weak focus limit cycles and bifurcations for bounded quadratic systems abstract we study the bounded quadratic systems with either two weak foci or a weak focus of order 2 from the first case we obtain 1 1 configuration of limit cycles and in the second case we prove that there is no limit cycle surrounding the weak focus of order 2 also we unfold the bounded quadratic system with a center and the maximum number of limit cycles in this case is two

bounded polynomial vector fields in r2 and rn abstract we characterize the bounded polynomial vector fields in r 2 additionally we provide a necessary condition but not sufficient which must be satisfied by bounded polynomial vector fields in r n

theories of causal ordering abstract this paper is a response to iwasaki and simon 14 which criticizes de kleer and brown 8 we argue that many of their criticisms particularly concerning causality modeling and stability originate from the difference of concerns between engineering and economics our notion of causality arises from considering the interconnections of components not equations when no feedback is present the ordering produced by our qualitative physics is similar to theirs however when feedback is present our qualitative physics determines a causal ordering around feedback loops as well causal ordering is a general technique not only applicable to qualitative reasoning therefore we also explore the relationship between causal ordering and propagation of constraints upon which the methods of qualitative physics are based

homoclinic loop and multiple limit cycle bifurcation surfaces this paper establishes the existence and analyticity of homoclinic loop bifurcation surfaces z and multiplicity two limit cycle bifurcation surfaces f for planar systems depending on two or more parameters it determines the side of or f on which limit cycles occur and it shows that if x and intersect then typically they do so at a flat contact

numerical computation of minimal polynomial bases a generalized resultant approach we propose a new algorithm for the computation of a minimal polynomial basis of the left kernel of a given polynomial matrix f s the proposed method exploits the structure of the left null space of generalized wolovich or sylvester resultants to compute row polynomial vectors that form a minimal polynomial basis of left kernel of the given polynomial matrix the entire procedure can be implemented using only orthogonal transformations of constant matrices and results to a minimal basis with orthonormal coefficients

computation of lyapunov values for two planar polynomial differential systems the complex formulas of computing the lyapunov values are investigated for two planar polynomial systems with a pair of pure imaginary eigenvalues the relations between the two systems and their standard forms are discussed and the transformation formulas are given the corresponding complex formulas of computing the lyapunov values are obtained the results obtained here can be applied to judging the type of singular points and the number of limit cycles in planar polynomial systems

the classification of 2 2 systems of non strictly hyperbolic conservation laws with application to oil recovery on considere un systeme 2 2 a une variable d espace u f u x 0 0 ou u u x t r 2 et f r 2 r 2 on montre que pour des equations hyperboliques des points ombilicaux sont typiquement isoles et que en un point ombilical isole df est typiquement diagonalisable on donne une classification des equations hyperboliques a points ombilicaux isoles

canard solutions at non generic turning points this paper deals with singular perturbation problems for vector fields on 2 dimensional manifolds canard solutions are solutions that starting near an attracting normally hyperbolic branch of the singular curve cross a turning point and follow for a while a normally repelling branch of the singular curve following the geometric ideas developed by dumortier and roussarie in 1996 for the study of canard solutions near a generic turning point we study canard solutions near non generic turning points characterization of manifolds of canard solutions is given in terms of boundary conditions their regularity properties are studied and the relation is described with the more traditional asymptotic approach it reveals that interesting information on canard solutions can be obtained even in cases where an asymptotic approach fails to work since the manifolds of canard solutions occur as intersection of center manifolds defined along respectively the attracting and the repelling branch of the singular curve we also study their contact and its relation to the control curve

bifurcation of limit cycles from centers and separatrix cycles of planar analytic systems this paper presents a survey of results on the bifurcation of limit cycles from centers and separatrix cycles of perturbed planar analytic systems and contributes some new results on the bifurcation of multiple limit cycles from centers and on the multiplicity of separatrix cycles of such systems the basic theme throughout the paper is that the number positions and multiplicities of the limit cycles that bifurcate under perturbations are related to the number positions and multiplicities of the zeros of the melnikov function for the system the general theory is illustrated by a number of examples from the literature some of which are extended to include new results

some theorems on the existence uniqueness and nonexistence of limit cycles for quadratic systems abstract given a quadratic system qs with a focus or a center at the origin we write it in the form y p 2 x y x dy q 2 x y where p 2 and q 2 are homogeneous polynomials of degree 2 if we define f x y x dy p 2 x y yq 2 x y and g x y xq 2 x y yp 2 x y we give results of existence nonexistence and uniqueness of limit cycles if f x y g x y does not change of sign then by using these results plus the properties on the evolution of the limit cycles of the semicomplete families of rotated vector fields we can study some particular families of qs i e the qs with a unique finite singularity and the bounded qs with either one or two finite singularities

compactification and desingularization of spaces of polynomial lienard equations abstract the paper deals with polynomial lienard equations of type m n i e planar vector fields associated to a scalar second order differential equation x f x x g x 0 with f and g polynomials of respective degree m and n it is shown that besides compactifying the phase plane or the lienard plane one can also compactify and desingularize the space of lienard equations of type m n for each m n separately by adding both singular perturbation problems and hamiltonian perturbation problems

regular and slow fast codimension 4 saddle node bifurcations abstract using geometric singular perturbation theory including the family blow up as one of the main techniques we prove that the cyclicity i e maximum number of limit cycles in both regular and slow fast unfoldings of nilpotent saddle node singularity of codimension 4 is 2 the blow up technique enables us to use the well known results for slow fast codimension 1 and 2 hopf bifurcations slow fast bogdanov takens bifurcations and slow fast codimension 3 saddle and elliptic bifurcations

