darboux method and search of invariants for the lotka volterra and complex quadratic systems the darboux method introduces algebraic solutions quite useful to obtain invariant and first integrals of polynomial differential systems here we study the 2d lotka volterra lvs and the complex quadratic system qs using straight lines for both and conics for the lvs the conditions needed to obtain these invariants are given and a study of the phase space portrait is done

comment on on the general solution for the modified emden type equation ddot x alpha x pdot x beta x 3 0 it is shown that the modified emden type equation in the title belongs to a class of lienard s equations that may be mapped into separable abel s equations and are therefore integrable first integrals for equations of this class are easily derived there are three of them corresponding to three distinct dynamical regimes and integrated to yield parametric representations of the corresponding solutions

modified emden type equation with dissipative term quadratic in velocity based on some physical observation we introduce a generalized modified emden type equation mee with a position dependent dissipative term which is quadratic in velocity unlike the usual mee the first integral of the proposed generalized mee is such that one can express the velocity of the system as a function of coordinate for all values of the parameters of the system this permits us to study the dynamical properties of the system using straightforward analytical methods the results presented in the phase diagram and plots of vector fields clearly delineate how does the presence of quadratic damping affect the motion of our nonlinear oscillator from the differential equation provided by the first integral of the generalized mee we have found an approximate analytical solution of the equation which reproduces the time variation of the corresponding numerical solution to a fair degree of accuracy

nonstandard conserved hamiltonian structures in dissipative damped systems nonlinear generalizations of damped harmonic oscillator in this paper we point out the existence of a remarkable nonlocal transformation between the damped harmonic oscillator and a modified emden type nonlinear oscillator equation with linear forcing x xx x3 x 0 which preserves the form of the time independent integral conservative hamiltonian and the equation of motion generalizing this transformation we prove the existence of nonstandard conservative hamiltonian structure for a general class of damped nonlinear oscillators including lienard type systems further using the above hamiltonian structure for a specific example namely the generalized modified emden equation x xqx x2q 1 0 where and q are arbitrary parameters the general solution is obtained through appropriate canonical transformations we also present the conservative hamiltonian structure of the damped mathews lakshmanan oscillator equation the associated lagrangian description for all the above systems is also briefly discussed

deterministic walks in random networks an application to thesaurus graphs abstract in a landscape composed of n randomly distributed sites in euclidean space a walker tourist goes to the nearest one that has not been visited in the last steps this procedure leads to trajectories composed of a transient part and a final cyclic attractor of period p the tourist walk presents a simple scaling with respect to and can be performed in a wide range of networks that can be viewed as ordinal neighborhood graphs as an example we show that graphs defined by thesaurus dictionaries share some of the statistical properties of low dimensional d 2 euclidean graphs and are easily distinguished from random link networks which correspond to the d limit this approach furnishes complementary information to the usual clustering coefficient and mean minimum separation length

symmetries of nonlinear differential equations and linearisation a non linear ordinary differential equation is linearisable if it possesses sl 3 r symmetry the conditions under which the abelian two dimensional subalgebras of sl 3 r are sufficient for linearisation are established

a group theoretical identification of integrable cases of the lienard type equation x f x x g x 0 i equations having nonmaximal number of lie point symmetries we carry out a detailed lie point symmetry group classification of the lienard type equation x f x x g x 0 where f x and g x are arbitrary smooth functions of x we divide our analysis into two parts in the present first part we isolate equations that admit lesser parameter lie point symmetries namely one two and three parameter symmetries and in the second part we identify equations that admit maximal eight parameter lie point symmetries in the former case the invariant equations form a family of integrable equations and in the latter case they form a class of linearizable equations under point transformations further we prove the integrability of all the equations obtained in the present paper through equivalence transformations either by providing the general solution or by constructing time independent hamiltonians several of these equations are being identified for the first time from the group theoretical analysis

solving second order ordinary differential equations by extending the prelle singer method we propose a method for solving second order ordinary differential equations odes which is based on the ideas behind the prelle singer ps procedure for first order odes while the ps procedure treats differential equations des of the form y p x y q x y with p and q polynomials whose coefficients lie in the field of complex numbers our method is applicable to des of the form y p x y y q x y y the key to our approach is to focus not on the final solution but on the first order invariants of the equation our method is an attempt to address algorithmically the solution of second order odes with solutions in terms of elementary functions

