synthesis of phase locking patterns in networks with cyclic group symmetries a theoretical model for the circular arrays of neurons that are found in cortical barrel structures is presented if the connection stencil is permutation invariant then the network has cyclic group symmetries the properties of cyclic groups are used to formulate an algorithm that synthesizes coupling stencils that are shown using the rotation vector method to guarantee specified patterns of phase locking in both symmetric and asymmetric arrays such patterns are related to stable combinations of firing frequencies and to memory storage in neural networks

finite element multiple mode approach to nonlinear free vibrations of shallow shells two finite element fe modal formulations for large amplitude free vibration of isotropic and arbitrary laminated composite shallow shells are presented the system equation of motion is formulated first in the physical structural node degrees of freedom dof then the system is transformed into two distinctly different sets of general duffing type modal equations based on 1 modal amplitudes of coupled linear bending and in plane modes where in plane inertia is included in the formulation and 2 modal amplitudes of linear bending modes only where the in plane inertia is neglected multiple modes and the first order transverse shear deformation are considered in the formulations a shallow shell finite element is developed as an extension from the triangular mindlin min3 plate element with the improved shear correction factor by tessler time numerical integration is employed to determine the nonlinear periodic frequency characteristics the inaccuracy in characterizing a shallow shell response with coupled linear bending and in plane modes is demonstrated and discussed by comparing with the fe solution in structural node dof study cases include isotropic and composite panels of different shallow shell geometries

asymptotic formation and orbital stability of phase locked states for the kuramoto model abstract we discuss the asymptotic formation and nonlinear orbital stability of phase locked states arising from the ensemble of non identical kuramoto oscillators we provide an explicit lower bound for a coupling strength on the formation of phase locked states which only depends on the diameters of natural frequencies and initial phase configurations we show that when the phases of non identical oscillators are distributed over the half circle and the coupling strength is sufficiently large the dynamics of kuramoto oscillators exhibits two stages transition and relaxation stages in a transition stage initial configurations shrink to configurations whose diameters are strictly less than 2 in a finite time and then the configurations tend to phase locked states asymptotically this improves previous results on the formation of phase locked states by chopra spong 2009 26 and ha ha kim 2010 27 where their attention were focused only on the latter relaxation stage we also show that the kuramoto model is l 1 contractive in the sense that the l 1 distance along two smooth kuramoto flows is less than or equal to that of initial configurations in particular when two initial configurations have the same averaged phases the l 1 distance between them decays to zero exponentially fast for the configurations with different phase averages we use the method of average adjustment and translation invariant of the kuramoto model to show that one solution converges to the translation of the other solution exponentially fast this establishes the orbital stability of the phase locked states our stability analysis does not employ any standard linearization technique around the given phase locked states but instead we use a robust l 1 metric functional as a lyapunov functional in the formation process of phase locked states we estimate the number of collisions between oscillators and lower upper bounds of the transversal phase differences

geometrically nonlinear free vibrations of simply supported isotropic thin circular plates nonlinear free axisymmetric vibration of simply supported isotropic circular plates is investigated by using the energy method and a multimode approach in plane deformation is included in the formulation lagrange s equations are used to derive the governing equation of motion using the harmonic balance method the equation of motion is converted into a nonlinear algebraic form the numerical iterative method of solution adopted here is the so called linearized updated mode method which permits the authors to obtain accurate results for vibration amplitudes up to three times the plate thickness the percentage of participation of each out of plane basic function to the deflection shape and to the bending stress at the plate centre and of each in plane basic function to the membrane stress at the centre are calculated in order to determine the minimum number of in and out of plane basic functions to be used in order to achieve a good accuracy of the model the nonlinear frequency the nonlinear fundamental mode shape and their associated nonlinear bending and membrane stresses are determined at large amplitudes of vibration the numerical results obtained here are presented and compared with available published results based on various approaches and with the single mode solution the limit of validity of the single mode approach is also investigated

finite element method for nonlinear free vibrations of composite plates a multimode time domain modal formulation based on the finite element method for large amplitude free vibration of thin composite plates is presented accurate frequency maximum deflection relations can be predicted for the fundamental and the higher nonlinear modes a modal participation is defined and accurate and convergent frequencies can be determined with minimum number of linear modes a procedure for the selection of initial conditions for periodic plate response is presented convergence of frequency with gridwork refinement and number of linear modes is studied the classical single mode elliptic function frequency solutions for simply supported beams and square plates are assessed examples of orthotropic and composite plates are given and the characteristics of nonlinear response are studied