limit cycles near hyperbolas in quadratic systems abstract in this paper we introduce the notion of infinity strip and strip of hyperbolas as organizing centers of limit cycles in polynomial differential systems on the plane we study a strip of hyperbolas occurring in some quadratic systems we deal with the cyclicity of the degenerate graphics d i 2 a from the programme set up in f dumortier r roussarie c rousseau hilbert s 16th problem for quadratic vector fields j differential equations 110 1994 86 133 to solve the finiteness part of hilbert s 16th problem for quadratic systems techniques from geometric singular perturbation theory are combined with the use of the bautin ideal we also rely on the theory of darboux integrability

slow fast bogdanov takens bifurcations abstract in this paper we study perturbations from planar vector fields having a line of zeros and representing a singular limit of bogdanov takens bt bifurcations we introduce among other precise definitions the notion of slow fast bt bifurcation and we provide a complete study of the bifurcation diagram and the related phase portraits based on geometric singular perturbation theory including blow up we get results that are valid on a uniform neighborhood both in parameter space and in the phase plane

global families of limit cycles of planar analytic systems the global behavior of any one parameter family of limit cycles of a planar analytic system x f x a depending on a parameter a e r is determined it is shown that any one parameter family of limit cycles belongs to a maximal one parameter family which is either open or cyclic if the family is open then it terminates as the parameter or the orbits become unbounded or it terminates at a critical point or on a compound separatrix cycle of the system this implies that the periods in a one parameter family of limit cycles can become unbounded only if the orbits become unbounded or if they approach a degenerate critical point or compound separatrix cycle of the system this is a more specific result for planar analytic systems than wintner s principle of natural termination for n dimensional systems where the periods can become unbounded in strange ways this work generalizes duff s results for one parameter families of limit cycles generated by a one parameter family of rotated vector fields in particular it is shown that the behavior at a nonsingular multiple limit cycle of any one parameter family of limit cycles is exactly the same as the behavior at a multiple limit cycle of a one parameter family of liniit cycles generated by a one parameter family of rotated vector fields

time analysis and entry exit relation near planar turning points abstract the paper deals with canard solutions at very general turning points of smooth singular perturbation problems in two dimensions we follow a geometric approach based on the use of c k normal forms centre manifolds and family blow up as we did in trans amer math soc to appear in trans amer math soc to appear we considered the existence of manifolds of canard solutions for given appropriate boundary conditions these manifolds need not be smooth at the turning point in this paper we essentially study the transition time along such manifolds as well as the divergence integral providing a structure theorem for these integrals as a consequence we get a nice structure theorem for the transition equation governing the canard solutions it permits to compare different control manifolds and to obtain a precise description of the entry exit relation of different canard solutions attention is also given to the special case in which the canard manifolds are smooth i e when formal canard solutions exist

a proof of perkoʼs conjectures for the bogdanov takens system abstract the bogdanov takens system has at most one limit cycle and in the parameter space it exists between a hopf and a saddle loop bifurcation curves the aim of this paper is to prove the perko s conjectures about some analytic properties of the saddle loop bifurcation curve moreover we provide sharp piecewise algebraic upper and lower bounds for this curve

multiple canard cycles in generalized lienard equations the paper treats multiple limit cycle bifurcations in singular perturbation problems of planar vector fields the results deal with any number of parameters proofs are based on the techniques introduced in canard cycles and center manifolds f dumortier and r roussarie 1996 mem amer math soc 121 the presentation is limited to generalized lienard equations ex x c x x c 0

bifurcation of limit cycles from quadratic isochrones abstract for a one parameter family of plane quadratic vector fields x e depending analytically on a small real parameter e we determine the number and position of the local families of limit cycles which emerge from the periodic trajectories surrounding an isochronous or linearizable center techniques are developed for treating the bifurcations of all orders and these are applied to prove the following results for the linear isochrone the maximum number of continuous families of limit cycles which can emerge is three for one class of nonlinear isochrones at most one continuous family of limit cycles can emerge whereas for all other nonlinear isochrones at most two continuous families of limit cycles can emerge moreover for each isochrone in one of these classes there are small perturbations such that the indicated maximum number of continuous families of limit cycles can be made to emerge from a corresponding number of arbitrarily prescribed periodic orbits within the period annulus of the isochronous center

shock wave admissibility for quadratic conservation laws abstract in this work we present a new approach to the study of the stability of admissible shock wave solutions for systems of conservation laws that change type the systems we treat have quadratic flux functions we employ the fundamental wave manifold w as a global framework to characterize shock waves that comply with the viscosity admissibility criterion points of w parametrize dynamical systems associated with shock wave solutions the region of w comprising admissible shock waves is bounded by the loci of structurally unstable dynamical systems explicit formulae are presented for the loci associated with saddle node hopf and bogdanov takens bifurcation and with straight line heteroclinic connections using melnikov s integral analysis we calculate the tangent to the homoclinic part of the admissibility boundary at bogdanov takens points of w furthermore using numerical methods we explore the heteroclinic loci corresponding to curved connecting orbits and the complete homoclinic locus we find the region of admissible waves for a generic two dimensional slice of the fundamental wave manifold and compare it with the set of shock points that comply with the lax admissibility criterion thereby elucidating how this criterion differs from viscous profile admissibility

