orbitally universal centers abstract in this paper we define when a polynomial differential system is orbitally universal and we show the relevance of this notion in the classical center problem i e in the problem of distinguishing between a focus and a center

hypergeometric first integrals of the duffing and van der pol oscillators abstract the autonomous duffing oscillator and its van der pol modification are known to admit time dependent first integrals for specific values of parameters this corresponds to the existence of darboux polynomials and in fact more can be shown that there exist liouvillian first integrals which do not depend on time they can be expressed in terms of the gauss and kummer hypergeometric functions and are neither analytic algebraic nor meromorphic a criterion for this to happen in a general dynamical system is formulated as well

synchronization in fractional order differential systems an symmetrically coupled system consisting of identical fractional order differential systems including chaotic and non chaotic systems is investigated in this paper such a coupled system has in its synchronous state a mode decomposition by which the linearized equation can be decomposed into motions transverse to and parallel to the synchronous manifold furthermore the decomposition can induce a sufficient condition on synchronization of the overall system which guarantees if satisfied that a group synchronization is achieved two typical numerical examples fractional brusselators and the fractional rossler system are used to verify the theoretical prediction the theoretical analysis and numerical results show that the lower the order of the fractional system the longer the time for achieving synchronization at a fixed coupling strength

an algebraic approach to the classification of centers in polynomial lienard systems abstract we consider the second order lienard system x f x x g x 0 where f x and g x are polynomials which we rewrite in the equivalent two dimensional form x y y g x yf x we show that the local question of whether a critical point of this system is a center can be expressed in terms of global conditions on f and g using these results we give a simple classification of all such centers we also address the problem of the coexistence of centers foci and limit cycles for these systems systems with degenerate centers are also considered

invariant algebraic curves and rational first integrals for planar polynomial vector fields we present three main results the first two provide sufficient conditions in order that a planar polynomial vector field in c2 has a rational first integral and the third one studies the number of multiple points that an invariant algebraic curve of degree n of a planar polynomial vector field of degree m can have in function of m and n

3d polynomial dynamical systems with elementary first integrals here we present a semi algorithm to find elementary first integrals of 3d polynomial dynamical systems it is a darboux type procedure that extends the method built by prelle and singer for 2d systems although it cannot deal with the general case the method presents a direct simple way to find elementary first integrals

diffusion process modeling by using fractional order models this paper deals with a concept and description of a rc network as an electro analog model of diffusion process which enables to simulate heat dissipation under different initial and boundary conditions it is based on well known analogy between heat and electrical conduction in the paper are compared analytical solution together with numerical solution and experimentally measured data for the first time a fractional order model of diffusion process and its modeling via lumped rc network has been used simple examples of simulations measurements and their comparison are shown

liouvillian first integrals of differential equations liouvillian functions are functions that are built up from rational functions using exponentiation integration and algebraic functions we show that if a system of differential equations has a generic solution that satisfies a liouvillian relation that is there is a liouvillian function of several variables vanishing on the curve defined by this solution then the system has a liouvillian first integral that is a nonconstant liouvillian function that is constant on solution curves in some nonempty open set we can refine this result in special cases to show that the first integral must be of a very special form

invariant algebraic curves of large degree for quadratic system in this paper we present for the first time examples of algebraic limit cycles and saddle loops of degree greater than 4 for planar quadratic systems in particular we give examples of algebraic limit cycles of degree 5 and 6 and algebraic saddle loops of degree 3 and 5 surrounding a strong focus we also give an example of an invariant algebraic curve of degree 12 for which the quadratic system has no darboux integrating factors or first integrals

efficient algorithms for computing rational first integrals and darboux polynomials of planar polynomial vector fields we present fast algorithms for computing rational first integrals with bounded degree of a planar polynomial vector field our approach builds upon a method proposed by ferragut and giacomini whose main ingredients are the calculation of a power series solution of a first order differential equation and the reconstruction of a bivariate polynomial annihilating this power series we provide explicit bounds on the number of terms needed in the power series this enables us to transform their method into a certified algorithm computing rational first integrals via systems of linear equations we then significantly improve upon this first algorithm by building a probabilistic algorithm with arithmetic complexity o n 2 omega and a deterministic algorithm solving the problem in at most o d 2n 2 omega 1 arithmetic operations where n denotes the given bound for the degree of the rational first integral and where d leq n is the degree of the vector field and omega the exponent of linear algebra we also provide a fast heuristic variant which computes a rational first integral or fails in o n omega 2 arithmetic operations by comparison the best previous algorithm uses at least d omega 1 n 4 omega 4 arithmetic operations we then show how to apply a similar method to the computation of darboux polynomials the algorithms are implemented in a maple package rationalfirstintegrals which is available to interested readers with examples showing its efficiency

exponential factors and darbouxian first integrals of the lorenz system in this article we characterize all the exponential factors and the darbouxian first integrals for the lorenz system s y x rx y xz z bz xy in the proofs we use the weight homogeneous polynomials and the method of characteristic curves for solving linear partial differential equations

darboux integrability and invariant algebraic curves for planar polynomial systems in this paper we study the normal forms of polynomial systems having a set of given generic invariant algebraic curves

