analysis of a numerical dynamic programming algorithm applied to economic models in this paper we develop a discretized version of the dynamic programming algorithm and study its convergence and stability properties we show that the computed value function converges quadratically to the true value function and that the computed policy function converges linearly as the mesh size of the discretization converges to zero further the algorithm is stable we also discuss several aspects of the implementation of our procedures as applied to some commonly studied growth models

stable cycles in a cournot duopoly model of kopel we consider a discrete map proposed by m kopel that models a nonlinear cournot duopoly consisting of a market structure between the two opposite cases of monopoly and competition the stability of the fixed points of the discrete dynamical system is analyzed synchronization of two dynamics parameters of the cournot duopoly is considered in the computation of stability boundaries formed by parts of codim 1 bifurcation curves we discover more on the dynamics of the map by computing numerically the critical normal form coefficients of all codim 1 and codim 2 bifurcation points and computing the associated two parameter codim 1 curves rooted in some codim 2 points it enables us to compute the stability domains of the low order iterates of the map we concentrate in particular on the second third and fourth iterates and their relation to the period doubling 1 3 and 1 4 resonant neimark sacker points

numerical normal forms for codim 2 bifurcations of fixed points with at most two critical eigenvalues in this paper we derive explicit formulas for the normal form coefficients to verify the nondegeneracy of eight codimension two bifurcations of fixed points with one or two critical eigenvalues these include all strong resonances as well as the degenerate flip and neimark sacker bifurcations applying our results to n dimensional maps one avoids the computation of the center manifold but one can directly evaluate the critical normal form coefficients in the original basis the formulas remain valid also for n 2 and allow one to avoid the transformation of the linear part of the map into jordan form the developed techniques are tested on two examples 1 a three dimensional map appearing in adaptive control 2 a periodically forced epidemic model we compute numerically the critical normal form coefficients for several codim 2 bifurcations occurring in these models to compute the required derivatives of the poincare map for the epidemic model the automatic differentiation package adol c is used

some ramifications of a theorem of boas and pollard concerning the completion of a set of functions in l 2 about fifty years ago r p boas and harry pollard proved that an orthonormal system that is completable by the adjunction of a finite number of functions also can be completed by multiplying the elements of the given system by a fixed bounded nonnegative measurable function in subsequent years several variations and extensions of this theorem have been given by a number of other investigators and this program is continued here a mildly surprising corollary of one of the results is that the trigonometric and walsh systems can be multiplicatively transformed into quasibases for l1 0 1 1 a curious connection between disparate mathematical ideas occurs in the theory of complete systems of real valued measurable functions the first of these notions is due to boas and pollard 3 who showed that certain incomplete systems in l may be transformed into complete systems by multiplying the elements of the system by a fixed bounded nonnegative measurable function subsequently talalyan 17 18 proved that a system is complete in measure on a set e if and only if for every positive e is complete in l ee where ee is a measurable subset of e that has measure greater than e e in addition talalyan showed that if has these properties so also does any family obtained from by deleting a finite number of its members later goffman and waterman 6 gave a new proof of this theorem and observed that it is always possible to make certain infinite deletions from a system that is complete in measure so as to leave a residual system that also satisfies this condition the somewhat surprising coincidence of these ideas was noted in 15 where the following result was established theorem a let e be a measurable set of finite positive measure and let n n 1 2 be a subset of l e then the following conditions are equivalent bp there exists a bounded measurable function m such that m n n 1 2 is complete in l e m is complete in measure on e received by the editors march 8 1995 and in revised form july 21 1995 1991 mathematics subject classification primary 42b65 42c15 46b15 41a30 41a58

numerical periodic normalization for codim 1 bifurcations of limit cycles explicit computational formulas for the coefficients of the periodic normal forms for all codim 1 bifurcations of limit cycles in generic autonomous odes are derived they include second order coefficients for the fold limit point bifurcation as well as third order coefficients for the flip period doubling and neimark sacker torus bifurcations the formulas are independent of the dimension of the phase space and involve solutions of certain boundary value problems on the interval 0 t where t is the period of the critical cycle as well as multilinear functions from the taylor expansion of the right hand sides near the cycle the formulas allow us to distinguish between sub and supercritical bifurcations in agreement with earlier asymptotic expansions of the bifurcating solutions our formulation makes it possible to use robust numerical boundary value algorithms based on orthogonal collocation rather than shooting techniques which greatly expands its applicability the actual implementation is described in detail we include three numerical examples in which codim 2 singularities are detected along branches of codim 1 bifurcations of limit cycles as zeros of the periodic normal form coefficients