on bifurcation delay an alternative approach using geometric singular perturbation theory abstract to explain the phenomenon of bifurcation delay which occurs in planar systems of the form x f x z z g x z z where f x 0 0 0 and g x 0 0 changes sign at least once on the x axis we use the exchange lemma in geometric singular perturbation theory to track the limiting behavior of the solutions using the trick of extending dimension to overcome the degeneracy at the turning point we show that the limiting attracting and repulsion points are given by the well known entry exit function and the minimum of z on the trajectory is of order exp 1 also we prove smoothness of the return map up to arbitrary finite order in

linearizations of matrix polynomials in newton bases we discuss matrix polynomials expressed in a newton basis and the associated polynomial eigenvalue problems properties of the generalized ansatz spaces associated with such polynomials are proved directly by utilizing a novel representation of pencils in these spaces also we show how the family of fiedler pencils can be adapted to matrix polynomials expressed in a newton basis these new newton fiedler pencils are shown to be strong linearizations and some computational aspects related to them are discussed

causality and model abstraction abstract much of science and engineering is concerned with characterizing processes by equations describing the relations that hold among parameters of objects and govern their behavior over time in formal descriptions of processes in terms of parameters and equations the notion of causality is rarely made explicit formal treatments of the foundations of sciences have avoided discussions of causation and spoken only of functional relations among variables nevertheless the notion of causality plays an important role in our understanding of phenomena even when we describe the behavior of a system formally in terms of acausal mathematical relations we often give an informal intuitive explanation of why the system behaves the way it does in terms of cause effect relations in this paper we will present an operational definition of causal ordering the definition allows us to extract causal dependency relations among variables implicit in a model of a system when a model is represented as a set of acausal mathematical relations our approach is based on the theory of causal ordering first presented by simon 22 the paper shows how to use the theory and its extension in reasoning about physical systems further the paper studies the relation of the theory to the problems of model aggregation

complete bifurcation diagram and global phase portraits of lienard differential equations of degree four abstract li and llibre in j differential equations 252 2012 3142 3162 proved that a lienard system of degree four d x d t y a x b x 2 c x 3 x 4 d y d t x has at most one limit cycle moreover the limit cycle is stable and hyperbolic if it exists based on their works the aim of this paper is to give the complete bifurcation diagram and global phase portraits in the poincare disc of this system further first we analyze the equilibria at both finity and infinity then a necessary and sufficient condition for existence of separatrix loop is founded by the rotation property moreover a necessary and sufficient condition of the existence of limit cycles is also obtained finally we show that the complete bifurcation diagram includes one hopf bifurcation surface and one bifurcation surface of separatrix loop

on melnikov functions of a homoclinic loop through a nilpotent saddle for planar near hamiltonian systems abstract the first order melnikov function of a homoclinic loop through a nilpotent saddle for general planar near hamiltonian systems is considered the asymptotic expansion of this melnikov function and formulas for its first coefficients are given the number of limit cycles which appear near the homoclinic loop is discussed by using the asymptotic expansion of the first order melnikov function an example is presented as an application of the main results

limit cycles in slow fast codimension 3 saddle and elliptic bifurcations abstract this paper deals with local bifurcations occurring near singular points of planar slow fast systems in particular it is concerned with the study of the slow fast variant of the unfolding of a codimension 3 nilpotent singularity the slow fast variant of a codimension 1 hopf bifurcation has been studied extensively before and its study has lead to the notion of canard cycles in the van der pol system similarly codimension 2 slow fast bogdanov takens bifurcations have been characterized here the singularity is of codimension 3 and we distinguish slow fast elliptic and slow fast saddle bifurcations we focus our study on the appearance on small amplitude limit cycles and rely on techniques from geometric singular perturbation theory and blow up

quadratic models for generic local 3 parameter bifurcations on the plane the first chapter deals with singularities occurring in quadratic planar vector fields we make distinction between singularities which as a general system are of finite codimension and singularities which are of infinite codimension in the sense that they are nonisolated or hamiltonian or integrable or that they have an axis of symmetry after a linear coordinate change or that they can be approximated by centers in the second chapter we provide quadratic models for all the known versal k parameter unfoldings with k 1 2 3 except for the nilpotent focus which cannot occur as a quadratic system we finally show that a certain type of elliptic points of codimension 4 does not have a quadratic versal unfolding introduction how complicated can local bifurcations be in quadratic systems or in other terms which phenomena can be described by germs of families of quadratic vector fields on the plane we do not give a complete answer to this question but we essentially treat two exemplary aspects the first one concerns the singularities themselves emphasizing their codimension as a quadratic system and as a general system the second one concerns the relation between the quadratic unfolding of the singularity and its versal unfolding as a general system the complete treatment of the first problem is described in chapter 1 we make a distinction between singularities having a finite codimension or having infinite codimension as a general system the singularities of the first kind are listed completely while those of the second kind are only checked to be nonisolated or isolated but hamiltonian or integrable or having an axis of symmetry after a linear coordinate change or approachable by centers these cases are of course not mutually distinct for the singularities of the first kind the quadratic codimension reveals to be the same as the general one it can reach at most 5 and this only happens if the vector field is homogeneous different kinds of singularities of codimension 4 however occur received by the editors december 1 1989 1980 mathematics subject classification 1985 revision primary 58fxx 34cxx

a system with three limit cycles appearing in a hopf bifurcation and dying in a homoclinic bifurcation the cusp of order 4 abstract in this paper we give an example of a family of polynomial vector fields with three limit cycles appearing simultaneously on a hopf bifurcation h of order 3 and vanishing simultaneously in a homoclinic loop bifurcation hl of order 3 the region with three limit cycles is a topological 3 simplex the system is a generalization of bogdanov s system at the same time we give the bifurcation diagram for the universal unfolding of the cusp of order 4 this bifurcation diagram is a cone it contains a cone on the bifurcation diagram with the three limit cycles inside a 3 simplex region plus saddle node and cusp bifurcations of lower order

