loop star decomposition of basis functions in the discretization of the efie a general and readily applicable scheme is presented for the determination of the basis functions that allow the decomposition of the surface current into a solenoidal part and a nonsolenoidal remainder the proposed approach brings into correspondence these two parts with two scalar functions and generates the known loop and star basis functions the completeness of the loop star basis is discussed employing the presented scheme the issue of the irrotational property of the nonsolenoidal functions is addressed

bifurcations with o 3 symmetry including applications to the benard problem this paper has two purposes to study from the singularity theory point of view certain bifurcation problems which commute with the five dimensional irreducible representation of the orthogonal group o 3 and to give some implications of this study for the benard problem in spherical geometry we shall now describe in general terms our results and compare them with the work of chossat 3 whose paper motivated our own interest in the problem the benard problem is concerned with convection in a viscous fluid when it is heated from below the fluid is assumed to be confined in a spherical shell of outer radius r and inner radius qro where q is near 0 3 this choice of q is partially motivated by considering convection within the molten layer of the core of the earth see busse l for further discussion we consider the binard problem in the boussinesq approximation in this model there is a trivial solution representing pure heat conduction radially outward as the temperature on the inner sphere that is the rayleigh number r is increased this trivial solution loses stability say at r r the benard problem is the study of the resulting bifurcation chossat studies this problem using the lyapunov schmidt reduction in the boussinesq approximation the fluid is driven from the pure conduction state by a term in the momentum equation involving the gravity vector g r and a term in the temperature equation involving the gradient of the conduction temperature vt assuming that the production of heat and the density are uniform throughout the shell these vectors have the form

generic bifurcation of hamiltonian systems with symmetry we study generic bifurcations of equilibria in one parameter hamiltonian systems with symmetry group on the generalized eigenvalues of the linearized system go through zero theorem 3 3 classifies expected actions of on the generalized eigenspace of this zero eigenvalue generic one degree of freedom symmetric systems is section 4 remarks concerning systems with more degrees of freedom are given in section 5

on the nature of the torus in the complex lorenz equations the complex lorenz equations are a nonlinear fifth order set of physically derived differential equations which exhibit an exact analytic limit cycle which subsequently bifurcates to a torus in this paper we build upon previously derived results to examine a connection between this torus at high and low r1 bifurcation parameter and between zero and nonzero r2 complexity parameter in so doing we are able to gain insight on the effect of the rotational invariance of the system and on how extra weak dispersion r2 0 affects the chaotic behavior of the real lorenz system which describes a weakly dissipative dispersive instability

solution regularity and smooth dependence for abstract equations and applications to hyperbolic pdes first we present a generalized implicit function theorem for abstract equations of the type f u 0 we suppose that u0 is a solution for 0 and that f is smooth for all but mainly we do not suppose that f u is smooth for all u we state conditions such that for all 0 there exists exactly one solution u u0 that u is smooth in a certain abstract sense and that the data to solution map u is smooth then we apply this to time periodic solutions of first order hyperbolic systems tuj aj x xuj bj t x u 0 and second order hyperbolic equations t2u a x 2 x2u b t x u tu xu 0 here we have to prevent small divisors from coming up moreover we need smooth dependence of bj and b on t to get smooth dependence of the solution on this is completely different to the case of parabolic pdes

heteroclinic orbits in a spherically invariant system the existence and stability of structurally stable heteroclinic cycles are discussed in a codimension 2 bifurcation problem with o 3 symmetry when the critical spherical modes l 1 and l 2 occur simultaneously several types of heteroclinic cycles are found which may explain aperiodic attractors found in numerical simulations for the onset of convection in a self gravitating fluid in a spherical shell

symmetry breaking and semilinear elliptic equations it is a readily observable fact that many physical and mathematical systems possess a degree of symmetry and that a study of this symmetry may give us valuable insight into their behaviour it is particularly interesting that symmetric systems exist which possess non symmetric solutions and where this solution branch arises from a symmetry breaking bifurcation on a branch of symmetric solutions in this paper we shall study an example of such a system namely we shall study the symmetry breaking bifurcations henceforth denoted as sbb s which occur on the radially symmetric solution branches of the following semilinear elliptic equation

classification and unfoldings of degenerate hopf bifurcations abstract this paper initiates the classification up to symmetry covariant contact equivalence of perturbations of local hopf bifurcation problems which do not satisfy the classical non degeneracy conditions the only remaining hypothesis is that i should be simple eigenvalues of the linearized right hand side at criticality then the lyapunov schmidt method allows a reduction to a scalar equation g x 0 where g x g x a definition is given of the codimension of g and a complete classification is obtained for all problems with codimension 3 together with the corresponding universal unfoldings the perturbed bifurcation diagrams are given for the cases with codimension 2 and for one case with codimension 3 for this last case one of the unfolding parameters is a modal parameter such that the topological codimension equals in fact 2 formulas are given for the calculation of the taylor coefficients needed for the application of the results and finally the results are applied to two simple problems a model of glycolytic oscillations and the fitzhugh nerve equations

