stochastic excitation of suspended cables involving three simultaneous internal resonances using monte carlo simulation abstract in the neighborhood of three simultaneous internal resonance conditions among four normal modes the response behavior of a suspended cable to random gaussian excitation is studied using computational simulations the study includes the interaction between the first two in plane and the first two out of plane modes when the first in plane mode is externally excited its response can act as a parametric excitation to the other three modes through nonlinear coupling the governed modal equations of motion are numerically simulated using a double precision initial value problem solver adam moulton algorithm the random excitation is generated numerically from a normal distribution using an inverse cumulative distribution function cdf technique the numerical results are processed to estimate response statistics such as mean squares power spectra probability density functions and autocorrelation functions the modes which are indirectly excited exhibit the well known phenomenon on off intermittency in the neighborhood of stochastic bifurcation furthermore in the neighborhood of internal resonances the directly excited mode is found to transfer energy to other modes under gaussian excitation the response coordinates of the four modes are essentially non gaussian narrow band random processes with a central frequency close to each mode natural frequency

complexity issues in natural gradient descent method for training multilayer perceptrons the natural gradient descent method is applied to train an n m 1 multilayer perceptron based on an efficient scheme to represent the fisher information matrix for an n m 1 stochastic multilayer perceptron a new algorithm is proposed to calculate the natural gradient without inverting the fisher information matrix explicitly when the input dimension n is much larger than the number of hidden neurons m the time complexity of computing the natural gradient is o n

integrability of a nonlinear evolution equation derived from isoperimetric plane curve motion we provide a geometrical interpretation for the series of transformations used by sakovich phys lett a 314 232 2003 to map the third order nonlinear evolution equation obtained by chou and qu physica d 162 9 2002 to the mkdv equation we also discuss its bi hamiltonian integrability as well as integrable equations associated with this system

responses of strongly non linear oscillator parametrically excited by random narrow band noise the principal response of a strongly van der pol duffing oscillator subjected to parametric random narrow band excitation is investigated the technique of modified lindstedt poincare mlp method is used to transform the strongly non linear system to a weak one by introducing a new expansion parameter and then the multiple scales method is applied to determine the modulation equations for amplitude and phase of the response of the system the effect of damping detuning and bandwidth on the dynamic behaviors such as stability bifurcation are examined by computing the maximum lyapunov exponent analytically also the numerical simulation is carried out to verify the analytical results and random jump phenomenon may be observed in the region of the parameters of the system the excellent agreement between theoretical results and numerical ones can be found immediately and so the present method in this paper is applicable to solve strongly non linear problems

on the symmetric modal interaction of the suspended cable three to one internal resonance abstract the two mode nonlinear response of a suspended cable subjected to the primary resonance is investigated and the three to one internal resonance is analyzed because the treatment of galerkin discrete models of spatially continuous systems with initial curvature may lead to erroneous quantitative or even qualitative results the method of multiple scales is applied to directly attack the nonlinear partial differential equation and the boundary conditions which leads to the modulation equations for the primary resonance of either the first or third symmetric mode the newton raphson method and the pseudo arclength scheme are used to obtain the frequency response curves and force response curves and the dynamic solutions of the modulation equations are also investigated

non linear galloping of sagged cables in 1 2 internal resonance the aeroelastic behaviour of a flexible elastic suspended cable driven by mean wind speed blowing perpendicularly to the cable s plane is investigated by applying the galerkin procedure to the partial differential equations of motion and using an in plane and an out of plane mode as shape functions a two d o f model is derived the discrete equations are coupled through quadratic and cubic terms arising both from geometric and aerodynamic effects the associated linear frequencies are assumed to be in an almost 1 2 ratio so that internal resonance occurs the multiple scale perturbation method is employed to obtain a set of three amplitude modulation equations whose coefficients depend on the mean wind speed which is assumed as control parameter two perturbative solutions are developed each based on a different assumption about the order of magnitude of the static displacements produced by steady state wind forces analytical results are then compared with direct numerical integrations of discrete non linear equations by performing a bifurcation analysis the existence of several equilibrium branches is proved the relative importance of geometric and aerodynamic non linearities is discussed through simplified models the influence on critical and postcritical behaviour of several parameters including geometrical cable parameters detuning and non symmetric flow effects is investigated the important role played by the steady state forces is highlighted