invariant surfaces and darboux integrability for non autonomous dynamical systems in the plane a novel method of finding and classifying irreducible invariant surfaces of non autonomous polynomial dynamical systems in the plane is presented the general structure of irreducible invariant surfaces and their cofactors is found the complete set of irreducible invariant surfaces for the classical forced duffing van der pol oscillator is obtained it is proved that the forced duffing van der pol oscillator possesses only one independent generalized darboux first integral provided that a constraint on the parameters is valid in other cases generalized darboux first integrals do not exist consequently the forced duffing van der pol oscillator is not integrable with two independent generalized darboux first integrals

reporting of subscores using multidimensional item response theory recently there has been increasing interest in reporting subscores this paper examines reporting of subscores using multidimensional item response theory mirt models e g reckase in appl psychol meas 21 25 36 1997 c r rao and s sinharay eds handbook of statistics vol 26 pp 607 642 north holland amsterdam 2007 beguin glas in psychometrika 66 471 488 2001 a mirt model is fitted using a stabilized newton raphson algorithm haberman in the analysis of frequency data university of chicago press chicago 1974 sociol methodol 18 193 211 1988 with adaptive gauss hermite quadrature haberman von davier lee in ets research rep no rr 08 45 ets princeton 2008 a new statistical approach is proposed to assess when subscores using the mirt model have any added value over i the total score or ii subscores based on classical test theory haberman in j educ behav stat 33 204 229 2008 haberman sinharay puhan in br j math stat psychol 62 79 95 2008 the mirt based methods are applied to several operational data sets the results show that the subscores based on mirt are slightly more accurate than subscore estimates derived by classical test theory

generalized rational first integrals of analytic differential systems abstract in this paper we mainly study the necessary conditions for the existence of functionally independent generalized rational first integrals of ordinary differential systems via the resonances the main results extend some of the previous related ones for instance the classical poincare s one poincare 1891 1897 16 the furta s one furta 1996 8 part of chen et al s ones chen et al 2008 4 and the shi s one shi 2007 18 the key point in the proof of our main results is that functionally independence of generalized rational functions implies the functionally independence of their lowest order rational homogeneous terms

lmi based sliding surface design for integral sliding mode control of mismatched uncertain systems we propose a linear matrix inequality lmi based sliding surface design method for integral sliding mode control of mismatched uncertain systems the uncertain system under consideration may have mismatched norm bounded uncertainties in the state matrix as well as the input matrix we give a sufficient condition for the existence of a sliding surface guaranteeing asymptotic stability of the full order sliding mode dynamics we also give an lmi characterization of the sliding surface together with an integral sliding mode control law guaranteeing the existence of a sliding mode from the initial time additionally we give an lmi condition of sliding surfaces guaranteeing the alpha stability constraint finally we give a simulation result to show the effectiveness of our method

brief paper pseudo state feedback stabilization of commensurate fractional order systems this paper addresses the problem of pseudo state feedback stabilization of commensurate fractional order systems fos in the proposed approach linear matrix inequalities lmi formalism is used to check if the pseudo state matrix eigenvalues belong to the fos stability region of the complex plane a review of lmi stability conditions is first proposed for fractional order 0 n 1 and 1 n 2 the paper then focuses particularly on the case 0 n 1 as the stability region is non convex and associated lmi condition is not as straightforward to obtain as in the case 1 n 2 a new lmi stability condition is thus proposed based on this condition a necessary and sufficient lmi method for the design of stabilizing controllers is given this method paves the way for extension to fos of various lmi based results among these possible extensions a first result on robust control of polytopic fractional order systems is given in this paper

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

finiteness theorems on hypersurfaces in partial differential algebraic geometry abstract hrushovski s generalization and application of jouanolou 1978 9 is here refined and extended to the partial differential setting with possibly nonconstant coefficient fields in particular it is shown that if x is a differential algebraic variety over a partial differential field f that is finitely generated over its constant field f 0 then there exists a dominant differential rational map from x to the constant points of an algebraic variety v over f 0 such that all but finitely many codimension one subvarieties of x over f arise as pull backs of algebraic subvarieties of v over f 0 as an application it is shown that the algebraic solutions to a first order algebraic differential equation over c t are of bounded height answering a question of eremenko two expected model theoretic applications to dcf 0 m are also given 1 lascar rank and morley rank agree in dimension two and 2 dimension one strongly minimal sets orthogonal to the constants are 0 categorical a detailed exposition of hrushovski s original unpublished theorem is included influenced by ghys 2000 5

polynomial inverse integrating factors for quadratic differential systems we characterize all the quadratic polynomial differential systems having a polynomial inverse integrating factor and provide explicit normal forms for such systems and for their associated first integrals we also prove that these families of quadratic systems have no limit cycles

global analytic first integrals for the real planar lotka volterra system we provide the complete classification of all lotka volterra systems of the form x x ax by c and y y ax by c in r2 having a global analytic first integral

