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

abelian integrals in unfoldings of codimension 3 singularities with nilpotent linear parts abstract this paper is concerned with the upper bound of the number of limit cycles in unfolding of codimension 3 planar singularities with nilpotent linear parts after making a central rescaling the problem reduces to a perturbation problem of a one parameter family of quadratic reversible systems as the parameter a 1 1 0 is rational except the case a 2 3 based on the chebyshev criterion for abelian integrals and a rationalizing transformation the problem could be solved theoretically to illustrate our approaches two particular cases corresponding to nilpotent codimension 3 saddle and elliptic case respectively are proved where the upper bound of the number of limit cycles is two

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

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

bifurcation of limit cycles from a 4 dimensional center in r m in resonance 1 n for every positive integer n 2 we consider the linear differential center x ax in rm with eigenvalues i n i and 0 with multiplicity m 4 we perturb this linear center inside the class of all polynomial differential systems of the form linear plus a homogeneous nonlinearity of degree n i e x ax ef x where every component of f x is a linear polynomial plus a homogeneous polynomial of degree n when the displacement function of order e of the perturbed system is not identically zero we study the maximal number of limit cycles that can bifurcate from the periodic orbits of the linear differential center in particular we give explicit upper bounds for the number of limit cycles

on the number of limit cycles near a homoclinic loop with a nilpotent singular point abstract in this article we study the expansion of the first melnikov function appearing by perturbing an integrable and reversible system with a homoclinic loop passing through a nilpotent singular point and obtain formulas for computing the first coefficients of the expansion based on these coefficients we obtain a lower bound for the maximal number of limit cycles near the homoclinic loop moreover as an application of our main results we consider a type of integrable and reversible polynomial systems obtaining at least 3 4 or 5 limit cycles respectively

counting real zeros of analytic functions satisfying linear ordinary differential equations abstract we suggest an explicit procedure to establish upper bounds for the number of real zeros of analytic functions satisfying linear ordinary differential equations with meromorphic coefficients if the equation formula has no singular points in a small neighborhooduof a real segmentk all the coefficientsaj t have absolute value aonuanda0 t 1 then any solution of this equation may have no more than a zeros onk where u k is a geometric constant depending only onkandu if the principal coefficienta0 t is nonconstant but its modulus is at leasta 0 somewhere onk then the number of real zeros onkof any solution analytic inu does not exceed a a with some u k

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

on the tate spectrum of mathrm tmf tmf at the prime 2 computations involving the mahowald invariant prompted mahowald and shick to develop the slogan the mahowald invariant of v n periodic homotopy is v n torsion while neither a proof nor a precise statement of this slogan appears in the literature numerous authors have offered computational evidence in support of its fundamental idea the mahowald invariant is closely related to his inverse limit description of the tate spectrum and computations have shown the tate spectrum of v n periodic cohomology theories to be v n torsion the purpose of this paper is to split the tate spectrum of mathrm tmf as a wedge of suspensions of mathrm ko providing yet another example in support of the slogan to the existing literature

parallelization of the lyapunov constants and cyclicity for centers of planar polynomial vector fields christopher in 2006 proved that under some assumptions the linear parts of the lyapunov constants with respect to the parameters give the cyclicity of an elementary center this paper is devote to establish a new approach namely parallelization to compute the linear parts of the lyapunov constants more concretely it is showed that parallelization computes these linear parts in a shorter quantity of time than other traditional mechanisms to show the power of this approach we study the cyclicity of the holomorphic center iz z 2 z 3 z n under general polynomial perturbations of degree n for n 13 we also exhibit that from the point of view of computation among the hamiltonian time reversible and darboux centers the holomorphic center is the best candidate to obtain high cyclicity examples of any degree for n 4 5 13 we prove that the cyclicity of the holomorphic center is at least n 2 n 2 this result give the highest lower bound for m 6 m 7 m 13 among the existing results where m n is the maximum number of limit cycles bifurcating from an elementary monodromic singularity of polynomial systems of degree n as a direct corollary we also obtain the highest lower bound for the hilbert numbers h 6 40 h 8 70 and h 10 108 because until now the best result was h 6 39 h 8 67 and h 10 100

