quadratic systems with center and their perturbations abstract we study the quadratic systems on a plane z i z az2 b z 2 cz2 z x iy we give a simple algebraic proof of the center conditions 0 and b 0 or a 1 2 b 1 or a b 1 c c or a 2 b 1 c 1 and of the theorem of bautin that the number of small amplitude limit cycles bifurcating from a center or a focus is not greater than 3 we present the bifurcational diagrams and phase portraits of the systems with center and describe the cyclicity precisely in each case here one mistake in bautin s work is revealed and some questions concerning focus numbers are answered we study also global not small limit cycles for a small perturbation of the center case with two invariant lines the lotka volterra system using abelian integrals we show that the number of zeroes of the corresponding abelian integral is 0 1 or 2 some attention is devoted to the problem of limit cycles for a small perturbation of a system with a two fold center when two center conditions hold the problem turns out to be equivalent to the problem of zeroes of some integral with polynomial coefficients

application of polynomial chaos in stability and control the polynomial chaos of wiener provides a framework for the statistical analysis of dynamical systems with computational cost far superior to monte carlo simulations it is a useful tool for control systems analysis because it allows probabilistic description of the effects of uncertainty especially in systems having nonlinearities and where other techniques such as lyapunov s method may fail we show that stability of a system can be inferred from the evolution of modal amplitudes covering nearly the full support of the uncertain parameters with a finite series by casting uncertain parameters as unknown gains we show that the separation of stochastic from deterministic elements in the response points to fast iterative design methods for nonlinear control

nodal solutions of boundary value problems of fourth order ordinary differential equations abstract we study the existence of nodal solutions of the fourth order two point boundary value problem y t y a t f y 0 t 1 y 0 y 1 y 0 y 1 0 where c 0 1 with t 2 on 0 1 a c 0 1 with a 0 on 0 1 and a t 0 on any subinterval of 0 1 f c r satisfies f u u 0 for all u 0 we give conditions on the ratio f s s at infinity and zero that guarantee the existence of nodal solutions the proof of our main results is based upon bifurcation techniques

boundary controllability of a hybrid system consisting in two flexible beams connected by a point mass we consider a hybrid system consisting of two flexible beams connected by a point mass the constant of rotational inertia is assumed to be nonzero in a previous paper we have proved that in the presence of the point mass the system is well posed in asymmetric spaces in which solutions have one more degree of regularity to one side of the mass we are interested in the problem of controllability when the control acts on the free extreme of one of the beams we prove that when the control time is large enough the system is exactly controllable in an asymmetric space this result is sharp the proofs combine classical techniques from asymptotic analysis and the theory of nonharmonic fourier series

a preparation theorem for codimension one foliations after gluing foliated complex manifolds we derive a preparation like theorem for singularities of codimension one foliations and planar vector fields in the real or complex setting without computation we retrieve and improve results of levinson moser for functions dufour zhitomirskii for nondegenerate codimension one foliations proving in turn the analyticity str oyzyna y zo ladek for non degenerate planar vector fields and bruno ecalle for saddle node foliations in the plane

some properties of eigenfunctions for the equation of vibrating beam with a spectral parameter in the boundary conditions abstract in this paper we consider a spectral problem for ordinary differential equations of fourth order with the spectral parameter contained in three of the boundary conditions we study the oscillatory properties of the eigenfunctions and using these properties we obtain sufficient conditions for the system of eigenfunctions of the problem in question to form a basis in the space l p 0 1 1 p after removing three functions

canard solutions at non generic turning points this paper deals with singular perturbation problems for vector fields on 2 dimensional manifolds canard solutions are solutions that starting near an attracting normally hyperbolic branch of the singular curve cross a turning point and follow for a while a normally repelling branch of the singular curve following the geometric ideas developed by dumortier and roussarie in 1996 for the study of canard solutions near a generic turning point we study canard solutions near non generic turning points characterization of manifolds of canard solutions is given in terms of boundary conditions their regularity properties are studied and the relation is described with the more traditional asymptotic approach it reveals that interesting information on canard solutions can be obtained even in cases where an asymptotic approach fails to work since the manifolds of canard solutions occur as intersection of center manifolds defined along respectively the attracting and the repelling branch of the singular curve we also study their contact and its relation to the control curve