estimating difficulty from polytomous categorical data a comprehensive analysis of difficulty for multiple choice items requires information at different levels the test the items and the alternatives this paper introduces a new parameterization of the nominal categories model ncm for analyzing difficulty at these three levels the new parameterization is referred to as the ne ncm and is statistically equivalent to the ncm the ne ncm is applied to a sample of responses from a logical analysis test the results suggest that the individuals execute a self terminated response process that is mostly determined by working memory load

algebraic limit cycles of degree 4 for quadratic systems abstract yablonskii differential equations 2 1996 335 and filipstov differential equations 9 1973 983 proved the existence of two different families of algebraic limit cycles of degree 4 in the class of quadratic systems it was an open problem to know if these two algebraic limit cycles where all the algebraic limit cycles of degree 4 for quadratic systems chavarriga a new example of a quartic algebraic limit cycle for quadratic sytems universitat de lleida preprint 1999 found a third family of this kind of algebraic limit cycles here we prove that quadratic systems have exactly four different families of algebraic limit cycles the proof provides new tools based on the index theory for algebraic solutions of polynomial vector fields

relationships between limit cycles and algebraic invariant curves for quadratic systems abstract algebraic limit cycles for quadratic systems started to be studied in 1958 up to now we know 7 families of quadratic systems having algebraic limit cycles of degree 2 4 5 and 6 here we present some new results on the limit cycles and algebraic limit cycles of quadratic systems these results provide sometimes necessary conditions and other times sufficient conditions on the cofactor of the invariant algebraic curve for the existence or nonexistence of limit cycles or algebraic limit cycles in particular it follows from them that for all known examples of algebraic limit cycles for quadratic systems those cycles are unique limit cycles of the system

fractional order systems and pi sup spl lambda d sup spl mu controllers dynamic systems of an arbitrary real order fractional order systems are considered the concept of a fractional order pi sup spl lambda d sup spl mu controller involving fractional order integrator and fractional order differentiator is proposed the laplace transform formula for a new function of the mittag leffler type made it possible to obtain explicit analytical expressions for the unit step and unit impulse response of a linear fractional order system with fractional order controller for both open and closed loops an example demonstrating the use of the obtained formulas and the advantages of the proposed pi sup spl lambda d sup spl mu controllers is given

on planar polynomial vector fields with elementary first integrals abstract we show that under rather general conditions a polynomial differential system having an elementary first integral already must admit a darboux first integral and we explicitly characterize the vector fields in this class we also investigate some exceptional cases i e equations admitting an elementary first integral but not a darboux first integral in particular we provide a rather detailed discussion of exceptional elementary first integrals built from algebraic functions of prime degree

multidimensional cat item selection methods for domain scores and composite scores theory and applications multidimensional computer adaptive testing mcat can provide higher precision and reliability or reduce test length when compared with unidimensional cat or with the paper and pencil test this study compared five item selection procedures in the mcat framework for both domain scores and overall scores through simulation by varying the structure of item pools the population distribution of the simulees the number of items selected and the content area the existing procedures such as volume segall in psychometrika 61 331 354 1996 kullback leibler information veldkamp van der linden in psychometrika 67 575 588 2002 minimize the error variance of the linear combination van der linden in j educ behav stat 24 398 412 1999 and minimum angle reckase in multidimensional item response theory springer new york 2009 are compared to a new procedure minimize the error variance of the composite score with the optimized weight proposed for the first time in this study the intent is to find an item selection procedure that yields higher precisions for both the domain and composite abilities and a higher percentage of selected items from the item pool the comparison is performed by examining the absolute bias correlation test reliability time used and item usage three sets of item pools are used with the item parameters estimated from real live cat data results show that volume and minimum angle performed similarly balancing information for all content areas while the other three procedures performed similarly with a high precision for both domain and overall scores when selecting items with the required number of items for each domain the new item selection procedure has the highest percentage of item usage moreover for the overall score it produces similar or even better results compared to those from the method that selects items favoring the general dimension using the general model segall in psychometrika 66 79 97 2001 the general dimension method has low precision for the domain scores in addition to the simulation study the mathematical theories for certain procedures are derived the theories are confirmed by the simulation applications

differential galois theory and non integrability of planar polynomial vector fields abstract we study a necessary condition for the integrability of the polynomials vector fields in the plane by means of the differential galois theory more concretely by means of the variational equations around a particular solution it is obtained a necessary condition for the existence of a rational first integral the method is systematic starting with the first order variational equation we illustrate this result with several families of examples a key point is to check whether a suitable primitive is elementary or not using a theorem by liouville the problem is equivalent to the existence of a rational solution of a certain first order linear equation the risch equation this is a classical problem studied by risch in 1969 and the solution is given by the risch algorithm in this way we point out the connection of the non integrability with some higher transcendent functions like the error function