families of invariants of the motion for the lotka volterra equations the linear polynomials family the modified carleman embedding method already introduced by the authors to find first integrals invariants of the motion of polynomial form to the lotka volterra system is described in detail and its efficiency to treat the n dimensional system proved using this method an extensive investigation is performed for polynomials of the first degree which allow a classification of the integrals in three families for some systems possessing one invariant it is possible to find a second invariant using rescaling methods they represent very restrictive solutions implying that there exists a great number of conditions among the equation s coefficients to satisfy a proof is given that the volterra invariants can be deduced as a limit finally the interesting properties of the solutions of these systems are studied in detail

on the general solution for the modified emden type equation in this paper we demonstrate that the modified equation of emden type mee is integrable either explicitly or by quadrature for any value of and we also prove that the mee possesses appropriate time independent hamiltonian functions for the full range of parameters and in addition we show that the mee is intimately connected with two well known nonlinear models namely the equation of force free duffing type oscillator and the two dimensional lotka volterra equation and thus the complete integrability of the latter two models can also be understood in terms of the mee

reply to comment on on the general solution for the modified emden type equation explicit implicit solutions for general parametric values of the modified emden equation and its generalizations were obtained recently by us using a modified prelle singer approach coupled with a hamiltonian formalism in his comment iacono has suggested an interesting alternate approach here we point out the limitations and incompleteness of the approach as well as some additional elementary approaches to the above equations

mixing patterns in networks we study assortative mixing in networks the tendency for vertices in networks to be connected to other vertices that are like or unlike them in some way we consider mixing according to discrete characteristics such as language or race in social networks and scalar characteristics such as age as a special example of the latter we consider mixing according to vertex degree i e according to the number of connections vertices have to other vertices do gregarious people tend to associate with other gregarious people we propose a number of measures of assortative mixing appropriate to the various mixing types and apply them to a variety of real world networks showing that assortative mixing is a pervasive phenomenon found in many networks we also propose several models of assortatively mixed networks both analytic ones based on generating function methods and numerical ones based on monte carlo graph generation techniques we use these models to probe the properties of networks as their level of assortativity is varied in the particular case of mixing by degree we find strong variation with assortativity in the connectivity of the network and in the resilience of the network to the removal of vertices

a nonlocal connection between certain linear and nonlinear ordinary differential equations extension to coupled equations identifying integrable coupled nonlinear ordinary differential equations odes of dissipative type and deducing their general solutions are some of the challenging tasks in nonlinear dynamics in this paper we undertake these problems and unearth two classes of integrable coupled nonlinear odes of arbitrary order to achieve these goals we introduce suitable nonlocal transformations in certain linear odes and generate the coupled nonlinear odes in particular we show that the problem of solving these classes of coupled nonlinear odes of any order effectively reduces to solving a single first order nonlinear ode we then describe a procedure to derive explicit general solutions for the identified integrable coupled odes when the above mentioned first order nonlinear ode reduces to a bernoulli equation the equations which we generate and solve include the two coupled versions of modified emden equations in second order coupled versions of chazy equations in third order and their variants higher d

the painleve test hidden symmetries and the equation y yy ky3 0 for general values of the parameter k the equation y yy ky3 0 can be reduced to quadrature via a lie algebraic approach either direct or through hidden symmetries for specific values of k mostly in 1 9 1 8 the solution can be expressed in parametric form for these values of k the equation passes the weak painleve test for some other values of k the equation passes the painleve test but the solution cannot in general be expressed parametrically

interplay of symmetries null forms darboux polynomials integrating factors and jacobi multipliers in integrable second order differential equations in this work we establish a connection between the extended prellesinger procedure with five other analytical methods which are widely used to identify integrable systems in the contemporary liter