bifurcation from rotating waves we consider abstract evolution equations which are invariant under a lie group we discuss bifurcation from trajectories that lie in the orbit under the lie group of a single point into trajectories in the orbit of a closed curve

heteroclinic cycles in dynamical systems with broken spherical symmetry in mathematics and physics the phrase symmetry breaking has two distinct meanings the first refers to the frequently observed phenomenon that a configuration of a physical system satisfying a law a set of equations which is invariant under a group of transformations may itself only be invariant under a subgroup of this group this is referred to as spontaneous symmetry breaking the second meaning refers to the process of explicitly adding symmetry breaking terms to the equations which describe the system this we call induced or forced symmetry breaking over the past decade spontaneous symmetry breaking has attracted a considerable amount of attention within the context of bifurcation and dynamical systems theory see for example 13 previous work on forced symmetry breaking has mainly concentrated on the persistance of equilibrium solutions under small symmetry breaking perturbations of symmetric systems 3 4 2 7 8 12 22 231 the main aim of this paper is to show that such perturbations may also give rise to more complex dynamical behavior in particular to heteroclinic cycles we first present a simple example of our constructions for the particular choices of groups g so 3 h so 2 and k t described below in order to illustrate these general ideas without too many technicalities we return to a rigorous discussion of these types of problems in section 2

normal forms resonances and meandering tip motions near relative equilibria of euclidean group actions the equivariant dynamics near relative equilibria to actions of noncompact finite dimensional lie groups g can be described by a skew product flow on a center manifold dot g g textbf a v dot v varphi v with g in g with v in a slice transverse to the group action and a v in the lie algebra of g we present a normal form theory near relative equilibria varphi v 0 0 in this general case for the specific case of the euclidean groups se n the skew product takes the form dot r r textbf r v qquad dot s r textbf s v qquad dot v varphi v with textbf r v in so n textbf s v in mathr n we give a precise meaning to the intuitive idea of tip motion of a meandering spiral it corresponds to the dynamics of s t this clarifies the notion of meander radii and drift resonance in the plane n 2 for illustration we discuss the unbounded tip motions associated with a weak focus in v on the verge of hopf bifurcation in the case of resonant hopf and rotation frequencies of the spiral and study resonant relative hopf bifurcation we also encounter random brownian tip motions for trajectories v t rightarrow gamma which become homoclinic for t rightarrow infty we conclude with some comments on the homoclinic tip shifts and drift resonance velocities in the bogdanov takens bifurcation which turn out to be small beyond any finite order

the numerical solution of nonlinear equations having several parameters part iii equations with z 2 symmetry the computation of symmetry breaking bifurcation points of nonlinear multiparameter problems with z 2 reflectional symmetry is considered the numerical approach is based on recent work in singularity theory which is used to construct systems of equations and inequalities characterising various types of symmetry breaking bifurcation points numerical continuation methods are then used to follow paths of symmetry breaking bifurcations and hence compute regions in parameter space for which a problem has qualitatively similar bifurcation diagrams the power of the numerical approach is illustrated by computations of axisymmetric flows in the finite taylor problem

on the use of cavity modes as basis functions in the full wave analysis of printed antennas magnetic wall cavity mwc modes are often employed as basis functions in the method of moments mom full wave analysis of printed antennas especially in its spectral domain version where the use of entire domain basis functions is particularly convenient but in some cases the te solenoidal modes are not included here we investigate on the effect of this exclusion explaining the nature of the te tm coupling presenting criteria to determine the necessary number of modes and giving numerical results for annular patches especially relevant in this context

symmetry breaking at non positive solutions of semilinear elliptic equations we consider symmetry breaking bifurcations at non positive radially symmetric solutions of semilinear elliptic equations on a ball with dirichlet boundary conditions for nonlinearities which are asymptotically affine linear we find solutions at which the symmetry breaks the kernel of the linearized equation at these solutions is an absolutely irreducible representation of the group o n for this kind of equation a transversality condition is satisfied if the perturbation of the affine linear problem is small enough thus we obtain by the equivariant branching lemma a large variety of isotropy subgroups of o n which occur as symmetries of the bifurcating solution branches

brief paper feedback control of limit cycle amplitudes from a frequency domain approach in this paper a problem in feedback control of limit cycle amplitudes is studied a graphical approach for this bifurcation control problem is developed by means of higher order harmonic balance approximations for both amplitude and frequency of the system oscillatory outputs the approach starts with the familiar hopf bifurcation mechanism and employs the second fourth and sixth order harmonic balance approximations to generate a sequence of graphical tests for convergence analysis of the system oscillatory outputs this sequential graphical testing leads to accurate approximations of limit cycles of small amplitudes in the system outputs degenerate hopf bifurcation theory is used to formulate an appropriate control objective of capturing small amplitude limit cycles which can avoid reaching unstable equilibria or other undesirable limit sets a rich cubic planar model is presented for illustration of the proposed control method