a global analysis of the bogdanov takens system this paper studies the bifurcations that occur in the bogdanov takens system dot x y qquad dot y x x 1 mu 1 y mu 2 xy both in the phase plane bf r 2 and at infinity i e on the poincare sphere bf s 2 for all parameter values mu 1 mu 2 in bf r 2 for all mu 2 ne 0 a unique limit cycle bifurcates from the origin at mu 1 0 and it expands monotonically until it intersects the saddle at 1 0 and forms a homoclinic loop at mu 1 h mu 2 the function h mu 2 describing the homoclinic loop bifurcation curve in bf r 2 is analytic for all mu 2 in bf r h 0 0 h 0 1 7 and h pm infty 0 the only other bifurcation for this system is a saddle saddle connection on bf s 2 which occurs when mu 1 1 mu 2 even for mu 1 mu 1 mu 2 1 mu 1 h mu 2 ne 0 this system is not structurally stable on bf s 2 due to the presence of nonhyperbolic critical points on the equ

singular perturbations and vanishing passage through a turning point abstract the paper deals with planar slow fast cycles containing a unique generic turning point we address the question on how to study canard cycles when the slow dynamics can be singular at the turning point we more precisely accept a generic saddle node bifurcation to pass through the turning point it reveals that in this case the slow divergence integral is no longer the good tool to use but its derivative with respect to the layer variable still is we provide general results as well as a number of applications we show how to treat the open problems presented in artes et al 2009 1 and dumortier and rousseau 2009 13 dealing respectively with the graphics di 2 a and df 1 a from dumortier et al 1994 14

multiple limit cycle bifurcation surfaces and global families of multiple limit cycles abstract sufficient conditions are given for the local exitence of multiplicity m limit cycle bifurcation surfaces cm of planar analytic systems depending on n parameters with n m 2 in the generic case the surfaces c2 c3 and c4 are the familiar saddle node cusp and swallow tail bifurcation surfaces respectively the author s termination principle for global one parameter families of simple limit cycles of relatively prime planar analytic systems is generalized to a termination principle for global one parameter families of multiple limit cycles which implies that the boundary of a global limit cycle bifurcation surface typically consists of hopf bifurcation surfaces and or homoclinic or heteroclinic loop bifurcation surfaces

analysis of degenerate hopf bifurcations for a nonlinear model of calcium metabolism results of the in viva studies related to calcium metabolism have been generally analysed within the framework of linear mathematical models e g classical compartmental analysis 11 this approach assumes the maintenance of the extracellular calcium concentration at a nearly constant value under physiological conditions plasma calcium homeostasis and the adequacy of a sum of exponential terms the analytical solution of a first order linear differential system to fit kinetic data such as the time course of tracer concentration nonlinear models which exhibit a self oscillatory behaviour have been proposed recently a model of the hormonal control of calcium homeostasis in chicken includes the consideration of a growth function and takes into account complex interrelationships among the calciumregulating hormones and between these hormones and specific metabolic functions 2 another model constructed in our laboratory is a self oscillating model for calcium metabolism in the rat it includes some nonlinear processes involved in bone calcium metabolism and adequately reproduces the kinetic behaviour of radiocalcium in plasma including circadian variations 3 in contrast to linear systems for which the simple dynamic behaviour patterns are predictable and their analytical solutions are commonly applied in calcium metabolism field it is wellestablished that nonlinear systems have potentially numerous and complex spatio temporal expressions excitability periodic oscillations chaos etc therefore the relevance of the proposed nonlinear models and their significance with regard to the control of calcium metabolism require a concomitant knowledge of the set of the dynamic patterns that in theory such nonlinear structures are able to produce the variety of the dynamic behaviours exhibited by the self oscillating model we have established in rat or rather its nonlinear sub structure 4 51 motivated this theoretical work our model of calcium metabolism has been constructed in the general context of compartmental formalism fig 1 briefly this model was based on the explicit consideration the nonlinear properties of the phase transition and nucleation mechanisms associated with bone calcification and their formulation through a single second order autocatalytic process 5 in addition to its ability to generate periodic temporal variations fitting to experimental data fig 2 the model correctly approximates the growth curve of calcium mass as a function of age and proposes a dynamic organization for bone calcium metabolism which distinguishes

causality in device behavior abstract this paper shows how formal characterizations of causality and of the method of comparative statics long used in econometrics thermodynamics and other domains can be applied to clarify and make rigorous the qualitative causal calculus recently proposed by de kleer and brown 2 the formalization shows exactly what assumptions are required to carry out causal analysis of a system of interdependent variables in equilibrium and to propagate disturbances through such a system the intuitive concepts of causality captured by de kleer and brown provide a rough approximation to the standard analytic techniques that are used in the treatment of simultaneous algebraic and differential equations

a dynamic systems perspective on qualitative simulation abstract this paper presents a simple geometric interpretation of qualitative simulation qs based on the phase space representation of dynamic systems theory qs consists of two steps transition analysis determines the sequence of qualitative states that a system traverses and global interpretation derives its long term behavior i recast transition analysis as a search problem in phase space and replace the assorted transition rules with two algebraic conditions the first condition determines transitions between arbitrarily shaped regions in phase space as opposed to qs which only handles n dimensional rectangles it also provides more accurate results by considering only the boundaries between regions the second condition determines whether nearby trajectories approach a fixed point asymptotically it obtains better results than qs by exploiting local stability properties i recast global interpretation of dissipative systems as a search for attractors in phase space and present a global interpretation strategy for the subset of these systems whose local behavior determines global behavior uniquely although limited in scope this strategy handles many systems that appear in the qs literature

