection rules for resolving degeneracy the method proposed in this paper does not instead the gass saaty parametric method applied to the objective cost form is used to choose the incoming column the pivot row choice among blocking rows i e those tied for pivot is arbitrary a simple anticycling device is used which avoids dual degeneracy of the parametrized objective with probability one tests were run on nine highly degenerate practical test problems ranging in size from small to large using minos software the standard simplex method required 31 195 iterations of which 20 504 or 66 were blocked pivots using the parametric scheme required 13 812 iterations of which 67 were blocked the reduction in the number of iterations for this set of highly degenerate test problems was 56 the cpu time reduction was 48

discrete rotating waves in a ring of coupled mechanical oscillators with strong damping by establishing the monotonicity in a ring of coupled oscillators with strong damping we show the global stability of the discrete rotating wave whose existence is demonstrated by applying the schauder fixed point theorem

on the node fractional iterates let m be a bounded convex set of r n let f m r n be a continuous non bijective map we study solutions g of the functional equation g m x g g m 1 x f x m 2 this functional equation represents the problem of the fractional iteration for f

a connection between fractional iteration and graph theory we consider the problem of the fractional iteration for n dimensional maps we have two objectives in this paper the first objective is to establish a connection between fractional iteration and graph theory the second objective is to make a study concerning the problem of fractional iteration for n dimensional maps

dynamics in a chain of overdamped pendula driven by constant torques this paper studies the dynamical behavior of a chain of overdamped pendula driven by constant torques with nearest neighbor coupling the coupling constant k is assumed to be 0 independent of n it is shown that when the system does not have equilibrium points the global attractor of this system is a one dimensional closed curve so no matter what input frequencies omega j are used the existence uniqueness and global stability of a limit cycle of second kind are proved therefore any solution will be frequency locked in the long time limit on the other hand if there are equilibrium points in the system any solution is bounded and converges to an equilibrium point

globally and locally attractive solutions for quasi periodically forced systems abstract we consider a class of differential equations x x g x f t with r d describing one dimensional dissipative systems subject to a periodic or quasi periodic diophantine forcing we study existence and properties of trajectories with the same quasi periodicity as the forcing for g x x 2 p 1 p n we show that when the dissipation coefficient is large enough there is only one such trajectory and that it describes a global attractor in the case of more general nonlinearities including g x x 2 describing the varactor equation we find that there is at least one trajectory which describes a local attractor

a computationally stable solution algorithm for linear programs an active area of research for linear programming is to construct initial simplex tableau that is close to the optimal solution and to improve its pivoting selection strategies efficiently in this paper we present a new approach to the problem of initialization and pivoting rules the algorithm is free from any artificial variables and artificial constraints known as the big m methods by supplying the simplex method without using any large numbers therefore the result is computationally stable and provides a better initial basis that reduces the number of pivoting iterations efficiency a three phase simplex type solution algorithm is developed for solving general linear programs in phase 1 greater than constraints are relaxed and the problem is solved starting at the origin if it is unbounded then the objective function is modified in order to have a bound feasible solution then in phase 2 the greater than constraints are introduced in a dual simplex framework following this in the case the original objective function was modified phase 3 restores optimality for the original objective function the algorithm is numerically stable and works with the original decision variables our initial experimental study indicates that the proposed algorithm is more efficient than both the primal and the dual simplex in terms of the number of iterations

polynomial expected behavior of a pivoting algorithm for linear complementarity and linear programming problems we show that a particular pivoting algorithm which we call the lexicographic lemke algorithm takes an expected number of steps that is bounded by a quadratic inn when applied to a random linear complementarity problem of dimensionn we present two probabilistic models both requiring some nondegeneracy and sign invariance properties the second distribution is concerned with linear complementarity problems that arise from linear programming in this case we give bounds that are quadratic in the smaller of the two dimensions of the linear programming problem and independent of the larger similar results have been obtained by adler and megiddo

parametric linear programming and anti cycling pivoting rules the traditional perturbation or lexicographic methods for resolving degeneracy in linear programming impose decision rules that eliminate ties in the simplex ratio rule and therefore restrict the choice of exiting basic variables bland s combinatorial pivoting rule also restricts the choice of exiting variables using ideas from parametric linear programming we develop anticycling pivoting rules that do not limit the choice of exiting variables beyond the simplex ratio rule that is any variable that ties for the ratio rule can leave the basis a similar approach gives pivoting rules for the dual simplex method that do not restrict the choice of entering variables