pattern selection in long wavelength convection abstract the long wavelength of the first instability in rayleigh benard convection between nearly thermally insulating horizontal plates is typical of a variety of physical systems the evolution of such an instability is described by the equation t 2 4 2 2 2 2 where is the planform function is the scaled rayleigh number and 1 the quantities represent the effects of finite biot number asymmetry in the boundary conditions at top and bottom and departures from the boussinesq approximation respectively the quantity when the original problem is self adjoint but otherwise planform selection is studied for 1 using equivariant bifurcation theory on the square lattice both rolls and squares can be stable depending on the parameters and the possible secondary bifurcations located near various codimension two singularities are analyzed on the hexagonal lattice the primary bifurcation is always degenerate when 0 of the six primary solution branches possible in this case the hexagon branch is stable when 0 both rolls and hexagons bifurcate supercritically but rolls are stable finally when and or 0 a hysteretic transition to h or h occurs depending on sgn k 2 c if stable h h coexist at larger amplitudes but if k 2 c 0 1 a further hysteretic transition takes place with increasing amplitude in which the hexagons are replaced by stable rolls

computations of limit cycles via higher order harmonic balance approximation the detection of limit cycles arising from hopf bifurcation phenomena by applying the harmonic balance method with different higher order approximations is discussed the results are presented using a graphical procedure that indicates clearly how the predictions of amplitude and frequency of a periodic solution can be improved by using higher and higher order approximations complete and explicit formulas for eighth order harmonic balance approximation are provided

bifurcating periodic solutions in rotationally symmetric systems rotationally symmetric systems governed by nonlinear ordinary differential equations and dependent on a parameter are analyzed for small bifurcating periodic solutions when the equilibrium configuration of the system loses stability at some critical value of the parameter with complex conjugate airs of eigenvalues crossing the imaginary axis two distinct possibilities are shown to exist in one case the symmetry of the system induces multiplicity of the complex pairs while in the second case despite the symmetry of the system only one complex pair crosses the imaginary axis for the case when two coincident complex pairs of eigenvalues cross the imaginary axis using the method of alternate problems it is shown that the system in general has two distinct bifurcating periodic solutions explicit expressions for the first approximations to these solutions are obtained using the general form of nonlinearities induced by the symmetry of the system conditions for stability of these solutions are also gi

a simple test for moment inequality models with an application to english auctions abstract testable predictions of many economic models involve inequality comparisons between transformations of nonparametric functionals we introduce an econometric test for these types of restrictions based on one sided l p statistics that adapt asymptotically to the contact sets without having to directly estimate them monte carlo experiments show that our test is less conservative than procedures based on least favorable configurations and has power comparable to other contact set based procedures as an application we test for interdependence of bidders valuations in ascending auctions using usfs timber auction data we reject the independent private values model in favor of a model of correlated private values

a minimal integrity basis for the elasticity tensor we definitively solve the old problem of finding a minimal integrity basis of polynomial invariants of the fourth order elasticity tensor c decomposing c into its so 3 irreducible components we reduce this problem to finding joint invariants of a triplet a b d where a and b are second order harmonic tensors and d is a fourth order harmonic tensor combining theorems of classical invariant theory and formal computations a minimal integrity basis of 297 polynomial invariants for the elasticity tensor is obtained for the first time

hopf bifurcation at a degenerate stationary pitchfork soit e un espace de hilbert reel on considere l existence de petites solutions periodiques pour une equation d evolution sur e donnee par du dt g u u e r avec g 0 0 r

applications of catastrophe theory to the physical sciences abstract catastrophe theory was introduced by rene thom in the late 1960 s as an attempt to model morphogentic changes in nature using ideas from topological dynamics and the theory of singularities of mappings thom envisaged a very general approach to topological changes in the solutions to parametrized systems of equations such as differential and difference equations and in particular discussed the special case of elementary catastrophe theory singularities of smooth real valued functions popular expositions have tended to overemphasize this special case but it remains the major source of ideas and methods here we survey the applications of catastrophe theory to the physical sciences physics chemistry engineering fluid mechanics etc for brevity we confine attention to an area lying between elementary and general catastrophe theory usually known as singularity theory this is the theory of singularities of smooth vector valued functions which mathematically is a straightforward though non trivial generalization of the real valued case in the last few years it has developed into a powerful and useful technique in several areas of theoretical physics notably optics and bifurcation theory equivariant catastrophe theory taking account of symmetry is likely to prove especially interesting

