smoothness of transition maps in singular perturbation problems with one fast variable abstract this paper deals with the smoothness of the transition map between two sections transverse to the fast flow of a singularly perturbed vector field one fast multiple slow directions orbits connecting both sections are canard orbits i e they first move rapidly towards the attracting part of a critical surface then travel a distance near this critical surface even beyond the point where the orbit enters the repelling part of the critical surface and finally repel away from the surface we prove that the transition map is smooth in a transcritical situation however where orbits from an attracting part of one critical manifold follow the repelling part of another critical manifold the smoothness of the transition map may be limited due to resonance phenomena that are revealed by blowing up the turning point we present a polynomial example in r 3

dynamic transcritical bifurcations in a class of slow fast predator prey models abstract we study the stability loss delay phenomenon in a dynamic transcritical bifurcation in a class of three dimensional prey and predator systems the dynamics of the predator is assumed to be slow comparatively to the dynamics of the prey as an application a well known model considered by clark will be discussed

on homogeneous and self dual algorithms for lcp we present some generalizations of a homogeneous and self dual linear programming lp algorithm to solving the monotone linear complementarity problem lcp again while it achieves the best known interior point iteration complexity the algorithm does not need to use any big m number and it detects lcp infeasibility by generating a certificate to our knowledge this is the first interior point and infeasible starting algorithm for the lcp with these desired features

a representation of an efficiency equivalent polyhedron for the objective set of a multiple objective linear program abstract associated with the objective set y of a linear k objective minimization problem is the efficiency equivalent polyhedron y y r k since y has the same efficient structure as y and since every extreme point of y is efficient this polyhedron provides a promising avenue for the analysis of the given multiple objective linear program molp however in order to fully explore this avenue a representation of y as a system of linear inequalities is needed in this paper an algorithm is given to construct a matrix h and a vector g such that y has the representation hy g and it is shown that no inequality in this representation is redundant the input data for the algorithm are a finite set of points of y containing the efficient extreme points and a finite set of recession directions for y containing the directions associated with unbounded efficient edges these data which can be obtained using standard molp software packages are used to form a polar polyhedron whose extreme points are precisely what is needed to define h and g

the existence of solitary wave solutions of delayed camassa holm equation via a geometric approach abstract this paper is concerned with the camassa holm equation which is a model for shallow water waves we first establish the existence of solitary wave solutions for the equation without delay and then we prove the existence of solitary wave solutions for the equation with a special local delay convolution kernel and a special nonlocal delay convolution kernel by using the method of dynamical system especially the geometric singular perturbation theory and invariant manifold theory according to the relationship between solitary wave and homoclinic orbit the camassa holm equation is transformed into the ordinary differential equations with fast variables by using the variable substitution it is proved that the equation with disturbance also possesses homoclinic orbit and there exists solitary wave solution of the delayed camassa holm equation

convergence behavior of interior point algorithms we show that most interior point algorithms for linear programming generate a solution sequence in which every limit point satisfies the strict complementarity condition these algorithms include all path following algorithms and some potential reduction algorithms the result also holds for the monotone complementarity problem if a strict complementarity solution exists in general the limit point is a solution that maximizes the number of its nonzero components among all solutions

on maximum bifurcation delay in real planar singularly perturbed vector fields abstract in singularly perturbed vector fields where the unperturbed vector field has a curve of singularities a critical curve orbits tend to be attracted towards or repelled away from this curve depending on the sign of the divergence of the vector field at the curve when at some point this sign bifurcates from negative to positive orbits will typically be repelled away immediately after passing the bifurcation point turning point atypical behaviour is nevertheless observed as well when orbits follow the critical curve for some distance after the turning point before they repel away from it a delay in the bifurcation is present interesting are systems that have a maximum bifurcation delay i e there is a point on the critical curve beyond which orbits cannot stay close to the critical curve this behaviour is known to appear in some systems in dimension 3 see e benoit ed dynamic bifurcations in lecture notes in mathematics vol 1493 springer verlag berlin 1991 and it is commonly believed that it is not an issue in real planar systems beside making the observation that it does appear in non analytic planar systems it is shown that whenever bifurcation delay appears it has no non trivial maximum for analytic planar vector fields the proof is based on the notion of family blow up at the turning point on formal power series in terms of blow up variables the study of their gevrey properties and analytic continuation of their borel transform these results complement existing results concerning the equivalence of local and global canard solutions in a fruchard r schafke overstability and resonance ann inst fourier grenoble 53 1 2003 227 264

