analytical solution of spatial elastica and its application to kinking problem abstract an analytical solution is presented for the spatially large deformation of a thin elastic rod spatial elastica which is naturally straight and uniform with equal principal stiffnesses and is subjected to terminal loads the elastica can suffer not only flexure and torsion as in the classical kirchhoff theory but also extension and shear the present solution is expressed in integral form and described in terms of only four parameters this solution clears the difficulty with the polar singularity in the use of euler angles hence the numerical analysis is possible for various boundary value problems with no limitation in this paper we study the post buckling behavior of an elastica under the terminal twist and uniaxial end shortening and give a theoretical explanation to commonly observed phenomena such as secondary bifurcation formation of a kink snap through behavior the contact problem is analyzed in the case where the elastica contacts with itself and forms a kink these results are available for other analysis e g based on finite element approximations

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

global hopf bifurcation of two parameter flows the behavior of center indices as introduced by j mallet paret j yorke is analyzed for two parameter flows the integer sum of center indices along a one dimensional curve in parameter space is called the h index a nonzero h index implies global hopf bifurcation the index h is not a homotopy invariant this fact is due to the occurrence of stationary points with an algebraically double eigenvalue zero which we call b points to each b point we assign an integer b index such that the h index relates to the b indices by a formula such as occurs in the calculus of residues

bifurcations of subharmonic solutions in reversible systems soit f r r n r r n une application c periodique cut on dit que l equation differentielle x f t x est reversible s il existe un operateur symetrique s sur r n tel que s 2 i de telles equations differentielles sont aussi appelees systemes avec la propriete e par rapport a s on etudie les bifurcations des solutions sous harmoniques de tels systemes reversibles

infinite period bifurcation and global bifurcation branches branches of periodic solutions which exhibit the alternatives of the global hopf bifurcation theorem are calculated for two general systems of differential equations in the first a branch of solu

a group theoretic approach to the global bifurcation analysis of an axially compressed cylindrical shell abstract the accurate prediction of the buckling load of thin shell structures is an important yet elusive goal it is particularly important in the aerospace industry where thin shell members are commonly used as structural elements due to a lack of adequate analytical results current practices in industry put heavy reliance on experimental testing and empirical data to supplement theoretical analysis see d bushnell computerized buckling analysis of shells martinus nijhoff dordrecht 1985 this paper focuses on recent results of the group theoretic approach to a numerical global postbuckling analysis of a perfect axially compressed cylindrical shell with built in end conditions the built in end conditions obviate the existence of a trivial membrane solution branch the example of an axially compressed cylindrical shell was chosen because it is well known that for thin shells the primary axisymmetric solution branch is riddled with closely spaced symmetry breaking bifurcation points in a numerical arc length continuation scheme the close proximity of the bifurcation points on the primary path manifests itself in severe ill conditioning of the tangent stiffness matrix group theory helps one systematically find an optimal set of basis vectors or symmetry modes which reflect the symmetry of a given solution path the immediate payoff in using these symmetry modes as basis vectors is that the tangent stiffness matrix block diagonalizes and the numerical ill conditioning is avoided thus an efficient and accurate technique for computing solution branches of a specific type and a subsequent diagnosis for symmetry breaking bifurcations is made relatively simple understanding the global behavior of the perfect structure is crucial in identifying critical imperfections and will ultimately diminish the heavy reliance on expensive experimental verification

ando s inequality for uniform submajorization abstract we discuss ando s inequality involving operator monotone functions and demonstrate that a natural strengthening of ando s result to uniform submajorization holds if and only if the operator monotone function has a very simple rational form

finite dimensional approximation of bifurcation problems in presence of symmetries we give here a brief summary of the results of 4 14 the aim of our work was not only the finite dimensional approximation of the specific problem studied in 3 but we also wanted to show how to adapt to discrete problems and how to use in this case some classical mathematical tools e g the splitting lemma of gromoll meyer magnus the singularity theory etc such considerations are developed in 5 in particular we prove in 5 that in most cases e g in the examples given in 6 7 9 10 the discrete bifurcation problem behaves as an imperfect bifurcation problem and we improve the methods of 12 another way to use universal unfoldings in the approxima tion of bifurcation problems is suggested by 2