infinitely many solutions of a symmetric dirichlet problem here q is a bounded domain in r with smooth boundary and f i r is c and satisfies certain growth conditions if m 1 and f is an even function hence f r r is an odd function then ambrosetti and rabinowitz l 21 proved the existence of infinitely many weak solutions of d using a symmetric version of the mountain pass theorem a similar result is due to michalek 3 who considered the case where m 21 rm s c and f is invariant under a free action of the cyclic group z p on c here z p is considered as a subgroup of s c c acting essentially via scalar multiplication on c in this paper we want to investigate which kind of symmetries guarantee the existence of infinitely many solutions of d let g be a compact lie group and p g o m an orthogonal representation of g on lrm then g acts on irm via the map g x ir ir g u y gv p g v we write v for the space rm considered as a g space do there exist infinitely many solutions of d if f is invariant under this symmetry i e f gu f u for all g e g and u e v without further assumptions on p the answer to this question is no in general for example if p g id for all g e g then there is no symmetry at all and one cannot expect infinitely many solutions more generally one should exclude the existence of fixed points vg u e vi gu u for all g e g 0 our first result says that there exist infinitely many solutions of d if the following condition is satisfied

nonlinear analysis of a maglev system with time delayed feedback control abstract this paper undertakes a nonlinear analysis of a model for a maglev system with time delayed feedback using linear analysis we determine constraints on the feedback control gains and the time delay which ensure stability of the maglev system we then show that a hopf bifurcation occurs at the linear stability boundary to gain insight into the periodic motion which arises from the hopf bifurcation we use the method of multiple scales on the nonlinear model this analysis shows that for practical operating ranges the maglev system undergoes both subcritical and supercritical bifurcations which give rise to unstable and stable limit cycles respectively numerical simulations confirm the theoretical results and indicate that unstable limit cycles may coexist with the stable equilibrium state this means that large enough perturbations may cause instability in the system even if the feedback gains are such that the linear theory predicts that the equilibrium state is stable

computing hopf bifurcations i this paper addresses the problems of detecting hopf bifurcations in systems of ordinary differential equations and following curves of hopf points in two parameter families of vector fields the established approach to this problem relies upon augmenting the equilibrium condition so that a hopf bifurcation occurs at an isolated regular point of the extended system we propose two new methods of this type based on classical algebraic results regarding the roots of polynomial equations and properties of kronecker products for matrices in addition to their utility as augmented systems for use with standard newton type continuation methods they are also particularly well adapted for solution by computer algebra techniques for vector fields of small or moderate dimension

bifurcation at multiple eigenvalues and stability of bifurcating solutions if k is a bounded linear operator from the real banach space u into the real banach space v and u r v has the value zero at 0 0 the existence and linear stability of the equilibrium solutions of the dynamical system k dudt u which are close to the origin in u r are studied it is assumed that u 0 0 u v is a freholm operator of index zero the only restriction on the dimension of the null space of u 0 0 and the order of vanishing at 0 0 of restricted to the null space of d 0 0 u r v is that they both be finite positive integers the main result gives conditions under which the equation which determines the equilibrium solutions in a neighborhood of the origin also determines the stability of these equilibrium solutions

hopf bifurcation and symmetry travelling and standing waves on the circle in this paper we consider hopf bifurcation in the presence of o 2 symmetry the system of reaction diffusion equations u t d u xx f u provided with periodic boundary conditions may serve as a model problem we prove the bifurcation of a torus of standing waves and two circles of travelling waves and we compute the stability with asymptotic phase of these periodic solutions giving explicit formulae finally we demonstrate how a small perturbation which breaks part of the symmetry leads to secondary bifurcation

a tangent space characterisation of the equivalence of germs for geometric subgroups of and it is shown that under the action of a geometric subgroup of and for a germ f satisfying a certain finiteness condition given a germ p if the tangent spaces of f and f p are equal for all t 0 1 then f and f p are equivalent

normal forms for generalised hopf bifurcations explicit coefficients of normal forms for generalised hopf bifurcations are calculated to the seventh but to save space they are given only to the fifth order near the critical point the real part of the third order coefficient reproduces at the critical point the third order focal value of andronov et al 1973

hopf bifurcation in the presence of symmetry on etablit une generalisation du theoreme de bifurcation de hopf pour des equations differentielles a symetrie

reduced stability of parameter dependent matrices abstract the stability properties of parameter dependent linear systems z a e z with a 0 block diag 0 f f a stable matrix and 0 of order n n are analyzed near e 0 by a reduction principle which amounts to considering the schur complement g e of a e some sufficient conditions are given for a e and g e to have the same stability properties the so called principle of reduced stability in terms of the asymptotic expansion of g e explicit necessary and sufficient conditions are given in the case f i n 2 in terms of the location of the spectrum of g e allowing for geometrical interpretation

functional limit theorems for u processes un processus u est une collection de statistiques u indicees par une famille de noyaux symetriques on utilise la theorie limite classique pour demontrer pour le processus u des analogues du theoreme central limite pour les processus empiriques