nonlinear vibration and dynamic stability analysis of composite plates large amplitude flexural vibration characteristics of composite plates under transverse harmonic pressure or periodic in plane load are investigated here using the shear deformable finite element method the nonlinear stiffness matrix is formulated based on von karman s assumptions to obtain the stiffness interaction between the in plane and bending degrees of freedom further the flexural motion of the plate is assumed to be harmonic and the in plane movement is assumed to be periodic the nonlinear matrix amplitude equation is obtained by employing galerkin s method the coupled nonlinear matrix amplitude equation in plane motion is coupled with flexural motion is solved to obtain 1 nonlinear free flexural vibration frequencies of isotropic and composite plates with different in plane boundary conditions 2 flexural vibration amplitudes of such plates under transverse harmonic pressure or periodic in plane load finally the time history analysis is carried out to understand the steady state or unsteady nature of the flexural vibration under different loading and boundary condition

computing hopf bifurcations ii three examples from neurophysiology in guckenheimer myers and sturmfels siam j numer anal 34 1997 we present algorithms for detecting hopf bifurcations in two parameter families of vector fields based on classical algebraic constructions in addition to their utility as augmented systems for use with standard newton type continuation methods they are shown to be particularly well adapted for solution by computer algebra techniques for vector fields of small or moderate dimension the present study examines the performance of these methods on test problems selected from models of current research interest in neurophysiology implementation issues are examined and the numerical properties of the proposed methods are compared with several alternative algorithms for hopf pathfollowing that appear in the literature

geometrical non linear steady state forced periodic vibration of plates part i model and convergence studies a model for the steady state geometrically non linear periodic vibration of thin rectangular plates under harmonic external excitation is presented the equations of motion in the time domain are derived by applying the principle of virtual work and the hierarchical finite element method hfem these equations are transformed into the frequency domain by the harmonic balance method hbm and are solved by a continuation method the convergence properties of the model are discussed by applying it to isotropic and to composite laminated plates

a combined modal finite element analysis technique for the dynamic response of a non linear beam to harmonic excitation abstract in this paper a method is proposed for modelling large deflection beam response involving multiple vibration modes significant savings in computational time can be obtained compared with the direct integration non linear finite element method the deflections from a number of static non linear finite element test cases are transformed into modal co ordinates using the modes of the underlying linear system regression analysis is then used to find the unknown coupled non linear modal stiffness coefficients the inclusion of finite element derived modal masses and an arbitrary damping model completes the governing non linear equations of motion the response of the beam to excitation of an arbitrary nature may then be found using time domain numerical integration of the reduced set of equations the work presented here extends upon the work of previous researchers to include non linearly coupled multi modal response the particular benefits of this approach are that no linearization is imposed and that almost any commercial finite element package may be employed without modification the proposed method is applied to the case of a homogeneous isotropic beam fully simply supported and fully clamped boundary conditions are considered for the free vibration case results are compared to those of previous researchers for the case of steady state harmonic excitation results are compared with the direct integration non linear finite element method using abaqus in all cases excellent agreement is obtained

non linear transient analysis of moderately thick laminated composite plates the non linear transient analysis of the shear deformable laminated composite plates subjected to step ramp and sinusoidal loading is presented the clamped simply supported free and their combinations non levy type of boundary conditions are considered the formulation is based on the mindlin first order shear deformation theory and von karman non linearity the methodology of the solution is based on the chebyshev series technique houbolt time marching scheme and quadratic extrapolation technique are used for the temporal discretization and linearization respectively the effects of magnitude and duration of loading rotary inertia in plane inertia b a a h lamination scheme and boundary conditions on the central displacement response are studied typical results are presented in dimensionless graphical forms for different parameters and loading conditions

the rotation number for almost periodic potentials we define and analyze the rotation number for the almost periodic schrodinger operatorl d2 dx2 q x we use the rotation number to discuss i the spectrum ofl ii its relation to the korteweg de vries equation