non linear dynamic analysis of the two dimensional simplified model of an elastic cable the non linear behavior of an elastic cable subjected to a harmonic excitation is investigated in this paper using garlerkin s method and method of multiple scales the discrete dynamical equations and a set of first order non linear differential equations are obtained a numerical simulation is used to obtain the steady state response and stable solutions finally the coupled dynamic features between the out planar pendulation and the in planar vibration of an elastic cable are analyzed

an investigation of multiple solutions yielded by the equivalent linearization method abstract the equivalent linearization method is a standard technique for the approximate solution of non linear random vibration problems it is known that in some cases the method can yield multiple stable solutions for the mean value and the standard deviation of the system response the validity and accuracy of this type of solution is investigated here by considering a single degree of freedom system for which the multiple solutions which are obtained may be compared with the exact response statistics it is found that the multiple solutions correspond to distinguishable states which occur in the response of the system a state is defined here as a form of response which persists for some time and whose local temporal statistics differ from the ensemble statistics of the complete response the accuracy of the equivalent linearization method is found to be variable

non linear string random vibration abstract an investigation is described of the concave shape generation mechanism in the displacement and its first derivative vibration velocity amplitude probability density functions of the narrow band random vibration of a non linear string three types of strings are studied in detail characterized as follows 1 constant damping factor and non linear stiffness 2 the damping factor an even function with respect to the displacement and its first derivative and non linear stiffness 3 the damping factor an even function of the displacement and its first derivative and linear stiffness it is verified by numerical calculation that for a response sample of relatively long duration of the order of eight representative periods of the oscillation the generation of the concave shape depends mainly upon 1 the saturation of the amplitude and the growth of higher harmonic oscillation due to the non linear stiffness or the even function damping and 2 a jump phenomenon in the input output relation

singularities affect dynamics of learning in neuromanifolds the parameter spaces of hierarchical systems such as multilayer perceptrons include singularities due to the symmetry and degeneration of hidden units a parameter space forms a geometrical manifold called the neuromanifold in the case of neural networks such a model is identified with a statistical model and a riemannian metric is given by the fisher information matrix however the matrix degenerates at singularities such a singular structure is ubiquitous not only in multilayer perceptrons but also in the gaussian mixture probability densities arma time series model and many other cases the standard statistical paradigm of the cramer rao theorem does not hold and the singularity gives rise to strange behaviors in parameter estimation hypothesis testing bayesian inference model selection and in particular the dynamics of learning from examples prevailing theories so far have not paid much attention to the problem caused by singularity relying only on ordinary statistical theories developed for regular nonsingular models only recently have researchers remarked on the effects of singularity and theories are now being developed this article gives an overview of the phenomena caused by the singularities of statistical manifolds related to multilayer perceptrons and gaussian mixtures we demonstrate our recent results on these problems simple toy models are also used to show explicit solutions we explain that the maximum likelihood estimator is no longer subject to the gaussian distribution even asymptotically because the fisher information matrix degenerates that the model selection criteria such as aic bic and mdl fail to hold in these models that a smooth bayesian prior becomes singular in such models and that the trajectories of dynamics of learning are strongly affected by the singularity causing plateaus or slow manifolds in the parameter space the natural gradient method is shown to perform well because it takes the singular geometrical structure into account the generalization error and the training error are studied in some examples

the short pulse hierarchy we study a new hierarchy of equations containing the short pulse equation which describes the evolution of very short pulses in nonlinear media and the elastic beam equation which describes nonlinear transverse oscillations of elastic beams under tension we show that the hierarchy of equations is integrable we obtain the two compatible hamiltonian structures we construct an infinite series of both local and nonlocal conserved charges a lax description is presented for both systems for the elastic beam equations we also obtain a nonstandard lax representation