multidimensional adaptive testing with optimal design criteria for item selection several criteria from the optimal design literature are examined for use with item selection in multidimensional adaptive testing in particular it is examined what criteria are appropriate for adaptive testing in which all abilities are intentional some should be considered as a nuisance or the interest is in the testing of a composite of the abilities both the theoretical analyses and the studies of simulated data in this paper suggest that the criteria of a optimality and d optimality lead to the most accurate estimates when all abilities are intentional with the former slightly outperforming the latter the criterion of e optimality showed occasional erratic behavior for this case of adaptive testing and its use is not recommended if some of the abilities are nuisances application of the criterion of as optimality or ds optimality which focuses on the subset of intentional abilities is recommended for the measurement of a linear combination of abilities the criterion of c optimality yielded the best results the preferences of each of these criteria for items with specific patterns of parameter values was also assessed it was found that the criteria differed mainly in their preferences of items with different patterns of values for their discrimination parameters

the hyperelliptic limit cycles of the lienard systems abstract for lienard systems x y y f m x y g n x with f m and g n real polynomials of degree m and n respectively in h zoladek algebraic invariant curves for the lienard equation trans amer math soc 350 1998 1681 1701 the author showed that if m 3 and m 1 n 2 m there always exist lienard systems which have a hyperelliptic limit cycle llibre and zhang j llibre xiang zhang on the algebraic limit cycles of lienard systems nonlinearity 21 2008 2011 2022 proved that the lienard systems with m 3 and n 5 have no hyperelliptic limit cycles and that there exist lienard systems with m 4 and 5 n 8 which do have hyperelliptic limit cycles so it is still an open problem to characterize the lienard systems which have an algebraic limit cycle in cases m 4 and m 1 n 2 m in this paper we will prove that there exist lienard systems with m 5 and m 1 n 2 m which have hyperelliptic limit cycles

on the stability of limit cycles for planar differential systems abstract we consider a planar differential system x p x y y q x y where p and q are c 1 functions in some open set u r 2 and d dt let be a periodic orbit of the system in u let f x y u r 2 r be a c 1 function such that p x y f x x y q x y f y x y k x y f x y where k x y is a c 1 function in u and x y f x y 0 we assume that if p u is such that f p 0 and f p 0 then p is a singular point we prove that 0 t p x q y t dt 0 t k t dt where t 0 is the period of as an application we take profit from this equality to show the hyperbolicity of the known algebraic limit cycles of quadratic systems

a class of polynomial planar vector fields with polynomial first integral abstract we give an algorithm for deciding whether a planar polynomial differential system has a first integral which factorizes as a product of defining polynomials of curves with only one place at infinity in the affirmative case our algorithm computes a minimal first integral in addition we solve the poincare problem for the class of systems which admit a polynomial first integral as above in the sense that the degree of the minimal first integral can be computed from the reduction of singularities of the corresponding vector field

darboux integrability and the inverse integrating factor abstract we mainly study polynomial differential systems of the form dx dt p x y dy dt q x y where p and q are complex polynomials in the dependent complex variables x and y and the independent variable t is either real or complex we assume that the polynomials p and q are relatively prime and that the differential system has a darboux first integral of the form h f 1 1 f p p exp h 1 g 1 n 1 1 exp h q g q n q q where the polynomials f i and g j are irreducible the polynomials g j and h j are coprime and the i and j are complex numbers when i 1 p and j 1 q prelle and singer proved that these systems have a rational integrating factor we improve this result as follows assume that h is a rational function which is not polynomial following to poincare we define the critical remarkable values of h then we prove that the system has a polynomial inverse integrating factor if and only if h has at most two critical remarkable values under some assumptions over the darboux first integral h we show first that the system has a polynomial inverse integrating factor and secondly that if the degree of the system is m the homogeneous part of highest degree of h is a multi valued function and the functions exp h j g j are exponential factors for j 1 q then the system has a polynomial inverse integrating factor of degree m 1 we also present versions of these results for real polynomial differential systems finally we apply these results to real polynomial differential systems having a darboux first integral and limit cycles or foci

a new approach for the study of limit cycles abstract we prove that star like limit cycles of any planar polynomial system can also be seen either as solutions defined on a given interval of a new associated planar non autonomous polynomial system or as heteroclinic solutions of a 3 dimensional polynomial system we illustrate these points of view with several examples one of the key ideas in our approach is to decompose the periodic solutions as the sum of two suitable functions as a first application we use these new approaches to prove that all star like reversible limit cycles are algebraic as a second application we introduce a function whose zeroes control the periodic orbits that persist as limit cycles when we perturb a star like reversible center as far as we know this is the first time that this question is solved in full generality somehow this function plays a similar role that an abelian integral for studying perturbations of hamiltonian systems

integrability of a sis model we prove that the classical model of an infectious disseise which never kills and which does not induce autoimmunity is integrable this model can be written as x bxy mx cy mk y bxy m c y with parameters b c k m r we provide the explicit expression of its first integrals and of the set of all its invariant algebraic curves

some new results on darboux integrable differential systems abstract we deal with complex planar differential systems having a darboux first integral h we present a definition of remarkable values and remarkable curves associated to h and characterize the existence of a polynomial inverse integrating factor for these systems furthermore we study the relation between the characteristic polynomial f and the inverse integrating factors of the system and show the importance of the numerator of the exponential factor of h in the construction of f