a relevant two qubit bell inequality inequivalent to the chsh inequality we computationally investigate the complete polytope of bell inequalities for two particles with small numbers of possible measurements and outcomes our approach is limited by pitowsky s connection of this problem to the computationally hard np problem despite this we find that there are very few relevant inequivalent inequalities for small numbers for example in the case with three possible 2 outcome measurements on each particle there is just one new inequality we describe mixed 2 qubit states which violate this inequality but not the chsh the new inequality also illustrates a sharing of bi partite non locality between three qubits something not seen using the chsh inequality it also inspires us to discover a class of bell inequalities with m possible n outcome measurements on each particle

bounding the number of zeros of certain abelian integrals abstract in this paper we prove a criterion that provides an easy sufficient condition in order for any nontrivial linear combination of n abelian integrals to have at most n k 1 zeros counted with multiplicities this condition involves the functions in the integrand of the abelian integrals and it can be checked in many cases in a purely algebraic way

root invariants in the adams spectral sequence let e be a ring spectrum for which the e adams spectral sequence converges we define a variant of mahowald s root invariant called the filtered root invariant which takes values in the e 1 term of the e adams spectral sequence the main theorems of this paper are concerned with when these filtered root invariants detect the actual root invariant and explain a relationship between filtered root invariants and differentials and compositions in the e adams spectral sequence these theorems are compared to some known computations of root invariants at the prime 2 we use the filtered root invariants to compute some low dimensional root invariants of v 1 periodic elements at the prime 3 we also compute the root invariants of some infinite v 1 periodic families of elements at the prime 3

the infinitesimal 16th hilbert problem in the quadratic case let h x y be a real cubic polynomial with four distinct critical values in a complex domain and let x h h y frac partial partial x h x frac partial partial y be the corresponding hamiltonian vector field we show that there is a neighborhood of x h in the space of all quadratic plane vector fields such that any x has at most two limit cycles

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

bifurcations of certain family of planar vector fields tangent to axes abstract the topological versality of the family v x x 1 x y y y 2 ax by cy 2 of vector fields on the plane is proven the asymptotic formulas of the bifurcational curves are given the proofs rely on an analysis of the limit cycles and the non abelian integrals along irrational curves

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

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

new family of abelian integrals satisfying chebyshev property abstract this paper is motivated by the chebyshev property of a family of abelian integrals along non algebraic level ovals a new criterion is established to study the chebyshev property of these abelian integrals as its applications we give three examples to show that the ordered sets defined by abelian integrals are ect systems moreover the number of the abelian integrals can be an arbitrary positive integer

the transverse dynamics of a holomorphic flow on considere un champ vectoriel polynomial homogene p z dans c n 1 les solutions du systeme d equations differentielles dz i dt p i z t c i 0 n feuillette c n 1 par des courbes holomorphes ayant des singularites le long du lieu commun p i z 0 on definit la notion de dynamique transverse du feuilletage

random periodic solutions of random dynamical systems abstract in this paper we give the definition of the random periodic solutions of random dynamical systems we prove the existence of such periodic solutions for a c 1 perfect cocycle on a cylinder using a random invariant set the lyapunov exponents and the pullback of the cocycle

numerical computation and continuation of invariant manifolds connecting fixed points a numerical method for the computation of an invariant manifold that connects two fixed points of a vector field in mathbb r n is given extending the results of an earlier paper comput appl math 26 1989 pp 159 170 by the authors basically a boundary value problem on the real line is truncated to a finite interval the method applies in particular to the computation of heteroclinic orbits the emphasis is on the systematic computation of such orbits by continuation using the fact that the linearized operator of our problem is fredholm in appropriate banach spaces the general theory of approximation of nonlinear problems is employed to show that the errors in the approximate solution decay exponentially with the length of the approximating interval several applications are considered including the computation of traveling wave solutions to reaction diffusion problems computations were done using the software package auto

low and high frequency oscillations in three dimensional food chain systems the dynamic behavior of a three trophic level food chain prey predator superpredator is analyzed in this paper prey is logistic while predator and superpredator have functional response of the holling type moreover the trophic levels are assumed to be characterized by increasing and quite diversified time responses using a singular perturbation approach explicit conditions for the persistence of the three populations are derived and the structure of the corresponding attractors is noted as well as the nature of transients the analysis shows that the system can have low frequency cycles due to the interactions between predator and superpredator nevertheless the interactions between prey and predator can give rise to high frequency oscillations which can arise during the transients toward the attractor or during the low frequency cycle this periodic burst of high frequency oscillations develops in particular when the two predators are fairly efficient these results note in a very concise w