non linear oscillations of a four degree of freedom model of a suspended cable under multiple internal resonance conditions abstract a four degree of freedom model able to capture the main phenomena which are likely to occur in the non planar finite dynamics of an elastic suspended cable subjected to external forcings and support motions is developed from the continuum equations it contains two in plane and two out of plane components one symmetric and one antisymmetric an order two multiple time scales solution is obtained for the case of primary external resonance and three simultaneous internal resonances cable at crossover frequency possible classes of steady state planar non planar unimodal multimodal regular motions of the system are identified and their linearized stability analysis is performed some results are presented to highlight the role played by key perturbations on the onset of various classes the rich behaviour of the system and the influence of some control parameters on the response

multiple time scaling of the response of a duffing oscillator to narrow band random excitation the response of a duffing oscillator to narrow band random excitation is considered two approaches multiple time scaling and stochastic averaging are used both lead to the same set of three state equations for the slowly varying local mean square response or envelope and two other variables the stationary solution of these equations is similar to results previously obtained by using statistical linearization the stationary and nonstationary from the three state equations results compare favourably with digital simulations random jumps are shown to occur if the excitation bandwidth is sufficiently small

on the nonlocal equations and nonlocal charges associated with the harry dym hierarchy a large class of nonlocal equations and nonlocal charges for the harry dym hierarchy is exhibited they are obtained from nonlocal casimirs associated with its bi hamiltonian structure the lax representation for some of these equations is also given

beam beam interaction models under narrow band random excitation abstract the aim of this paper is to continue our previous investigations on beam beam interaction models in particle accelerators by considering these models under narrow band random excitation in this investigation the multiple scale and the moment methods are used the equations of modulation of the amplitude and the phase are derived and perturbation techniques are used to seek approximate steady state solutions two special cases are considered to illustrate this study and excellent agreement is found between analytical and numerical results other cases can be similarly studied the effects of noise and detuning parameters are examined numerically and theoretically and we find that the system demonstrates a diffused limit cycle as the intensity of noise increases the numerical results show that the multiple scale method is effective for these stochastic systems

tri hamiltonian duality between solitons and solitary wave solutions having compact support a simple scaling argument shows that most integrable evolutionary systems which are known to admit a bi hamiltonian structure are in fact governed by a compatible trio of hamiltonian structures we demonstrate how their recombination leads to integrable hierarchies endowed with nonlinear dispersion that supports compactons solitary wave solutions having compact support or cusped and or peaked solitons a general algorithm for effecting this duality between classical solitons and their nonsmooth counterparts is illustrated by the construction of dual versions of the modified korteweg de vries equation the nonlinear schr odinger equation the integrable boussinesq system used to model the two way propagation of shallow water waves and the ito system of coupled nonlinear wave equations these hierarchies include a remarkable variety of interesting integrable nonlinear differential equations textcopyright 1996 the american physical society

modelling and transient planar dynamics of suspended cables with moving mass in this study the 3d nonlinear equations of motion of the suspended cable with moving mass are obtained via the hamilton principle and its transient linear planar dynamics is investigated considering the quasi static assumption the condensed planar model accounting for the effect of the moving mass is derived and it is then discretized by choosing the static deflection and sine series as shape functions it is shown that this expansion shows good convergence features the newmark method is used to investigate the transient response the effects of the inertia force mass sag and velocity of the moving mass on the transient dynamics of the suspended cable are systematically investigated finally the horizontal tension of the suspended cable and the case of sequentially moving masses are examined

nonlinear interactions and chaotic dynamics of suspended cables with three to one internal resonances abstract nonlinear planar oscillations of suspended cables subjected to external excitations with three to one internal resonances are investigated at first the galerkin method is used to discretize the governing nonlinear integral partial differential equation then the method of multiple scales is applied to obtain the modulation equations in the case of primary resonance the equilibrium solutions the periodic solutions and chaotic solutions of the modulation equations are also investigated the newton raphson method and the pseudo arclength path following algorithm are used to obtain the frequency force response curves the supercritical hopf bifurcations are found in these curves choosing these bifurcations as the initial points and applying the shooting method and the pseudo arclength path following algorithm the periodic solution branches are obtained at the same time the floquet theory is used to determine the stability of the periodic solutions numerical simulations are used to illustrate the cascades of period doubling bifurcations leading to chaos at last the nonlinear responses of the two degree of freedom model are investigated