connected and alternating vectors polyhedra and algorithms given a graphg v e letas s l be the edge set incidence vectors of its nontrivial connected subgraphs

the entry exit function and geometric singular perturbation theory this work was partly supported by fwo project g093910 de maesschalck and nsf grant dms 1211707 schecter we thank the referee for a careful reading of the paper which led to a number of clarifications and corrections

tauberian properties for monomial summability with applications to pfaffian systems abstract in this paper we will show that monomial summability processes with respect to different monomials are not compatible except in the trivial case of a convergent series we will apply this fact to the study of solutions of pfaffian systems with normal crossings focusing in the implications of the complete integrability condition on these systems

error bound and convergence analysis of matrix splitting algorithms for the affine variational inequality problem consider the affine variational inequality problem it is shown that the distance to the solution set from a feasible point near the solution set can be bounded by the norm of a natural residual at that point this bound is then used to prove linear convergence of a matrix splitting algorithm for solving the symmetric case of the problem this latter result improves upon a recent result of luo and tseng that further assumes the problem to be monotone

interior point trajectories and a homogeneous model for nonlinear complementarity problems over symmetric cones we study the continuous trajectories for solving monotone nonlinear mixed complementarity problems over symmetric cones while the analysis in l faybusovich positivity 1 1997 pp 331 357 depends on the optimization theory of convex log barrier functions our approach is based on the paper of monteiro and pang math oper res 23 1998 pp 39 60 where a vast set of conclusions concerning continuous trajectories is shown for monotone complementarity problems over the cone of symmetric positive semidefinite matrices as an application of the results we propose a homogeneous model for standard monotone nonlinear complementarity problems over symmetric cones and discuss its theoretical aspects consequently we show the existence of a path having the following properties a the path is bounded and has a trivial starting point without any regularity assumption concerning the existence of feasible or strictly feasible solutions b any accumulation point of the path is a solution of the homogeneous model c if the original problem is solvable then every accumulation point of the path gives us a finite solution d if the original problem is strongly infeasible then under the assumption of lipschitz continuity any accumulation point of the path gives us a finite certificate proving infeasibility

summability of canard heteroclinic saddle connections abstract for a given real analytic slow fast system x e f x y e y g x y e that admits a slow fast saddle and that satisfies some mild assumptions the borel summability properties of the saddle separatrix tangent in the direction of the critical curve are investigated 1 summability is shown it is also shown that slow fast saddle connections of canard type have summability properties in contrast to the typical lack of borel summability for canard solutions of general equations

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

error bounds for monotone linear complementarity problems we give a bound on the distance between an arbitrary point and the solution set of a monotone linear complementarity problem in terms of a condition constant that depends on the problem data only and a residual function of the violations of the complementary problem conditions by the point considered when the point satisfies the linear inequalities of the complementarity problem the residual consists of the complementarity condition plus its square root this latter term is essential and without it the error bound cannot hold we also show that another natural residual that has been employed to bound errors for strictly monotone linear complementarity problems fails to bound errors for the monotone case considered here

a logarithmic descent direction algorithm for the quadratic knapsack problem abstract the quadratic knapsack problem is an np hard optimization problem with many diverse applications in industrial and management engineering however computational complexities still remain in the quadratic knapsack problem in this study a logarithmic descent direction algorithm is proposed to approximate a solution to the quadratic knapsack problem the proposed algorithm is based on the karush kuhn tucker necessary optimality condition and the damped newton method the convergence of the algorithm is proven and the numerical results indicate its effectiveness

regularization around a generic codimension one fold fold singularity abstract this paper is devoted to study the generic fold fold singularity of filippov systems on the plane its unfoldings and its sotomayor teixeira regularization we work with general filippov systems and provide the bifurcation diagrams of the fold fold singularity and their unfoldings proving that under some generic conditions is a codimension one embedded submanifold of the set of all filippov systems the regularization of this singularity is studied and its bifurcation diagram is shown in the visible invisible case the use of geometric singular perturbation theory has been useful to give the complete diagram of the unfolding specially the appearance and disappearance of periodic orbits that are not present in the filippov vector field in the case of a linear regularization we prove that the regularized system is equivalent to a general slow fast system studied by krupa and szmolyan 10

