random matrix eigenvalue problems in structural dynamics natural frequencies and mode shapes play a fundamental role in the dynamic characteristics of linear structural systems considering that the system parameters are known only probabilistically we obtain the moments and the probability density functions of the eigenvalues of discrete linear stochastic dynamic systems current methods to deal with such problems are dominated by mean centred perturbation based methods here two new approaches are proposed the first approach is based on a perturbation expansion of the eigenvalues about an optimal point which is best in some sense the second approach is based on an asymptotic approximation of multidimensional integrals a closed form expression is derived for a general rth order moment of the eigenvalues two approaches are presented to obtain the probability density functions of the eigenvalues the first is based on the maximum entropy method and the second is based on a chi square distribution both approaches result in simple closed form expressions which can be easily calculated the proposed methods are applied to two problems and the analytical results are compared with monte carlo simulations it is expected that the small randomness assumption usually employed in mean centred perturbation based methods can be relaxed considerably using these methods

numerical normal forms for codim 2 bifurcations of fixed points with at most two critical eigenvalues in this paper we derive explicit formulas for the normal form coefficients to verify the nondegeneracy of eight codimension two bifurcations of fixed points with one or two critical eigenvalues these include all strong resonances as well as the degenerate flip and neimark sacker bifurcations applying our results to n dimensional maps one avoids the computation of the center manifold but one can directly evaluate the critical normal form coefficients in the original basis the formulas remain valid also for n 2 and allow one to avoid the transformation of the linear part of the map into jordan form the developed techniques are tested on two examples 1 a three dimensional map appearing in adaptive control 2 a periodically forced epidemic model we compute numerically the critical normal form coefficients for several codim 2 bifurcations occurring in these models to compute the required derivatives of the poincare map for the epidemic model the automatic differentiation package adol c is used

stable cycles in a cournot duopoly model of kopel we consider a discrete map proposed by m kopel that models a nonlinear cournot duopoly consisting of a market structure between the two opposite cases of monopoly and competition the stability of the fixed points of the discrete dynamical system is analyzed synchronization of two dynamics parameters of the cournot duopoly is considered in the computation of stability boundaries formed by parts of codim 1 bifurcation curves we discover more on the dynamics of the map by computing numerically the critical normal form coefficients of all codim 1 and codim 2 bifurcation points and computing the associated two parameter codim 1 curves rooted in some codim 2 points it enables us to compute the stability domains of the low order iterates of the map we concentrate in particular on the second third and fourth iterates and their relation to the period doubling 1 3 and 1 4 resonant neimark sacker points

a remark on chaotic behavior in adaptive control systems this note deals with the control of a linear discrete time system designed on the assumptions that we do not have explicit knowledge of the order of the plant the objective is to cause the output to track a given constant reference y when the initial conditions are in a neighborhood of y in previous papers the presence of chaos was conjectured for an adaptive scheme where the plant to be controlled was a second order one undermodeled by a first order one we show that with an adequate choice of the parameter estimator local regulation with exponential convergence of both the output and the parameter is obtained

multiparametric bifurcation analysis of a basic two stage population model in this paper we investigate long term dynamics of the most basic model for stage structured populations in which the per capita transition from the juvenile into the adult class is density dependent the model is represented by an autonomous system of two nonlinear differential equations with four parameters for a single population we find that the interaction of intra adult competition and intra juvenile competition gives rise to multiple attractors one of which can be oscillatory a detailed numerical study reveals a rich bifurcation structure for this two dimensional system originating from a degenerate bogdanov takens bt bifurcation point when one parameter is kept constant depending on the value of this fixed parameter the corresponding triple critical equilibrium has either an elliptic sector or it is a topological focus which is demonstrated by the numerical normal form analysis it is shown that the canonical unfolding of the codimension three bt point reveals the underlying dynamics of

computational procedure for a fast calculation of eigenvectors and eigenvalues of structures with random properties abstract the inherent uncertainties in geometry material properties etc of engineering structures can be represented by stochastic models where the parameters are described by probabilistic laws results from any analysis based on stochastic models inherit probabilistic information as well which can be used e g for reliability analysis particularly in linear dynamics of structures the calculation and analysis of random eigenvalues and eigenvectors is crucial a very flexible however computationally intensive way to analyze such systems is direct monte carlo simulation mcs in this paper procedures are shown which allow a significant reduction of computational efforts of the simulation using a subspace iteration scheme with optimally selected start vectors the procedures are not restricted to any kind of sampling technique

performance of the taylor series method for odes daes this paper revisits the use of the taylor series method for the numerical integration of odes and daes the numerical method is implemented using an efficient variable step variable order scheme several numerical tests comparing with well established numerical codes are presented

on the c8 3 regularisation of simultaneous binary collisions in the collinear 4 body problem abstract a conjecture on the nature of the simultaneous binary collision is explored in the collinear 4 body problem it is known that any attempt to remove the singularity via block regularisation will result in a regularised flow that is no more than c 8 3 differentiable with respect to initial conditions through a blow up of the singularity this loss of differentiability is investigated and a new proof of the c 8 3 regularity is provided in the process it is revealed that the collision manifold consists of two manifolds of normally hyperbolic saddle singularities which are connected by a manifold of heteroclinics by utilising recent work on transitions near such objects and their normal forms an asymptotic series of the transition past the singularity is explicitly computed it becomes remarkably apparent that the finite differentiability at 8 3 is due to the inability to construct a set of integrals local to the simultaneous binary collision the finite differentiability is shown to be independent of the value of the masses