the integration of three dimensional lotka volterra systems the general solutions of many three dimensional lotka volterra systems previously known to be at least partially integrable are constructed with the aid of special functions examples include certain abc and may leonard systems the special functions used are elliptic and incomplete beta functions in some cases the solution is parametric with the independent and dependent variables expressed as functions of a new time variable this auxiliary variable satisfies a nonlinear third order differential equation of a generalized schwarzian type and results of carton lebrun on the equations of this type that have the painleve property are exploited so as to produce solutions in closed form for several especially difficult lotka volterra systems the solutions are expressed in terms of painleve transcendents an appendix on incomplete beta functions and closed form expressions for their inverses is included

exact solutions of coupled lienard type nonlinear systems using factorization technique general solutions of nonlinear ordinary differential equations odes are in general difficult to find although powerful integrability techniques exist in the literature for this purpose it has been shown that in some scalar cases particular solutions may be found with little effort if it is possible to factorize the equation in terms of first order differential operators in our present study we use this factorization technique to address the problem of finding solutions of a system of general two coupled lienard type nonlinear differential equations we describe a generic algorithm to identify specific classes of lienard type systems for which solutions may be found we demonstrate this method by identifying a class of two coupled equations for which the particular solution can be found by solving a bernoulli equation this class of equations include coupled generalization of the modified emden equation we further deduce the general solution of a class of coupled odes using the factorization proced

nonlocal symmetries of a class of scalar and coupled nonlinear ordinary differential equations of any order in this paper we devise a systematic procedure to obtain nonlocal symmetries of a class of scalar nonlinear ordinary differential equations odes of arbitrary order related to linear odes through nonlocal relations the procedure makes use of the lie point symmetries of the linear odes and the nonlocal connection to deduce the nonlocal symmetries of the corresponding nonlinear odes using these nonlocal symmetries we obtain reduction transformations and reduced equations to specific examples we find that the reduced equations can be explicitly integrated to deduce the general solutions for these cases we also extend this procedure to coupled higher order nonlinear odes with specific reference to second order nonlinear odes

dynamics of a completely integrable n coupled lienard type nonlinear oscillator we present a system of n coupled lienard type nonlinear oscillators which is completely integrable and possesses n time independent and n time dependent explicit integrals in a special case it becomes maximally superintegrable and admits 2n 1 time independent integrals the results are illustrated for the n 2 and arbitrary number cases general explicit periodic with frequency independent of amplitude and quasi periodic solutions as well as decaying type frontlike solutions are presented depending on the signs and magnitudes of the system parameters though the system is of a nonlinear damped type our investigations show that it possesses a hamiltonian structure and that under a contact transformation it is transformable to a system of uncoupled harmonic oscillators

on the complete integrability and linearization of certain second order nonlinear ordinary differential equations a method for finding general solutions of second order nonlinear ordinary differential equations by extending the prelle singer ps method is briefly discussed we explore integrating factors integrals of motion and the general solution associated with several dynamical systems discussed in the current literature by employing our modifications and extensions of the ps method we also introduce a novel way of deriving linearizing transformations from the first integrals to linearize the second order nonlinear ordinary differential equations to free particle equations we illustrate the theory with several potentially important examples and show that our procedure is widely applicable

nonlocal symmetries of riccati and abel chains and their similarity reductions we study nonlocal symmetries and their similarity reductions of riccati and abel chains our results show that all the equations in riccati chain share the same form of nonlocal symmetry the similarity reduced nth order ordinary differential equation ode n 2 3 4 in this chain yields n 1 th order ode in the same chain all the equations in the abel chain also share the same form of nonlocal symmetry which is different from the one that exist in riccati chain but the similarity reduced nth order ode n 2 3 4 in the abel chain always ends at the n 1 th order ode in the riccati chain we describe the method of finding general solution of all the equations that appear in these chains from the nonlocal symmetry

modelling hierarchical and modular complex networks division and independence abstract we introduce a growing network model which generates both modular and hierarchical structure in a self organized way to this end we modify the barabasi albert model into the one evolving under the principles of division and independence as well as growth and preferential attachment pa a newly added vertex chooses one of the modules composed of existing vertices and attaches edges to vertices belonging to that module following the pa rule when the module size reaches a proper size the module is divided into two and a new module is created the karate club network studied by zachary is a simple version of the current model we find that the model can reproduce both modular and hierarchical properties characterized by the hierarchical clustering function of a vertex with degree k c k being in good agreement with empirical measurements for real world networks