a multiplicity of localised buckling modes for twisted rod equations the kirchhoff love equations governing the spatial equilibria of long thin elastic rods subject to end tension and moment are reviewed and used to examine the existence of localized buckling solutions the effects of shear and axial extension are not considered but the model does additionally allow for nonlinear constitutive laws under the assumption of infinite length the dynamical phase space analogy allows one to use techniques from dynamical systems theory to characterize many possible equilibrium paths localizing solutions correspond to homoclinic orbits of the dynamical system under non dimensionalization the twisted rod equations are shown to depend on a single load parameter and the bifurcation behaviour of localizing solutions of this problem is investigated using analytical and numerical techniques first in the case of a rod with equal principal bending stiffnesses where the equilibrium equations are completely integrable a known one parameter family of localizing solutions is computed for a variety of subcritical loads load deflection diagrams are computed for this family and certain materially nonlinear constitutive laws are shown to make little difference to the qualitative picture the breaking of the geometrical circular symmetry destroys complete integrability and in particular breaks the non transverse intersection of the stable and unstable manifolds of the trivial steady state the resulting transverse intersection which is already known to lead to spatial chaos is explicitly demonstrated to imply a multitude of localized buckling modes a sample of primary and multi modal solutions are computed numerically aided by the reversibility of the differential equations finally parallels are drawn with the conceptually simpler problem of a strut resting on a nonlinear elastic foundation for which much more information is known about the global behaviour of localized buckling modes

bifurcation in classical bipolar transistor oscillator circuits the results of a systematic asymptotic analysis of two classical oscillator circuits are presented in this paper both circuits are composed of passive components and a single bipolar transistor they are each described by a third order system of nonlinear ordinary differential equations that is singularly perturbed in the limit varepsilon to 0 here varepsilon is the inverse of the circuit s q the two timing method is used to construct the asymptotic behavior of the solutions as varepsilon to 0 these solutions are shown to depend upon amplitudes that evolve on a slow time scale according to amplitude equations these amplitude equations are analyzed and the results are physically interpreted

systematic description of imperfect bifurcation behavior of symmetric systems abstract a systematic method is presented for describing experimental curves of force vs strain of a system with regular polygonal dihedral group symmetry subject to bifurcation behavior with an aim toward overcoming the following problems 1 it is difficult to judge whether the system is undergoing bifurcation or not 2 the perfect behavior of the system cannot be known due to the presence of initial imperfections 3 those curves are often qualitatively different from bifurcation diagrams predicted by mathematics the tools employed are the asymptotic theory for imperfect bifurcation such as the koiter law and the stochastic theory of initial imperfections the former theory is extended in this paper to the system with regular polygonal symmetry to present asymptotic laws for recovering perfect curves with reference to the experimental ones these laws are formulated for physically observable displacements instead of the variables in the mathematical bifurcation diagrams in order to make them readily applicable to the experimental curves the stochastic theory is combined with an asymptotic law to develop a means to identify the multiplicity of the bifurcation point the systematic method for describing the experimental curves developed in this manner is applied to the bifurcation analysis of regular polygonal truss domes to testify its validity furthermore this method is applied to the shear behavior of cylindrical sand specimens to show that they in fact are undergoing bifurcation and in turn to demonstrate the importance of a viewpoint of bifurcation in the study of shear behavior of materials the need of a dual viewpoint of bifurcation and plasticity in the study of constitutive relationship of materials is emphasized to conclude the paper

on the persistence of quasi periodic invariant tori for double hopf bifurcation of vector fields abstract we analyze the persistence of quasi periodic invariant 2 and 3 tori for the double hopf hopf hopf bifurcation by using the kam method we prove that in a sufficiently small neighborhood of the bifurcation point the full system has quasi periodic 2 tori for most of the parameter sets where its truncated normal form possesses 2 tori under appropriate conditions we obtain that the full system also has quasi periodic 3 tori for most parameters near the hopf bifurcation curve of its truncated normal form and along the direction of the bifurcation and these 3 tori bifurcate from invariant 2 tori we also give concrete formulas on the existence of quasi periodic invariant 2 and 3 tori which are based on coefficients of the truncated normal form

classification and unfolding of degenerate hopf bifurcations with o 2 symmetry no distinguished parameter abstract the hopf bifurcation in the presence of o 2 symmetry is considered when the bifurcation breaks the symmetry the critical imaginary eigenvalues have multiplicity two and generically there are two primary branches of periodic orbits which bifurcate simultaneously in applications these correspond to rotating traveling waves and standing waves using equivariant singularity theory a classification of all such bifurcations up to and including codimension three is presented no distinguished parameter is assumed the universal unfoldings reveal the existence of both 2 tori and 3 tori corresponding to quasiperiodic waves with two and three independent frequencies respectively