an improved algorithm for computing topological entropy a new algorithm is presented for computing the topological entropy of a unimodal map of the interval the accuracy of the algorithm is discussed and some graphs of the topological entropy which are obtained using the algorithm are displayed

bifurcation in model reference adaptive control systems abstract we study the effect of unmodeled dynamics on the performance of a simple model reference adaptive control mechanism exact stability boundaries are computed and different routes to chaotic behavior and global instability are described typically we observed a region of period doubling and hopf bifurcations similar to those observed in the ziegler nichols approach to pid control design the analysis applies to higher order and certain nonlinear systems as well

explicit transversality conditions and local bifurcation diagrams for bogdanov takens bifurcation on center manifolds abstract this application oriented study is concerned with the derivation of parameter dependent normal forms for the codimension two bogdanov takens bifurcation of n dimensional m parameterized systems on the basis of the homological method in the case of an enduring equilibrium simple formulas are obtained for the transformation of parameters enabling the formulation of explicit transversality conditions and bifurcation diagrams to at most the second order moreover in z 2 symmetric systems the calculation can be further limited within certain subspaces in the general case existing results are re derived and a revision necessary for determining the bifurcation diagrams to the second order is then provided these results facilitate the derivation of normal forms check of transversality and depiction of bifurcation diagrams for the bogdanov takens bifurcation

sandwich theorems for set functions abstract in the present paper the following problems are treated when do there exist a a precontent and b a precharge which lies between two given set functions the obtained results yield among others several well known theorems such as bondareva s core theorem kelley s intersection number theorem and sandwich theorems of kranz gol dberg and frank as immediate consequences

a note on the equilibria of an economic model with local competition a la cournot guirao and rubio 2010 12 introduce an economic model which generalizes the classical duopoly of cournot type where the competitors are located around a circle or a line and each firm competes a la cournot with its right and left neighboring for the case of having three and four players we describe completely the bifurcations of equilibria in terms of the production costs of each firm and we study the stability of them moreover for the case of four players we provide some information on the two periodic orbits of the system

on bertrand duopoly game with differentiated goods utility function of the form u q 1 q 2 q 1 q 2 is introduced new dynamical bertrand games are derived stability conditions of equilibrium points are discussed chaos and bifurcation behavior are studied the paper investigates a dynamic bertrand duopoly with differentiated goods in which boundedly rational firms apply a gradient adjustment mechanism to update their price in each period the demand functions are derived from an underlying ces utility function we investigate numerically the dynamical properties of the model we consider two specific parameterizations for the ces function and study the nash equilibrium and its local stability in the models the general finding is that the nash equilibrium becomes unstable as the speed of adjustment increases the nash equilibrium loses stability through a period doubling bifurcation and the system eventually becomes chaotic either through a series of period doubling bifurcations or after a neimark sacker bifurcation

bifurcation intermittent chaos and multi stability in a two stage cournot game with r d spillover and product differentiation abstract in this paper a dynamical two stage cournot duopoly game with r d spillover effect and product differentiation is established the stability of all the equilibrium points is studied using jury criterion and then the stability condition is given the direction of flip bifurcation is given by using central manifold theorem and norm form theory by numerical simulation two routes to chaos are studied through 2 d bifurcation diagram one of which is flip bifurcation and the other is neimark sacker bifurcation at period 2 point the system will lose its stability as the speed of adjustment increases in addition a common nonlinear phenomenon named intermittent chaos is observed in the built model two types of intermittent chaos are displayed in time series plots and phase diagrams one of which is pm iii intermittent chaos and the other is induced by crisis and then four types of coexistence of attractors are illustrated through basin of attraction which are coexistence of periodic attractor and chaotic attractor coexistence of multiple chaotic attractors coexistence of periodic attractor quasi periodic attractor and chaotic attractor and coexistence of multiple periodic attractors respectively