lie symmetries and invariants of the lotka volterra system in this paper we use the lie symmetry method for finding rational and transcendental symmetry transformations and invariants for the 3d lotka volterra system

riccati parameter solutions of nonlinear second order odes it has been proven by rosu and cornejo perez rosu and cornejo perez 2005 phys rev e 71 046607 cornejo perez and rosu 2005 prog theor phys 114 533 that for some nonlinear second order odes it is a very simple task to find one particular solution once the nonlinear equation is factorized with the use of two first order differential operators here it is shown that an interesting class of parametric solutions is easy to obtain if the proposed factorization has a particular form which happily turns out to be the case in many problems of physical interest the method that we exemplify with a few explicitly solved cases consists in using the general solution of the riccati equation which contributes with one parameter to this class of parametric solutions for these nonlinear cases the riccati parameter serves as a growth parameter from the trivial null solution up to the particular solution found through the factorization procedure

the complexity of chinese syntactic dependency networks abstract this paper proposes how to build a syntactic network based on syntactic theory and presents some statistical properties of chinese syntactic dependency networks based on two chinese treebanks with different genres the results show that the two syntactic networks are small world networks and their degree distributions obey a power law the finding that the two syntactic networks have the same diameter and different average degrees path lengths clustering coefficients and power exponents can be seen as an indicator that complexity theory can work as a means of stylistic study the paper links the degree of a vertex with a valency of a word the small world with the minimized average distance of a language that reinforces the explanations of the findings from linguistics

unusual lienard type nonlinear oscillator a lienard type nonlinear oscillator of the form x kxx k2 9 x3 lambda1x 0 which may also be considered as a generalized emden type equation is shown to possess unusual nonlinear dynamical properties it is shown to admit explicit nonisolated periodic orbits of conservative hamiltonian type for lambda1 0 these periodic orbits exhibit the unexpected property that the frequency of oscillations is completely independent of amplitude and continues to remain as that of the linear harmonic oscillator this is completely contrary to the standard characteristic property of nonlinear oscillators interestingly the system though appears deceptively a dissipative type for lambda1 or 0 does admit a conserved hamiltonian description where the characteristic decay time is also independent of the amplitude the results also show that the criterion for conservative hamiltonian system in terms of divergence of flow function needs to be generalized

detecting communities in large networks we develop an algorithm to detect community structure in complex networks the algorithm is based on spectral methods and takes into account weights and link orientation since the method detects efficiently clustered nodes in large networks even when these are not sharply partitioned it turns to be specially suitable for the analysis of social and information networks we test the algorithm on a large scale data set from a psychological experiment of word association in this case it proves to be successful both in clustering words and in uncovering mental association patterns

connection between the existence of first integrals and the painleve property in two dimensional lotka volterra and quadratic systems taking advantage of the considerable amount of work done in the search for first integrals invariants for the two dimensional lotka volterra system and the quadratic system lvs and qs we compare the relations needed to exhibit invariants one for the lvs at least three for the qs to the two conditions of the painleve test index and compatibility we find that eventually restricting the invariants to those which are analytic all exponents integers and thereby adding new constraints these constraints always coalesce with the two painleve conditions we conclude that straightforward application of the painleve test picks up only these simple analytic invariants and that possession of the painleve property is too strong a condition for the existence of the invariants

a class of second order differential equations and related first order systems a class of second order nonlinear differential equations which arises in several branches of mathematical physics is considered it is shown that equations of this class may be factorised into first order equations of riccati type conditions are obtained on the coefficient functions of the second order equations for the first order equations to be of matrix riccati form whose solutions have a finite superposition property the factorisation into first order equations is then not unique and there is an alternative first order set of equations whose solutions do not have this superposition property a second order equation arising in the theory of pellet fusion processes is investigated in detail solutions are obtained when the corresponding first order equations are of matrix riccati from and shown to be equivalent to solutions derived by alternative methods lagrangian systems giving rise to equations of the class are also considered