computational methods for bifurcation problems with symmetries with special attention to steady state and hopf bifurcation points we show how group theoretical methods can be employed to utilize the symmetry of a bifurcation problem in numerical computations we extend the approach by werner 1988 by presenting methods for the detection of bifurcation points and the computation of multiple hopf points the essential numerical point is the utilization of certain reduced instead of full systems involving appropriate subgroups of the underlying symmetry group the group theoretical tool is an a priori knowledge of the interaction of certain subgroups 0 and of at possibly multiple steady state or hopf bifurcation points minimal 0 breaking bifurcation we introduce a bifurcation graph which shows graphically this a priori information its edges represent possible symmetry breaking bifurcations our analysis follows the lines of golubitsky stewart and schaeffer 1988 but it is aimed to numerical applications we have chosen a 4 box brusselator model in order to explain our notions and ideas and to discuss the numerical procedure

lock on to tape like behaviour in the torsional buckling of anisotropic rods abstract through nonlinear normal form analysis of the equilibrium equations developed in a neighbourhood of the straight rod solution we show the existence of a phase transition from mildly anisotropic to full tape like buckling in non symmetric rods subject to terminal loads provided the material has a sufficiently low poisson s ratio for a solid elliptical rod with a poisson s ratio of 0 2 the critical point is found to occur at a cross sectional aspect ratio of just over 3 for larger aspect ratios the rod locks on to a one twist per wave buckling mode with no internal twist in the mildly anisotropic regime there turn out to be two physically distinct pairs of symmetric so called primary localised buckling modes differing in the phase of the internal twist these four modes homoclinic orbits remain of the full circle of localised buckling modes existing in the symmetric case after breaking of the circular symmetry of the rod s cross section in the strongly anisotropic regime only the energetically favourable pair of flat buckling modes survives

merging of resonance tongues abstract this paper considers the resonance behavior found near a saddle node hopf bifurcation in a flow the global structure of resonance tongues is examined and it is shown that in the absence of certain symmetries different resonance tongues can merge with one another the position and shape of the resonance tongues are seen to be related to the presence of homoclinic and heteroclinic bifurcations of two saddle foci the implications of these results for the normal form analysis of the saddle node hopf bifurcation are discussed

synchronization and fluctuations coupling a finite number of stochastic units it is well established that ensembles of globally coupled stochastic oscillators may exhibit a nonequilibrium phase transition to synchronization in the thermodynamic limit infinite number of elements in fact since the early work of kuramoto mean field theory has been used to analyze this transition in contrast work that directly deals with finite arrays is relatively scarce in the context of synchronization and yet it is worth noting that finite number effects should be seriously taken into account since in general the limits n where n is the number of units and t where t is time do not commute mean field theory implements the particular choice first n and then t here we analyze an ensemble of three state coupled stochastic units which has been widely studied in the thermodynamic limit we formally address the finite n problem by deducing a fokker planck equation that describes the system we compute the steady state solution of this fokker planck equation that is finite n but t we use this steady state to analyze the synchronic properties of the system in the framework of the different order parameters that have been proposed in the literature to study nonequilibrium transitions

functions of operators under perturbations of class sp this is a continuation of our paper 2 we prove that for functions f in the holder class r and 1 p the operator f a f b belongs to sp whenever a and b are self adjoint operators with a b sp we also obtain sharp estimates for the schatten von neumann norms f a f b sp in terms of a b sp and establish similar results for other operator ideals we also estimate schatten von neumann norms of higher order differences j 0m 1 m j mj f a jk we prove that analogous results hold for functions on the unit circle and unitary operators and for analytic functions in the unit disk and contractions then we find necessary conditions on f for f a f b to belong to sq under the assumption that a b sp we also obtain schatten von neumann estimates for quasicommutators f a r rf b and introduce a spectral shift function and find a trace formula for operators of the form f a k 2f a f a k

an algebraic multigrid method with guaranteed convergence rate we consider the iterative solution of large sparse symmetric positive definite linear systems we present an algebraic multigrid method which has a guaranteed convergence rate for the class of nonsingular symmetric m matrices with nonnegative row sum the coarsening is based on the aggregation of the unknowns a key ingredient is an algorithm that builds the aggregates while ensuring that the corresponding two grid convergence rate is bounded by a user defined parameter for a sensible choice of this parameter it is shown that the recursive use of the two grid procedure yields a convergence independent of the number of levels provided that one uses a proper amli cycle on the other hand the computational cost per iteration step is of optimal order if the mean aggregate size is large enough this cannot be guaranteed in all cases but is analytically shown to hold for the model poisson problem for more general problems a wide range of experiments suggests that there are no complexity issues and further demonstrates the robustness of the method the experiments are performed on systems obtained from low order finite difference or finite element discretizations of second order elliptic partial differential equations pdes the set includes two and three dimensional problems with both structured and unstructured grids some of them with local refinement and or reentering corner and possible jumps or anisotropies in the pde coefficients