optimal synergetic control for fractional order systems abstract nowadays the control of fractional order system is one of the most popular topics in control theory recent studies have demonstrated the interest of fractional calculus both for systems modeling in many areas of science and engineering and for robust controller design thus several research contributions have been devoted to the extension of control theory to fractional order systems synergetic control was introduced in power electronics and other industrial processes the benefit of this control scheme has been recognized for both integer order linear and nonlinear systems in this paper a fractional order synergetic control for fractional order systems is proposed both linear and nonlinear cases are considered the macro variable is defined by the fractional order integral of state variables optimality and stability properties are analyzed a numerical example is investigated to confirm the effectiveness of the proposed method

darboux integrating factors inverse problems abstract we discuss planar polynomial vector fields with prescribed darboux integrating factors in a nondegenerate affine geometric setting we establish a reduction principle which transfers the problem to polynomial solutions of certain meromorphic linear systems and show that the space of vector fields with a given integrating factor modulo a subspace of explicitly known standard vector fields has finite dimension for several classes of examples we determine this space explicitly

liouvillian first integrals of quadratic linear polynomial differential systems abstract for a large class of quadratic linear polynomial differential systems with a unique singular point at the origin having non zero eigenvalues we classify the ones which have a liouvillian first integral and we provide the explicit expression of them

some criteria for the existence of limit cycles for quadratic vector fields abstract in this paper we consider real quadratic systems we present new criteria for the existence and uniqueness of limit cycles for such systems by using darbouxian particular solutions some results are based on the study of such systems in cp 2 we also generalize the well know result of bautin on the nonexistence of limit cycles for quadratic lotka volterra systems

state variables and transients of fractional order differential systems fractional order differentiation is generally considered as the basis of fractional calculus but the real basis is in fact fractional order integration and particularly the fractional integrator because definition and properties of fractional differentiation and of fractional differential systems rely essentially on fractional integration we present the frequency distributed model of the fractional integrator and its finite dimension approximation the simulation of fdss based on fractional integrators leads to the definition of fds internal state variables which are the state variables of the fractional integrators as a generalization of the integer order case the initial condition problem has been an open problem for a long time in fractional calculus we demonstrate that the frequency distributed model of the fractional integrator provides a solution to this problem through the knowledge of its internal state beyond the solution of this fundamental problem mastery of the integrator internal state allows the analysis and prediction of fractional differential system transients moreover the finite dimension approximation of the fractional integrator provides an efficient technique for practical simulation of fdss and analysis of their transients with a particular insight into the interpretation of initial conditions as illustrated by numerical simulations laplace transform equations and initial conditions of the caputo and the riemann liouville derivatives are used to formulate the free responses of fdes because usual equations are wrong the corresponding free responses do not fit with real transients we demonstrate that revised equations including the initial state vector of the fractional integrator used to perform differentiation provide corrected free responses which match with real transients as exhibited by numerical simulations

algorithmic derivation of centre conditions the conditions for a critical point of a polynomial differential system to be a centre are of particular significance because of the frequency with which they are required in applications we demonstrate how computer algebra can be effectively employed in the search for necessary and sufficient conditions for critical points of such systems to be centres we survey recent developments and illustrate our approach by means of examples

a note on the use of the lambert w function in the stability analysis of time delay systems the lambert w function is defined to be the multivalued inverse of the function w we w z this function has been used in an extremely wide variety of applications including the stability analysis of fractional order as well as integer order time delay systems the latter application is based on taking the mth power and or nth root of the transcendental characteristic equation tce and representing the roots of the derived tce s in terms of w functions in this note we re examine such an application of using the lambert w function through actually computing the root distributions of the derived tces of some chosen orders it is found that the rightmost root of the original tce is not necessarily a principal branch lambert w function solution and that a derived tce obtained by taking the mth power of the original tce introduces superfluous roots to the system with these observations some deficiencies displayed in the literature chen y q moore k l 2002a analytical stability bound for delayed second order systems with repeating poles using lambert w function automatica 38 5 891 895 chen y q moore k l 2002b analytical stability bound for a class of delayed fractional order dynamic systems nonlinear dynamics 29 1 4 191 200 are pointed out moreover we clarify the correct use of lambert w function to stability analysis of a class of time delay systems this will actually enlarge the application scope of the lambert w function which is becoming a standard library function for various commercial symbolic software packages to time delay systems

sur les integrales premieres dans la classe de nilsson d equations differentielles holomorphes we study two classes of holomorphic differential equations the first one is constitued by elements admitting solutions defined in an algebraic way the so called liouville class and the second of elements admitting solutions defined in an analytic way the nilsson class we build up links between these two classes using special properties of the holonomy and its results on the monodromy

on liouvillian integrability of the first order polynomial ordinary differential equations abstract recently the authors provided an example of an integrable liouvillian planar polynomial differential system that has no finite invariant algebraic curves see gine and llibre 2012 8 in this note we prove that if a complex differential equation of the form y a 0 x a 1 x y a n x y n with a i x polynomials for i 0 1 n a n x 0 and n 2 has a liouvillian first integral then it has a finite invariant algebraic curve so this result applies to riccati and abel polynomial differential equations we shall prove that in general this result is not true when n 1 i e for linear polynomial differential equations