transverse vibrations of elastically connected double string complex system part i free vibrations this paper analyzes the forced transverse vibrations of an elastically connected double string complex continuous system the general solutions of forced vibrations of strings subjected to arbitrarily distributed continuous loads are found by using the method of expansion in a series of the mode shape functions different cases of exciting loadings are analyzed the action of stationary harmonic loads and moving concentrated forces is considered vibrations caused by the harmonic exciting forces are discussed and conditions of resonance and dynamic vibration absorption are formulated thus the string type dynamic absorber can be used to suppress the excessive vibrations of corresponding string systems the dynamic string type damper is a new concept of a continuous dynamic vibration absorber cdva it is shown that a corresponding two degree of freedom discrete system is an analogue of an elastically connected double string complex system theoretical analysis is illustrated by a numerical example

bounded input bounded output stability of a damped nonlinear string in this paper we prove that the displacement of a damped nonlinear string under bounded distributed forces is bounded the proof is achieved by showing that an energy like lyapunov function corresponding to the string is bounded

statistical dynamics of on line independent component analysis the learning dynamics of on line independent component analysis is analysed in the limit of large data dimension we study a simple hebbian learning algorithm that can be used to separate out a small number of non gaussian components from a high dimensional data set the de mixing matrix parameters are confined to a stiefel manifold of tall orthogonal matrices and we introduce a natural gradient variant of the algorithm which is appropriate to learning on this manifold for large input dimension the parameter trajectory of both algorithms passes through a sequence of unstable fixed points each described by a diffusion process in a polynomial potential choosing the learning rate too large increases the escape time from each of these fixed points effectively trapping the learning in a sub optimal state in order to avoid these trapping states a very low learning rate must be chosen during the learning transient resulting in learning time scales of o n2 or o n3 iterations where n is the data dimension escape from each sub optimal state results in a sequence of symmetry breaking events as the algorithm learns each source in turn this is in marked contrast to the learning dynamics displayed by related on line learning algorithms for multilayer neural networks and principal component analysis although the natural gradient variant of the algorithm has nice asymptotic convergence properties it has an equivalent transient dynamics to the standard hebbian algorithm

multiple internal resonances and non planar dynamics of shallow suspended cables to the harmonic excitations abstract in the present study the nonlinear response of a shallow suspended cable with multiple internal resonances to the primary resonance excitation is investigated the method of multiple scales is applied directly to the nonlinear equations of motion and associated boundary conditions to obtain the modulation equations and approximate solutions of the cable frequency response curves and force response curves are used to study the equilibrium solution and its stability the effects of the excitation amplitude on the frequency response curves of the cable are also analyzed moreover the chaotic dynamics of the shallow suspended cable is investigated by means of numerical simulations

dynamics of learning in multilayer perceptrons near singularities the dynamical behavior of learning is known to be very slow for the multilayer perceptron being often trapped in the plateau it has been recently understood that this is due to the singularity in the parameter space of perceptrons in which trajectories of learning are drawn the space is riemannian from the point of view of information geometry and contains singular regions where the riemannian metric or the fisher information matrix degenerates this paper analyzes the dynamics of learning in a neighborhood of the singular regions when the true teacher machine lies at the singularity we give explicit asymptotic analytical solutions trajectories both for the standard gradient sgd and natural gradient ngd methods it is clearly shown in the case of the sgd method that the plateau phenomenon appears in a neighborhood of the critical regions where the dynamical behavior is extremely slow the analysis of the ngd method is much more difficult because the inverse of the fisher information matrix diverges we conquer the difficulty by introducing the blow down technique used in algebraic geometry the ngd method works efficiently and the state converges directly to the true parameters very quickly while it staggers in the case of the sgd method the analytical results are compared with computer simulations showing good agreement the effects of singularities on learning are thus qualitatively clarified for both standard and ngd methods

on the blow up structure for the generalized periodic camassa holm and degasperis procesi equations considered herein are the generalized camassa holm and degasperis procesi equations in the spatially periodic setting the precise blow up scenarios of strong solutions are derived for both of equations several conditions on the initial data guaranteeing the development of singularities in finite time for strong solutions of these two equations are established the exact blow up rates are also determined finally geometric descriptions of these two integrable equations from non stretching invariant curve flows in centro equiaffine geometries pseudo spherical surfaces and affine surfaces are given