resonant parametric perturbations of the hopf bifurcation abstract the effects of periodic parametric perturbations on a system undergoing hopf bifurcation are studied in detail of primary interest is the case of resonance between the hopf bifurcation frequency and the perturbation frequency method of alternative problems is used to obtain the small nonlinear periodic solutions it is shown that for a certain range of parameters the hopf solution is modified to give rise to jump response as well as isolated solutions for some parameter combinations stable solutions can get unstable and may bifurcate into aperiodic or amplitude modulated motions

symmetry groups of attractors for maps equivariant under the action of a finite group on n the possible symmetries of fixed points are known and correspond to the isotropy subgroups this paper investigates the possible symmetries of arbitrary possibly chaotic attractors and finds that the necessary conditions of melbourne dellnitz golubitsky 15 are sufficient at least for continuous maps

application of numerical regularization options to the integral equation analysis of printed antennas in this letter a novel numerical technique is applied to the analysis of printed antennas the technique transforms the first kind electric field integral equation efie into a regularized second kind equation via extraction of two singular frequency independent operators and their numerical inversion two options are presented in which the inversion is based on the eigenfunctions of these singular operators that appear to be a generalization of the tm and te modes of a magnetic wall cavity coupled by the edge singularities the example of application shows favorable convergence properties

new stability results for patterns in a model of long wavelength convection abstract we consider the transition from a spatially uniform state to a steady spatially periodic pattern in a partial differential equation describing long wavelength convection e knobloch pattern selection in long wavelength convection physica d 41 1990 450 479 this both extends existing work on the study of rolls squares and hexagons and demonstrates how recent generic results for the stability of spatially periodic patterns may be applied in practice we find that squares even if stable to roll perturbations are often unstable when a wider class of perturbations is considered we also find scenarios where transitions from hexagons to rectangles can occur in some cases we find that near onset more exotic spatially periodic planforms are preferred over the usual rolls squares and hexagons

little groups of irreps of o 3 so 3 and the infinite axial subgroups a new chain criterion for the determination of little groups is presented this result distinguishes massive rotationally inequivalent irrep basis functions and allows for multiple branching paths consequently it is suitable for application to lie groups applied to the groups o 3 so 3 d h d c v c h and c the result enables the enumeration of all possible little groups together with their associated basis functions these results are relevant to the determination of the symmetry of a material from its linear and nonlinear optical properties and to the choices of order parameters for symmetry breaking in liquid crystals

a truncated floquet wave diffraction method for the full wave analysis of large phased arrays ii generalization to 3 d cases for pt i see ibid vol 48 no 3 p 594 600 2000 this paper deals with the generalization to three dimensional 3 d arrays of the truncated floquet wave tfw diffraction method for the full wave analysis of large arrays this generalization potentially includes arrays consisting of microstrip excited slots cavity backed apertures and patches the formulation is carried out first by deriving an appropriate fringe integral equation ie and next by defining entire domain basis functions in terms of global array functions shaped as tfw diffracted rays whose analytical expression is derived on the basis of prototype canonical problems the efficiency and the accuracy of this method is demonstrated by comparison with the results of an element by element full wave approach for a rectangular slot array

symmetries and pattern selection in rayleigh benard convection abstract this paper describes the process of pattern selection between rolls and hexagons in rayleigh benard convection with reflectional symmetry in the horizontal midplane this symmetry is a consequence of the boussinesq approximation provided the boundary conditions are the same on the top and bottom plates all possible local bifurcation diagrams assuming certain non degeneracy conditions are found using only group theory the results are therefore applicable to other systems with the same symmetries rolls hexagons or a new solution regular triangles can be stable depending on the physical system rolls are stable in ordinary rayleigh benard convection the results are compared to those of buzano and golubitsky 1 without the midplane reflection symmetry the bifurcation behavior of the two cases is quite different and a connection between them is established by considering the effects of breaking the reflectional symmetry finally the relevant experimental results are described

stability of solutions bifurcating from multiple eigenvalues let x and y be real banach spaces and g x r be a twice continuously differentiate function which is not necessarily linear suppose g u0 0 0 and the dimension of the null space of gu u0 0 is m where 1 m usually s u g u 0 in a neighborhood of u0 0 consists of a finite number of curves emanating from u0 0 we will determine the stability of points u in s i e the maximum of the real parts of the spectrum of gu u for each u s using a general perturbation theorem of kato our results contain as a special case the stability theorems of crandall and rabinowitz for the case m 1 we will also tie our stability theorems together with some bifurcation results of decker and keller finally we apply our results to systems of reaction diffusion equations