analysis of bifurcations of limit cycles with lyapunov exponents and numerical normal forms in this paper we focus on the combination of normal form and lyapunov exponent computations in the numerical study of the three codim 2 bifurcations of limit cycles with dimension of the center manifold equal to 4 or to 5 in generic autonomous odes the normal form formulas are independent of the dimension of the phase space and involve solutions of certain linear boundary value problems the formulas allow one to distinguish between the complicated bifurcation scenarios which can happen near these codim 2 bifurcations where 3 tori and 4 tori can be present we apply our techniques to the study of a known laser model a novel model from population biology and a model of mechanical vibrations these models exhibit limit point neimark sacker period doubling neimark sacker and double neimark sacker bifurcations lyapunov exponents are computed to numerically confirm the results of the normal form analysis in particular with respect to the existence of stable invariant tori of various dimensions conversely the normal forms are essential to understand the significance of the lyapunov exponents

nonadditive set functions on a finite set and linear inequalities a set function is a function whose domain is the power set of a set which is assumed to be finite in this paper we treat a possibly nonadditive set function i e a set function which does not satisfy necessarily additivity a b a b fora b as an element of the linear space on the power set then some of the famous classes of set functions are polyhedral in that linear space i e expressed by a finite number of linear inequalities we specify the sets of the coefficients of the linear inequalities for some classes of set functions then we consider the following three problems a the domain extension problem for nonadditive set functions b the sandwich problem for nonadditive set functions and c the representation problem of a binary relation by a nonadditive set function i e the problem of nonadditive comparative probabilities

extremality of submodular functions abstract we consider the space of all set functions defined on a finite set s this space is a linear space whose dimension is 2 s the set of all polymatroid functions is a convex cone in that space we investigate conditions for a polymatroid function to be extremal in the cone to handle the extremal problem of set functions easily we treat the mobius inverse forms of set functions moreover we give the notions of compressedness and uniformity which are satisfied by matroid functions we discuss the extremality on compressed and uniform submodular functions

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

on the dynamics of kopel s cournot duopoly model abstract in this paper we study the duopoly model proposed by m kopel 26 where two firms compete a la cournot and the reaction curves have a higher degree of coupling in the sense that firms have to make their choices simultaneously we will make a descriptive analysis of the two dimensional model making an approach through particular situations on the other hand when the firms are homogeneous a one dimensional invariant subset is present in the model we will give an analytical proof of the existence of a topological chaotic behavior for a wide range of parameter values and we will study when chaotic synchronization and collusion occur

totally balanced games and games of flow a class of characteristic function games arising from maximum flow problems is introduced and is shown to coincide with the class of totally balanced games the proof relies on the max flow min cut theorem of ford and fulkerson and on the observation that the class of totally balanced games is the span of the additive games with the minimum operation

a sandwich theorem for precharges abstract in the present paper the following question is treated given two arbitrary set functions on an arbitrary set system when can one find a precharge that lies in between a number of necessary and sufficient conditions for the existence of such a precharge is derived and it is shown how these conditions can be simplified in special cases

on the existence of universal series by trigonometric system abstract in this paper we prove the following let t be a continuous function increasing in 0 and 0 0 then there exists a series of the form k c k e ikx with k c k 2 c k c k c k with the following property for each 0 a weighted function x 0 x 1 x 0 2 x 1 can be constructed so that the series is universal in the weighted space l 1 0 2 with respect to rearrangements

analysis of codimension 2 bifurcations for high dimensional discrete systems using symbolic computation methods this article reports an algebraic criterion of the eigenvalue assignment transversality condition and non resonance condition for fold n s bifurcations by means of symbolic computation methods we propose an algorithmic approach for systematically analyzing codimension 2 bifurcations for high dimensional discrete systems the effectiveness of the proposed symbolic approach is verified by experiments in particular the flip and fold n s bifurcations of a five dimensional discrete dynamical system with a washout filter feedback controller are analyzed