null boundary controllability of a one dimensional heat equation with an internal point mass and variable coefficients in this paper we consider a linear hybrid system which is composed by two non homogeneous rods connected by a point mass with dirichlet boundary conditions on the left end and a boundary control acts on the right end we prove that this system is null controllable with dirichlet or neumann boundary controls our approach is mainly based on a detailed spectral analysis together with the moment method in particular we show that the associated spectral gap in both cases dirichlet or neumann boundary controls is positive without further conditions on the coefficients other than the regularities

multi resolution analysis of wiener type uncertainty propagation schemes a multi resolution analysis mra is applied to an uncertainty propagation scheme based on a generalized polynomial chaos pc representation the mra relies on an orthogonal projection of uncertain data and solution variables onto a multi wavelet basis consisting of compact piecewise smooth polynomial functions the coefficients of the expansion are computed through a galerkin procedure the mra scheme is applied to the simulation of the lorenz system having a single random parameter the convergence of the solution with respect to the resolution level and expansion order is investigated in particular results are compared to two monte carlo sampling strategies demonstrating the superiority of the mra for more complex problems however the mra approach may require excessive cpu times adaptive methods are consequently developed in order to overcome this drawback two approaches are explored the first is based on adaptive refinement of the multi wavelet basis while the second is based on adaptive block partitioning of the space of random variables computational tests indicate that the latter approach is better suited for large problems leading to a more efficient flexible and parallelizable scheme

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

control of a network of euler bernoulli beams abstract the aim is to study the boundary controllability of a system modeling the vibrations of a network of n euler bernoulli beams connected by n vibrating point masses using the classical hilbert uniqueness method the control problem is reduced to the obtention of an observability inequality the solution is then expressed in terms of fourier series so that it is also enough to show that the distance between two consecutive large eigenvalues of the spatial operator involved in this evolution problem is superior to a minimal fixed value this property called spectral gap holds as soon as the roots of a function denoted by f and giving the asymptotic behaviour of the eigenvalues are all simple for a network of n 2 different beams this assumption on the multiplicity of the roots of f denoted by a is proved to be satisfied and controllability follows for higher values of n a numerical approach allows one to prove a in many situations and no counterexample has been found but the problem of giving a general proof of controllability remains open

nodal solutions of a nonlinear eigenvalue problem of the euler bernoulli equation abstract we are concerned with determining values of r for which there exist nodal solutions of the boundary value problems of the euler bernoulli equation p x u q x u r f x u u 0 x 0 l u 0 cos p u 0 sin 0 u 0 0 u l cos p u l sin 0 u l 0 where 0 2 p c 2 0 l and q c 1 0 l the proof of our main result is based upon bifurcation techniques

stochastic model reduction for chaos representations abstract this paper addresses issues of model reduction of stochastic representations and computational efficiency of spectral stochastic galerkin schemes for the solution of partial differential equations with stochastic coefficients in particular an algorithm is developed for the efficient characterization of a lower dimensional manifold occupied by the solution to a stochastic partial differential equation spde in the hilbert space spanned by wiener chaos a description of the stochastic aspect of the problem on two well separated scales is developed to enable the stochastic characterization on the fine scale using algebraic operations on the coarse scale with such algorithms at hand the solution of spde s becomes both computationally manageable and efficient moreover a solid foundation is thus provided for the adaptive error control in stochastic galerkin procedures different aspects of the proposed methodology are clarified through its application to an example problem from solid mechanics

on the simplicity of eigenvalues of two nonhomogeneous euler bernoulli beams connected by a point mass abstract in this paper we consider a linear system modeling the vibrations of two nonhomogeneous euler bernoulli beams connected by a point mass this system is generated by the following equations x y t t t x x y x x t x x x q x y x t x x 0 t 0 x 1 0 0 1 m y t t t 0 t y t x x 0 t y t x x 0 t 0 with hinged boundary conditions at both ends where m 0 x 0 x 0 q x 0 and t y x y x x x q x y x for x 1 0 0 1 we prove that all the associated eigenvalues n n 1 are algebraically simple furthermore the corresponding eigenfunctions n n 1 satisfy n 1 t n 1 0 and n 1 t n 1 0 for all n 1 these results are essential for the proof of the exact boundary controllability problem related to this system by means of the non harmonic fourier series method