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

maximal violation of the collins gisin linden massar popescu inequality for infinite dimensional states we present a much simplified version of the collins gisin linden massar popescu inequality for the 2 ifmmode times else texttimes fi 2 ifmmode times else texttimes fi d bell scenario numerical maximization of the violation of this inequality over all states and measurements suggests that the optimal state is far from maximally entangled while the best measurements are the same as conjectured best measurements for the maximally entangled state for very large values of d the inequality seems to reach its minimal value given by the probability constraints this gives numerical evidence for a tight quantum bell inequality or generalized csirelson inequality for the 2 ifmmode times else texttimes fi 2 ifmmode times else texttimes fi ensuremath infty scenario

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

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

canard phenomenon for an sis epidemic model with nonlinear incidence abstract by using the singular perturbation theory especially on canard cycles the canard phenomenon for an sis epidemic model with nonlinear incidence is investigated the phenomenon suggests the existence of disease outbreak in the infection transmission it is proved that the cyclicity of any possible slow fast cycle 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 slow fast cycle by small perturbations we also indicate the regions in parameter space where the corresponding slow fast cycle has cyclicity at most one or at most two

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

polynomial systems a lower bound for the hilbert numbers let h n be the maximum possible number of limit cycles of systems dot x p x y dot y q x y where p and q are polynomials of degree at most n we are concerned with the rate of growth of h n as n increases it is known that h n geq kn 2 for some constant k in this paper we show that h n grows at least as rapidly as n 2 log n

on local behavior of holomorphic functions along complex submanifolds of ℂ n in this paper we establish some general results on local behavior of holomorphic functions along complex submanifolds of n as a corollary we present multi dimensional generalizations of an important result of coman and poletsky on bernstein type inequalities on transcendental curves in 2

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

center conditions at infinity for abel differential equations an abel differential equation y p x y 2 q x y 3 is said to have a center at a set a a 1 a r of complex numbers if y a 1 y a 2 y a r for any solution y x with the initial value y a 1 small enough the polynomials p q are said to satisfy the polynomial composition condition on a if there exist polynomials p q and w such that p p and q q are representable as p x p w x q x q w x and w a 1 w a 2 w a r we show that for wide ranges of degrees of p and q restricted only by certain assumptions on the common divisors of these degrees the composition condition provides a very accurate approximation of the center one up to a finite number of configurations not accounted for to our best knowledge this is the first general i e not restricted to small degrees of p and q or to a very special form of these polynomials result in the center problem for abel equations as an important intermediate result we show that at infinity according to an appropriate projectivization of the parameter space the center conditions are given by a system of the moment equations of the form as a1 p k q 0 s 2 r k 0 1

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

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

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

the number and stability of limit cycles for planar piecewise linear systems of node saddle type abstract the objective of this paper is to study the number and stability of limit cycles for planar piecewise linear pwl systems of node saddle type with two linear regions firstly we give a thorough analysis of limit cycles for lienard pwl systems of this type proving one is the maximum number of limit cycles and obtaining necessary and sufficient conditions for the existence and stability of a unique limit cycle these conditions can be easily verified directly according to the parameters in the systems and play an important role in giving birth to two limit cycles for general pwl systems in this step the tool of a bendixon like theorem is successfully employed to derive the existence of a limit cycle secondly making use of the results gained in the first step we obtain parameter regions where the general pwl systems have at least one at least two and no limit cycles respectively in addition for the general pwl systems some sufficient conditions are presented for the existence and stability of a unique one and exactly two limit cycles respectively finally some numerical examples are given to illustrate the results and especially to show the existence and stability of two nested limit cycles

entropie geometrique des feuilletages on definit et on etudie l entropie geometrique d un feuilletage generalisant directement l entropie topologique d un flot

an application of the separation principle for detecting slow fast limit cycles in a three dimensional system a geometric method called the separation principle is used in this paper for detecting limit cycles in a slow fast dynamical system describing a food chain with trophic levels with diversified time responses the analysis is very simple and based on singular perturbation arguments the limit cycle resulting from the method is composed of a concatenation of catastrophic transitions occuring at different speeds

