symbolic dynamics and characterization of complexity abstract symbolic dynamics deals with robust properties of dynamics without digging into numbers for one dimensional maps there have been some new results periodic window theorem generalized composition rule construction of median words without using harmonics and antiharmonics incorporation of discrete symmetry to analyze symmetry breaking and symmetry restoration etc this new understanding has been applied to ordinary differential equations in particular the systematics of stable periodic solutions in the lorenz model has been shown to be given basically by symbolic dynamics of the cubic map symbolic dynamics may be used to extract invariant characteristics from time series its relation with grammatical complexity will also be commented on

generic representation theory of quivers with relations abstract the irreducible components of varieties parametrizing the finite dimensional representations of a finite dimensional algebra are explored with regard to both their geometry and the structure of the modules they encode provided that the base field is algebraically closed and has infinite transcendence degree over its prime field we establish the existence and uniqueness not up to isomorphism but in a strong sense to be specified of a generic module for any irreducible component c that is of a module which displays all categorically defined generic properties of the modules parametrized by c the crucial point of the existence statement a priori almost obvious lies in the description of such a module in a format accessible to representation theoretic techniques our approach to generic modules over path algebras modulo relations by way of minimal projective resolutions is largely constructive it is explicit for large classes of algebras of wild type we follow with an investigation of the properties of such generic modules in terms of quiver and relations the sharpest specific results on all fronts are obtained for truncated path algebras that is for path algebras of quivers modulo ideals generated by all paths of a fixed length this class of algebras extends the comparatively thoroughly studied hereditary case displaying many novel features

opinion dynamics of modified hegselmann krause model in a group based population with heterogeneous bounded confidence abstract continuous opinion dynamics in a group based population with heterogeneous bounded confidences is considered in this paper a slightly modified hegselmann krause model is proposed and agents are classified into three categories open minded moderate minded and closed minded agents while the whole population is divided into three subgroups accordingly we study how agents of each category and the population size can affect opinion dynamics it is observed that the number of final opinion clusters is dominated by the closed minded agents open minded agents cannot contribute to forming opinion consensus and the existence of open minded agents may diversify the final opinions instead for the fixed population size and proportion of closed minded agents the relative size of the largest final opinion cluster varies along concave parabola like curve as the proportion of open minded agents increases and there is a tipping point when the number of open minded agents is almost equal to that of moderate minded agents for the fixed proportion of the three categories in the population as the population size becomes larger the number of final opinion clusters will reach a plateau some of the results are different from the previous studies

why nonlocal recursion operators produce local symmetries new results and applications it is well known that integrable hierarchies in 1 1 dimensions are local while the recursion operators that generate these hierarchies usually contain nonlocal terms we resolve this apparent discrepancy by providing simple and universal sufficient conditions for a nonlocal recursion operator in 1 1 dimensions to generate a hierarchy of local symmetries these conditions are satisfied by virtually all recursion operators known today and are much easier to verify than those found in earlier work we also give explicit formulae for the nonlocal parts of higher recursion hamiltonian and symplectic operators of integrable systems in 1 1 dimensions using these two results we prove under some natural assumptions the maltsev novikov conjecture stating that higher hamiltonian symplectic and recursion operators of integrable systems in 1 1 dimensions are weakly nonlocal i e the coefficients of these operators are local and these operators contain at most one integration operator in each term

an algorithmic approach to collective behavior the emergence of collective structure from the decentralized interaction of autonomous agents remains with notable exceptions a mystery while powerful tools from dynamics and statistical mechanics have been brought to bear sometimes with great success an algorithmic perspective has been lacking viewing collective behavior through the lens of natural algorithms offers potential benefits this article examines the merits and challenges of an algorithmic approach to the emergence of collective order