riesz basis approach to the stabilization of a flexible beam with a tip mass using an abstract condition of riesz basis generation of discrete operators in the hilbert spaces we show in this paper that a sequence of generalized eigenfunctions of an euler bernoulli beam equation with a tip mass under boundary linear feedback control forms a riesz basis for the state hilbert space in the meanwhile an asymptotic expression of eigenvalues and the exponential stability are readily obtained the main results of siam j control optim 36 1998 pp 1962 1986 are concluded as a special case and the additional conditions imposed there are removed

slow fast bogdanov takens bifurcations abstract in this paper we study perturbations from planar vector fields having a line of zeros and representing a singular limit of bogdanov takens bt bifurcations we introduce among other precise definitions the notion of slow fast bt bifurcation and we provide a complete study of the bifurcation diagram and the related phase portraits based on geometric singular perturbation theory including blow up we get results that are valid on a uniform neighborhood both in parameter space and in the phase plane

time analysis and entry exit relation near planar turning points abstract the paper deals with canard solutions at very general turning points of smooth singular perturbation problems in two dimensions we follow a geometric approach based on the use of c k normal forms centre manifolds and family blow up as we did in trans amer math soc to appear in trans amer math soc to appear we considered the existence of manifolds of canard solutions for given appropriate boundary conditions these manifolds need not be smooth at the turning point in this paper we essentially study the transition time along such manifolds as well as the divergence integral providing a structure theorem for these integrals as a consequence we get a nice structure theorem for the transition equation governing the canard solutions it permits to compare different control manifolds and to obtain a precise description of the entry exit relation of different canard solutions attention is also given to the special case in which the canard manifolds are smooth i e when formal canard solutions exist

oscillation properties for the equation of vibrating beam with irregular boundary conditions abstract in this paper we study the oscillatory properties for the eigenfunctions of some fourth order eigenvalue problems where the boundary conditions are irregular in the sense of the classification of s janczewski oscillation theorems for the differential boundary value problems of the fourth order ann of math 29 1928 521 542 in this case we show that these oscillatory properties are different from those of the sturm liouville problem

sturm theory for the equation of vibrating beam abstract in this paper we develop an extension of the classical sturm theory c sturm sur une classe d equations a derivee partielle j math pures appl 1 1836 373 444 to study the oscillation properties for the eigenfunctions of some fourth order two point boundary value problems on the interval 0 1 we are mainly interested in the case when these problems have negative eigenvalues induced by the sign of the parameters in the boundary conditions in particular we give an asymptotic estimate of the number of zeros in 0 1 of the first eigenfunction in terms of the variation of parameters in the boundary conditions

cyclicity of a fake saddle inside the quadratic vector fields abstract this paper concerns the study of small amplitude limit cycles that appear in the phase portrait near an unfolded fake saddle singularity this degenerate singularity is also known as an impassable grain the canonical form of the unperturbed vector field is like a degenerate flow box near the singularity the phase portrait consists of parallel fibers all but one of which have no singular points and at the singular fiber there is one node we demonstrate different techniques in order to show that the cyclicity is bigger than or equal to two when the canonical form is quadratic

ingredients for a general purpose stochastic finite elements implementation abstract the stochastic finite element method has seen a number of incarnations over the past few years ranging from perturbation based methods and neumann expansion based methods to methods for evaluating bounds on the response variability and more recent functional expansions combined with error minimization techniques of these only the last class of methods provides an explicit expression to the solution of the problem as opposed to evaluating a global norm of this solution and can thus be viewed as a rational extension of concepts developed for the deterministic finite element method the main obstacle in developing a substantial user community of various stochastic finite element methods has been the lack of a general purpose formalism for addressing issues of general engineering interest this paper presents the main ingredients for developing a general purpose version of the spectral stochastic finite element method