separatrix and limit cycles of quadratic systems and dulac s theorem separatrix cycles for a planar quadratic vector field are studied the results obtained are used to show that in any bounded region of the plane a quadratic vector field has at most a finite number of limit cycles

local bifurcations and a survey of bounded quadratic systems abstract this paper presents a survey of the known results for bounded quadratic systems as well as a study of the local bifurcations that occur at critical points of such systems it is shown that the only finite codimension bifurcations that occur at a critical point of a bounded quadratic system are the saddle node and the hopf takens bifurcations of codimensions 1 and 2 and the bogdanov takens bifurcations of codimensions 2 and 3 furthermore it is shown that whenever a bounded quadratic system has one of these critical points then a full generic unfolding of the critical point exists in the class of bounded quadratic systems finally we give a complete list of those limit periodic sets whose finite cyclicity still needs to be established in order to obtain the existence of a finite upper bound for the number of limit cycles that can occur in a bounded quadratic system

a neural network model of causality this paper proposes a model for commonsense causal reasoning based on the basic idea of neural networks after an analysis of the advantages and limitations of existing accounts of causality a fuzzy logic based formalism fel is proposed that takes into account the inexactness and the cumulative evidentiality of commonsense causal reasoning overcoming the limitations of existing accounts analyses concerning how fel handles various aspects of commonsense causal reasoning are performed in an abstract way fel can be implemented naturally in a neural connectionist network this work also serves to link rule based reasoning with neural network models in that a rule encoding scheme fel is equated directly to a neural network model

block minimal bases l ifications of matrix polynomials abstract the standard way of solving a polynomial eigenvalue problem associated with a matrix polynomial starts by embedding the matrix coefficients of the polynomial into a matrix pencil known as a strong linearization this process transforms the problem into an equivalent generalized eigenvalue problem however there are some situations in which is more convenient to replace linearizations by other low degree matrix polynomials this has motivated the idea of a strong l ification of a matrix polynomial which is a matrix polynomial of degree at most l having the same finite and infinite elementary divisors and the same numbers of left and right minimal indices as the original matrix polynomial we present in this work a novel method for constructing strong l ifications of matrix polynomials of size m n and grade d when l d and l divides nd or md this method is based on a family called strong block minimal bases matrix polynomials and relies heavily on properties of dual minimal bases we show how strong block minimal bases l ifications can be constructed from the coefficients of a given matrix polynomial p we also show that these l ifications satisfy many desirable properties for numerical applications they are strong l ifications regardless of whether p is regular or singular the minimal indices of the l ifications are related to those of p via constant uniform shifts and eigenvectors and minimal bases of p can be recovered from those of any of the strong block minimal bases l ifications in the special case where l divides d we introduce a subfamily of strong block minimal bases matrix polynomials named block kronecker matrix polynomials which is shown to be a fruitful source of companion l ifications

block kronecker linearizations of matrix polynomials and their backward errors we introduce a new family of strong linearizations of matrix polynomials which we call block kronecker pencils and perform a backward stability analysis of complete polynomial eigenproblems these problems are solved by applying any backward stable algorithm to a block kronecker pencil such as the staircase algorithm for singular pencils or the qz algorithm for regular pencils this analysis allows us to identify those block kronecker pencils that yield a computed complete eigenstructure which is exactly that of a slightly perturbed matrix polynomial these favorable pencils include the famous fiedler linearizations which are just a very particular case of block kronecker pencils thus our analysis offers the first proof available in the literature of global backward stability for fiedler pencils in addition the theory developed for block kronecker pencils is much simpler than the theory available for fiedler pencils especially in the case of rectangular matrix polynomials the global backward error analysis in this work presents for the first time the following key properties it is a rigorous analysis valid for finite perturbations i e it is not a first order analysis it provides precise bounds it is valid simultaneously for a large class of linearizations and it establishes a framework that may be generalized to other classes of linearizations these features are related to the fact that block kronecker pencils are a particular case of the new family of strong block minimal bases pencils which include certain perturbations of block kronecker pencils this allows us to extend the results in this paper to other contexts

automatic qualitative analysis of dynamic systems using piecewise linear approximations abstract this paper explores automating the qualitative analysis of physical systems it describes a program called plr for piecewise linear reasoner that formalizes an analysis strategy employed by experts plr takes parameterized ordinary differential equations as input and produces a qualitative description of the solutions for all initial values it approximates intractable nonlinear systems with piecewise linear ones analyzes the approximations and draws conclusions about the original systems it chooses approximations that are accurate enough to reproduce the essential properties of their nonlinear prototypes yet simple enough to be analyzed completely and efficiently it derives additional properties such as boundedness or periodicity by theoretical methods plr s analysis depends on abstract properties of systems rather than on specific numeric values this makes its conclusions more robust and enables it to handle parameterized equations transparently i demonstrate plr on several common nonlinear systems and on published examples from mechanical engineering

a note on the correctness of the causal ordering algorithm in this paper we examine in detail the algorithm of simon h a simon causal ordering and identifiability in w c hood t c koopmans eds studies in econometric method cowles commission for research in economics monograph no 14 john wiley sons inc new york 1953 pp 49 74 chapter iii called the causal ordering algorithm coa used for constructing the causal ordering of a system given a complete specification of the system in terms of a set of structural equations that govern the variables in the system this algorithm constructs a graphical characterization of the model in a form that we call a partial causal graph simon argued in h a simon causal ordering and identifiability in w c hood t c koopmans eds studies in econometric method cowles commission for research in economics monograph no 14 john wiley sons inc new york 1953 pp 49 74 chapter iii and subsequent papers that a graph so generated explicates causal structure among variables in the model we formalize this claim further by proving that any causal model based on a one to one correspondence between equations and variables must be consistent with the coa