iterations by odd functions with two extrema abstract let i 1 1 and fi i be continuous piecewise monotone and odd with two extrema a periodic orbit is called symmetric if x is in the orbit when x is in the orbit a periodic orbit which is not symmetric is called asymmetric the first result of this paper proves an ordering of the periods for the symmetric orbits there are two possibilities depending on how f behaves in a neighbourhood of 0 the second result of this paper proves that for a one parameter family of odd functions with negative schwarzian derivative there are three different types of nondegenerate bifurcations saddle node period doubling pitchfork and period preserving pitchfork the last type of bifurcation occurs exactly when a symmetric orbit bifurcates to two asymmetric orbits

inertial hegselmann krause systems we derive an energy bound for inertial hegselmann krause hk systems which are a variant of the classic hk model that allows agents to change weights as they please at each step we use the bound to prove the convergence of hk systems with static agents which settles a widely believed conjecture this paper also introduces anchored hk systems and show their equivalence to the symmetric heterogeneous model

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

the dynamics of off center reflection dynamical properties of a two parameter circle map called off center reflection are studied certain symmetry breaking phenomena in the bifurcation process are illustrated and discussed

opinion evolution analysis for short range and long range deffuant weisbuch models abstract in this paper we propose and then analyze two generalized deffuant weisbuch dw models the generalized models extend the conventional dw model by taking multiple choices in two different ways first we demonstrate the almost sure convergence of the agent opinions for the short range multi choice dw dynamics when only the opinions within confidence regions may be count in then we analyze dynamical behavior about the long range multi choice dw model when some opinions out of the confidence ranges are considered with a weighted combination moreover both theoretical and simulation results show that the dynamical behaviors of the two models are totally different

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

cyclic star products and universalities in symbolic dynamics of trimodal maps abstract cyclic star products for the triple superstable kneading tssk sequences are presented for symbolic dynamics of trimodal maps of endomorphisms on the interval feigenbaum s metric universalities in unimodal maps are generalized to trimodal maps four equal value universal convergent rates a c a c and three universal scaling factors c d e are first obtained

the generalized milnor thurston conjecture and equal topological entropy class in symbolic dynamics of order topological space of three letters this paper presents an answer to an open problem in the dynamical systems of three letters the generalized milnor thurston conjecture on the existence of infinitely many plateaus of topological entropy in the two dimensional parameter plane the concept of equal topological entropy class is introduced by the dual star product which is a generalization of the derrida gervois pomeau star product to the symbolic dynamics of three letters for the endomorphisms on the interval the algebraic rules established by the dual star products for the doubly superstable kneading sequences are equivalent to the normal factorization of the milnor thurston characteristic polynomials moreover the classification theory of symbolic primitive and compound sequences based on the topological conjugacy in the meaning of equal entropy is completed in the topological space 3 of three letters

on open orbits and their complements 1 1 we fix an algebraically closed field k and a quiver q without oriented cycles whose vertex set q we identify with 1 n for a in the set q of arrows of q we denote by ta and ha the tail and the head of a respectively a representation x of q consists of a family x i in qo of finite dimensional k vector spaces and a family x a x ta x ha a e q1 of k linear maps a morphism f x y between two representations of q is a family f i x i y i in qo of k linear maps satisfying f ha ox a y a of ta for all arrows a we denote the category of representations of q by mod q it is an abelian category of global dimension at most 1 we will denote the only possibly non trivial extension group ext by ext the dimension vector of a representation x of q is the vector dim x dim x l dim x n in n and the dimension of x is the natural number dim x dim x 1 dim x n the support of x is the full subquiver of q whose vertices are i x i zo for a vector d d d e n we define r q d to be the set of representations x of q such that x i k for ie qo thus r q d is the product r q d n hom kdru kdhe