uniqueness of algebraic limit cycles for quadratic systems abstract we know five different families of algebraic limit cycles in quadratic systems one of degree 2 and four of degree 4 moreover if there are other families of algebraic limit cycles for quadratic systems then their degrees must be larger than 4 it is known that if a quadratic system has an algebraic limit cycle of degree 2 then this is the unique limit cycle of the system the first main goal of this paper is to prove that if a quadratic system has an algebraic limit cycle of degree 4 then this is the unique limit cycle of the system for the yablonskii and filiptsov algebraic limit cycles the problem of proving their uniqueness has been open since 1966 and 1973 respectively the second main goal is to provide sufficient conditions in order that all the limit cycles of a quadratic system to be algebraic

quadratic systems with a polynomial first integral a complete classification in the coefficient space r12 abstract in this paper we are going to apply the invariant theory to give invariant conditions on the coefficients of any non degenerate quadratic system in order to determine if it has or not a polynomial first integral without using any normal form we obtain that the existence of polynomial first integral is directly related with the fact that all the roots of a convenient cubic polynomial are rational and negative the coefficients of this cubic polynomial are invariants related with some geometric properties of the system

letter to the editor integrating first order differential equations with liouvillian solutions via quadratures a semi algorithmic method here we present a semi algorithmic method to deal with rational first order ordinary differential equations with liouvillian solutions this method is based on the knowledge of the general structure for the integrating factor for such equations

bounding the number of remarkable values via jouanolou s theorem abstract in this article we bound the number of remarkable values of a polynomial vector field the proof is short and based on jouanolou s theorem about rational first integrals of planar polynomial derivations our bound is given in term of the size of a newton polygon associated to the vector field we prove that this bound is almost reached

darboux theory of integrability in the sparse case darboux s theorem and jouanolou s theorem deal with the existence of first integrals and rational first integrals of a polynomial vector field these results are given in terms of the degree of the polynomial vector field here we show that we can get the same kind of results if we consider the size of a newton polytope associated to the vector field furthermore we show that in this context the bound is optimal

analysing the structure of the integrating factors for first order ordinary differential equations with liouvillian functions in the solution here we demonstrate a theorem concerning the general structure of the integrating factor for first order ordinary differential equations whose solutions contain liouvillian functions this result assures the generality of a method presented in a forthcoming paper extending the usual prelle singer approach

synchronization of chaotic fractional order systems via active sliding mode controller abstract in this paper we propose a controller based on active sliding mode theory to synchronize chaotic fractional order systems in master slave structure master and slave systems may be identical or different based on stability theorems in the fractional calculus analysis of stability is performed for the proposed method finally three numerical simulations synchronizing fractional order lu lu systems synchronizing fractional order chen chen systems and synchronizing fractional order lu chen systems are presented to show the effectiveness of the proposed controller the simulations are implemented using two different numerical methods to solve the fractional differential equations

dynamics of the polynomial differential systems with homogeneous nonlinearities and a star node abstract we consider the class of polynomial differential equations x x p n x y y y q n x y in r 2 where p n x y and q n x y are homogeneous polynomials of degree n 1 and 0 i e the class of polynomial differential systems with homogeneous nonlinearities with a star node at the origin we prove that these systems are darboux integrable moreover for these systems we study the existence and non existence of limit cycles surrounding the equilibrium point located at the origin

on adjoint symmetry equations integrating factors and solutions of nonlinear odes we consider the role of the adjoint equation in determining explicit integrating factors and first integrals of nonlinear odes in chandrasekar et al 2006 j math phys 47 023508 the authors have used an extended version of the prelle singer method for a class of nonlinear odes of the oscillator type in particular we show that their method actually involves finding a solution of the adjoint symmetry equation next we consider a coupled second order nonlinear ode system and derive the corresponding coupled adjoint equations we illustrate how the coupled adjoint equations can be solved to arrive at a first integral

darboux integrability for 3d lotka volterra systems we describe the improved darboux theory of integrability for polynomial ordinary differential equations in three dimensions using this theory and computer algebra we study the existence of first integrals for the three dimensional lotka volterra systems only working up to degree two with the invariant algebraic surfaces and the exponential factors we find the major part of the known first integrals for such systems and in addition we find three new classes of integrability the method used is of general interest and can be applied to any polynomial ordinary differential equations in arbitrary dimension

a note on liouvillian integrability abstract there was the belief that if a planar polynomial differential system has a liouvillian first integral then it has some finite invariant algebraic curves in this note we provide a counterexample to this belief

inverse problems for multiple invariant curves planar polynomial vector fields which admit invariant algebraic curves darboux integrating factors or darboux first integrals are of special interest in the present paper we solve the inverse problem for invariant algebraic curves with a given multiplicity and for integrating factors under generic assumptions regarding the multiple invariant algebraic curves involved in particular we prove in this generic scenario that the existence of a darboux integrating factor implies darboux integrability furthermore we construct examples where the genericity assumption does not hold and indicate that the situation is different for these

sliding mode control of uncertain fractional order systems with delay abstractthis paper presents a sliding mode control smc based framework to design a stabilizing controller for uncertain fractional order time delay systems fotds using the lambert w function te