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

switching to nonhyperbolic cycles from codim 2 bifurcations of equilibria in odes the paper provides full algorithmic details on switching to the continuation of all possible codim 1 cycle bifurcations from generic codim 2 equilibrium bifurcation points in n dimensional odes we discuss the implementation and the performance of the algorithm in several examples including an extended lorenz 84 model and a laser system

a singular value inequality for block matrices abstract let a be an n n submatrix of a nonsingular n m n m matrix m and let s be the inverse of the schur complement of a in m let p min m n we obtain upper bounds on the p smallest singular values of a in terms of the corresponding ones in s and vice versa

a semi analytical approach to the non linear dynamic response problem of beams at large vibration amplitudes part ii multimode approach to the steady state forced periodic response abstract the semi analytical approach to the non linear dynamic response of beams based on multimode analysis has been presented in part i of this series of papers azrar et al 1999 journal of sound and vibration 224 183 207 1 the mathematical formulation of the problem and single mode analysis have been studied the objective of this paper is to take advantage of applying this semi analytical approach to the large amplitude forced vibrations of beams various types of excitation forces such as harmonic distributed and concentrated loads are considered the governing equation of motion is obtained and can be considered as a multi dimensional form of the duffing equation using the harmonic balance method the equation of motion is converted into non linear algebraic form techniques of solution based on iterative incremental procedures are presented the non linear frequency and the non linear modes are determined at large amplitudes of vibration the basic function contribution coefficients to the displacement response for various beam boundary conditions are calculated the percentage of participation for each mode in the response is presented in order to appraise the relation to higher modes contributing to the solution also the percentage contributions of the higher modes to the bending moment near to the clamps are given in order to determine accurately the error introduced in the non linear bending stress estimated by different approximations solutions obtained in the jump phenomena region have been determined by a careful selection of the initial iteration at each frequency the non linear deflection shapes in various regions of the solution the corresponding axial force ratios and the bending moments are presented in order to follow the behaviour of the beam at large vibration amplitudes the numerical results obtained here for the non linear forced response are compared with those from the linear theory with available non linear results based on various approaches and with the single mode analysis

defining functions for multiple hopf bifurcations let a u alpha f u u alpha u in r n alpha in r k be a family of real n times n matrices arising as the jacobian matrices of equilibrium solutions to the dynamical system stackrel u f u alpha an equilibrium point is called a hopf point if a has a conjugate pair of pure imaginary eigenvalues pm i omega omega 0 it is called a double hopf point if there are two such pairs pm i omega 1 pm i omega 2 and a 1 1 resonant double hopf point if in addition omega 1 omega 2 defining functions are obtained for the numerical detection computation and continuation of hopf double hopf and 1 1 resonant double hopf points they are based on a combination of matrix biproduct and bordered matrix methods example computations are done in a fairly realistic and complicated neural model problem with n 13 and k 29 however to make the methods applicable to large scale problems e g discretized boundary value problems we reduce the state space to a subspace that essentially contains the generalized eigenspaces of the eigenvalues with largest real part

large amplitude finite element flexural vibration of plates stiffened plates large amplitude free flexural vibration of stiffened and unstiffened plates has been studied by using the finite element method an isoparametric quadratic plate bending element has been used both for the plate and the stiffener the dynamic version of von karman s field equations has been adopted and the formulation has been done in the total lagrangian coordinate system the in plane deformation and inertia have been taken into account the resulting nonlinear equations have been solved by the direct iteration technique using a linear mode shape as the starting vector the stiffener has been elegantly modeled so that it can be placed anywhere within the plate element and it need not follow the nodal lines this has increased the flexibility of the mesh generation considerably the arbitrary orientation and eccentricity of the stiffener have been incorporated in the formulation the shear deformation has been incorporated according to mindlin s hypothesis stiffened and unstiffened plates that have various boundary conditions have been analyzed and the results have been compared with those available in the literature

non linear finite element modal approach for the large amplitude free vibration of symmetric and unsymmetric composite plates a theoretical analysis is presented for the large amplitude vibration of symmetric and unsymmetric composite plates using the non linear finite element modal reduction method the problem is first reduced to a set of duffing type modal equations using the finite element modal reduction method the main advantage of the proposed approach is that no updating of the non linear stiffness matrices is needed without loss of generality accurate frequency ratios for the fundamental mode and the higher modes of a composite plate at various values of maximum deflection are then determined by using the runge kutta numerical integration scheme the procedure for obtaining proper initial conditions for the periodic plate motions is very time consuming thus an alternative scheme the harmonic balance method is adopted and assessed as it was employed to formulate the large amplitude free vibration of beams in a previous study and the results agreed well with the elliptic solution the numerical results that are obtained with the harmonic balance method agree reasonably well with those obtained with the runge kutta method the contribution of each linear mode to the maximum deflection of a plate can also be obtained the frequency ratios for isotropic and composite plates at various maximum deflections are presented and convergence of frequencies with the number of finite elements number of linear modes and number of harmonic terms is also studied copyright 2005 john wiley sons ltd