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

top stable degenerations of finite dimensional representations i abstractgiven a finite dimensional representation m of a finite dimensional algebra two hierarchies of degenerations of m are analyzed in the context of their natural orders the poset of those degenerations of m which share the top m jm with m here j denotes the radical of the algebra and the sub poset of those which share the full radical layering gl j l m j l 1 mgr l 0 with m in particular the article addresses the existence of proper top stable or layer stable degenerations more generally it addresses the sizes of the corresponding posets including bounds on the lengths of saturated chains as well as structure and classification

a stabilization theorem for dynamics of continuous opinions abstract a stabilization theorem for processes of opinion dynamics is presented the theorem is applicable to a wide class of models of continuous opinion dynamics based on averaging like the models of hegselmann krause and weisbuch deffuant the analysis detects self confidence as a driving force of stabilization

dirac s theory of constraints in field theory and the canonical form of hamiltonian differential operators a simple algorithm for constructing the canonical form of hamiltonian systems of evolution equations with constant coefficient hamiltonian differential operators is given the result of the construction is equivalent to the canonical system derived using dirac s theory of constraints from the corresponding degenerate lagrangian

universality and self similarity in the bifurcations of circle maps abstract the bifurcation structure in a two parameter family of circle maps is considered these maps have a topological degree that may be different from one a generalization of the rotation number is given and symmetries of the bifurcations in parameter space are described continuity arguments are used to establish the existence of periodic orbits by plotting the locus of parameter values associated with superstable cycles self similar bifurcations are found these bifurcations are a generalization of the familiar period doubling cascade in maps with one extrema to two parameter maps with two extrema finally a scheme for the global organization of bifurcation in these maps is proposed

top stable degenerations of finite dimensional representations ii abstract let be a finite dimensional algebra over an algebraically closed field we exhibit slices of the representation theory of that are always classifiable in stringent geometric terms namely we prove that for any semisimple object t mod the class of those modules with fixed dimension vector say d and top t which do not permit any proper top stable degenerations possesses a fine moduli space this moduli space modulimax d t is a projective variety despite classifiability up to isomorphism the targeted collections of modules are representation theoretically rich indeed any projective variety arises as modulimax d t for suitable choices of d and t in tandem we give a structural characterization of the finite dimensional representations that have no proper top stable degenerations

a genealogy for finite kneading sequences of bimodal maps on the interval we generate all the finite kneading sequences of one of the two kinds of bimodal map on the interval building each sequence uniquely from a pair of shorter ones there is a single pair at generation 0 with members of length 1 concomitant with this genealogy of kneading sequences is a unified genealogy of all the periodic orbits see figure 0

on degenerations and extensions of finite dimensional modules abstract we derive a cancellation theorem for degenerations of modules that says in particular that projective or injective common direct summands can always be neglected combining the cancellation result with the existence of almost split sequences we characterize the orbit closure of a module living on preprojective components by the fact that the dimension of the homomorphism space to any other module does not decrease for representation directed algebras whence in particular for path algebras of dynkin quivers we provide an alternative proof which shows in addition that any minimal degenerationnofmcomes from an exact sequence with middle termmwhose end terms add up ton by a careful examination the same is true for degenerations of matrix pencils having used so far the existence of certain extensions to obtain degenerations we then turn the tables and use degenerations to produce a lot of interesting short exact sequences in particular we show that any non simple indecomposable over a tame quiver is an extension of an indecomposable and a simple

on approximations and ergodicity classes in random chains we study the limiting behavior of a random dynamic system driven by a stochastic chain our interest is in the chains that are not necessarily ergodic but are decomposable into ergodic classes to investigate the conditions under which the ergodic classes of a model can be identified we introduce and study an l1 approximation and infinite flow graph of the model we show that the l1 approximations of random chains preserve certain limiting behavior using the l1 approximations we show how the connectivity of the infinite flow graph is related to the structure of the ergodic groups of the model our main result of this paper provides conditions under which the ergodicity groups of the model can be identified by considering the connected components in the infinite flow graph we provide two applications of our main result to random networks namely broadcast over time varying networks and networks with random link failure