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

maximal non classicality in multi setting bell inequalities the discrepancy between maximally entangled states and maximally non classical quantum correlations is well known but still not well understood we aim to investigate the relation between quantum correlations and entanglement in a family of bell inequalities with n settings and d outcomes using analytical as well as numerical techniques we derive both maximal quantum violations and violations obtained from maximally entangled states furthermore we study the most non classical quantum states in terms of their entanglement entropy for large values of d and many measurement settings interestingly we find that the entanglement entropy behaves very differently depending on whether n 2 or n 2 when n 2 the entanglement entropy is a monotone function of d and the most non classical state is far from maximally entangled whereas when n 2 the entanglement entropy is a non monotone function of d and converges to that of the maximally entangled state in the limit of large d

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

transversal heteroclinic and homoclinic orbits in singular perturbation problems abstract in this paper we give a geometric construction of heteroclinic and homoclinic orbits for singularly perturbed differential equations by using methods from invariant manifold theory we show that transversal intersection of stable and unstable manifolds of the reduced problem implies the existence of transversal heteroclinic or homoclinic orbits of the singularly perturbed problem we derive analytical conditions for transversality we show how these results can be used to prove the existence of heteroclinic and homoclinic orbits in singularly perturbed problems which depend on additional parameters we describe a configuration which implies transversal intersection of the stable and unstable manifolds of periodic orbits and the associated chaotic dynamics

uniqueness of limit cycles for sewing planar piecewise linear systems abstract this paper proves the uniqueness of limit cycles for sewing planar piecewise linear systems with two zones separated by a straight line and only one singularity of monodromic type the proofs are based in an extension of rolle s theorem for dynamical systems on the plane

continuation and bifurcations numerical techniques and applications bifurcation to rotating waves from non trivial steady states use of approximate inertial manifolds in bifurcation calculations understanding steady state bifurcation diagrams for a model reaction diffusion system bifurcations chaos and self organization in reaction diffusion systems eigenvalue problems with the symmetry of a group and bifurcations steady state steady state mode interaction in nonlinear equations with z2 symmetry symbolic computation and bifurcation methods bifurcation analysis a combined numerical and analytical approach to the numerical solution of symmetry breaking bifurcation problems a computational method and path following for periodic solutions with symmetry global bifurcations and their numerical computation computation of invariant manifold bifurcations a method for homoclinic and heteroclinic continuation in two and three dimensions the global attractor under discretisation the numerical detection of hopf bifurcation points a newton like method for simple bifurcation problems with application to large sparse systems aspects of continuation software interactive system for studies in nonlinear dynamics linlbf a program for continuation and bifurcation analysis of equilibria up to codimension three on the topology of three dimensional separations a guide for classification the construction of cubature formulae using continuation and bifurcation software determining an organizing center for passive optical systems optimization by continuation stability of marangoni convection in a microgravity environment computing with reaction diffusion systems applications in image processing bifurcation of codimension 2 for a discrete map bifurcation of periodic solutions in pde s numerical techniques and applications abstract bifurcation and chaos in chua s circuit family abstract continuation and collocation for parameter dependent boundary value problems abstract porous medium combustion abstract block elimination and the computation of simple turning points abstract bifurcation into gaps in the essential spectrum abstract application of numerical continuation in aerospace problems abstract application of a reduced basis method in structural analysis abstract some applications of bifurcation theory in engineering abstract higher order predictors in numerical path following schemes abstract