multimodal oscillations in systems with strong contraction one and two parameter families of flows in r 3 near an andronov hopf bifurcation ahb are investigated in this work we identify conditions on the global vector field which yield a rich family of multimodal orbits passing close to a weakly unstable saddle focus and perform a detailed asymptotic analysis of the trajectories in the vicinity of the saddle focus our analysis covers both cases of sub and supercritical ahb for the supercritical case we find that the periodic orbits born from the ahb are bimodal when viewed in the frame of coordinates generated by the linearization about the bifurcating equilibrium if the ahb is subcritical it is accompanied by the appearance of multimodal orbits which consist of long series of nearly harmonic oscillations separated by large amplitude spikes we analyze the dependence of the interspike intervals which can be extremely long on the control parameters in particular we show that the interspike intervals grow logarithmically as the boundary between regions of sub and supercritical ahb is approached in the parameter space we also identify a window of complex and possibly chaotic oscillations near the boundary between the regions of sub and supercritical ahb and explain the mechanism generating these oscillations this work is motivated by the numerical results for a finite dimensional approximation of a free boundary problem modeling solid fuel combustion c 2007 elsevier b v all rights reserved

abstract forced symmetry breaking and forced frequency locking of modulated waves we consider abstract forced symmetry breaking problems of the typef x y it is supposed that for all the mapsf are equivariant with respect to the action of a compact lie group thatf x0 0 0 and hence thatf x 0 0 for all elementsxof the group orbit o x0 ofx0 we look for solutionsxwhich bifurcate from the solution family o x0 as andymove away from 0and zero respectively especially we describe the number of different solutionsx for fixed control parameters andy their dynamic stability and their asymptotic behavior forytending to zero further generalizations are given to problems of the typef x y x finally our results are applied to a forced frequency locking problem of the typex t f x t y t here it is supposed that the vector fieldsf ares1 equivariant that the unperturbed equationx f x 0 has an orbitally stable modulated wave solution and that the forcingy t is a modulated wave

coincident bifurcation of equilibrium and periodic solutions of evolution equations abstract bifurcation of equilibrium and periodic solutions of nonlinear evolution equations is considered in the neighbourhood of an equilibrium solution for which the corresponding linear problem admits both non zero equilibrium and non constant periodic solutions these solutions of the linear problem are related to those of the nonlinear equation by deriving bifurcation equations possessing a simple symmetry property this results in a simplification of the bifurcation analysis illustrated by a discussion of two important special cases exhibiting secondary bifurcation of periodic solutions author

full wave spectral analysis and design of annular patch antenna with electromagnetically coupled microstrip feed line this paper presents a full wave analysis and the ensuing design considerations of the annular patch antenna fed by an electromagnetically coupled microstrip line the numerical analysis employs the spectral domain version of the integral equation with the method of moments formulation the basis functions on the ring patch are the eigenmodes of the corresponding magnetic wall cavity and subsectional rooftop functions are employed on the access line both tm and te modes are used along with more efficient collective solenoidal modes mode sorting criteria are given to ensure uniform convergence efficient schemes are presented for the calculation of the spectral domain reaction integrals along with criteria to avoid calculation of negligible mode to mode couplings based on the spectral analysis of the matrix condition number very good agreement is shown with measured data and design considerations are given for matching the radiator

u processes rates of convergence on introduit un nouveau processus stochastique une collection de statistiques u indicees par une famille de noyaux symetriques on obtient des conditions pour la convergence presque sure uniforme d une suite de tels processus

periodic solutions in an array of coupled fitzhugh nagumo cells abstract we analyse the dynamics of an array of n 2 identical cells coupled in the shape of a torus each cell is a 2 dimensional ordinary differential equation of fitzhugh nagumo type and the total system is z n z n symmetric the possible patterns of oscillation compatible with the symmetry are described the types of patterns that effectively arise through hopf bifurcation are shown to depend on the signs of the coupling constants under conditions ensuring that the equations have only one equilibrium state

turing patterns with o 3 symmetry abstract we perform an explicit center manifold reduction for the general steady state bifurcation of a chemical reaction diffusion system in an arbitrary compact domain in particular we focus on those systems with two chemical species in domains with o 3 symmetry we illustrate with the brusselator on the spherical shell and produce explicit computations of the solution branches and their stabilities for the low l bifurcations because of the phenomenon of parameter collapse we can create universal bifurcation diagrams for even l this work is an extension of that done in physica d 132 1999 339 362

hopf bifurcation for semilinear dissipative hyperbolic systems abstract we consider boundary value problems for semilinear hyperbolic systems of the type t u j a j x x u j b j x u 0 x 0 1 j 1 n with smooth coefficient functions a j and b j such that b j x 0 0 for all x 0 1 r and j 1 n we state conditions for hopf bifurcation i e for existence local uniqueness up to phase shifts smoothness and smooth dependence on of time periodic solutions bifurcating from the zero stationary solution furthermore we derive a formula which determines the bifurcation direction the proof is done by means of a liapunov schmidt reduction procedure for this purpose fredholm properties of the linearized system and implicit function theorem techniques are used