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

canard cycles for predator prey systems with holling types of functional response abstract by using the singular perturbation theory developed by dumortier and roussarie and recent work of de maesschalck and dumortier we study the canard phenomenon for predator prey systems with response functions of holling types we first develop a formula for computing the slow divergence integrals by using the formula we prove that for the systems with the response function of holling types iii and iv the cyclicity of any limit periodic set is at most two that is at most two families of hyperbolic limit cycles or at most one family of limit cycles with multiplicity two can bifurcate from the limit periodic set by small perturbations we also indicate the regions in parameter space where the corresponding limit periodic set has cyclicity at most one or at most two

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

multi layer canard cycles and translated power functions abstract the paper deals with two dimensional slow fast systems and more specifically with multi layer canard cycles these are canard cycles passing through n layers of fast orbits with n 2 the canard cycles are subject to n generic breaking mechanisms and we study the limit cycles that can be perturbed from the generic canard cycles of codimension n we prove that this study can be reduced to the investigation of the fixed points of iterated translated power functions

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

multisummability of formal solutions of singular perturbation problems abstract we investigate the summability of the unique formal power series solution of a singular perturbation problem for certain linear systems of ordinary meromorphic differential equations using a new type of summability of power series in several here two variables

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

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

polyhedral polarity defined by a general bilinear inequality letx andy be finite dimensional vector spaces over the real numbers let be a binary relation betweenx andy given by a bilinear inequality the polar of a subsetp ofx is the set of all elements ofy which are related by to all elements ofp the polar of a subset ofy is defined similarly the polar of the polar ofp is called the closure ofp andp is called closed ifp equals its closure we describe the polar of any finitely generated setp as the solution set of a finite system of linear inequalities and describe the closure ofp as a finitely generated set the closed polyhedra are characterized in terms of defining systems of linear inequalities and also in terms of the relationship of the polyhedronp with its recessional cone and with certain subsets ofx andy determined by the relation six classes of bilinear inequalities are distinguished in the characterization of closed polyhedra

uncertainty transformation via hopf bifurcation in fast slow systems propagation of uncertainty in dynamical systems is a significant challenge here we focus on random multiscale ordinary differential equation models in particular we study hopf bifurcation in the fast subsystem for random initial conditions we show that a random initial condition distribution can be transformed during the passage near a delayed dynamic hopf bifurcation i to certain classes of symmetric copies ii to an almost deterministic output iii to a mixture distribution with differing moments and iv to a very restricted class of general distributions we prove under which conditions the cases i iv occur in certain classes vector fields

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

cyclicity of a fake saddle inside the quadratic vector fields abstract this paper concerns the study of small amplitude limit cycles that appear in the phase portrait near an unfolded fake saddle singularity this degenerate singularity is also known as an impassable grain the canonical form of the unperturbed vector field is like a degenerate flow box near the singularity the phase portrait consists of parallel fibers all but one of which have no singular points and at the singular fiber there is one node we demonstrate different techniques in order to show that the cyclicity is bigger than or equal to two when the canonical form is quadratic

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

borel summability of divergent solutions for singular first order partial differential equations with variable coefficients part i abstract under a certain restriction singular first order linear partial differential equations of nilpotent type with two variables are divided into two classes in the previous paper part i we dealt with the one class and comprehended that there was a close affinity between the borel summability of divergent solutions and global analytic continuation properties for coefficients in this part ii we give a similar consideration on the other class more precise global estimates than those given in part i for coefficients will be required to prove the borel summability of divergent solutions

analysis of a slow fast system near a cusp singularity this paper studies a slow fast system whose principal characteristic is that the slow manifold is given by the critical set of the cusp catastrophe our analysis consists of two main parts first we recall a formal normal form suitable for systems as the one studied here afterwards taking advantage of this normal form we investigate the transition near the cusp singularity by means of the blow up technique our contribution relies heavily in the usage of normal form theory allowing us to refine previous results

