response of an sdof system to periodic external and parametric excitations with random phase modulations a single degree of freedom system is considered which is excited simultaneously by external and parametric time variant loadings both these dynamic loads are assumed to be periodic in time with some random disorder or phase modulations response analysis is made for this system by using stochastic averaging with subsequent application of the method of moments to the resulting set of the shortened ito sdes the results of steady state mean square response analysis are presented in terms of the corresponding amplification factor illustrating its dependence on excitation system bandwidth ratios and detuning between expected excitation frequencies and the system s natural frequency

transition functions paths and path integrals the main theme of this expository paper is the relation between analysis and probability in the context of diffusion theory section 1 discusses in rather heuristic fashion the very satisfying solution to the problem of describing diffusion processes which kolmogorov achieved via pde theory the theory of partial differential equations and his criterion for path continuity section 2 describes how ito calculus totally transformed the subject by allowing us to construct the sample paths of a diffusion process x by solving an sde stochastic differential equation driven by brownian motion of course sdes have great intrinsic importance too as noisy perturbations of nonlinear dynamical systems though 2 begins heuristically the mathematics is then tightened up this paper is after all a tribute to the man whose greatest contribution to science is his setting probability theory on a rigorous foundation once kolmogorov s precise language is available 2 then takes a quick sight seeing trip through some of the great developments by doob ito and their successors in an age in which so many do simulations of ito equations i have explained precisely in the briefest possible fashion what the exact theory is it is easy and usable you will just have time for a snapshot of how brownian motion on the orthonormal frame bundle is linked to index theorems and of what the malliavin calculus is about you will however be advised on guide books on these and other areas including physicists favourites large deviations measure valued diffusions etc so that you can later explore at your leisure confession the paper consists very largely of selected tracks remixed from the album rogers williams 1987 diffusions markov processes and martingales chichester wiley tributes to kolmogorov s work in probability and statistics have appeared in every book ever written on probability and in ann probability 17 1989 815 964 ann statist 18 1990 987 1031 bull lond math soc 22 1990 31 100 teor veroyatnost primenen 34 1989 no 1 usp mat nauk 43 1988 no 6 the official biography by shiryaev will be a wonderful volume for me this paper is a further expression of my thanks to kolmogorov and as he would have wished to levy doob and ito too

moment lyapunov exponent and stability index for linear conservative system with small random perturbation asymptotic expansion series for the moment lyapunov exponent and stability index are constructed and justified for the two dimensional linear stochastic system close to a harmonic oscillator as an example a one degree of freedom mechanical system parametrically excited in stiffness and damping is considered and several terms of the expansion are obtained

asymptotic analysis of the lyapunov exponent and rotation number of the random oscillator and application we construct asymptotic expansions for the exponential growth rate lyapunov exponent and rotation number of the random oscillator when the noise is large small rapidly varying or slowly varying we then apply our results to problems in the stability of the random oscillator the spectrum of the one dimensional random schr6dinger operator and wave propagation in a one dimensional random medium

non linear flexural flexural torsional extensional dynamics of beams i formulation abstract the non linear differential equations of motion and boundary conditions for euler bernoulli beams able to experience flexure along two principal directions and thus flexure in any direction in space torsion and extension are formulated the beam s material is assumed to be hookean but its properties may vary along its span the nonlinearities present in the differential equations include contributions from the curvature expression and from inertia terms a set of differential equations with polynomial nonlinearities to cubic order suitable for a perturbation analysis of the motion is also developed and the validity of the inextensional approximation is assessed the equations developed here reduce to those for an inextensional beam in part ii of this paper a specific example of application is analyzed and the results obtained are compared with those available in the literature where several non linear terms have been neglected a priori

stochastic averaging of strongly non linear oscillators under combined harmonic and white noise excitations abstract a stochastic averaging procedure of strongly non linear oscillators subject to external and or parametric excitations of both harmonic and white noise forces is developed by using the so called generalized harmonic functions the procedure is applied to a duffing oscillator with hardening stiffness under both external harmonic excitation and external and parametric excitations of white noises the averaged fokker planck kolmogrov equation is solved by using the path integration method based on the stationary joint probability density of amplitude and phase obtained by using the stochastic averaging and the path integration the stochastic jump of the duffing oscillator under combined harmonic and white noise excitations and its bifurcation as the system parameters frequency ratio strength of the non linearity amplitude of harmonic excitation and intensity of white noise change are examined for the first time