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

widely separated frequencies in coupled oscillators with energy preserving quadratic nonlinearity in this paper we present an analysis of a system of coupled oscillators suggested by atmospheric dynamics we make two assumptions for our system the first assumption is that the frequencies of the characteristic oscillations are widely separated and the second is that the nonlinear part of the vector field preserves the distance to the origin using the first assumption we prove that the reduced normal form of our system has an invariant manifold which exists for all values of the parameters this invariant manifold cannot be perturbed away by including higher order terms in the normal form using the second assumption we view the normal form as an energy preserving three dimensional system which is linearly perturbed restricting ourselves to a small perturbation the flow of the energy preserving system is used to study the flow in general we present a complete study of the flow of the energy preserving system and its bifurcations using these results we return to the dissipative system and provide the condition for having a hopf bifurcation of one of the two equilibria of the perturbed system we also numerically follow the periodic solution created via the hopf bifurcation and find a sequence of period doubling and fold bifurcations also a torus or neimark sacker bifurcation

semiring induced valuation algebras exact and approximate local computation algorithms local computation in join trees or acyclic hypertrees has been shown to be linked to a particular algebraic structure called valuation algebra there are many models of this algebraic structure ranging from probability theory to numerical analysis relational databases and various classical and non classical logics it turns out that many interesting models of valuation algebras may be derived from semiring valued mappings in this paper we study how valuation algebras are induced by semirings and how the structure of the valuation algebra is related to the algebraic structure of the semiring in particular c semirings with idempotent multiplication induce idempotent valuation algebras and therefore permit particularly efficient architectures for local computation also important are semirings whose multiplicative semigroup is embedded in a union of groups they induce valuation algebras with a partially defined division for these valuation algebras the well known architectures for bayesian networks apply we also extend the general computational framework to allow derivation of bounds and approximations for when exact computation is not feasible

canard cycles in the presence of slow dynamics with singularities we study the cyclicity of limit periodic sets that occur in families of vector flelds of slow fast type the limit periodic sets are formed by a fast orbit and a curve of singularities containing a unique turning point at this turning point a stability change takes place on one side of the turning point the dynamics point strongly towards the curve of singularities on the other side the dynamics point away from the curve of singularities the presence of periodic orbits in a perturbation is related to the presence of canard orbits passing near this turning point i e orbits that stay close to the curve of singularities despite the exponentially strong repulsion near this curve all existing results deal with a non zero slow movement permitting to get a good estimate of the cyclicity by considering the slow divergence integral along the curve of singularities in this paper we study what happens when the slow dynamics exhibit singularities in particular our study includes the cyclicity of the slow fast 2 saddle cycle formed by a regular saddle connection the fast part and a part of the curve of singularities the slow part we see that the relevant information is no longer merely contained in the slow divergence integral this paper concerns the study of the cyclicity of limit periodic sets in a quite general class of slow fast vector flelds on a 2 manifold m we are interested in families of vector flelds x possibly depending on other parameters as well where the unperturbed fast vector fleld x0 has a curve of singular points called a critical curve we call a point p on normally attracting resp normally repelling when dx0 p has a strictly negative resp strictly positive eigenvalue corresponding to an eigendirection not tangent to when has both normally attracting points and normally repelling points it may occur that x0 has orbits connecting two such points let f be such a fast orbit so that the limit and fi limit lie on we study the limit periodic set lf formed by f and the piece of going from the limit of f to the fi limit of f of course the nonzero eigenvalue of dx0 p must bifurcate along this piece of assume that this happens in a unique point p called a turning point one also says normal hyperbolicity of x0 is lost in p in such a situation it is possible that the limit periodic set perturbs into one or more isolated periodic orbits such orbits are called canard cycles

computation of normal forms for high dimensional non linear systems and application to non planar non linear oscillations of a cantilever beam a new and efficient computation of the normal forms is developed in this paper for high dimensional non linear systems and the computational method is applied to non planar non linear oscillations of a cantilever beam the method developed here has the advantage of directly calculating the coefficients of the normal forms and the associated near identity non linear transformations for three different cases that is 1 the case of two pairs of pure imaginary eigenvalues 2 the case of one non semisimple double zero and a pair of pure imaginary eigenvalues and 3 the case of two non semisimple double zero eigenvalues the final partial differential equations of various resonant cases appear in a canonical form whose solutions can be conveniently obtained using polynomial equations with the aid of the maple software a symbolic program for computing the normal forms of high dimensional non linear systems is given comparing the method developed here with other methods of computing the normal forms it is understood that we may respectively obtain the normal forms the coefficients of the normal forms and the associated near identity non linear transformations for three resonant cases by using a same main maple symbolic program moreover the method is easy to apply to engineering problems the normal forms of the averaged equations and their coefficients for non planar non linear oscillations of the cantilever beam under combined parametric and forcing excitations are calculated for two resonant cases