the lindstedt poincare technique as an algorithm for computing periodic orbits the lindstedt poincare technique in perturbation theory is used to calculate periodic orbits of perturbed differential equations it uses a nearby periodic orbit of the unperturbed differential equation as the first approximation we derive a numerical algorithm based upon this technique for computing periodic orbits of dynamical systems the algorithm unlike the lindstedt poincare technique does not require the dynamical system to be a small perturbation of a solvable differential equation this makes it more broadly applicable the algorithm is quadratically convergent it works with equal facility as examples show irrespective of whether the periodic orbit is attracting or repelling or a saddle one of the examples presents what is possibly the most accurate computation of hill s orbit of lunation since its justly celebrated discovery in 1878

numerical continuation of bifurcations of limit cycles in matlab rm cl matcont and matcont are matlab continuation packages for the interactive numerical study of a range of parameterized nonlinear dynamical systems in particular odes matcont is an interactive graphical package and rm cl matcont is a commandline version both packages allow us to compute curves of equilibria limit points hopf points limit cycles flip fold and torus bifurcation points of limit cycles we discuss computational details of the continuation of limit cycles and flip fold and torus bifurcations of limit cycles in matcont and rm cl matcont using orthogonal collocation instead of the more commonly used fully extended systems we use minimally extended systems we further describe the use of the matlab sparse matrix routines and the initialization and adaptation of the bordering vectors that are essential in minimally extended systems finally we compare the use of the minimally and the fully extended systems in the matlab environment

global generation of pluricanonical and adjoint linear series on smooth projective threefolds the purpose of this paper is to show how the cohomological techniques developed by kawamata reid shokurov and others lead to some effective and practical results of reider type on freeness of linear series on smooth complex projective threefolds in recent years there has been a great deal of interest in the geometric properties of pluricanonical and adjoint linear systems on surfaces and higherdimensional varieties among other things one wants to understand as explicitly as possible when the systems in question are base point free or very ample modem work in this area goes back to kodaira kod and bombieri bmb who studied pluricanonical maps of surfaces of general type more recently many of their results have emerged as special cases of a celebrated theorem of reider rdr reider uses vector bundle techniques to show that if b is a nef line bundle on a surface x such that b2 5 then kx b is globally generated unless there exist certain special curves c c x such that b c 1 he also obtains analogous criteria for kx b to be very ample a cohomological approach to these theorems based on miyaoka s vanishing theorem for zariski decompositions was given by sakai sak2

transient response analysis of systems with different damping models a response of systems with different damping models to transient excitations is studied using the finite element method for this purpose an efficient method using fast fourier transform has been developed this method is based on the trigonometrical representation of the input signals and matrix of the transfer functions as an example systems with viscous and viscoelastic structural and external damping are examined two forms of viscous damping are available in the present implementation raleigh damping and element damping the viscoelastic material behaviour is represented by the complex modulus model the present implementation gives the possibility to preserve exactly the frequency dependence of the storage and loss moduli of viscoelastic materials logarithmic decrements are determined using the steady state vibrations of systems to characterize their damping properties test problems and numerical examples are given to demonstrate the validity and application of the approach suggested in this paper

vector bundles of rank 2 and linear systems on algebraic surfaces let s be a smooth complex algebraic surface and let l be a divisor on s it is well known that the linear system il ksi where ks is the canonical divisor plays an important role in understanding the geometry of s several authors see for example 8 11 studied various properties of this system our paper treats the same subject but the point of view is different from the works mentioned above the philosophy underlying our approach is almost classical points on s more generally effective 0 cycles in special position with respect to il ksi see definition below contain information about the geometry of s this point of view was recently revived in 5 also 10 in fact this paper contains the technique of going from the effective 0 cycle z in special position with respect to il ksi to the geometry of s for the sake of completeness we give a definition and state a theorem which can be found in 5 10

a robust indirect adaptive control approach this paper considers the robust design of an indirect adaptive control approach which is applicable when the unknown parameters of a linear time invariant plant lie in a known convex set throughout which no unstable pole zero cancellation occurs in order to achieve the robustness the use of a relative dead zone in the adaptive law is proposed it is shown that with a suitably designed relative dead zone the adaptive control system is globally stable even in the presence of small unmodeled plant uncertainties

a galerkin finite element technique for stochastic field problems a new finite element method is developed to analyse the structures with more than one parameter behaving in a stochastic manner the galerkin weighted residual method is used as a generalization this paper treats the random eigenvalue problems arising when the material property values of the structural systems are having stochastic fluctuations i e the measurement errors can be accounted for when probabilistic modelling is used the free vibration problem of a stochastic beam whose young s modulus and mass density are distributed stochastically is considered using galerkin s weighted residual procedure a stochastic finite element method is developed and implemented to arrive at a random algebraic eigenvalue problem the stochastic characteristics of eigensolutions are derived in terms of the stochastic material property variations numerical examples are given it is demonstrated that through this formulation the finite element discretization need not be dependent on the characteristics of stochastic fields describing the fluctuations in the material property values further calculation of covariances between stiffness elements mass elements etc is carried out in a relatively simple manner and is more general in nature

non linear dynamics in adaptive control chaotic and periodic stabilization abstract in this paper the non linear features of adaptive control algorithms are highlighted using a simple linear discrete time example based on classical design it is demonstrated that in the presence of undermodelling errors non linear phenomena in the feedback gain such as limit cycles and even chaos arise despite these complicated dynamics robust stabilization of the plant can still occur for the class of undermodelling errors considered the set of plants stabilizable by this adaptive controller is completely characterized

