the mathematics of eigenvalue optimization optimization problems involving the eigenvalues of symmetric and nonsymmetric matrices present a fascinating mathematical challenge such problems arise often in theory and practice particularly in engineering design and are amenable to a rich blend of classical mathematical techniques and contemporary optimization theory this essay presents a personal choice of some central mathematical ideas outlined for the broad optimization community i discuss the convex analysis of spectral functions and invariant matrix norms touching briefly on semidefinite representability and then outlining two broader algebraic viewpoints based on hyperbolic polynomials and lie algebra analogous nonconvex notions lead into eigenvalue perturbation theory the last third of the article concerns stability for polynomials matrices and associated dynamical systems ending with a section on robustness the powerful and elegant language of nonsmooth analysis appears throughout as a unifying narrative thread

discovering multiple lyapunov functions for switched hybrid systems in this paper we analyze local asymptotic stability of switched hybrid systems whose subsystems have polynomial vector fields by discovering multiple lyapunov functions in quadratic forms we start with an algebraizable sufficient condition for the existence of quadratic multiple lyapunov functions then since different discrete modes are considered we apply real root classification together with a projection operator to underapproximate this sufficient condition step by step arriving at a set of semialgebraic sets which only involve the coefficients of the pre assumed multiple lyapunov function note that for each step we additionally use the information on the different discrete modes to optimize our intermediate computation results afterward we compute a sample point in the corresponding semialgebraic set for the coefficients resulting in a multiple lyapunov function finally we test our approach on some examples using a prototypical implementation and compare it with the generic quantifier e

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

subdifferentials faces and dual matrices abstract a characterization of the dual matrices for the unitarily invariant norms is given moreover the connection between the dual matrices the subdifferentials of matrix norms and the faces of the unit ball of matrices is presented

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

on the extremal structure of least upper bound norms and their dual abstract given finite dimensional real or complex banach spaces e and f with norms e r and f r we denote by n the least upper bound norm induced on l e f some results are given on the extremal structures of b the unit ball of n of its polar b and of b which is the polar of the unit ball of the least upper bound norm n the exposed faces the extreme points and a large family of other faces of b and b are presented it turns out that b is a subset of b the set of tangency points of the surfaces of b and b is completely determined and represented as the union of the exposed faces of b which are normal to rank one mappings we determine sharp bounds on the ranks of mappings in these exposed faces

on approximation problems with zero trace matrices abstract we consider some approximation problems in the linear space of complex matrices with respect to unitarily invariant norms we deal with special cases of approximation of a matrix by zero trace matrices moreover some characterizations of zero trace matrices are given by means of matrix approximation problems

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

on the number of limit cycles for perturbed pendulum equations abstract we consider perturbed pendulum like equations on the cylinder of the form x sin x e s 0 m q n s x x s where q n s are trigonometric polynomials of degree n and study the number of limit cycles that bifurcate from the periodic orbits of the unperturbed case e 0 in terms of m and n our first result gives upper bounds on the number of zeros of its associated first order melnikov function in both the oscillatory and the rotary regions these upper bounds are obtained expressing the corresponding abelian integrals in terms of polynomials and the complete elliptic functions of first and second kind some further results give sharp bounds on the number of zeros of these integrals by identifying subfamilies which are shown to be chebyshev systems

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

on the number of zeros of abelian integrals a constructive solution of the infinitesimal hilbert sixteenth problem we prove that the number of limit cycles generated from nonsingular energy level ovals periodic trajectories in a small non conservative perturbation of a hamiltonian polynomial vector field on the plane is bounded by a double exponential of the degree of the fields this solves the long standing infinitesimal hilbert 16th problem

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

characterization of the subdifferential of some matrix norms abstract a characterization is given of the subdifferential of matrix norms from two classes orthogonally invariant norms and operator or subordinate norms specific results are derived for some special cases

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

on the characterization of the extremal points of the unit sphere of matrices abstract we give a characterization of the extremal points of the unit sphere of matrices for the unitarily invariant norms we also investigate the properties of the dual matrices moreover we show that a result given by lau and riha is not correct