adaptive decision dynamics bifurcations multistability and chaos abstract in this paper we propose a model describing the dynamical process of decision and opinion formation of two economic homogeneous interacting and boundedly rational agents the decisional process represented in our model is given by an adaptive adjustment mechanism in which two agents take into account the difference between their own opinion and the opinion of the other agent the smaller that difference the larger the weight given to the comparison of the opinions by means of an auxiliary variable describing the distance between the opinions we obtain a one dimensional dynamical system for which we investigate via analytical and numerical tools the stability of the unique steady state its bifurcations as well as the existence of a globally absorbing interval and of chaotic dynamics we also investigate multistability phenomena i e the presence of coexisting attractors finally we relax the assumption of homogeneity between agents and we show that there is a strong correspondence between the dynamic behaviors in the scenarios with and without homogeneity

a strange recursion operator demystified we show that a new integrable two component system of kdv type studied by karasu kalkanli et al 2004 acta appl math 83 85 94 is bi hamiltonian and its recursion operator which has a highly unusual structure of nonlocal terms can be written as a ratio of two compatible hamiltonian operators found by us using this we prove that the system in question possesses an infinite hierarchy of local commuting generalized symmetries and conserved quantities in involution and the evolution systems corresponding to these symmetries are bi hamiltonian as well we also show that upon introduction of suitable nonlocal variables the nonlocal terms of the recursion operator under study can be written in the usual form with the integration operator d 1x appearing in each term at most once

opinion dynamics with decaying confidence application to community detection in graphs we study a class of discrete time multi agent systems modeling opinion dynamics with decaying confidence we consider a network of agents where each agent has an opinion at each time step the agents exchange their opinion with their neighbors and update it by taking into account only the opinions that differ from their own less than some confidence bound this confidence bound is decaying an agent gives repetitively confidence only to its neighbors that approach sufficiently fast its opinion essentially the agents try to reach an agreement with the constraint that it has to be approached no slower than a prescribed convergence rate under that constraint global consensus may not be achieved and only local agreements may be reached the agents reaching a local agreement form communities inside the network in this paper we analyze this opinion dynamics model we show that communities correspond to asymptotically connected components of the network and give an algebraic characterization of communities in terms of eigenvalues of the matrix defining the collective dynamics finally we apply our opinion dynamics model to address the problem of community detection in graphs we propose a new formulation of the community detection problem based on eigenvalues of normalized laplacian matrix of graphs and show that this problem can be solved using our opinion dynamics model we consider three examples of networks and compare the communities we detect with those obtained by existing algorithms based on modularity optimization we show that our opinion dynamics model not only provides an appealing approach to community detection but that it is also effective

topological invariants and renormalization of lorenz maps we prove that the invariants of the topological semiconjugation of lorenz maps with transformations remains constant on the renormalization archipelagoes and analyze how the dynamics on the archipelagoes depends on its structure

the convergence of bird flocking we bound the time it takes for a group of birds to stabilize in a standard flocking model each bird averages its velocity with its neighbors lying within a fixed radius we resolve the worst case complexity of this natural algorithm by providing asymptotically tight bounds on the time to equilibrium we reduce the problem to two distinct questions in computational geometry and circuit complexity

low order hamiltonian operators having momentum abstract we describe all fifth order hamiltonian operators in one dependent and one independent variable that possess momentum i e for which there exists a hamiltonian associated with translation in the independent variable similar results for first and third order hamiltonian operators were obtained earlier by mokhov

min max time consensus tracking with communication guarantee in this paper a collection of double integrator agents with bounded inputs is considered communication is possible between any two agents only if the inter agent distance is less than a fixed threshold a special node referred to as the leader generates an unknown reference trajectory to which all the other agents are required to converge in the shortest possible time assuming the initial communication graph to be connected a directed spanning tree rooted at the leader is identified using a local algorithm the dynamics of any two agents connected by an edge in the selected tree are modeled as a time optimal pursuit evasion game while maintaining the inter agent communication link using the corresponding feedback saddle point strategies local min max time control laws for each pair of agents are formulated for the selected tree the proposed collection of local min max time control strategies is shown to be the communication preserving global min max time strategy