a criticality result for polycycles in a family of quadratic reversible centers abstract we consider the family of dehomogenized loud s centers x y x 1 x x d x 2 f y 2 y where d f r 2 and we study the number of critical periodic orbits that emerge or disappear from the polycycle at the boundary of the period annulus this number is defined exactly the same way as the well known notion of cyclicity of a limit periodic set and we call it criticality the previous results on the issue for the family x r 2 distinguish between parameters with criticality equal to zero regular parameters and those with criticality greater than zero bifurcation parameters a challenging problem not tackled so far is the computation of the criticality of the bifurcation parameters which form a set b of codimension 1 in r 2 in the present paper we succeed in proving that a subset of b has criticality equal to one

a dynamical system with hopf bifurcations and catastrophes a well known second order positive dynamical system describing the dynamics of a renewable resource together with its consumers prey predator model is first considered such a dynamical system is recognized to be a subtle hopf bifurcation without catastrophes then the model is slightly modified by including a control parameter which represents the indirect exploitation of the resource this new system is shown to have a much richer and interesting dynamics in particular catastrophes of the fold type hysteresis may arise moreover when the control value corresponding to the hopf bifurcation is greater than the lower threshold of the hysteresis one of the two transitions characterizing the catastrophe is a transition between a stableequilibrium and a limit cycle the results are illustrated in the control phase portrait

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

bell inequalities for arbitrarily high dimensional systems we develop a novel approach to bell inequalities based on a constraint that the correlations exhibited by local variable theories must satisfy this is used to construct a family of bell inequalities for bipartite quantum systems of arbitrarily high dimensionality which are strongly resistant to noise in particular our work gives an analytic description of previous numerical results and generalizes them to arbitrarily high dimensionality

computer assisted techniques for the verification of the chebyshev property of abelian integrals abstract we develop techniques for the verification of the chebyshev property of abelian integrals these techniques are a combination of theoretical results analysis of asymptotic behavior of wronskians and rigorous computations based on interval arithmetic we apply this approach to tackle a conjecture formulated by dumortier and roussarie in f dumortier r roussarie birth of canard cycles discrete contin dyn syst 2 2009 723 781 which we are able to prove for q 2

multipartite bell type inequalities for arbitrary numbers of settings and outcomes per site we introduce a single general representation incorporating in a unique manner all bell type inequalities for a multipartite correlation scenario with an arbitrary number of settings and any spectral type of outcomes at each site specifying this general representation for correlation functions we prove that the form of any correlation bell type inequality does not depend on spectral types of outcomes in particular on their numbers at different sites and is determined only by extremal values of outcomes at each site we also specify the general form of bounds in bell type inequalities on joint probabilities our approach to the derivation of bell type inequalities is universal concise and can be applied to a multipartite correlation experiment with outcomes of any spectral type discrete or continuous we in particular prove that for an n partite quantum state possibly infinite dimensional admitting the setting lhv description the mermin klyshko inequality holds for any two bounded quantum observables per site not necessarily dichotomic

lower bounds for the hilbert number of polynomial systems let h m denote the maximal number of limit cycles of polynomial systems of degree m it is called the hilbert number the main part of hilbert s 16th problem posed in 1900 is to find its value the problem is still open even for m 2 however there have been many interesting results on the lower bound of it for m 2 in this paper we give some new lower bounds of this number the results obtained in this paper improve all existing results for all m 7 based on some known results for m 3 4 5 6 in particular we obtain that h m grows at least as rapidly as 12ln2 m 2 2ln m 2 for all large m

genericity conditions for finite cyclicity of elementary graphics abstract in this paper we explore the khovanskii method for proving the finite cyclicity of elementary graphics and how it can be applied in practice the genericity conditions needed in that case form a proper subset of the usual methods moreover some of the conditions are non intrinsic and can be artificially created by action of the automorphism groups preserving the normal forms near the singularities and their action on regular transitions hence we introduce an extension of the method which treats the usual functional pfaffian systems together with the admissible changes of coordinates in the functional equations

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

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

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

hidden variables joint probability and the bell inequalities it is shown that the following statements about a quantum correlation experiment are mutually equivalent 1 there is a deterministic hidden variables model for the experiment 2 there is a factorizable stochastic model 3 there is one joint distribution for all observables of the experiment returning the experimental probabilities 4 there are well defined compatible joint distributions for all pairs and triples of commuting and noncommuting observables 5 the bell inequalities hold