limit cycles in slow fast codimension 3 saddle and elliptic bifurcations abstract this paper deals with local bifurcations occurring near singular points of planar slow fast systems in particular it is concerned with the study of the slow fast variant of the unfolding of a codimension 3 nilpotent singularity the slow fast variant of a codimension 1 hopf bifurcation has been studied extensively before and its study has lead to the notion of canard cycles in the van der pol system similarly codimension 2 slow fast bogdanov takens bifurcations have been characterized here the singularity is of codimension 3 and we distinguish slow fast elliptic and slow fast saddle bifurcations we focus our study on the appearance on small amplitude limit cycles and rely on techniques from geometric singular perturbation theory and blow up

polynomial chaos expansions for optimal control of nonlinear random oscillators the polynomial chaos decomposition of stochastic variables and processes is implemented in conjunction with optimal polynomial control of nonlinear dynamical systems the procedure is demonstrated on a base excited system whereby ground motion is modeled as a stochastic process with a specified correlation function and is approximated by its karhunen loeve expansion an adaptive scheme for stochastic approximation with polynomial chaos bases is proposed which is based on a displacement velocity norm and is applied to the identification of phase orbits of nonlinear oscillators this approximation is then integrated in the design of an optimal polynomial controller allowing for the efficient estimation of statistics and probability density functions of quantities of interest numerical investigations are carried out employing the polynomial chaos expansions and the lyapunov asymptotic stability condition based control policy the results reveal that the performance as gaged by probabilistic quantities of interest of the controlled oscillators is greatly improved a comparative study is also presented against the classical stochastic optimal control whereby statistical linearization based lqg is employed to design the optimal controller it is remarked that the proposed polynomial chaos expansion is a preferred approach to the optimal control of nonlinear random oscillators

polynomial chaos representation of spatio temporal random fields from experimental measurements two numerical techniques are proposed to construct a polynomial chaos pc representation of an arbitrary second order random vector in the first approach a pc representation is constructed by matching a target joint probability density function pdf based on sequential conditioning a sequence of conditional probability relations in conjunction with the rosenblatt transformation in the second approach the pc representation is obtained by having recourse to the rosenblatt transformation and simultaneously matching a set of target marginal pdfs and target spearman s rank correlation coefficient srcc matrix both techniques are applied to model an experimental spatio temporal data set exhibiting strong non stationary and non gaussian features the data consists of a set of oceanographic temperature records obtained from a shallow water acoustics transmission experiment 1 the measurement data observed over a finite denumerable subset of the indexing set of the random process is treated as a collection of observed samples of a second order random vector that can be treated as a finite dimensional approximation of the original random field a set of properly ordered conditional pdfs that uniquely characterizes the target joint pdf in the first approach and a set of target marginal pdfs and a target srcc matrix in the second approach are estimated from available experimental data digital realizations sampled from the constructed pc representations based on both schemes capture the observed statistical characteristics of the experimental data with sufficient accuracy the relative advantages and disadvantages of the two proposed techniques are also highlighted

the wiener askey polynomial chaos for stochastic differential equations we present a new method for solving stochastic differential equations based on galerkin projections and extensions of wiener s polynomial chaos specifically we represent the stochastic processes with an optimum trial basis from the askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error several continuous and discrete processes are treated and numerical examples show substantial speed up compared to monte carlo simulations for low dimensional stochastic inputs

spectral parameter power series for fourth order sturm liouville problems a general solution of the fourth order sturm liouville equation is presented in the form of a spectral parameter power series spps the uniform convergence of the series is proved and the coefficients of the series are calculated explicitly through a recursive intergration procedure based on the spps representation characteristic equations for spectral problems arising in mechanics and elasticity theory are obtained and it is shown that the spectral problems reduce to computation of zeros of corresponding analytic functions of the spectral parameter given by their taylor series expansions this leads to a simple and efficient numerical method for solving the spectral problems for fourth order sturm liouville equations several examples of application are discussed