reaching consensus on rumors abstract an important contribution in sociophysics is the galam s model of rumors spreading this model provides an explanation of rumors spreading in a population and explains some interesting social phenomena such as the diffusion of hoaxes in this paper the model has been reformulated as a markov process highlighting the stochastic nature of the phenomena this formalization allows us to derive conditions for consensus to be reached and for the existence of some interesting phenomena such as the emergence of impasses the proposed formulation allows a deeper and more comprehensive analysis of the diffusion of rumors

convergence properties of the heterogeneous deffuant weisbuch model the deffuant weisbuch dw model is a bounded confidence opinion dynamics model that has attracted much recent interest despite its simplicity and appeal the dw model has proved technically hard to analyze and its most basic convergence properties easy to observe numerically are only conjectures this paper solves the convergence problem for the heterogeneous dw model with the weighting factor not less than 1 2 we establish that for any positive confidence bounds and initial values the opinion of each agent will converge to a limit value almost surely and the convergence rate is exponential in mean square moreover we show that the limiting opinions of any two agents either are the same or have a distance larger than the confidence bounds of the two agents finally we provide some sufficient conditions for the heterogeneous dw model to reach consensus

the total s energy of a multiagent system we introduce the total s energy of a multiagent system with time dependent links this provides a new analytical perspective on bidirectional agreement dynamics which we use to bound the convergence rates of dynamical systems for synchronization flocking opinion dynamics and social epistemology

opinion fluctuations and disagreement in social networks we study a tractable opinion dynamics model that generates long run disagreements and persistent opinion fluctuations our model involves an inhomogeneous stochastic gossip process of continuous opinion dynamics in a society consisting of two types of agents 1 regular agents who update their beliefs according to information that they receive from their social neighbors and 2 stubborn agents who never update their opinions and might represent leaders political parties or media sources attempting to influence the beliefs in the rest of the society when the society contains stubborn agents with different opinions the belief dynamics never lead to a consensus among the regular agents instead beliefs in the society fail to converge almost surely the belief profile keeps on fluctuating in an ergodic fashion and it converges in law to a nondegenerate random vector the structure of the graph describing the social network and the location of the stubborn agents within it shape the opinion dynamics the expected belief vector is proved to evolve according to an ordinary differential equation coinciding with the kolmogorov backward equation of a continuous time markov chain on the graph with absorbing states corresponding to the stubborn agents and hence to converge to a harmonic vector with every regular agent s value being the weighted average of its neighbors values and boundary conditions corresponding to the stubborn agents beliefs expected cross products of the agents beliefs allow for a similar characterization in terms of coupled markov chains on the graph describing the social network we prove that in large scale societies which are highly fluid meaning that the product of the mixing time of the markov chain on the graph describing the social network and the relative size of the linkages to stubborn agents vanishes as the population size grows large a condition of homogeneous influence emerges whereby the stationary beliefs marginal distributions of most of the regular agents have approximately equal first and second moments

a new type of period doubling bifurcations in one dimensional transformations with two extrema abstract we consider the iteration of a one dimensional transformation x f x symmetrical with the origin and with two symmetrical extrema there exists a sequence of period doubling bifurcations where period 2k cycles are created at parameter values k and which is not of the feigenbaum type we show that the k s sequence converge toward a limit value as a double exponential function of k

transition to topological chaos for circle maps abstract many biperiodic flows can be modelled by maps of a circle to itself for such maps the transition from zero to positive topological entropy can be achieved in several ways we describe all the possible routes for smooth circle maps and discuss the relevance of our results to the transition to chaos for two frequency systems

classification results and the darboux theorem for low order hamiltonian operators hamiltonian operators and their behavior under differential substitutions are studied scalar hamiltonian operators are classified up to fifth order and it is shown that all such operators may be obtained from the first order gardner operator dx by differential substitutions thus proving an infinite dimensional darboux theorem for hamiltonian systems of evolution equations corresponding to such operators