analysis of natural gradient descent for multilayer neural networks natural gradient descent ngd is an on line algorithm for redefining the steepest descent direction an analysis of ngd for training a multilayer neural network is presented using statistical mechanics the performance can be significantly improved using ngd algorithm and can be used for both the transient and asymptotic stages of learning

non linear responses of suspended cables to primary resonance excitations we investigate the non linear forced responses of shallow suspended cables we consider the following cases 1 primary resonance of a single in plane mode and 2 primary resonance of a single out of plane mode in both cases we assume that the excited mode is not involved in an autoparametric resonance with any other mode we analyze the system by following two approaches in the first we discretize the equations of motion using the galerkin procedure and then apply the method of multiple scales to the resulting system of non linear ordinary differential equations to obtain approximate solutions discretization approach in the second we apply the method of multiple scales directly to the non linear integral partial differential equations of motion and associated boundary conditions to determine approximate solutions direct approach we then compare results obtained with both approaches and discuss the influence of the number of modes retained in the discretization procedure on the predicted solutions

natural gradient works efficiently in learning when a parameter space has a certain underlying structure the ordinary gradient of a function does not represent its steepest direction but the natural gradient does information geometry is used for calculating the natural gradients in the parameter space of perceptrons the space of matrices for blind source separation and the space of linear dynamical systems for blind source deconvolution the dynamical behavior of natural gradient online learning is analyzed and is proved to be fisher efficient implying that it has asymptotically the same performance as the optimal batch estimation of parameters this suggests that the plateau phenomenon which appears in the backpropagation learning algorithm of multilayer perceptrons might disappear or might not be so serious when the natural gradient is used an adaptive method of updating the learning rate is proposed and analyzed

parameter identifiability in statistical machine learning a review this review examines the relevance of parameter identifiability for statistical models used in machine learning in addition to defining main concepts we address several issues of identifiability closely related to machine learning showing the advantages and disadvantages of state of the art research and demonstrating recent progress first we review criteria for determining the parameter structure of models from the literature this has three related issues parameter identifiability parameter redundancy and reparameterization second we review the deep influence of identifiability on various aspects of machine learning from theoretical and application viewpoints in addition to illustrating the utility and influence of identifiability we emphasize the interplay among identifiability theory machine learning mathematical statistics information theory optimization theory information geometry riemann geometry symbolic computation bayesian inference algebraic geometry and others finally we present a new perspective together with the associated challenges

random superharmonic and subharmonic response multiple time scaling of a duffing oscillator narrowband random third order superharmonic and one third order subharmonic response of a duffing oscillator is discussed a multiple time scale approach is used the work is a more rigorous approach than that already published on random superharmonic response by the authors and the results obtained corroborate the previous ones in particular random jumps in amplitude can occur but are less likely to occur as the bandwidth of the excitation is increased the results are further corroborated by comparison with digital simulation in addition new results on random subharmonic response are also presented obtained by using the same approach

response of a duffing oscillator to combined deterministic harmonic and random excitation the response of duffing oscillator to combined deterministic harmonic and random excitation is investigated the method of harmonic balance and the method of stochastic averaging are used to determine the response of the system theoretical analyses and numerical simulations show that when the intensity of the random excitation increases the non trivial steady state solution may change from a limit cycle to a diffused limit cycle under some conditions the system may have two steady state solutions and jumps may exist

phase plane for narrow band random excitation of a duffing oscillator abstract the response of a duffing oscillator to narrow band random excitation is consedered it is shown that multiple values of the response can occur at some frequencies for very narrow excitation bandwidths but these reduce to a single value as the bandwidth is increased time histories of quantities such as mean square displacement are obtained from the time dependent fokker planck equation smoothed time histories of mean square velocity and displacement are plotted on a phase plane the phase plane can show either one sink or two sinks and a saddle point as in the sinusoidal case but again excitation bandwith determines whether or not multiple sinks occur

stochastic response and stability of the duffing oscillator under narrowband excitation the duffing equation under narrowband gaussian excitation is studied almost sure asymptotic stability analysis is used to discuss the realizability of multivalued steady states the theoretical predictions are supported by numerical simulation results