numerical normalization techniques for all codim 2 bifurcations of equilibria in ode s explicit computational formulas for the coefficients of the normal forms for all codim 2 equilibrium bifurcations of equilibria in autonomous odes are derived they include second order coefficients for the bogdanov takens bifurcation third order coefficients for the cusp and fold hopf bifurcations and coefficients of the fifth order terms for the generalized hopf bautin and double hopf bifurcations the formulas are independent on the dimension of the phase space and involve only critical eigenvectors of the jacobian matrix of the right hand sides and its transpose as well as multilinear functions from the taylor expansion of the right hand sides at the critical equilibrium the normal form coefficients for the fold hopf bifurcation in the new lorenz model are computed using the derived formulas proving the existence of a nontrivial invariant set in the system

computing periodic orbits and their bifurcations with automatic differentiation this paper formulates several algorithms for the direct computation of periodic orbits as solutions of boundary value problems the algorithms emphasize the use of coarse meshes and high orders of accuracy convergence theorems are given in the limit of increasing order with a fixed mesh the algorithms are implemented with the use of matlab and adol c a software package for automatic differentiation automatic differentiation enables accurate computation of high order derivatives of functions without the truncation errors inherent in finite difference calculations we embed the algorithms in a continuation framework and extend them to compute saddle node bifurcations of periodic orbits directly we present data from numerical studies of four test problems making some comparisons with other methods for computing periodic orbits these results demonstrate that high order methods based upon automatic differentiation are capable of high precision with small meshes

social significance of community structure statistical view community structure analysis is a powerful tool for social networks that can simplify their topological and functional analysis considerably however since community detection methods have random factors and real social networks obtained from complex systems always contain error edges evaluating the significance of a partitioned community structure is an urgent and important question in this paper integrating the specific characteristics of real society we present a framework to analyze the significance of a social community the dynamics of social interactions are modeled by identifying social leaders and corresponding hierarchical structures instead of a direct comparison with the average outcome of a random model we compute the similarity of a given node with the leader by the number of common neighbors to determine the membership vector an efficient community detection algorithm is proposed based on the position of the nodes and their corresponding leaders then using a log likelihood score the tightness of the community can be derived based on the distribution of community tightness we establish a connection between p value theory and network analysis and then we obtain a significance measure of statistical form finally the framework is applied to both benchmark networks and real social networks experimental results show that our work can be used in many fields such as determining the optimal number of communities analyzing the social significance of a given community comparing the performance among various algorithms etc

on the concept of attractor this note proposes a definition for the concept of attractor based on the probable asymptotic behavior of orbits the definition is sufficiently broad so that every smooth compact dynamical system has at least one attractor

detecting simple dynamics in cournot like models we introduced the so called cournot like models i e n dimensional discrete dynamical systems which constitute the mathematical environment for modeling some economic and biological processes the main aim of this work is to present a characterization of the dynamical simplicity for these types of systems through the property to have zero topological entropy cournot like systems generalize the well known economic situation of competition in a duopolistic market introduces by cournot in 1838

periodicity versus chaos in one dimensional dynamics we survey recent results in one dimensional dynamics and as an application we derive rigorous basic dynamical facts for two standard models in population dynamics the ricker and the hassell families we also informally discuss the concept of chaos in the context of one dimensional discrete time models first we use the model case of the quadratic family for an informal exposition we then review precise generic results before turning to the population models our focus is on typical asymptotic behavior seen for most initial conditions and for large sets of maps parameter sets corresponding to different types of attractors are described in particular it is shown that maps with strong chaotic properties appear with positive frequency in parameter space in our population models natural measures asymptotic distributions and their stability properties are considered

a simple global characterization for normal forms of singular vector fields we derive a new global characterization of the normal forms of amplitude equations describing the dynamics of competing order parameters in degenerate bifurcation problems using an appropriate scalar product in the space of homogeneous vector polynomials we show that the resonant terms commute with the group generated by the adjoint of the original critical linear operator this leads to a very efficient constructive method to compute both the nonlinear coefficients and the unfolding of the normal form explicit examples and results obtained when there are additional symmetries are also presented