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

recent progress in the theory of formal solutions for ode and pde in this article we describe the theory of multisummability of formal power series and discuss its application to power series solutions of ordinary and partial differential equations in particular we give some new application to a singular perturbation of a regular singular point

relaxation oscillation and canard explosion abstract we give a geometric analysis of relaxation oscillations and canard cycles in singularly perturbed planar vector fields the transition from small hopf type cycles to large relaxation cycles which occurs in an exponentially thin parameter interval is described as a perturbation of a family of singular cycles the results are obtained by means of two blow up transformations combined with standard tools of dynamical systems theory the efficient use of various charts is emphasized the results are applied to the van der pol equation

canards heteroclinic and homoclinic orbits for a slow fast predator prey model of generalized holling type iii abstract for a classical ratio dependent predator prey model with the generalized holling type iii functional response it was previously investigated in 20 by hsu and huang for global stability of an equilibrium and in 21 by huang ruan and song for subcritical hopf and bogdanov takens bifurcations here in this model when prey reproduces much faster than predator by using geometric singular perturbation theory we achieve much richer new dynamical phenomena than the existing ones such as the existence of canard cycles canard explosion and relaxation oscillations heteroclinic and homoclinic orbits cyclicity of slow fast cycles and the coexistence of the hopf cycle and the relaxation oscillation on global stability of the equilibrium we also provide less restricted conditions than the existing ones

polarities given by systems of bilinear inequalities we extend the concept of polar pairs of cones to polyhedra using binary relations given by systems of bilinear inequalities in particular we consider those which give natural generalizations of the theory of blocking and antiblocking of fulkerson these extensions preserve the important relations between vertices and facets of polar pairs

an extension of borel laplace methods and monomial summability abstract in this paper we will show that monomial summability can be characterized using borel laplace like integral transformations depending of a parameter 0 s 1 we will apply this result to prove 1 summability in a monomial of formal solutions of a family of partial differential equations

slow divergence integral and its application to classical lienard equations of degree 5 abstract the slow divergence integral is a crucial tool to study the cyclicity of a slow fast cycle for singularly perturbed planar vector fields in this paper we deduce a useful form for this integral in order to apply it to various problems as an example we use it to prove that the slow divergence integral along any non degenerate slow fast cycle for singular perturbations of classical lienard equations of degree 5 has at most one zero and the zero is simple if it exists hence the cyclicity of any non degenerate slow fast cycle in this class of equations is at most 2 up to now there were many interesting results about lienard equations of degree 3 4 and 6 but almost nothing is known about degree 5 the result in this paper can be seen as a first stage to study the uniform property for classical lienard equations of degree 5

asymptotic behaviour of solutions near a turning point the example of the brusselator equation abstract the brusselator equation is an example of a singularly perturbed differential equation with an additional parameter it has two turning points at x 0 and x 1 we study some properties of so called canard solutions that remain bounded in a full neighbourhood of 0 and in the largest possible domain the main goal is the complete asymptotic expansion of the difference between two values of the additional parameter corresponding to such solutions for this purpose we need a study of behaviour of the solutions near a turning point here we prove that for a large class of equations if 0 is a turning point of order p any solution y not exponentially large has in some sector centred at 0 an asymptotic behaviour when e 0 of the form y n x e e n where e p 1 e for x e x with x large enough but independent of e in the brusselator case we moreover compute a stokes constant for a particular nonlinear differential equation