numerical and dimensional investigation of two phase countercurrent imbibition in porous media in this paper we introduce a numerical solution of the problem of two phase immiscible flow in porous media in the first part of this work we present the general conservation laws for multiphase flows in porous media as outlined in the literature for the sake of completion where we emphasize the difficulties associated with these equations in their primitive form and the fact that they are generally unclosed the second part concerns the 1d computation for dimensional and non dimensional cases and a theoretical analysis of the problem under consideration a time scale based on the characteristic velocity is used to transform the macroscopic governing equations into a non dimensional form the resulting dimensionless governing equations involved some important dimensionless physical parameters such as bond number bo capillary number ca and darcy number da numerical experiments on the bond number effect is performed for two cases gravity opposing and assisting the theoretical analysis illustrates that common formulations of the time scale forces the coefficient da 1 2 ca to be equal to one while formulation of dimensionless time based on a characteristic velocity allows the capillary and darcy numbers to appear in the dimensionless governing equation which leads to a wide range of scales and physical properties of fluids and rocks the results indicate that the buoyancy effects due to gravity force take place depending on the location of the open boundary

global hopf bifurcation in a simple climate model the mathematical structure of a simple climate model is investigated the model is governed by a system of two nonlinear autonomous differential equations for the evolution in time of global temperature t and meridional ice sheet extent l the system s solutions are studied by a combination of qualitative reasoning with explicit calculations both analytical and numerical for plausible values of the physical parameters a branch of periodic solutions obtains which is both orbitally and structurally stable the amplitude of the stable periodic solutions in t and l correspond roughly to that obtained from proxy records of quaternary glaciation cycles the period of these solutions increases along the branch until it becomes infinite while the amplitude of the limiting solution is finite the limiting solution is a homoclinic orbit formed by the reconnecting separatrix of a saddle the exchange of stability between the branch of periodic solutions and the steady solution from which it arises is studied

bifurcations of mode equations spin waves abstract we study the bifurcations of the spin wave equations that describe the parametric pumping of collective modes in magnetic media we discuss the following dynamical phenomena i sequential excitation of modes via zero eigenvalue bifurcations ii hopf bifurcations followed or not by feigenbaum cascades of period doublings iii local and global homoclinic phenomena two new organizing centers for routes to chaos are identified in the classification given by guckenheimer and holmes one is the codimension two local bifurcation with one pair of imaginary eigenvalues and a zero eigenvalue to which many dynamical consequences are known the second global homoclinic bifurcations associated to splitting of separatrices in the limit where the system can be considered a hamiltonian subjected to weak dissipation and forcing

a codimension two bifurcation with a third order picard fuchs equation abstract using a third order picard fuchs equation we show that a certain two parameter family of planar vectorfields for parameter values in a certain cone has a unique limit cycle which is born from a hopf bifurcation and dies in a saddle connection this removes a superfluous hypothesis in theorem 3 2 chapter 13 of s n chow and j k hale methods of bifurcation theory springer verlag new york 1982

invariance under the dihedral group and application to bifurcation problems soit un groupe d ordre agissant lineairement sur r s et soient 1 2 t des generateurs homogenes de r x 1 x s l algebre de tous les polynomes invariants a s variables si f est une fonction invariante de classe c m sur r s avec 0 m alors il existe une fonction f c p r t telle que f f 1 t avec p sup 0 m s 1 on considere ici le cas s 2 d n le groupe diedre d ordre 2n avec n 2

solving global problem by considering multitude of local problems application to fluid flow in anisotropic porous media using the multipoint flux approximation abstract in this work we apply the experimenting pressure field approach to the numerical solution of the single phase flow problem in anisotropic porous media using the multipoint flux approximation we apply this method to the problem of flow in saturated anisotropic porous media in anisotropic media the component flux representation requires generally multiple pressure values in neighboring cells e g six pressure values of the neighboring cells is required in two dimensional rectangular meshes this apparently results in the need for a nine points stencil for the discretized pressure equation 27 points stencil in three dimensional rectangular mesh the coefficients associated with the discretized pressure equation are complex and require longer expressions which make their implementation prone to errors in the experimenting pressure field technique the matrix of coefficients is generated automatically within the solver a set of predefined pressure fields is operated on the domain through which the velocity field is obtained apparently such velocity fields do not satisfy the mass conservation equations entailed by the source sink term and boundary conditions from which the residual is calculated in this method the experimenting pressure fields are designed such that the residual reduces to the coefficients of the pressure equation matrix