on the principle of reduced stability abstract the principle of reduced stability says that the stability of bifurcating stationary or periodic solutions is given by the finite dimensional bifurcation equation obtained by the method of lyapunov schmidt to be more precise the linearized stability is governed by the linearization of the bifurcation equation about the bifurcating branch of solutions and in particular by the signs of the real parts of the perturbation of the eigenvalues along this branch this principle is true for simple eigenvalue bifurcation whereas it may be false for higher dimensional bifurcation equations a condition for the validity of that principle is given a counterexample shows that it cannot be dropped in general

on a numerical liapunov schmidt spectral method and its application to biological pattern formation spectral expansions are used to provide a basis which preserves continuous symmetries we show that spectral methods satisfy the conditions for convergence of numerical liapunov schmidt methods an explicit algorithm for the calculation of stationary bifurcation scenarios near primary instabilities in general continuous symmetric equations is given the above convergence is extended to gamma equivalent discrete and original bifurcation scenarios the method is applied to a biologically motivated reaction diffusion system with spherical symmetry forming patterns a specific singularity of a generic steady state bifurcation is investigated in detail

inference for first price auctions with guerre perrigne and vuong s estimator abstract we consider inference on the probability density of valuations in the first price sealed bid auctions model within the independent private value paradigm we show the asymptotic normality of the two step nonparametric estimator of guerre et al 2000 gpv and propose an easily implementable and consistent estimator of the asymptotic variance we prove the validity of the pointwise percentile bootstrap confidence intervals based on the gpv estimator lastly we use the intermediate gaussian approximation approach to construct bootstrap based asymptotically valid uniform confidence bands for the density of the valuations

a consistent nonparametric test of affiliation in auction models abstract we propose a new nonparametric test of affiliation a strong form of positive dependence with independence as a special knife edge case the test is consistent against all departures from the null of affiliation and its null distribution is standard normal like most nonparametric tests a sample size dependent input parameter is needed we provide an informal procedure for choosing the input parameter and evaluate the test s performance using a simulation study our test can be used to test the fundamental assumptions of the auctions literature we implement our test empirically using the outer continental shelf ocs auction data

a principle of reduced stability for reaction diffusion equations a semilinear elliptic boundary value problem au f x u 0 withfu x u bounded below can be shown to be equivalent to a finite dimensional problem b c 0 rd c rd in the sense that their solution sets which are not necessarily singletons are in a one to one correspondence c u u c the functionb c is called the bifurcation function it is shown that for any solutionu c the number of negative resp zero eigenvalues of the matrixbc c is identical to the number of negative resp zero eigenvalues of the linearized elliptic operatorav fu x u c v this results in a version of the principle of reduced stability for the problemut au f x u 0 and its reductionc b c 0

complete symmetry classification and compact matrix representations for 3d strain gradient elasticity abstract strain gradient elasticity sge is now often used in mechanics and physics owing to its capability to model some non classical phenomena such as size effects in materials and structures however certain fundamental questions about it have not yet received complete responses in its linear setting the constitutive law of sge is characterized by a fourth order elasticity tensor a fifth order one and a sixth order one even if the matrix representations for the 3d fourth and sixth order elasticity tensors are available for all possible symmetry classes the counterparts for the 3d fifth order tensor whose presence is unavoidable for materials with non centrosymmetric microstructure are still lacking in addition although the symmetry classes for each of the fourth fifth and sixth order elasticity tensor spaces are known the symmetry classes for these tensor spaces as a whole have never been reported and clarified in the literature the present work solves these two fundamental problems preventing the full understanding and exploitation of the linear constitutive law of sge precisely the matrix representations of the fifth order tensor for all of its 29 symmetry classes are provided in a compact and well structured way further linear sge is shown to possess 48 symmetry classes and for each of these symmetry classes the matrix representations of its fourth fifth and sixth order tensors are now available

a bifurcation theorem for potential operators abstract a sufficient condition for bifurcation of real solutions of g u 0 where g r d e d e is given in terms of the crossing number of the eigenvalue perturbation of the linearization along the trivial solutions 0 this number describes how many eigenvalues leave the negative real axis wen the parameter varies from negative to positive values whereas in general only an odd crossing number entails bifurcation for potential operators in a hilbert space e a nonzero crossing number is sufficient for bifurcation the proof uses a lyapunov schmidt reduction and conley s index theory

an example of symmetry breaking to heteroclinic cycles is orbitally asymptotically stable suppose now that we consider a small system symmetry breakingperturbation of 1 1 z f z g z 1 2 where is small and g is only 2 equivariant where 2 1 is a lie subgroup when is sufficiently small normal hyperbolicity guarantees that there isa perturbed flow invariant manifold x