dimension reduction of karhunen loeve expansion for simulation of stochastic processes abstract conventional karhunen loeve expansions for simulation of stochastic processes often encounter the challenge of dealing with hundreds of random variables for breaking through the barrier a random function embedded karhunen loeve expansion method is proposed in this paper the updated scheme has a similar form to the conventional karhunen loeve expansion both involving a summation of a series of deterministic orthonormal basis and uncorrelated random variables while the difference from the updated scheme lies in the dimension reduction of karhunen loeve expansion through introducing random functions as a conditional constraint upon uncorrelated random variables the random function is expressed as a single elementary random variable orthogonal function in polynomial format non gaussian variables or trigonometric format non gaussian and gaussian variables for illustrative purposes the simulation of seismic ground motion is carried out using the updated scheme numerical investigations reveal that the karhunen loeve expansion with random functions could gain desirable simulation results in case of a moderate sample number except the hermite polynomials and the laguerre polynomials it has the sound applicability and efficiency in simulation of stochastic processes besides the updated scheme has the benefit of integrating with probability density evolution method readily for the stochastic analysis of nonlinear structures

exact controllability and stabilization of a vibrating string with an interior point mass in this article we examine the problems of boundary control and stabilization for a one dimensional wave equation with interior point masses we show that singularities in waves are smoothed one order as they cross a point mass thus in the case of one interior point mass with e g l 2 dirichlet control at the left end the most general reachable space from 0 that one can expect is l 2 times h 1 to the left of the mass and h 1 times l 2 to the right of the mass we show that this is in fact the optimal result modulo certain compatibility conditions several related results for both control and stabilization of such systems are also given

normal forms near a symmetric planar saddle connection abstract the present paper studies vector fields of the form x q 2 o 1 x 2 1 x 2 o y y p x o 1 x 2 y o y 2 which contain a separatrix connection between hyperbolic saddles with opposite eigenvalues where the connection is fixed smooth semi local normal forms are provided in vicinity of the connection both in the resonant and non resonant case first a formal conjugacy is constructed near the separatrix then a smooth change of coordinates is realized by generalizing known local results near the hyperbolic points

normal linearization and transition map near a saddle connection with symmetric resonances abstract we consider a heteroclinic connection in a planar system between two symmetric hyperbolic saddles of which the eigenvalues are resonant starting with a c normal form defined globally near the connection we normally linearize the vector field by using finitely smooth tags of logarithmic form we furthermore define an asymptotic entry exit relation and we discuss on two examples how to deal with counting limit cycles near a limit periodic set involving such a connection

canard cycles in the presence of slow dynamics with singularities we study the cyclicity of limit periodic sets that occur in families of vector flelds of slow fast type the limit periodic sets are formed by a fast orbit and a curve of singularities containing a unique turning point at this turning point a stability change takes place on one side of the turning point the dynamics point strongly towards the curve of singularities on the other side the dynamics point away from the curve of singularities the presence of periodic orbits in a perturbation is related to the presence of canard orbits passing near this turning point i e orbits that stay close to the curve of singularities despite the exponentially strong repulsion near this curve all existing results deal with a non zero slow movement permitting to get a good estimate of the cyclicity by considering the slow divergence integral along the curve of singularities in this paper we study what happens when the slow dynamics exhibit singularities in particular our study includes the cyclicity of the slow fast 2 saddle cycle formed by a regular saddle connection the fast part and a part of the curve of singularities the slow part we see that the relevant information is no longer merely contained in the slow divergence integral this paper concerns the study of the cyclicity of limit periodic sets in a quite general class of slow fast vector flelds on a 2 manifold m we are interested in families of vector flelds x possibly depending on other parameters as well where the unperturbed fast vector fleld x0 has a curve of singular points called a critical curve we call a point p on normally attracting resp normally repelling when dx0 p has a strictly negative resp strictly positive eigenvalue corresponding to an eigendirection not tangent to when has both normally attracting points and normally repelling points it may occur that x0 has orbits connecting two such points let f be such a fast orbit so that the limit and fi limit lie on we study the limit periodic set lf formed by f and the piece of going from the limit of f to the fi limit of f of course the nonzero eigenvalue of dx0 p must bifurcate along this piece of assume that this happens in a unique point p called a turning point one also says normal hyperbolicity of x0 is lost in p in such a situation it is possible that the limit periodic set perturbs into one or more isolated periodic orbits such orbits are called canard cycles