the transport of nanoparticles in subsurface with fractured anisotropic porous media numerical simulations and parallelization abstract the flow of fluids through fractured porous media has been an important topic in the research of subsurface flow the several orders of magnitude in size between the fractures and the rock matrix causes difficulties for simulating such flow scenario the fluid velocities in fractures are also several orders of magnitude higher than that in the rock matrix due to high permeability and porosity if there exists pollutant such as nanoparticles in the fluids the pollutant may be transported rapidly and the rock matrix s properties near the fractures are hence changed in this research we simulate the transport phenomena of nanoparticles in the fluid flow through fractured porous media the permeability fields which contain different anisotropy angles are considered in the simulation fractures are represented explicitly by volumetric grid cells and the numerical algorithm is parallelized in order to reduce the simulation time we investigate the effect of the appearance of fractures and rotated anisotropy on the transport of nanoparticles particles deposition entrapment and detachment the results show that flow directions are affected by the direction of anisotropy and the transport of nanoparticles in the fractures is significantly faster than that in rock matrix due to high fluid velocities the direction of anisotropy distorted the pressure field and changed the fluid flow directions which determined the time needed for the pollutant front to reach the fractures the parallel efficiency of the overall algorithm is also discussed and the experimental results show that it is deeply affected by the performance of the multigrid solver

time reversal symmetry in dynamical systems a survey abstract in this paper we survey the topic of time reversal symmetry in dynamical systems we begin with a brief discussion of the position of time reversal symmetry in physics after defining time reversal symmetry as it applies to dynamical systems we then introduce a major theme of our survey namely the relation of time reversible dynamical sytems to equivariant and hamiltonian dynamical systems we follow with a survey of the state of the art on the theory of reversible dynamical systems including results on symmetric periodic orbits local bifurcation theory homoclinic orbits and renormalization and scaling some areas of physics and mathematics in which reversible dynamical systems arise are discussed in an appendix we provide an extensive bibliography on the topic of time reversal symmetry in dynamical systems

interaction of steady and periodic bifurcating modes with imperfection effects in reaction diffusion systems asymptotic solutions for two dimensional reaction diffusion systems are obtained under suitable conditions a trivial solution independent of both time and space is assumed to exist and the linearized system about this solution is considered the analysis is restricted to the case where this linearized system possesses one real eigenvalue and a complex pair of eigenvalues all with real part approximately equal to zero steady periodic and doubly periodic bifurcating solutions are found with special attention given to the effects of imperfection in one of the parameters

stability and transitions of the second grade poiseuille flow abstract in this study we consider the stability and transitions for the poiseuille flow of a second grade fluid which is a model for non newtonian fluids we restrict our attention to perturbation flows in an infinite pipe with circular cross section that are independent of the axial coordinate we show that unlike the newtonian 0 case in the second grade model 0 case the time independent base flow exhibits transitions as the reynolds number r exceeds the critical threshold r c 8 505 1 2 where is a material constant measuring the relative strength of second order viscous effects compared to inertial effects at r r c we find that the transition is either continuous or catastrophic and a small amplitude time periodic flow with 3 fold azimuthal symmetry bifurcates the time period of the bifurcated solution tends to infinity as r tends to r c our numerical calculations suggest that for low values the system prefers a catastrophic transition where the bifurcation is subcritical we also show that there is a reynolds number r e with r e r c such that for r r e the base flow is globally stable and attracts any initial disturbance with at least exponential speed we show that the gap between r e and r c vanishes quickly as increases

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

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

non commutative banach function spaces in this paper we survey some aspects of the theory of non commutative banach function spaces that is spaces of measurable operators associated with a semi finite von neumann algebra these spaces are also known as non commutative symmetric spaces the theory of such spaces emerged as a common generalization of the theory of classical commutative rearrangement invariant banach function spaces in the sense of w a j luxemburg and a c zaanen and of the theory of symmetrically normed ideals of bounded linear operators in hilbert space in the sense of i c gohberg and m g krein these two cases may be considered as the two extremes of the theory in the first case the underlying von neumann algebra is the commutative algebra l on some measure space with integration as trace in the second case the underlying von neumann algebra is b the algebra of all bounded linear operators on a hilbert space with standard trace important special cases of these non commutative spaces are the non commutative l p spaces which correspond in the commutative case with the usual l p spaces on a measure space and in the setting of symmetrically normed operator ideals they correspond to the schatten p classes mathfrak s p