three dimensional gravity capillary waves on water small surface tension case abstract this paper considers three dimensional gravity capillary waves on water of finite depth which are uniformly translating in a horizontal propagating direction and periodic in a transverse direction the exact euler equations are formulated as a spatial dynamic system in which the variable used for the propagating direction is a time like variable the existence of the solutions of the system is determined by two non dimensional constants the bond number b and the froude number f which in turn give the number of eigenvalues on the imaginary axis of the complex plane for the corresponding linearized operator around a uniform flow assume that f 2 c 1 is the curve in the b plane on which the first two eigenvalues for three dimensional waves collide at the imaginary axis and the intersection point of c 1 with 1 is b 1 0 in this paper the case for 0 b b 1 and b near c 1 is considered a center manifold reduction technique and a normal form analysis are applied to show that the dynamical system can be reduced to a system of ordinary differential equations using the existence of a homoclinic orbit connecting to a two dimensional periodic called generalized solitary wave thereafter solution for the reduced system it is shown that such a generalized solitary wave solution persists for the original system by applying a perturbation method and adjusting some appropriate constants

hybrid uncertain natural frequency analysis for structures with random and interval fields abstract this paper presents a robust non deterministic free vibration analysis for engineering structures involving hybrid yet spatially dependent uncertain system parameters distinguished from the conventional hybrid uncertain eigenvalue problem the concept of interval field is enclosed with random field model such that both the stochastic and non stochastic representations of the spatial dependency of the uncertainties are simultaneously incorporated within a unified non deterministic free vibration analysis in order to determine the probabilistic characteristics i e means and standard deviations of the extremities of structural natural frequencies an extended unified interval stochastic sampling x uiss method is implemented for the purpose of effective hybrid uncertain free vibration analysis by meticulously blending sharpness promised interval eigenvalue analysis with stochastic sampling techniques the stochastic profiles i e probability density functions pdfs and the cumulative distribution functions cdfs of the extreme bounds of the structural natural frequencies can be rigorously established by utilizing the adequate statistical inference methods the applicability and effectiveness of the proposed computational framework are evidently demonstrated through the numerical investigations on various practically motivated engineering structures

global control with application to bifurcating power systems a multilevel nonlinear control scheme that is capable of system control over wide ranges of operating conditions is proposed the first control level confines the system operation to some reference segments in its state space each reference segment is further subdivided using the bifurcation analysis for establishing the bounds of the state space of the system with different control requirements the principles of global control lead to a hierarchical control structure which combines regional linear or nonlinear controllers smoothly the scheme is illustrated for control of a power system known to have diverse dynamical behavior the control will accomplish desired steady state fixed point or small oscillation operation of the power system for a wide range of loading levels

numerical periodic normalization for codim 1 bifurcations of limit cycles explicit computational formulas for the coefficients of the periodic normal forms for all codim 1 bifurcations of limit cycles in generic autonomous odes are derived they include second order coefficients for the fold limit point bifurcation as well as third order coefficients for the flip period doubling and neimark sacker torus bifurcations the formulas are independent of the dimension of the phase space and involve solutions of certain boundary value problems on the interval 0 t where t is the period of the critical cycle as well as multilinear functions from the taylor expansion of the right hand sides near the cycle the formulas allow us to distinguish between sub and supercritical bifurcations in agreement with earlier asymptotic expansions of the bifurcating solutions our formulation makes it possible to use robust numerical boundary value algorithms based on orthogonal collocation rather than shooting techniques which greatly expands its applicability the actual implementation is described in detail we include three numerical examples in which codim 2 singularities are detected along branches of codim 1 bifurcations of limit cycles as zeros of the periodic normal form coefficients

computing periodic orbits with high accuracy this paper introduces a new family of algorithms for computing periodic orbits of vector fields these global methods achieve high accuracy with relatively coarse discretizations of periodic orbits through the use of automatic differentiation high degree taylor series expansions of trajectories are computed at mesh points on a fixed mesh we construct closed curves that converge smoothly to periodic orbits as the degree of the taylor series expansions increase the algorithms have been implemented in matlab together with the use of the automatic differentiation code adol c numerical tests of our codes are compared with auto calculations using the hodgkin huxley equations as a test problem

hopf bifurcation at 1 1 resonance on etudie un probleme concernant la bifurcation de hopf pour une famille a 1 parametre d equations differentielles dans un cas de resonance

a numerical criterion for very ample line bundles let x be a projective algebraic manifold of dimension n and let l be an ample line bundle over x we give a numerical criterion ensuring that the adjoint bundle kx l is very ample the sufficient conditions are expressed in terms of lower bounds for the intersection numbers l y over subvarieties y of x in the case of surfaces our criterion gives universal bounds and is only slightly weaker than i reider s criterion when dimx 3 and codimy 2 the lower bounds for l y involve a numerical constant which depends on the geometry of x by means of an iteration process it is finally shown that 2kx ml is very ample form 12n our approach is mostly analytic and based on a combination of hormander s l estimates for the operator lelong number theory and the aubin calabi yau theorem