continuation techniques and interactive software for bifurcation analysis of odes and iterated maps physics abstract we present a numerical technique for the analysis of local bifurcations which is based on the continuation of structurally unstable invariant sets in a suitable phase parameter space the invariant sets involved in our study are equilibrium points and limit cycles of autonomous odes periodic solutions of time periodic nonautonomous odes fixed points and periodic orbits of iterated maps the more general concept of a continuation strategy is also discussed it allows the analysis of various singularities of generic systems and of their mutual relationships the approach is extended to codimension three singularities we introduce several bifurcation functions and show how to use them to construct well posed continuation problems the described continuation technique is supported by an interactive graphical program called locbif we discuss briefly the concepts of the locbif interface and give some examples of typical applications

stability of phase locking in a ring of unidirectionally coupled oscillators we discuss the dynamic behaviour of a finite group of phase oscillators unidirectionally coupled in a ring the dynamics are based on the kuramoto model in the case of identical oscillators all phase locking solutions and their stability properties are obtained for nonidentical oscillators it is proven that there exist phase locking solutions for sufficiently strong coupling an algorithm to obtain all phase locking solutions is proposed these solutions can be classified into classes each with its own stability properties the stability properties are obtained by means of a novel extension of gershgorin s theorem one class of stable solutions has the property that all phase differences between neighbouring cells are contained in contrary to intuition a second class of stable solutions is established with exactly one of the phase differences contained in the stability results are extended from sinusoidal interconnections to a class of odd functions to conclude a connection with the field of active antenna arrays is made generalizing some results earlier obtained in this field

non linear forced vibrations of plates by an asymptotic numerical method abstract non linear forced vibrations of thin elastic plates have been investigated by an asymptotic numerical method anm various types of harmonic excitation forces such as distributed and concentrated are considered using the harmonic balance method and hamilton s principle the equation of motion is converted into an operational formulation based on the finite element method a starting point corresponding to a non linear solution associated to a given frequency and amplitude of excitation is computed applying perturbation techniques in the vicinity of this solution the non linear governing equation obtained is transformed into a sequence of linear problems having the same stiffness matrix employing one matrix inversion a large number of terms of the perturbation series of the displacement and frequency can be easily computed with a small computation time iterations of this method lead to a powerful path following technique comprehensive numerical tests for forced vibrations of plates subjected to time harmonic lateral excitations are reported

matrix manifolds and the jordan structure of the bialternate matrix product the bialternate product of matrices was introduced at the end of the 19th century and recently revived as a computational tool in problems where real matrices with conjugate pairs of pure imaginary eigenvalues are important i e in stability theory and hopf bifurcation problems we give a complete description of the jordan structure of the bialternate product 2a in of an n n matrix a thus extending several partial results in the literature we use these results to obtain regular local defining systems for some manifolds of matrices which occur naturally in applications in particular for manifolds with resonant conjugate pairs of pure imaginary eigenvalues such defining systems can be used analytically to obtain local parameterizations of the manifolds or numerically to set up newton systems with local quadratic convergence we give references to explicit numerical applications and implementations in software we expect that the analysis provided in this paper can be used to further improve such implementations

computation of cusp singularities for operator equations and their discretizations we discuss the direct calculation of cusp singularities as solutions of a minimally augmented defining system which is nonsingular under the canonical cusp conditions the underlying operator equation may have any finite fredholm index and bounds on the discretization error are derived for the case of projection methods

computation of takens bogdanov type bifurcations with arbitrary codimension let dot u f u alpha u 0 u 0 describe a dynamical system where u in r n is the state variable alpha in r k is a parameter variable and f u alpha takes values in r n here n can be large but k is usually small say leqslant 10 a solution u alpha to the equilibrium equations f u alpha 0 is a takens bogdanov type bifurcation of codimension m if the jacobian f u u alpha contains an m dimensional jordan block corresponding to a zero eigenvalue and has no other eigenvalues equal to zero this includes the cases of turning point bifurcation m 1 takens bogdanov bifurcation m 2 and triple point bifurcation m 3 a unified direct method is proposed based on ideas of a griewank and g w reddien in the case of takens bogdanov points for both the detection and computation of parameter values a at which thkens bogdanov type bifurcations of any given codimension occur as an application we compute tkens bogdanov type bifurcations with codi