stochastic linear quadratic regulators with indefinite control weight costs ii in part i of this paper s chen x li and x zhou siam j control optim 36 1998 pp 1685 1702 an optimization model of stochastic linear quadratic regulators lqrs with indefinite control cost weighting matrices is proposed and studied in this sequel the problem of solving lqr models with system diffusions dependent on both state and control variables which is left open in part i is tackled first the solvability of the associated stochastic riccati equations sres is studied in the normal case namely all the state and control weighting matrices and the terminal matrix in the cost functional are nonnegative definite with at least one positive definite which in turn leads to an optimal state feedback control of the lqr problem in the general indefinite case the problem is decomposed into two optimal lqr problems one with a forward dynamics and the other with a backward dynamics the well posedness and solvability of the original lqr problem are then obtained by solving these two subproblems and an optimal control is explicitly constructed examples are presented to illustrate the results

stochastic linear quadratic regulators with indefinite control weight costs this paper considers optimal minimizing control of stochastic linear quadratic regulators lqrs the assumption that the control weight costs must be positive definite inherited from the determ

properties of natural frequencies and harmonic bending vibrations of a rod at one end of which is concentrated inertial load abstract in this paper we consider a spectral problem that describes the bending vibrations of a homogeneous rod in cross sections of which the longitudinal force acts the left end of which is fixed and on the right end an inertial mass is concentrated we give a general characteristic of the location of the eigenvalues on the real axis we study the structure of root spaces and oscillation properties of eigenfunctions we investigate the basic properties in the space l p 1 p of the system of eigenfunctions of this problem

versal unfoldings of predator prey systems with ratio dependent functional response abstract in this paper we study the versal unfolding of a predator prey system with ratio dependent functional response near a degenerate equilibrium in order to obtain all possible phase portraits for its perturbations we first construct the unfolding and prove its versality and degeneracy of codimension 2 then we discuss all its possible bifurcations including transcritical bifurcation hopf bifurcation and heteroclinic bifurcation give conditions of parameters for the appearance of closed orbits and heteroclinic loops and describe the bifurcation curves phase portraits for all possible cases are presented

exact controllability for string with attached masses we consider the problem of boundary control for a vibrating string with n interior point masses we assume the control is at the left end and the string is fixed at the right end singularities in waves are smoothed out to one order as they cross a point mass we characterize the reachable set for a l 2 control the control problem is reduced to a moment problem which is then solved using the theory of exponential divided differences in tandem with unique shape and velocity controllability results

hilbert s 16th problem for quadratic vector fields abstract we discuss here a systematic approach towards a positive answer to hilbert s 16th problem for quadratic systems namely the existence of a uniform bound for the number of limit cycles of a quadratic system the method is the following describe the limit periodic sets surrounding the origin in a family of quadratic vector fields and prove that they have finite cyclicity in this paper we give the list of all graphics and degenerate graphics that should be considered and describe their general features we also indicate how to find or where to find concrete examples of these limit periodic sets

regularity and exact controllability for a beam with piezoelectric actuator we consider an initial and boundary value problem modelling the vibrations of a bernoulli euler beam with an attached piezoelectric actuator we show that the sobolev regularity of the solution is by frac 1 2 epsilon higher than that one obtains by simply using the sobolev regularity of the control term the main results concern the dependence of the space of exactly controllable initial data on the location of the actuator our approach is based on the hilbert uniqueness method introduced by lions em contr olabilit e exacte des syst emes distribu es masson paris 1988 combined with some results from the theory of diophantine approximation

sturm liouville problems with boundary conditions rationally dependent on the eigenparameter ii necessary and sufficient conditions are given for two sequences n and n to be the eigenvalues and norming constants of a sturm liouville boundary value problem y qy y y 0 cos y 0 sin and y 1 f y 1 where f is a rational function of herglotz nevanlinna type it is also proved that q and f are uniquely determined by the sequences n and n