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

statistics of vibration energy flow in randomly parametered trusses the problem of determining the ensemble averages of vibration energy flow in harmonically driven trusses with random parameter variations is considered the mass elasticity damping and length of the truss members are modelled as random variables the uncertainty associated with member lengths result in randomness in truss geometry four different linear damping models for truss members namely strain rate dependent viscous hysteretic models and velocity dependent viscous hysteretic models are considered the analysis employs random dynamic stiffness matrices and monte carlo simulation procedures a comparison of the ensemble averages of subsystem energies obtained using this approach and those using statistical energy analysis sea formalism is also made the energy coefficient matrix arising in sea studies is determined using the direct dynamic stiffness matrix approach the paper illustrates the relative importance of alternative damping models and alternative sources of system randomness on the behavior of spectra of subsystem vibration energies in a 13 member truss the uncertainties in randomness in truss geometry and the choice of damping model are shown to significantly affect the energy statistics

bifurcation in model reference adaptive control systems abstract we study the effect of unmodeled dynamics on the performance of a simple model reference adaptive control mechanism exact stability boundaries are computed and different routes to chaotic behavior and global instability are described typically we observed a region of period doubling and hopf bifurcations similar to those observed in the ziegler nichols approach to pid control design the analysis applies to higher order and certain nonlinear systems as well

parametric instability of stochastic columns columns which have stochastically distributed young s modulus and mass density and are subjected to deterministic periodic axial loadings are considered the general case of a column supported on a winkler elastic foundation of random stiffness and also on discrete elastic supports which are also random is considered material property fluctuations are modeled as independent one dimensional univariate homogeneous real random fields in space in addition to autocorrelation functions or their equivalent power spectral density functions the input random fields are characterized by scale of fluctuations or variance functions for their second order properties the foundation stiffness coefficient and the stiffnesses of discrete elastic supports are treated to constitute independent random variables the system equations of boundary frequencies are obtained using bolotin s method for deterministic systems stochastic fem is used to obtain the discrete system with random as well as periodic coefficients statistical properties of boundary frequencies are derived in terms of input parameter statistics a complete covariance structure is obtained the equations developed are illustrated using a numerical example employing a practical correlation structure

analysis of discrete vibratory systems with parameter uncertainties part i eigensolution abstract a new analytical method is proposed for the estimation of eigensolutions for undamped and proportionally damped discrete vibratory systems when the system parameters are uncertain or random variables given several simplifying assumptions a direct product technique is used to estimate the first and second moments of eigensolutions in terms of the moments of system parameter matrices the proposed methodology is reasonably accurate and computationally fast when compared with existing methods such as the first perturbation method and the monte carlo simulation application of the new method is demonstrated through several single and multi degree of freedom systems it is seen that the standard deviation predictions match well with the results yielded by the monte carlo simulation unlike the first order perturbation method especially when moderately large random fluctuations are considered the impulse response issue is addressed in the companion paper part ii

brief detecting period doubling bifurcation an approximate monodromy matrix approach a quasi analytical approach is developed for detecting period doubling bifurcation emerging near a hopf bifurcation point the new algorithm employs higher order harmonic balance approximations hbas to compute the monodromy matrix useful for the study of limit cycle bifurcations prediction of the period doubling bifurcation is accomplished very accurately by using this algorithm along with a detailed approximation error analysis without using numerical integration of the dynamical system an example is given to illustrate the results

on fujita s freeness conjecture in dimension 5 abstract let x be a smooth projective variety of dimension 5 and a an integral ample divisor on x we show that k x k a is basepoint free for any integer k 6

explicit transversality conditions and local bifurcation diagrams for bogdanov takens bifurcation on center manifolds abstract this application oriented study is concerned with the derivation of parameter dependent normal forms for the codimension two bogdanov takens bifurcation of n dimensional m parameterized systems on the basis of the homological method in the case of an enduring equilibrium simple formulas are obtained for the transformation of parameters enabling the formulation of explicit transversality conditions and bifurcation diagrams to at most the second order moreover in z 2 symmetric systems the calculation can be further limited within certain subspaces in the general case existing results are re derived and a revision necessary for determining the bifurcation diagrams to the second order is then provided these results facilitate the derivation of normal forms check of transversality and depiction of bifurcation diagrams for the bogdanov takens bifurcation

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

random eigenvalues algebraic methods and structural dynamic models a review of the literature of the study of structural vibration problems with random parameters is provided there have been several approaches to this problem these have been encapsulated in this paper for the benefit of those who need to assess such possibilities in particular the algebraic theory of random variables is delineated here with the application in mind being the determination or estimation of the statistics of the eigenvalues of linear dynamical systems various transformation techniques are summarized and discussed with simple examples

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

normal forms and nonlinear symmetries we give some general theorems and extensions of previous results concerning the problem of transforming an algebra of vector fields into poincare normal form by means of a unifying algebraic language we show the possibility of obtaining either a parallel or joint normal form of the vector fields in a definite way which simplifies the construction of normal forms providing a precise restriction on their structure the application to finite dimensional dynamical systems and their lie point symmetries is also discussed

higher order reverse automatic differentiation with emphasis on the third order it is commonly assumed that calculating third order information is too expensive for most applications but we show that the directional derivative of the hessian d 3 f x cdot d d3f x d can be calculated at a cost proportional to that of a state of the art method for calculating the hessian matrix we do this by first presenting a simple procedure for designing high order reverse methods and applying it to deduce several methods including a reverse method that calculates d 3f x cdot d d3f x d we have implemented this method taking into account symmetry and sparsity and successfully calculated this derivative for functions with a million variables these results indicate that the use of third order information in a general nonlinear solver such as halley chebyshev methods could be a practical alternative to newton s method furthermore high order sensitivity information is used in methods for robust aerodynamic design an efficient high order differentiation tool could facilitate the use of similar methods in the design of other mechanical structures