a method to tackle first order ordinary differential equations with liouvillian functions in the solution the usual prelle singer ps approach misses many first order ordinary differential equations presenting liouvillian functions in the solution lfoodes we point out why and propose a method extending the ps method to solve a class of these previously unsolved lfoodes although our method does not cover all the lfoodes it maintains the semi decision nature of the usual ps method

integrability and algebraic limit cycles for polynomial differential systems with homogeneous nonlinearities abstract we consider the class of polynomial differential equations x x y p n x y y x y q n x y where p n and q n are homogeneous polynomials of degree n these systems have a focus at the origin if 0 and have either a center or a focus if 0 inside this class we identify a new subclass of darbouxian integrable systems having either a focus or a center at the origin additionally under generic conditions such darbouxian integrable systems can have at most one limit cycle and when it exists is algebraic for the case n 2 and 3 we present new classes of darbouxian integrable systems having a focus

polynomial first integrals of quadratic vector fields abstract we classify all quadratic polynomial differential systems having a polynomial first integral and provide explicit normal forms for such systems and for their first integrals

integrability degenerate centers and limit cycles for a class of polynomial differential systems we consider the class of polynomial differential equations email protected p n x y p n 1 x y p n 2 x y email protected q n x y q n 1 x y q n 2 x y for n 1 and where p i and q i are homogeneous polynomials of degree i these systems have a linearly zero singular point at the origin if n 2 inside this class we identify a new subclass of darboux integrable systems and some of them having a degenerate center i e a center with linear part identically zero moreover under additional conditions such darboux integrable systems can have at most one limit cycle we provide the explicit expression of this limit cycle

on the complete integrability and linearization of nonlinear ordinary differential equations ii third order equations we introduce a method for finding general solutions of third order nonlinear differential equations by extending the modified prelle singer method we describe a procedure to deduce all the integrals of motion associated with the given equation so that the general solution follows straightforwardly from these integrals the method is illustrated with several examples further we propose a powerful method of identifying linearizing transformations the proposed method not only unifies all the known linearizing transformations systematically but also introduces a new and generalized linearizing transformation in addition to the above we provide an algorithm to invert the non local linearizing transformation through this procedure the general solution for the original nonlinear equation can be obtained from the solution of the linear ordinary differential equation

local first integrals for systems of differential equations the main purpose of this paper is to provide some sufficient conditions for a system of differential equations to have local first integrals in a certain neighbourhood of a singularity our results generalize those given in kwek et al 2003 z angew math phys 54 26 and li et al 2003 z angew math phys 54 235

explicit non algebraic limit cycles for polynomial systems we consider a system of the form email protected p n x y xr m x y email protected q n x y yr m x y where p n x y q n x y and r m x y are homogeneous polynomials of degrees n n and m respectively with n

algebraic limit cycles in polynomial systems of differential equations using elementary tools we construct cubic polynomial systems of differential equations with algebraic limit cycles of degrees 4 5 and 6 we also construct a cubic polynomial system of differential equations having an algebraic homoclinic loop of degree 3 moreover we show that there are polynomial systems of differential equations of arbitrary degree that have algebraic limit cycles of degree 3 as well as give an example of a cubic polynomial system of differential equations with two algebraic limit cycles of degree 4

kullback leibler information and its applications in multi dimensional adaptive testing this paper first discusses the relationship between kullback leibler information kl and fisher information in the context of multi dimensional item response theory and is further interpreted for the two dimensional case from a geometric perspective this explication should allow for a better understanding of the various item selection methods in multi dimensional adaptive tests mat which are based on these two information measures the kl information index ki method is then discussed and two theorems are derived to quantify the relationship between ki and item parameters due to the fact that most of the existing item selection algorithms for mat bear severe computational complexity which substantially lowers the applicability of mat two versions of simplified kl index ski built from the analytical results are proposed to mimic the behavior of ki while reducing the overall computational intensity

quasi homogeneous linearization of degenerate vector fields abstract we characterize the analytic planar vector fields orbitally equivalent to its quasi homogeneous leader term by means the existence of a class of inverse integrating factors such a class of inverse integrating factors is determined by providing a normal form of two dimensional scalar functions this fact allows us to give some relevant criteria on t linearization analytical integrability and characterization of the centers for several families of planar vector fields we study the systems whose leader term is quadratic or cubic and we analyze also a class of nilpotent systems

algebraic particular integrals integrability and the problem of the center in this work we clarify the global geometrical phenomena corresponding to the notion of center for plane quadratic vector fields we first show the key role played by the algebraic particular integrals of degrees less than or equal to three in the theory of the center these curves control the changes in the systems as parameters vary the bifurcation diagram used to prove this result is realized in the natural topological space for the situation considered namely the real four dimensional projective space next we consider the known four algebraic conditions for the center for quadratic vector fields one of them says that the system is hamiltonian a condition which has a clear geometric meaning we determine the geometric meaning of the remaining other three algebraic conditions i ii iii