quasi lie schemes and emden fowler equations the recently developed theory of quasi lie schemes is studied and applied to investigate several equations of emden type and a scheme to deal with them and some of their generalizations is given as a first result we obtain t dependent constants of the motion for particular instances of emden equations by means of some of their particular solutions previously known results are recovered from this new perspective some t dependent constants of the motion for equations of emden type satisfying certain conditions are recovered finally new exact particular solutions are given for certain cases of emden equations

chinese character structure analysis based on complex networks abstract in this paper chinese character networks are modelled using complex networks theory we analyze statistical properties of the networks and find that character networks also display two important features as other real networks i e small world feature and the non poisson distribution these results indicate that the discovered features of chinese character structure reflect the combinatorial nature of chinese characters we also simulate the formation of chinese phono semantic characters using bipartite graph theory the bipartite graph model generates non poisson distributions and disassortative mixing as the empirical networks which effectively explain the origin and formation of phono semantic characters

chinese lexical networks the structure function and formation abstract in this paper chinese phrases are modeled using complex networks theory we analyze statistical properties of the networks and find that phrase networks display some important features not only small world and the power law distribution but also hierarchical structure and disassortative mixing these statistical traits display the global organization of chinese phrases the origin and formation of such traits are analyzed from a macroscopic chinese culture and philosophy perspective it is interesting to find that chinese culture and philosophy may shape the formation and structure of chinese phrases to uncover the structural design principles of networks network motif patterns are studied it is shown that they serve as basic building blocks to form the whole phrase networks especially triad 38 feed forward loop plays a more important role in forming most of the phrases and other motifs the distinct structure may not only keep the networks stable and robust but also be helpful for information processing the results of the paper can give some insight into chinese language learning and language acquisition it strengthens the idea that learning the phrases helps to understand chinese culture on the other side understanding chinese culture and philosophy does help to learn chinese phrases the hub nodes in the networks show the close relationship with chinese culture and philosophy learning or teaching the hub characters hub linking phrases and phrases which are meaning related based on motif feature should be very useful and important for chinese learning and acquisition

symmetry singularities and integrability in complex dynamics ii rescaling and time translation in two dimensional systems the explicit integrability of second order ordinary differential equations invariant under time translation and rescaling is investigated quadratic systems generated from the linearisable version of this class of equations are analysed to determine the relationship between the painleve and singularity properties of the different systems the transformation contains a parameter and for critical values intimately related to the possession of the painleve property in the parent second order equation one finds a difference from the generic behaviour this study is a prelude to a full discussion of the class of transformations which preserve the painleve property in the construction of quadratic systems from scalar nth order ordinary differential equations invariant under time invariant under time translation and rescaling

euclidean distance between syntactically linked words we study the euclidean distance between syntactically linked words in sentences the average distance is significantly small and is a very slowly growing function of sentence length we consider two nonexcluding hypotheses a the average distance is minimized and b the average distance is constrained support for a comes from the significantly small average distance real sentences achieve the strength of the minimization hypothesis decreases with the length of the sentence support for b comes from the very slow growth of the average distance versus sentence length furthermore b predicts under ideal conditions an exponential distribution of the distance between linked words a trend that can be identified in real sentences

resolutions relation modules and schur multipliers for categories abstract we show that the construction in group cohomology of the gruenberg resolution associated to a free presentation and the resulting relation module can be copied in the context of representations of categories we establish five term exact sequences in the cohomology of categories and go on to show that the schur multiplier of the category has properties which generalize those of the schur multiplier of a group

pseudofractal scale free web we find that scale free random networks are excellently modeled by simple deterministic graphs our graph has a discrete degree distribution degree is the number of connections of a vertex which is characterized by a power law with exponent gamma 1 ln 3 ln 2 properties of this compact structure are surprisingly close to those of growing random scale free networks with gamma in the most interesting region between 2 and 3 we succeed to find exactly and numerically with high precision all main characteristics of the graph in particular we obtain the exact shortest path length distribution for a large network ln n 1 the distribution tends to a gaussian of width approximately sqrt ln n centered at l approximately ln n we show that the eigenvalue spectrum of the adjacency matrix of the graph has a power law tail with exponent 2 gamma