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

uncertainty in structural dynamics experimental validation of a wishart random matrix model in the modeling of complex dynamical systems high resolution finite element models are routinely adopted to reduce the discretization error such an approach may fail to enhance the confidence in simulation based predictions when the dynamical systems exhibit significant variability in the parameterization and description of the mathematical model due to parametric data uncertainty and model uncertainty in contrast to parametric uncertainty model uncertainty poses significant challenges as no explicit parameter is available a priori to characterize its behavior in the current investigation a set of sprung mass oscillators randomly attached to a baseline system namely a homogeneous cantilever plate gives rise to model uncertainty contrary to model parametric uncertainty traditionally tackled using stochastic finite element method sfem the model uncertainty arising from randomly attached sprung masses gives rise to a new variant of dynamical system for every sample the feasibility of adopting a wishart random matrix to represent such ensemble of dynamical systems derived from model perturbation is investigated in this paper the experiments conducted on a vibrating plate with randomly attached sprung mass oscillators demonstrate that the wishart random matrix model may provide a reasonable representation of model uncertainty in linear dynamical systems in the medium and high frequency region

analysis of bifurcations of limit cycles with lyapunov exponents and numerical normal forms in this paper we focus on the combination of normal form and lyapunov exponent computations in the numerical study of the three codim 2 bifurcations of limit cycles with dimension of the center manifold equal to 4 or to 5 in generic autonomous odes the normal form formulas are independent of the dimension of the phase space and involve solutions of certain linear boundary value problems the formulas allow one to distinguish between the complicated bifurcation scenarios which can happen near these codim 2 bifurcations where 3 tori and 4 tori can be present we apply our techniques to the study of a known laser model a novel model from population biology and a model of mechanical vibrations these models exhibit limit point neimark sacker period doubling neimark sacker and double neimark sacker bifurcations lyapunov exponents are computed to numerically confirm the results of the normal form analysis in particular with respect to the existence of stable invariant tori of various dimensions conversely the normal forms are essential to understand the significance of the lyapunov exponents

an approximate solution scheme for the algebraic random eigenvalue problem a reduced basis formulation is presented for the efficient solution of large scale algebraic random eigenvalue problems this formulation aims to improve the accuracy of the first order perturbation method and also allow the efficient computation of higher order statistical moments of the eigenparameters in the present method the two terms of the first order perturbation approximation for the eigenvector are used as basis vectors for ritz analysis of the governing random eigenvalue problem this leads to a sequence of reduced order random eigenvalue problems to be solved for each eigenmode of interest since only two basis vectors are used to represent each eigenvector explicit expressions for the random eigenvalues and eigenvectors can readily be derived this enables the statistics of the random eigenparameters and the forced response to be efficiently computed numerical studies are presented for free and forced vibration analysis of a linear stochastic structural system it is demonstrated that the reduced basis method gives better results as compared to the first order perturbation method

matrix models for beta ensembles this paper constructs tridiagonal random matrix models for general 0 hermite gaussian and laguerre wishart ensembles these generalize the well known gaussian and wishart models for 1 2 4 furthermore in the cases of the laguerre ensembles we eliminate the exponent quantization present in the previously known models we further discuss applications for the new matrix models and present some open problems

sensitivity analysis of odes daes using the taylor series method this paper studies the applicability of the taylor method for the sensibility analysis of odes and daes extended automatic differentiation rules are introduced for the calculus of partial derivatives of taylor series the numerical method is implemented using an efficient variable step variable order scheme finally some numerical tests are presented showing the benefits of the formulation

discretizing dynamical systems with generalized hopf bifurcations we consider the discretizations of parameter dependent continuous time dynamical systems we show that the general one step methods shift a generalized hopf bifurcation and turn it into a generalized neimark sacker point we analyze the effect of discretization methods on the emanating hopf curve in particular we obtain estimates for the eigenvalues of the discretized system along this curve a detailed analysis of the discretized first lyapunov coefficient is also given the results are illustrated by a numerical example dynamical consequences are discussed

stability of discrete model reference adaptive control revisited abstract we demonstrate that the model reference adaptive controller is robust with respect to small model plant mismatch the sign and a lower bound on the high frequency gain are known and the control parameters are constrained to belong to a compact set which contains the parameters of a stabilizing controller data normalization and deadzones do not play a role in the analysis a constructive proof is used to develop bounds for the allowable gain of the model mismatch when the algorithm state is far from equilibrium the signals are measured by a decaying exponential the rate of convergence is estimated an expression which can be used to calculate the size of an overbounding set is developed its size scales uniformly with respect to the external disturbances the inverse of the adaptive gain and the reference the ideal case is obtained as these tend to zero this result holds true even when there is a model mismatch

numerical taylor expansions of invariant manifolds in large dynamical systems in this paper we develop a numerical method for computing higher order local approximations of invariant manifolds such as stable unstable or center manifolds near steady states of a dynamical system the underlying system is assumed to be large in the sense that a large sparse jacobian at the equilibrium occurs for which only a linear black box solver and a low dimensional invariant subspace is available but for which methods like the qr algorithm are considered to be too expensive our method is based on an analysis of the multilinear sylvester equations for the higher derivatives which can be solved under certain nonresonance conditions these conditions are weaker than the standard gap conditions on the spectrum which guarantee the existence of the invariant manifold the final algorithm requires the solution of several large linear systems with a bordered jacobian to these systems we apply a block elimination method recently developed by govaerts and pryce 12 14