the period function of classical lienard equations in this paper we study the number of critical points that the period function of a center of a classical lienard equation can have centers of classical lienard equations are related to scalar differential equations x x f x x 0 with f an odd polynomial let us say of degree 2 1 we show that the existence of a finite upperbound on the number of critical periods only depending on the value of can be reduced to the study of slow fast lienard equations close to their limiting layer equations we show that near the central system of degree 2 1 the number of critical periods is at most 2 2 we show the occurrence of slow fast lienard systems exhibiting 2 2 critical periods elucidating a qualitative process behind the occurrence of critical periods it all provides evidence for conjecturing that 2 2 is a sharp upperbound on the number of critical periods we also show that the number of critical periods multiplicity taken into account is always even

the gevrey asymptotics in the case of singular perturbations abstract in this paper we present results in the gevrey asymptotics which correspond to some existing results concerning asymptotic solutions in the poincare asymptotics of singularly perturbed ordinary differential equations the main idea is based on a characterization of the gevrey flat functions and a characterizaton of the gevrey asymptotic expansions as it had been already exhibited by y sibuya 1990 lecture notes in pure and applied math vol 124 pp 393 401 dekker new york and p f hsieh and y sibuya 1999 basic theory of ordinary differential equations chap xi sect xi 2 and chap xii sect xii 5 springer verlag berlin

slow divergence integral on a mobius band abstract the slow divergence integral has proved to be an important tool in the study of slow fast cycles defined on an orientable two dimensional manifold e g r 2 the goal of our paper is to study 1 canard cycle and 2 canard cycle bifurcations on a non orientable two dimensional manifold e g the mobius band by using similar techniques our focus is on smooth slow fast models with a hopf breaking mechanism the same results can be proved for a jump breaking mechanism and non generic turning points the slow fast bifurcation problems on the mobius band require the study of the 2 return map attached to such 1 and 2 canard cycles we give a simple sufficient condition expressed in terms of the slow divergence integral for the existence of a period doubling bifurcation near the 1 canard cycle we also prove the finite cyclicity property of singular 1 and 2 homoclinic loops regular 1 homoclinic loops of finite codimension have been studied by guimond

classical lienard equations of degree n 6 can have n 12 2 limit cycles abstract based on geometric singular perturbation theory we prove the existence of classical lienard equations of degree 6 having 4 limit cycles it implies the existence of classical lienard equations of degree n 6 having at least n 1 2 2 limit cycles this contradicts the conjecture from lins de melo and pugh formulated in 1976 where an upperbound of n 1 2 limit cycles was predicted this paper improves the counterexample from dumortier panazzolo and roussarie 2007 by supplying one additional limit cycle from degree 7 on and by finding a counterexample of degree 6 we also give a precise system of degree 6 for which we provide strong numerical evidence that it has at least 3 limit cycles

gevrey properties of the asymptotic critical wave speed in a family of scalar reaction diffusion equations abstract we consider front propagation in a family of scalar reaction diffusion equations in the asymptotic limit where the polynomial degree of the potential function tends to infinity we investigate the gevrey properties of the corresponding critical propagation speed proving that the formal series expansion for that speed is gevrey 1 with respect to the inverse of the degree moreover we discuss the question of optimal truncation finally we present a reliable numerical algorithm for evaluating the coefficients in the expansion with arbitrary precision and to any desired order and we illustrate that algorithm by calculating explicitly the first ten coefficients our analysis builds on results obtained previously in f dumortier n popovic t j kaper the asymptotic critical wave speed in a family of scalar reaction diffusion equations j math anal appl 326 2 2007 1007 1023 and makes use of the blow up technique in combination with geometric singular perturbation theory and complex analysis while the numerical evaluation of the coefficients in the expansion for the critical speed is based on rigorous interval arithmetic

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

on a homogeneous algorithm for the monotone complementarity problem we present a generalization of a homogeneous self dual linear programming lp algorithm to solving the monotone complementarity problem mcp the algorithm does not need to use any big m parameter or two phase method and it generates either a solution converging towards feasibility and comple mentarity simultaneously or a certificate proving infeasibility moreover if the mcp is polynomially solvable with an interior feasible starting point then it can be polynomially solved without using or knowing such information at all to our knowledge this is the first interior point and infeasible starting algorithm for solving the mcp that possesses these desired features preliminary computational results are presented