quasi periodic self excited travelling waves for damped beam equations abstract a two dimensional beam equation is considered it is shown that for a dense set of model parameters the presence of linear and nonlinear damping terms of opposite sign produce small amplitude quasi periodic travelling waves which are continuations of two branches of rotating waves lyapunov schmidt reduction and only the classical implicit function theorem are used

the bifurcation structure of the holling tanner model for predator prey interactions using two timing the holling tanner model for predator prey systems has two hopf bifurcation points for certain parameters the dependence of the environmental parameters on the underlying bifurcation structure is uncovered using two timing emphasis is on how the bifurcation diagram changes as the hopf bifurcation points separate two degenerate cases require a modification of conventional two timing when the two hopf bifurcation points nearly coalesce the two stable periodic solution branches are shown to be connected as a ratio of linear growth rates varies the hopf bifurcation points separate further and one limit cycle becomes unstable this situation can correspond to an outbreak in populations the modified two timing analysis analytically captures the unstable and stable limit cycles of the new branch

bifurcation of solutions of the laser equations abstract we prove the existence and stability of bifurcating periodic and quasiperiodic solutions of the equations describing a ring laser

interaction of periodic and stationary bifurcation from multiple eigenvalues on considere des equations d evolution non lineaires du dt g u o dependant d un parametre reel dans un espace de hilbert reel e

relative equilibria of point vortices on the sphere we prove the existence of many different symmetry types of relative equilibria for systems of identical point vortices on a non rotating sphere the proofs use the rotational symmetry group so 3 and the resulting conservation laws the time reversing reflectional symmetries in o 3 and the finite symmetry group of permutations of identical vortices results include both global existence theorems and local results on bifurcations from equilibria a more detailed study is made of relative equilibria which consist of two parallel rings with n vortices in each rotating about a common axis the paper ends with discussions of the bifurcation diagrams for systems of 3 6 identical vortices

an explicit expression of the first liapunov and period constants with applications in this paper we study systems in the plane having a critical point with pure imaginary eigenvalues and we search for effective conditions to discern whether this critical point is a focus or a center in the case of it being a center we look for additional conditions in order to be isochronous we stress that the essential differences between the techniques used in this work and the more usual ones are basically two the elimination of the integration constants when we consider primitives of functions see also remark 3 2 and the fact that we maintain the complex notation in the whole study thanks to these aspects we reach with relative ease an expression of the first three liapunov constants v3 v5 andv7 and of the first two period ones p2andp4 for a general system as far as we know this is the first time that a general and compact expression ofv7has been given moreover the use of a computer algebra system is only needed in the computation ofv7andp4 these results are applied to give a classification of centers and isochronous centers for certain families of differential equations

applications of the blowing up construction and algebraic geometry to bifurcation problems a generalization of the morse lemma to vector valued functions is proved by a blowing up argument this is combined with a theorem from algebraic geometry on the number of real solutions of a system of homogeneous equations of even degree to yield a new bifurcation theorem bifurcation in a one or multi parameter problem is guaranteed if the leading term is of even degree it is often two and satisfies a regularity condition applications are given to nonlinear eigenvalue problems and to the hopf bifurcation

shapes and cycles arising at the steady bifurcation with icosahedral symmetry abstract this paper analyses the steady state bifurcation with icosahedral symmetry the equivariant branching lemma is used to predict the generic bifurcating solution branches corresponding to each irreducible representation of the icosahedral group i h the relevant amplitude equations are deduced from the equivariance condition and used to investigate the stability of bifurcating solutions it is found that the bifurcation with icosahedral symmetry can lead to competition between two fold three fold and five fold symmetric structures and between solutions with tetrahedral three fold and two fold symmetry stable heteroclinic cycles between solutions with d 2 z symmetry are found to exist in one of the irreps the theoretical scenarios are compared with the observed behaviour of icosahedral viruses and nanoclusters

classification and unfoldings of 1 2 resonant hopf bifurcation in this paper we study the bifurcations of periodic solutions from an equilibrium point of a differential equation whose linearization has two pairs of simple pure imaginary complex conjugate eigenvalues which are in 1 2 ratio this corresponds to a hopf hopf mode interaction with 1 2 resonance as occurs in the context of dissipative mechanical systems using an approach based on liapunov schmidt reduction and singularity theory we give a framework in which to study these problems and their perturbations in two cases no distinguished parameter and one distinguished bifurcation parameter we give a complete classification of the generic cases and their unfoldings

pattern selection with o 3 symmetry abstract for each irreducible representation of so 3 and o 3 we determine up to conjugacy all isotropy subgroups and identify in particular the maximal isotropy subgroups each isotropy subgroup corresponds to a possible planform for the spherical benard problem using an equivariant branching lemma of cicogna 1 we prove for each of these representations the existence of solutions corresponding to a number of different planforms thus extending substantially the work of busse 2 3 and sattinger 4 we also give a useful criterion for showing when solutions obtained by the equivariant branching lemma must be unstable