drift instabilities and chaos in forecasting and adaptive decision theory abstract bifurcation theory shows that policy adaptation and the rational expectations hypothesis of macro economics can be used to explain unpredictability rapid changes in solution structure and chaos in decision problems with uncertainty structural errors lead to catastrophic instability and forecasts become irrelevant short evaluation horizon and the application of measures designed to give quick response give multiperiodicity and chaos finally wrong interpretations of the context lead to global bifurcations laminar drift complacency and chaotic bursting the discrete map representing these dynamics is of interest in its own right it is non invertible and displays bifurcation behaviour not commonly seen in systems derived from physical considerations

small amplitude chaos and ergodicity in adaptive control abstract the theory for continuous iterated maps on the real line is used to study the dynamics of a continuous family of one parameter adaptive systems we establish sufficient and necessary conditions for uniform asymptotic stability and ultimate boundedness exact bifurcation boundaries and sufficient conditions for the existence of chaotic attractors and ergodicity of the estimated parameter it is argued that the chaotic dynamics are structurally stable that they persist in higher dimensional adaptive control systems and that they do not necessarily lead to poor performance when the controller is modified to deal with the infinite drift and small divisor problems a novel modification involving a sign change in the adaptive law is proposed

homoclinic orbits in reversible systems and their applications in mechanics fluids and optics abstract this survey article reviews the theory and application of homoclinic orbits to equilibria in reversible continuous time dynamical systems where the homoclinic orbit and the equilibrium are both reversible the focus is on even order reversible systems in four or more dimensions local theory generic argument and global existence theories are examined for each qualitatively distinct linearisation several recent results such as coalescence caused by non transversality and the reversible orbit flip bifurcation are covered a number of open problems are highlighted applications are reviewed to systems arising from a variety of disciplines with the aid of numerical methods three examples are presented in detail one of which is infinite dimensional

paper some design guidelines for discrete time adaptive controllers there have been many algorithms proposed for adaptive control which will provide globally asymptotically stable controllers if some stringent conditions on the plant are met the conditions on the plant cannot be met in practice as all plants will contain high frequency unmodeled dynamics this paper uses a linearization analysis of a nonlinear adaptive controller to demonstrate analytically some design guidelines which alleviate some of the problems associated with adaptive control in the presence of unmodeled dynamics the points made are further demonstrated by simulation results

hyperbolic periodic orbits in nongradient systems and small noise induced metastable transitions abstract small noise can induce rare transitions between metastable states which can be characterized by maximum likelihood paths mlps nongradient systems contrast gradient systems in that mlp does not have to cross the separatrix at a saddle point but instead possibly at a point on a hyperbolic periodic orbit a numerical approach for identifying such unstable periodic orbits is proposed based on string method in a special class of nongradient systems orthogonal type there are provably local mlps that cross such saddle point or hyperbolic periodic orbit and the separatrix crossing location determines the associated local maximum of transition rate in general cases however the separatrix crossing may not determine a unique local maximum of the rate as we numerically observed a counter example in a sheared 2d space allen cahn spde it is a reasonable conjecture that there are always local mlps associated with each attractor on the separatrix such as saddle point or hyperbolic periodic orbit our numerical experiments did not disprove so

analysis of eigenvalues and modal interaction of stochastic systems an eigenvalue spectral analysis of stochastic engineering systems is presented a comparative numerical study between approximations based on monte carlo sampling a taylor series based perturbation approach and the polynomial chaos representation is conducted it is observed that the polynomial chaos representation gives more accurate estimates of the statistical moments than the perturbation method especially for the higher modes the differences of accuracy in the two methods are more pronounced as the system variability increases moreover the chaos expansion gives a more detailed probabilistic description of the eigenvalues and the eigenvectors in addition a method for representing the statistical modal overlapping is presented that characterizes the statistical interaction between the various modes

an algorithm for melnikov functions and application to a chaotic rotor in this work we study a dynamical system with a complicated nonlinearity which describes oscillation of a turbine rotor and give an algorithm to compute melnikov functions for analysis of its chaotic behavior we first derive the rotor model whose nonlinear term brings difficulties to investigating the distribution and qualitative properties of its equilibria this nonlinear model provides a typical example of a system for which the homoclinic and heteroclinic orbits cannot be analytically determined in order to apply melnikov s method to make clear the underlying conditions for chaotic motion we present a generic algorithm that provides a systematic procedure to compute melnikov functions numerically substantial analysis is done so that the numerical approximation precision at each phase of the computation can be guaranteed using the algorithm developed in this paper it is straightforward to obtain a sufficient condition for chaotic motion under damping and periodic external excitation whenever the rotor parameters are given

vsvo formulation of the taylor method for the numerical solution of odes this paper presents an analysis of the taylor method for the numerical solution of odes when a very high precision of the solution is required some theoretical properties of the taylor method are considered from the practical point of view a variable stepsize variable order vsvo scheme is presented and its utility is discussed with several examples to reach the goal of high precision the use of multiprecision libraries is considered finally some numerical tests based on the test problems given in 1 and on a set of important problems in dynamical systems and astrodynamics are presented showing the benefits of the vsvo formulation especially for high precision demands compared with a well established runge kutta code