polynomial zigzag matrices dual minimal bases and the realization of completely singular polynomials minimal bases of rational vector spaces are a well known and important tool in systems theory if minimal bases for two subspaces of rational n space are displayed as the rows of polynomial matrices z 1 lambda k times n and z 2 lambda m times n respectively then z 1 and z 2 are said to be dual minimal bases if the subspaces have complementary dimension i e k m n and z 1 lambda z 2 t lambda 0 in other words each z j lambda provides a minimal basis for the nullspace of the other it has long been known that for any dual minimal bases z 1 lambda and z 2 lambda the row degree sums of z 1 and z 2 are the same in this paper we show that this is the only constraint on the row degrees thus characterizing the possible row degrees of dual minimal bases the proof is constructive making extensive use of a new class of sparse structured polynomial matrices that we have baptized zigzag matrices another application of these polynomial zigzag matrices is the constructive solution of the following inverse problem for minimal indices given a list of left and right minimal indices and a desired degree d does there exist a completely singular matrix polynomial i e a matrix polynomial with no elementary divisors whatsoever of degree d having exactly the prescribed minimal indices we show that such a matrix polynomial exists if and only if d divides the sum of the minimal indices the constructed realization is simple and explicitly displays the desired minimal indices in a fashion analogous to the classical kronecker canonical form of singular pencils

robustness and perturbations of minimal bases ii the case with given row degrees abstract this paper studies generic and perturbation properties inside the linear space of m m n polynomial matrices whose rows have degrees bounded by a given list d 1 d m of natural numbers which in the particular case d 1 d m d is just the set of m m n polynomial matrices with degree at most d thus the results in this paper extend to a much more general setting the results recently obtained in 29 only for polynomial matrices with degree at most d surprisingly most of the properties proved in 29 as well as their proofs remain to a large extent unchanged in this general setting of row degrees bounded by a list that can be arbitrarily inhomogeneous provided the well known sylvester matrices of polynomial matrices are replaced by the new trimmed sylvester matrices introduced in this paper the following results are presented among many others in this work 1 generically the polynomial matrices in the considered set are minimal bases with their row degrees exactly equal to d 1 d m and with right minimal indices differing at most by one and having a sum equal to i 1 m d i and 2 under perturbations these generic minimal bases are robust and their dual minimal bases can be chosen to vary smoothly

temporal constraint satisfaction on causal models abstract causal models play a fundamental role in many intelligent problem solving activities such as diagnosis or simulation although the notion of causation is strictly related to the notion of time temporal knowledge and temporal reasoning have been often disregarded in problem solvers operating on causal models in this page we propose an approach to temporal constraint satisfaction on causal models the approach departs significantly from traditional approaches to temporal constraint satisfaction in our approach in fact we use synchronization mechanisms for dealing with events caused by a conjunction of other events in the paper we analyze the computation complexity of temporal constraint satisfaction on causal models showing that such a form of reasoning is intrinsically intractable i e it is intractable even if one limits the attention to a simple form of causal model and of consistency constraints however we show that it is possible to define sufficient conditions for temporal consistency that can be checked in polynomial time interestingly such conditions allow us to relate our approach to traditional approaches to temporal constraint satisfaction

robustness and perturbations of minimal bases abstract polynomial minimal bases of rational vector subspaces are a classical concept that plays an important role in control theory linear systems theory and coding theory it is a common practice to arrange the vectors of any minimal basis as the rows of a polynomial matrix and to call such matrix simply a minimal basis very recently minimal bases as well as the closely related pairs of dual minimal bases have been applied to a number of problems that include the solution of general inverse eigenstructure problems for polynomial matrices the development of new classes of linearizations and l ifications of polynomial matrices and backward error analyses of complete polynomial eigenstructure problems solved via a wide class of strong linearizations these new applications have revealed that although the algebraic properties of minimal bases are rather well understood their robustness and the behavior of the corresponding dual minimal bases under perturbations have not yet been explored in the literature as far as we know therefore the main purpose of this paper is to study in detail when a minimal basis m is robust under perturbations i e when all the polynomial matrices in a neighborhood of m are minimal bases and in this case how perturbations of m change its dual minimal bases in order to study such problems a new characterization of whether or not a polynomial matrix is a minimal basis in terms of a finite number of rank conditions is introduced and based on it we prove that polynomial matrices are generically minimal bases with very specific properties in addition some applications of the results of this paper are discussed

the cyclicity of triangles and segments in quadratic systems abstract the paper deals with quadratic systems having separatrix contour in the form of a triangle or a conic segment the limit cycles appearing near this configuration after quadratic perturbation is studied the maximal number of limit cycles is 3 for the bounded triangle and 2 for the unbounded triangle and for the bounded segment

causality tests and observationally equivalent representations of econometric models abstract this article deals with the problem of observational equivalence of alternative econometric equation systems on the same data as a practical obstacle to econometricians success in applying definitions of causality by simon 1953 strotz and wold 1960 and granger 1969 to economic time series