deformations of surfaces associated with integrable gauss mainardi codazzi equations using the formulation of the immersion of a two dimensional surface into the three dimensional euclidean space proposed recently a mapping from each symmetry of integrable equations to surfaces in r3 can be established we show that among these surfaces the sphere plays a unique role indeed under the rigid su 2 rotations all integrable equations are mapped to a sphere furthermore we prove that all compact surfaces generated by the infinitely many generalized symmetries of the sine gordon equation are homeomorphic to a sphere we also find some new weingarten surfaces arising from the deformations of the modified kurteweg de vries and of the nonlinear schrodinger equations surfaces can also be associated with the motion of curves we study curve motions on a sphere and we identify a new integrable equation characterizing such a motion for a particular choice of the curve velocity

response of a strongly non linear oscillator to narrowband random excitations abstract the principal resonance of a van der pol duffing oscillator subject to narrowband random excitations has been studied by introducing a new expansion parameter e e e u 0 the method of multiple scales is adapted for the strongly non linear system the behavior of steady state responses together with their stability and the effects of system damping and the detuning and magnitude of the random excitation on steady state responses are analyzed in detail theoretical analyses are verified by some numerical results it is found that when the random noise intensity increases the steady state solution may change form a limit cycle to a diffused limit cycle and the system may have two different stable steady state solutions for the same excitation under certain conditions the results obtained for the strongly non linear oscillator complement previous results in the literature for weakly non linear systems

parametric oscillations of a non linear string abstract a study of the nature of periodic oscillations bifurcation of vibrations and growth of chaos is presented with the negative stiffness parametric oscillations of a non linear stretched string used as a model the influences of the variables on the behavior of the oscillations are analyzed the variables studied are the damping factor the frequency ratio the non linear term the parametric driving amplitudeqand the linear stiffness p the oscillations of this system were approximated and calculated by computer these results were analyzed in several ways trajectories poincare maps probability density functions of displacement and velocity root mean square responses and lyapunov exponents these pictorial representations were in good agreement with the independently calculated map of chaotic periodic regions over p q it was found that the magnitudes of the linear stiffness and the parametric driving amplitude are the most important determinants of the presence of chaos additionally the damping factor was found to influence the appearance of chaos varying the non linear term was found to have only a minimal effect on the appearance of chaos similarly the frequency ratio was found not to effect the occurrence of chaos

integrable equations arising from motions of plane curves the motion of plane curves in klein geometry is studied it is shown that the kdv harry dym sawada kotera burgers the defocusing mkdv hierarchies the camassa holm and the kaup kupershmidt equation naturally arise from the motions of plane curves in sl 2 sim 2 sa 2 and a 2 geometries these local and nonlocal dynamics conserve global geometric quantities of curves such as perimeter and enclosed area motions of curves in euclidean special linear and similarity geometries corresponding to the traveling wave solutions of the mkdv kdv and burgers equations are discussed

the korteweg de vries hierarchy as dynamics of closed curves in the plane the korteweg de vries modified korteweg de vries and harry dym hierarchies of integrable systems are shown to be equivalent to a hierarchy of chiral shape dynamics of closed curves in the plane these purely local dynamics conserve an infinite number of global geometric properties of the curves such as perimeter and enclosed area several techniques used to study these integrable systems are shown to have simple differential geometric interpretations connections with incompressible inviscid fluid flow in two dimensions are suggested

non linear resonance in strings under narrow band random excitation part i planar response and stability non linear planar response of a string to planar narrow band random excitation is investigated in this paper a response equation for the mean square deflection 2 is obtained under a single mode approximation by using the equivalent linearization technique it is shown that the response is triple valued as in the case of harmonic excitation if the centre frequency of excitation lies in a certain specified range the triple valued response occurs only if the excitation bandwidth is smaller than a critical value crit which is a monotonically increasing function of the intensity of excitation an approximate method of investigating the almost sure asymptotic stability of the solution is presented and regions of instability in the 2 plane have been charted it is shown that planar response can become unstable either due to an unbounded growth of the in plane component of motion or due to a spontaneous appearance of an out of plane component