strategic influence in social networks we consider a model of influence with a set of non strategic agents and two strategic agents the non strategic agents have initial opinions and are linked through a simply connected network they update their opinions as in the degroot model the two strategic agents have fixed opinions 1 and 0 respectively and are characterized by the magnitude of the impact they can exert on non strategic agents each strategic agent forms a link with one non strategic agent in order to alter the average opinion that eventually emerges in the network this procedure defines a zero sum game whose players are the two strategic agents and whose strategy set is the set of non strategic agents we focus on the existence and the characterization of equilibria in pure strategy in this setting simple examples show that the existence of a pure strategy equilibrium does depend on the structure of the network the characterization of equilibrium we obtain emphasizes on the one hand the influenceability of target agents and on the other hand their centrality whose natural measure in our context defines a new concept related to betweenness centrality that we call intermediacy we also show that in the case where the two strategic agents have the same impact symmetric equilibria emerge as natural solutions whereas in the case where the impacts are uneven the strategic players generally have differentiated equilibrium targets the high impacts agent focusing on centrality and the low impact agent on influenceability

product of random stochastic matrices the paper deals with the convergence properties of the products of random row stochastic matrices the limiting behavior of such products is studied from a dynamical system point of view in particular by appropriately defining a dynamic associated with a given sequence of random row stochastic matrices we prove that the dynamics admits a class of time varying lyapunov functions including a quadratic one then we discuss a special class of stochastic matrices a class p which plays a central role in this work we then study cut balanced chains and using some geometric properties of these chains we characterize the stability of a subclass of cut balanced chains as a special consequence of this stability result we obtain an extension of a central result in the non negative matrix theory stating that for any aperiodic and irreducible row stochastic matrix a the limit limk ak exists and it is a rank one stochastic matrix we show that a generalization of this result holds not only for sequences of stochastic matrices but also for independent random sequences of such matrices

homological domino effects and the first finitistic dimension conjecture counterexamples to the first finitistic dimension conjecture are constructed in fact for any fieldk and any integerm 2 there exist finite dimensionalk algebras such that the little finitistic dimension of ism while the big finitistic dimension ism 1 our examples are monomial relation algebras within this class of algebras the big and little finitistic dimensions cannot differ by more than 1 the analysis of the examples is based on a sharp picture of arbitrary second syzygies over monomial relation algebras

diffusive influence systems influence systems seek to model how influence broadly defined spreads across a dynamic network we build a general analytical framework which we then use to prove that while turing complete influence dynamics of the diffusive type is almost surely asymptotically periodic in addition to resolving the dynamics of a widely used family of multiagent systems we introduce a general renormalization method for the bifurcation analysis of multiagent systems

darboux theorem for hamiltonian differential operators it is proved that any one dimensional first order hamiltonian differential operator can be put into constant coefficient form by a suitable change of variables consequently there exist canonical variables for any such hamiltonian operator in the course of the proof a complete characterization of all first order hamiltonian differential operators as well as the general formula for the behavior of a hamiltonian operator under a change of variables involving both the independent and the dependent variables are found

two parameter analysis of the scaling behavior at the onset of chaos tricritical and pseudo tricritical points abstract we discuss the so called tricritical points at the border of the period doubling transition to chaos and examine to what extent the associated universality applies to 2d dissipative maps as a concrete example the ikeda map is studied together with its 1d analog for the approximate 1d map the tricritical points appear as the terminal points of feigenbaum s critical curves in the parameter plane for the 2d map the same type of critical behavior does not occur in a rigorous sense it may be observed as a kind of intermediate asymptotics however when one considers a finite number of period doublings we refer to the associated points in the parameter plane as pseudo tricritical for the ikeda map we present estimates of the number of period doublings after which the departure from the tricritical universality becomes essential

well posedness of the limiting equation of a noisy consensus model in opinion dynamics abstract this paper establishes the global well posedness of the nonlinear fokker planck equation for a noisy version of the hegselmann krause model the equation captures the mean field behavior of a classic multiagent system for opinion dynamics we prove the global existence uniqueness nonnegativity and regularity of the weak solution we also exhibit a global stability condition which delineates a forbidden region for consensus formation this is the first nonlinear stability result derived for the hegselmann krause model