paper instability analysis and improvement of robustness of adaptive control the effects of unmodeled high frequency dynamics and bounded disturbances on stability and performance of adaptive control schemes are analyzed five possible types of instability mechanisms parameter drift linear instability fast adaptation instability high frequency instability and throughput instability are analyzed using simple examples a procedure is used to construct lyapunov like functions for a modified adaptive controller applied to a dominant plant of relative degree one in the presence of parasitics and disturbances and obtain sufficient conditions under which none of the five types of instability can occur the modified scheme is robust in the sense that it guarantees the existence of a large region of attraction from which all the trajectories remain bounded and the state errors converge exponentially to a much smaller residual set the size of the region of attraction depends on the speed of parasitics in such a way that as the parasitics become infinitely fast the region of attraction becomes the whole space

a simple global characterization for normal forms of singular vector fields we derive a new global characterization of the normal forms of amplitude equations describing the dynamics of competing order parameters in degenerate bifurcation problems using an appropriate scalar product in the space of homogeneous vector polynomials we show that the resonant terms commute with the group generated by the adjoint of the original critical linear operator this leads to a very efficient constructive method to compute both the nonlinear coefficients and the unfolding of the normal form explicit examples and results obtained when there are additional symmetries are also presented

analysis of codimension 2 bifurcations for high dimensional discrete systems using symbolic computation methods this article reports an algebraic criterion of the eigenvalue assignment transversality condition and non resonance condition for fold n s bifurcations by means of symbolic computation methods we propose an algorithmic approach for systematically analyzing codimension 2 bifurcations for high dimensional discrete systems the effectiveness of the proposed symbolic approach is verified by experiments in particular the flip and fold n s bifurcations of a five dimensional discrete dynamical system with a washout filter feedback controller are analyzed

modal approaches for the stochastic finite element analysis of structures with material and geometric uncertainties this paper presents two efficient modal approaches as an alternative to direct formulations for the time harmonic dynamic analysis of structures with random material and shape parameters in both approaches the structural eigenproblem is solved and complemented by a sensitivity analysis to the random parameters the modal perturbation stochastic finite element method sfem then condenses the response sensitivities to assess the response variability the mixed perturbation monte carlo sfem assesses the response statistics by sampling the structural eigenmodes according to the perturbation estimation of their probability distribution functions pdfs geometric uncertainties are handled through an appropriate shape parameterisation and a shape design sensitivity analysis two numerical applications examine both approaches in terms of accuracy variability level and computational requirements the applications involve a plate bending problem with random young modulus or edge length and a plate with a random flatness default particular observations related to the influence of the parameter pdfs in simulation based methods are also provided

numerical normalization techniques for all codim 2 bifurcations of equilibria in ode s explicit computational formulas for the coefficients of the normal forms for all codim 2 equilibrium bifurcations of equilibria in autonomous odes are derived they include second order coefficients for the bogdanov takens bifurcation third order coefficients for the cusp and fold hopf bifurcations and coefficients of the fifth order terms for the generalized hopf bautin and double hopf bifurcations the formulas are independent on the dimension of the phase space and involve only critical eigenvectors of the jacobian matrix of the right hand sides and its transpose as well as multilinear functions from the taylor expansion of the right hand sides at the critical equilibrium the normal form coefficients for the fold hopf bifurcation in the new lorenz model are computed using the derived formulas proving the existence of a nontrivial invariant set in the system

a solution of the random eigenvalue problem by crossing theory abstract a method is developed for finding probabilistic descriptors of the eigenvalues of stochastic real valued symmetric matrices the method is based on the mean rate at which the characteristic polynomial of a stochastic matrix crosses level zero the proposed approach can also be applied when additional information is available on the value of the first or the first few eigenvalues several examples are presented in the paper to demonstrate the application and usefulness of the method

computing periodic orbits and their bifurcations with automatic differentiation this paper formulates several algorithms for the direct computation of periodic orbits as solutions of boundary value problems the algorithms emphasize the use of coarse meshes and high orders of accuracy convergence theorems are given in the limit of increasing order with a fixed mesh the algorithms are implemented with the use of matlab and adol c a software package for automatic differentiation automatic differentiation enables accurate computation of high order derivatives of functions without the truncation errors inherent in finite difference calculations we embed the algorithms in a continuation framework and extend them to compute saddle node bifurcations of periodic orbits directly we present data from numerical studies of four test problems making some comparisons with other methods for computing periodic orbits these results demonstrate that high order methods based upon automatic differentiation are capable of high precision with small meshes

the p and h p versions of the finite element method basic principles and properties in the classical form of the finite element method called the hversion piecewise polynomials of fixed degree p are used and the mesh size h is decreased for accuracy in this paper we discuss the fundamental theoretical ideas behind the relatively recent p version and h p version in the p version a fixed mesh is used and p is allowed to increase the h p version combines both approaches the authors describe and explain the basic properties and characteristics of these newer versions especially in areas where their behavior is significantly different from that of the h version simplified proofs of key concepts are included and computational illustrations of several results are provided a benchmark comparison between the various versions in included

numerical methods for the generalized hopf bifurcation the presence of a generalized hopf gh point hopf point where the first lyapunov coefficient vanishes deeply influences the qualitative behavior of a dynamical system further information on this behavior can be obtained from the second lyapunov coefficient which depends on derivatives of order up to five we describe two computational procedures implemented in the software package content to compute and numerically continue gh points this is applied to a biochemical model and two examples from neurobiology we make some comparisons between the two methods for computing gh points and discuss briefly the merits of three methods for computing curves of hopf points