the problem of distinguishing between a center and a focus for nilpotent and degenerate analytic systems abstract in this work we study the centers of planar analytic vector fields which are limit of linear type centers it is proved that all the nilpotent centers are limit of linear type centers and consequently the poincare liapunov method to find linear type centers can be also used to find the nilpotent centers moreover we show that the degenerate centers which are limit of linear type centers are also detectable with the poincare liapunov method

chaos synchronization of the chua system with a fractional order chaos synchronization of two identical chua systems with the same fractional order is studied by utilizing the pecora carroll pc method the active passive decomposition pad method the one way coupling method and the bidirectional coupling one the sufficient conditions for achieving synchronization between these two systems are derived via the laplace transformation theory numerical simulations show the effectiveness of the theoretical analyses

non nested configuration of algebraic limit cycles in quadratic systems this work deals with algebraic limit cycles of planar polynomial differential systems of degree two more concretely we show among other facts that a quadratic vector field cannot possess two non nested algebraic limit cycles contained in different irreducible invariant algebraic curves

liouvillian integration and bernoulli foliations analytic foliations in the 2 dimensional complex projective space with algebraic invariant curves are studied when the holonomy groups of these curves are solvable it is shown that such a condition leads to the existence of a liouville type first integral and under generic extra conditions it is proven that these foliations can be defined by bernoulli equations

centennial history of hilbert s 16th problem the second part of hilbert s 16th problem deals with polynomial differential equations in the plane it remains unsolved even for quadratic polynomials there were several attempts to solve it that failed yet the problem inspired significant progress in the geometric theory of planar differential equations as well as bifurcation theory normal forms foliations and some topics in algebraic geometry the dramatic history of the problem as well as related developments are presented below 1 the problem and its counterparts what may be said about the number and location of limit cycles of a planar polynomial vector field of degree n the limit cycle is an isolated closed orbit of a vector field this second part of hilbert s 16th problem appears to be one of the most persistent in the famous hilbert list h second only to the riemann function conjecture traditionally hilbert s question is split into three each one requiring a stronger answer problem 1 is it true that a planar polynomial vector field has but a finite number of limit cycles problem 2 is it true that the number of limit cycles of a planar polynomial vector field is bounded by a constant depending on the degree of the polynomials only the bound on the number of limit cycles in problem 2 is denoted by h n and known as the hilbert number linear vector fields have no limit cycles hence h 1 0 it is still unknown whether or not h 2 exists problem 3 give an upper bound for h n a solution to any of these problems implies a solution for the previous ones only the first problem is solved now the positive answer was established in e92 i91 there are analytic counterparts of problems 1 and 2 received by the editors december 2001 2000 mathematics subject classification primary 34cxx 34mxx 37f75

algorithms and methods in differential algebra abstract founded by j f ritt differential algebra is a true part of algebra so that constructive and algorithmic problems and methods appear in this field in this talk i do not intend to give an exhaustive survey of algorithmic aspects of differential algebra but i only propose some examples to give an insight of the state of knowledge in this domain some problems are known to have an effective solution others have an efficient effective solution which is implemented in recent computer algebra systems and the decidability of some others is still an open question which does not prevent computations from leading to interesting results liouville s theory of integration in finite terms and risch s theorem are examples of problems that computer algebra systems now deal with very efficiently implementation work by m bronstein in what concerns linear differential equations of arbitrary order a basis for the vector space of all liouvillian solutions can in principle be computed effectively thanks to a theorem of singer s 17 22 the complexity bound is actually awful and a lot of work is done or in progress especially by m singer and f ulmer to give realistic algorithms 20 21 for third order linear differential equations existence of liouvillian first integrals is a way to make precise the notion of integrability of vector fields even in the simplest case of three dimensional polynomial vector fields no decision procedure is known for this existence nevertheless explicit computations with computer algebra yield interesting solutions for special examples in this case the process of looking for so called darboux curves can only be called a method but not an algorithm for a given degree this search is a classical algebraic elimination process but no bound is known on the degree of the candidate polynomials this paper insists on the search of liouvillian first integrals of polynomial vector fields and a new result is given the generic absence of such liouvillian first integrals for factorisable polynomial vector fields in three variables

on the remarkable values of the rational first integrals of polynomial vector fields abstract the remarkable values for polynomial vector fields in the plane having a rational first integral were introduced by poincare he was mainly interested in their algebraic aspects here we are interested in their dynamic aspects i e how they contribute to the phase portrait of the system to its separatrices to its singular points etc the relationship between remarkable values and dynamics mainly takes place through the inverse integrating factor

on a classification of polynomial differential operators with respect to the type of first integrals abstract this paper gives a classification of polynomial differential operators x x 1 x 1 x 2 1 x 2 x 1 x 2 2 i x i the classification is defined through an order derived from x let x x y be the associated differential polynomial the order is defined as the order of a differential ideal that is an essential extension of x the main result shows the order can only be four possible values 0 1 2 3 or furthermore when the order is finite the essential extension x a where a is a differential polynomial with coefficients obtained through a rational solution of a partial differential equation given explicitly by coefficients of x when the order is infinite the extension is identical with x in addition if and only if the order is 0 1 or 2 the associated polynomial differential equation has liouvillian first integrals examples and connections with godbillon vey sequences are also discussed