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

characterization of best approximations in normed linear spaces of matrices by elements of finite dimensional linear subspaces abstract a general characterization theorem of best approximations in normed linear spaces is specialized to the linear space of real n n matrices endowed with the spectral norm

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

ilyashenko algebras based on transserial asymptotic expansions abstract we construct a hardy field that contains ilyashenko s class of germs at of almost regular functions found in 12 as well as all log exp analytic germs this implies non oscillatory behaviour of almost regular germs with respect to all log exp analytic germs in addition each germ in this hardy field is uniquely characterized by an asymptotic expansion that is an le series as defined by van den dries et al 7 as these series generally have support of order type larger than the notion of asymptotic expansion itself needs to be generalized

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

convex analysis on cartan subspaces in 1937 von neumann 31 gave a famous characterization of unitarily invariant matrix norms that is norms f on cp q satisfying f uxv f x for all unitary matrices u and v and matrices x in cp q his result states that such norms are those functions of the form g where the map x cp q 7 x r has components the singular values 1 x 2 x p x of x assuming p q and g is a norm on rp invariant under sign changes and permutations of components furthermore he showed the respective dual norms satisfy

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

the 16th hilbert problem restricted to circular algebraic limit cycles abstract we prove the following two results first every planar polynomial vector field of degree s with s invariant circles is darboux integrable without limit cycles second a planar polynomial vector field of degree s admits at most s 1 invariant circles which are algebraic limit cycles in particular we solve the 16th hilbert problem restricted to algebraic limit cycles given by circles because a planar polynomial vector field of degree s has at most s 1 algebraic limit cycles given by circles and this number is reached

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

bifurcations of limit cycles in complex networks of hamiltonian systems and lloyd s conjecture abstract this paper is motivated by hilbert s sixteenth problem concerning the number and distribution of limit cycles in planar polynomial vector fields we propose an original method inspired by the complex systems framework in order to construct a family of hamiltonian systems admitting non degenerate centers located on the vertices of a planar network we introduce two novel polynomial perturbations adapted to those systems with the aim to produce a high number of limit cycles the first one is oriented along the gradient of the unperturbed system and the second one is again constructed with a complex systems approach using the melnikov integral we reach in a small number of particular cases the lower bound of cubical order for the hilbert number h n conjectured by lloyd in 1988

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

exponentially improved asymptotic expansions for resonances of an elliptic cylinder scattering by an elliptic cylinder is considered asymptotic expansions for regge poles and resonances are derived from the uniform asymptotic expansions of mathieu functions and modified mathieu functions constructed by applying the langer olver method in addition asymptotic expansions for resonances are exponentially improved by emphasizing the role of the symmetries of the scatterer the splitting up of resonances is then explained in terms of the symmetry breaking o 2 2v

limit cycles of quadratic systems abstract in this paper the global qualitative analysis of planar quadratic dynamical systems is established and a new geometric approach to solving hilbert s sixteenth problem in this special case of polynomial systems is suggested using geometric properties of four field rotation parameters of a new canonical system which is constructed in this paper we present a proof of our earlier conjecture that the maximum number of limit cycles in a quadratic system is equal to four and their only possible distribution is 3 1 v a gaiko global bifurcation theory and hilbert s sixteenth problem kluwer boston 2003 besides applying the wintner perko termination principle for multiple limit cycles to our canonical system we prove in a different way that a quadratic system has at most three limit cycles around a singular point focus and give another proof of the same conjecture

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

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

on the geometry of chemical reaction networks lyapunov function and large deviations in an earlier paper we proved the validity of large deviations theory for the particle approximation of quite general chemical reaction networks in this paper we extend its scope and present a more geometric insight into the mechanism of that proof exploiting the notion of spherical image of the reaction polytope this allows to view the asymptotic behavior of the vector field describing the mass action dynamics of chemical reactions as the result of an interaction between the faces of this polytope in different dimensions we also illustrate some local aspects of the problem in a discussion of wentzell freidlin theory together with some examples

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

properties of linear approximations of matrices in the spectral norm abstract the properties of linear approximations of a matrix are presented with respect to the spectrum norm we describe the connection between all spectral approximations of a given matrix and we give a sufficient condition under which the spectral approximation is unique

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

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

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

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

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