a solution of the random eigenvalue problem by a dimensional decomposition method summary this paper presents a dimensional decomposition method for obtaining probabilistic descriptors of realvalued eigenvalues of positive semi definite random matrices the method involves a novel function decomposition allowing lower variate approximations of eigenvalues lower dimensional numerical integration for statistical moments and lagrange interpolation facilitating efficient monte carlo simulation for probability density functions compared with commonly used perturbation and recently developed asymptotic methods no derivatives of eigenvalues are required by the new method developed results of numerical examples from structural dynamics indicate that the decomposition method provides excellent estimates of moments and probability densities of eigenvalues for various cases including closely spaced modes and large statistical variations of input copyright 2006 john wiley sons ltd

random eigenvalue problems in structural dynamics experimental investigations an experimental investigation of uncertainty in natural frequencies of linear structures estimated from measured frequency response function frf under dynamic loading is presented experiments were conducted on one hundred nominally identical realizations of a fixed fixed beam each realisation was obtained by placin g twelve identical masses at random spatial locations generated by a computer along the length of the beam the total random mass is about 2 of the total mass of the beam these experiments contain parametric uncertainty in the mass matrix and hence may be useful for the validation of many random eigenvalue analysis and prediction methods currently available to structural dynamicist predictions from monte carlo simulation of deterministic finite elements model are compared with experiments it is co ncluded that the method of estimation of natural frequencies from frfs and the spatial location of the measurements has significant influence upon the first two moments mean and standard deviation of the natural frequency ensemble furthermore whilst the monte carlo simulation estimates of the mean and standard deviation are in reasonable agreement with experiments at higher frequencies the probability density function differ appreciably within the limits of the sample size investigated in this study

eigenvalue density of linear stochastic dynamical systems a random matrix approach abstract eigenvalue problems play an important role in the dynamic analysis of engineering systems modeled using the theory of linear structural mechanics when uncertainties are considered the eigenvalue problem becomes a random eigenvalue problem in this paper the density of the eigenvalues of a discretized continuous system with uncertainty is discussed by considering the model where the system matrices are the wishart random matrices an analytical expression involving the stieltjes transform is derived for the density of the eigenvalues when the dimension of the corresponding random matrix becomes asymptotically large the mean matrices and the dispersion parameters associated with the mass and stiffness matrices are necessary to obtain the density of the eigenvalues in the frameworks of the proposed approach the applicability of a simple eigenvalue density function known as the mar enko pastur mp density is investigated the analytical results are demonstrated by numerical examples involving a plate and the tail boom of a helicopter with uncertain properties the new results are validated using an experiment on a vibrating plate with randomly attached spring mass oscillators where 100 nominally identical samples are physically created and individually tested within a laboratory framework

subadjunction of log canonical divisors ii we extend a subadjunction formula of log canonical divisors as in kawamata contemp math 207 1997 79 88 to the case when the codimension of the minimal center is arbitrary by using the positivity of the hodge bundles

fold pitchfork bifurcation for maps with z 2 symmetry in pipe flow this study aims to provide a better understanding of recently identified transition scenarios exhibited by traveling wave solutions in pipe flow this particular family of solutions are invariant under certain reflectional symmetry transformations and they emerge from saddle node bifurcations within a two dimensional parameter space characterized by the length of the pipe and the reynolds number the present work precisely provides a detailed analysis of a codimension two saddle node bifurcation arising in discrete dynamical systems maps with z2 symmetry normal form standard techniques are applied in order to obtain the reduced map up to cubic order all possible bifurcation scenarios exhibited by this normal form are analyzed in detail finally a qualitative comparison of these scenarios with the ones observed in the aforementioned hydrodynamic problem is provided

non linear stability analysis of a complex rotor stator contact system in this paper a non linear strategy based on the centre manifold the rational approximants and the alternating frequency time domain method has been developed in order to study the non linear dynamical behaviour of a system in the neighbourhood of a critical steady state equilibrium point the stability analysis and the non linear dynamics of a complex braking system with a non linear rotor stator contact are presented moreover one of the most important steps of this paper is the determination of the non linear behaviour and the limit cycle amplitudes of this complex system in order to conduct this study the dynamic response is evaluated by using applying the centre manifold the rational approximants and the alternating frequency time domain method that permit to obtain rapidly and efficiently the non linear behaviour of the system the dynamic response obtained by applying this method is compared with that evaluated through numerical integration

amplitude equations for systems with competing instabilities we describe how to derive ordinary differential equations to predict the time dependence of a system governed by partial differential equations when that system is near to the polycritical condition for the onset of several instabilities the method is illustrated for the general case of two competing instabilities and then a specific physical example of this case is worked out in detail the basic idea is to extend the krylov bogolyubov mitropolsky method of nonlinear mechanics to bifurcation problems

iterative solution of the random eigenvalue problem with application to spectral stochastic finite element systems a new algorithm for the computation of the spectral expansion of the eigenvalues and eigenvectors of a random non symmetric matrix is proposed the algorithm extends the deterministic inverse power method using a spectral discretization approach the convergence and accuracy of the algorithm is studied for both symmetric and non symmetric matrices the method turns out to be efficient and robust compared to existing methods for the computation of the spectral expansion of random eigenvalues and eigenvectors