spectral equivalence of matrix polynomials and the index sum theorem abstract the concept of linearization is fundamental for theory applications and spectral computations related to matrix polynomials however recent research on several important classes of structured matrix polynomials arising in applications has revealed that the strategy of using linearizations to develop structure preserving numerical algorithms that compute the eigenvalues of structured matrix polynomials can be too restrictive because some structured polynomials do not have any linearization with the same structure this phenomenon strongly suggests that linearizations should sometimes be replaced by other low degree matrix polynomials in applied numerical computations motivated by this fact we introduce equivalence relations that allow the possibility of matrix polynomials with coefficients in an arbitrary field to be equivalent with the same spectral structure but have different sizes and degrees these equivalence relations are directly modeled on the notion of linearization and consequently inherit the simplicity applicability and most relevant properties of linearizations simultaneously though they are much more flexible in the possible degrees of equivalent polynomials this flexibility allows us to define in a unified way the notions of quadratification and l ification to introduce the concept of companion form of arbitrary degree and to provide concrete and simple examples of these notions that generalize in a natural and smooth way the classical first and second frobenius companion forms the properties of l ifications are studied in depth in this process a fundamental result on matrix polynomials the index sum theorem is recovered and extended to arbitrary fields although this result is known in the systems theory literature for real matrix polynomials it has remained unnoticed by many researchers it establishes that the sum of the finite and infinite partial multiplicities together with the left and right minimal indices of any matrix polynomial is equal to the rank times the degree of the polynomial the index sum theorem turns out to be a key tool for obtaining a number of significant results on the possible sizes and degrees of l ifications and companion forms on the minimal index preservation properties of companion forms of arbitrary degree as well as on obstructions to the existence of structured companion forms for structured matrix polynomials of even degree this paper presents many new results blended together with results already known in the literature but extended here to the most general setting of matrix polynomials of arbitrary sizes and degrees over arbitrary fields therefore we have written the paper in an expository and self contained style that makes it accessible to a wide variety of readers

constructing strong l ifications from dual minimal bases abstract we provide an algorithm for constructing strong l ifications of a given matrix polynomial p of degree d and size m n using only the coefficients of the polynomial and the solution of linear systems of equations a strong l ification of p is a matrix polynomial of degree l having the same finite and infinite elementary divisors and the same numbers of left and right minimal indices as the original matrix polynomial p all explicit constructions of strong l ifications introduced so far in the literature have been limited to the case where l divides d though recent results on the inverse eigenstructure problem for matrix polynomials show that more general constructions are possible based on recent developments on dual polynomial minimal bases we present a general construction of strong l ifications for wider choices of the degree l namely when l divides one of nd or md and d l in the case where l divides nd respectively md the strong l ifications we construct allow us to easily recover the minimal indices of p in particular we show that they preserve the left resp right minimal indices of p and the right resp left minimal indices of the l ification are the ones of p increased by d l each moreover in the particular case l divides d the new method provides a companion l ification that resembles very much the companion l ifications already known in the literature

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

example of a quadratic system with two cycles appearing in a homoclinic loop bifurcation abstract we give here a planar quadratic differential system depending on two parameters there is a curve in the space corresponding to a homoclinic loop bifurcation hlb the bifurcation is degenerate at one point of the curve and we get a narrow tongue in which we have two limit cycles this is the first example of such a bifurcation in planar quadratic differential systems we propose also a model for the bifurcation diagram of a system with two limit cycles appearing at a singular point from a degenerate hopf bifurcation and dying in a degenerate hlb this model shows a deep duality between degenerate hopf bifurcations and degenerate hlbs we give a bound for the maximal number of cycles that can appear in certain simultaneous hopf and homoclinic loop bifurcations we also give an example of quadratic system depending on three parameters which has at one place a degenerate hopf bifurcation of order 3 and at another place a hopf bifurcation of order 2 together with a hlb we characterize the planar quadratic systems which are integrable in the neighbourhood of a homoclinic loop

metamorphoses of phase portraits of vector field in the case of symmetry of order 4 to investigate the topological structure of hopf bifurcations in a periodically time dependent system one usually makes use of its normal form which can be reduced into two dimensional space by using center manifold method 1 2 the truncated normal form is usually time independent and the vector field induced by such normal form is invariant with respect to some group of symmetry it is well known that the nondegenerate bifurcations of phase portraits with the symmetry of order other than 4 have been well investigated 3 4 but the obtained results about the bifurcations in the case of order 4 are still partial and far from complete

hilbert s 16th problem for quadratic vector fields abstract we discuss here a systematic approach towards a positive answer to hilbert s 16th problem for quadratic systems namely the existence of a uniform bound for the number of limit cycles of a quadratic system the method is the following describe the limit periodic sets surrounding the origin in a family of quadratic vector fields and prove that they have finite cyclicity in this paper we give the list of all graphics and degenerate graphics that should be considered and describe their general features we also indicate how to find or where to find concrete examples of these limit periodic sets

bifurcation of nonclassical viscous shock profiles from the constant state we determine the bifurcation from the constant solution of nonclassical transitional and overcompressive viscous shock profiles in regions of strict hyperbolicity whereas classical shock waves in systems of conservation laws involve a single characteristic field nonclassical waves involve two fields in an essential way this feature is reflected in the viscous profile differential equation which undergoes codimension three bifurcation of the kind studied by dumortier et al as opposed to the codimension one bifurcation occurring in the classical case we carry out a complete bifurcation analysis for systems of two quadratic conservation laws with constant strictly parabolic viscosity matrices by reducing to a canonical form introduced by fiddelaers we show that all such systems except possibly those on a codimension one variety in parameter space give rise to nonclassical shock waves and we classify the number and types of their bifurcation points one consequence of our analysis is that weak transitional waves arise in pairs with profiles forming a 2 cycle configuration previously shown to lead to nonuniqueness of riemann solutions and to nontrivial asymptotic dynamics of the conservation laws another consequence is that appearance of weak nonclassical waves is necessarily associated with change of stability in constant solutions of the parabolic system of conservation laws rather than with change of type in the associated hyperbolic system