singular perturbations of the holling i predator prey system with a focus abstract we study the occurrence of limit cyles in a model for two species predator prey interaction referred to as holling i this model is based on the phenomenological fitting of the so called functional response function to observed data in nature by a continuous piecewise differentiable function of the original models presented by holling it is the only model for which the possible asymptotic behaviour of the prey and predator densities has not been determined yet we extend the work of liu who showed that for certain values of parameters of the holling i system at least two nested limit cycles will occur surrounding a focus his case corresponds to a bifurcation problem where limit cycles are created from a system with a continuum of singularities i e a singular perturbation problem with slow fast solutions we prove that exactly two hyperbolic limit cycles occur after perturbation

gevrey normal forms of vector fields with one zero eigenvalue abstract we study normal forms of isolated singularities of vector fields in r n or c n when all eigenvalues of the linear part of the vector field are nonzero one can eliminate all so called nonresonant terms from the equation provided some spectral condition like siegel is satisfied in this paper we discuss the case where there is one zero eigenvalue in that case siegel s condition is not satisfied and show that the formal normalizing transformations are either convergent or divergent of at most gevrey type in some cases we show the summability of the normalizing transformations which leads to the existence of analytic normal forms in complex sectors around the singularity

gevrey properties of real planar singularly perturbed systems abstract by applying geometric techniques to real analytic singularly perturbed vector fields on the plane we develop a way to give a bound on the gevrey type of the taylor development of canard manifolds at degenerate planar turning points by blowing up the phase space at the turning point we find asymptotic estimates even when such expansions w r t traditional phase space variables do not exist the asymptotic estimates are then used to give a sufficient and necessary condition on the existence of local canard solutions

monomial summability and doubly singular differential equations abstract in this work we consider systems of differential equations that are doubly singular i e that are both singularly perturbed and exhibit an irregular singular point if the irregular singular point is at the origin they have the form e x r 1 d y d x f x e y f 0 0 0 0 with f analytic in some neighborhood of 0 0 0 if the jacobian d f d y 0 0 0 is invertible we show that the unique bivariate formal solution is monomially summable i e summable with respect to the monomial t e x r in a new sense that will be defined as a preparation poincare asymptotics and gevrey asymptotics in a monomial are studied

verification for existence of solutions of linear complementarity problems abstract recently g alefeld x chen and f potra numer math 83 1999 265 315 presented a verification method for solutions of linear complementarity problems lcps this paper is an attempt to obtain more useful information from the output of this verification method in particular existing results can only claim the nonexistence of solutions in a given interval we will use the farkas lemma to check if the interval contains a negative certification which shows the nonexistence of solutions in the whole space r n moreover we will study how to choose a good nonsingular matrix a in the interval operator for p 0 matrix linear complementarity problems we report numerical results to illustrate the efficiency of the proposed technique

singularly perturbed linear two point boundary value problems finding asymptotic solutions to two point boundary value problems for linear singularly perturbed second order ordinary differential equations is now quite well understood troublesome exceptions involving turning points known as problems of boundary layer resonance have been studied by many experts since the initial work of ackerberg and o malley stud appl math 49 1970 pp 277 295 this significant work is surveyed here closely related literature involves the canard phenomena new generally more geometric approaches including gevrey asymptotics and blowup promise improved understanding of such sensitive asymptotics and their generalizations

sungular hopf bifurcation to relaxation oscillations this paper is a continuation of our study of singular hopf bifurcations to relaxation oscillations siam journal on applied mathematics 46 1986 pp 721 739 in our previous analysis a system of two first order nonlinear equatins is investigated and it is shown how the bifurcation problem can be reduced to a weakly perturbed conservative system of two first order equations in this paper a method based on analyzing nearly conservative quantities was developed to find the bifurcation diagram of the periodic solutions emerging from singular hopf bifurcation points specifically the subcritical hopf bifurcation of the fitzhugh nagumo equations is analyzed the extension from the earlier bifurcation analysis is achieved by performing a perturbation analysis from the first integral of the leading order nonlinear oscillator at a critical value of a parameter the first integral exists up to an additional order in the perturbed equation this condition permits the resolution of features determined by h