properties of second order ordinary differential equations invariant under time translation and self similar transformation the general form of a second order ordinary differential equation invariant under time translation and self similarity is obtained together with its solution in parametric form next two representatives of two different families of such equations are studied in terms of rescaled phase space variables the first example has a simple physical interpretation and its limit cycle properties and period are easily derived similar results are found for the second example but the physical interpretation is less obvious the invariant equations are used as a basis to study the asymptotic behavior of related noninvariant equations thereby underlining the critical value nature of the parameters for which a self similar solution sss exists

why social networks are different from other types of networks we argue that social networks differ from most other types of networks including technological and biological networks in two important ways first they have nontrivial clustering or network transitivity and second they show positive correlations also called assortative mixing between the degrees of adjacent vertices social networks are often divided into groups or communities and it has recently been suggested that this division could account for the observed clustering we demonstrate that group structure in networks can also account for degree correlations we show using a simple model that we should expect assortative mixing in such networks whenever there is variation in the sizes of the groups and that the predicted level of assortative mixing compares well with that observed in real world networks

scale free characteristics of random networks the topology of the world wide web the world wide web forms a large directed graph whose vertices are documents and edges are links pointing from one document to another here we demonstrate that despite its apparent random character the topology of this graph has a number of universal scale free characteristics we introduce a model that leads to a scale free network capturing in a minimal fashion the self organization processes governing the world wide web c 2000 elsevier science b v all rights reserved pacs 84 35 i 64 60 fr 87 23 ge

a simple and unified approach to identify integrable nonlinear oscillators and systems in this paper we consider a generalized second order nonlinear ordinary differential equation ode of the form x k1xq k2 x k3x2q 1 k4xq 1 1x 0 where ki s i 1 2 3 4 1 and q are arbitrary parameters which includes several physically important nonlinear oscillators such as the simple harmonic oscillator anharmonic oscillator force free helmholtz oscillator force free duffing and duffing van der pol oscillators modified emden type equation and its hierarchy generalized duffing van der pol oscillator equation hierarchy and so on and investigate the integrability properties of this rather general equation we identify several new integrable cases for arbitrary value of the exponent q q r the q 1 and q 2 cases are analyzed in detail and the results are generalized to arbitrary q our results show that many classical integrable nonlinear oscillators can be derived as subcases of our results and significantly enlarge the list of integrable equations that exists in the contemporary literature t

social network growth with assortative mixing networks representing social systems display specific features that put them apart from biological and technological ones in particular the number of links attached to a node is positively correlated to that of its nearest neighbours we develop a model that reproduces this feature starting from microscopical mechanisms of growth the statistical properties arising from the simulations are in good agreement with those of the real world social networks of scientists co authoring papers in condensed matter physics moreover the model highlights the determinant role of correlations in shaping the network s topology

on the singularity analysis of ordinary differential equations invariant under time translation and rescaling the pt is applied to the general second and third order ordinary differential equations invariant under the two symmetries associated with time translation and rescaling in order to investigate their solvability and global integrability the effect of the two symmetries on the compatibility conditions is determined and we show that generally these conditions are automatically a consequence of the resonance condition use is made of truncated laurent series both in ascending and descending powers as an example the case of the generalized chazy equation is presented

thesaurus as a complex network abstract a thesaurus is one out of many possible representations of term or word connectivity the terms of a thesaurus are seen as the nodes and their relationship as the links of a directed graph the directionality of the links retains all the thesaurus information and allows the measurement of several quantities this has lead to a new term classification according to the characteristics of the nodes for example nodes with no links in no links out etc using an electronic available thesaurus we have obtained the incoming and outgoing link distributions while the incoming link distribution follows a stretched exponential function the lower bound for the outgoing link distribution has the same envelope of the scientific paper citation distribution proposed by tsallis and albuquerque eur phys j b 13 2000 777 however a better fit is obtained by simpler function which is the solution of ricatti s differential equation we conjecture that this differential equation is the continuous limit of a stochastic growth model of the thesaurus network we also propose a new manner to arrange a thesaurus using the inversion method

first integrals for the modified emden equation q α t q qn 0 it is shown that the modified emden equation q t q qn 0 possesses first integrals for functions t other than kt 1 the function t is obtained explicitly in the case n 3 and parametrically for other n 2 the case n 2 is seen to be particularly difficult to solve