moment lyapunov exponents of the stochastic parametrical hill s equation abstract the lyapunov exponent and moment lyapunov exponents of hill s equation with frequency and damping coefficient fluctuated by white noise stochastic process are investigated a perturbation approach is used to obtain explicit expressions for these exponents in the presence of small intensity noises the results are applied to the study of the almost sure and the moment stability of the stationary solutions of the thin simply supported beam subjected to axial compressions and time varying damping which are small intensity stochastic excitations

entropy formula for random transformations we exhibit random strange attractors with random sinai bowen ruelle measures for the composition of independent random diffeomorphisms

almost sure stability of linear stochastic differential equations with jumps under the nondegenerate condition as in the diffusion case see 14 21 6 the linear stochastic jump diffusion process projected on the unit sphere is a strong feller process and has a unique invariant measure which is also ergodic using the relation between the transition probabilities of jump diffusions and the corresponding diffusions due to 22 the unique deterministic lyapunov exponent can be represented by the furstenberg khas minskii formula as an integral over the sphere or the projective space with respect to the ergodic invariant measure so that the almost sure asymptotic stability of linear stochastic systems with jumps depends on its sign the critical case of zero lyapunov exponent is discussed and a large deviations result for asymptotically stable systems is further investigated several examples are treated for illustration

almost sure stability of dynamical systems under combined harmonic and stochastic excitations abstract the almost sure asymptotic stability and the associated sample behavior of mechanical and structural systems under combined harmonic and stochastic excitations are examined the method of extended stochastic averaging developed by the author is used to obtain a lower dimensional markov processes for multi degree of freedom structures subjected to both harmonic and stochastic excitations provided that the intensities of these excitations are small the approach in this paper formulated for the study of almost sure asymptotic stability is based on the well developed concept of lyapunov exponents the largest lyapunov exponent is obtained explicitly by using the lower dimensional approximate ito equations the results from this analysis are applied to study the almost sure asymptotic stability of uncoupled flapping motion of rotor blades in forward flight under the effect of atmospheric turbulence

a limit shape theorem for periodic stochastic dispersion we consider the evolution of a connected set on the plane carried by a space periodic incompressible stochastic flow while for almost every realization of the stochastic flow at time t most of the particles are at a distance of order t away from the origin there is a measure zero set of points that escape to infinity at the linear rate we study the set of points visited by the original set by time t and show that such a set when scaled down by the factor of t has a limiting nonrandom shape c 2004 wiley periodicals inc

a formula connecting sample and moment stability of linear stochastic systems given the linear stochastic system dot x a xi t x x in mathbb r d xi t some nice diffusion sample and moment stability of x equiv 0 are studied via the sample lyapunov exponent lambda and the lyapunov exponent of the pth mean g p of a solution generalizing a result of molchanov for the undamped linear oscillator it is shown that g p is a smooth convex function of p in mathbb r with g 0 0 and g 0 lambda for trace a 0 a second zero of g at p d is detected this allows one to draw simple but quite complete conclusions about the relation between sample and moment stability

blow up of h1 solution for the nonlinear schrodinger equation we consider the blow up of the solution in h1 for the following nonlinear schrodinger equation i tu u u p 1u x rn t 0 u 0 x u0 x x rn t 0 where n 2 and 1 4 n p min n 2 n 2 5 we prove that if the initial data u0 in h1 are radially symmetric and have negative energy then the solution of in h1 blows up in finite time we do not assume that xu0 l2 and therefore our result is the generalization of the results of glassey 4 and m tsutsumi 18 for the radially symmetric case

moment lyapunov exponents of a two dimensional system under bounded noise parametric excitation the moment lyapunov exponents of a two dimensional system under bounded noise parametric excitation are studied in this paper the method of regular perturbation is applied to obtain weak noise expansions of the moment lyapunov exponent lyapunov exponent and stability index in terms of the small fluctuation parameter

response of a cantilever model with a surface crack under basal white noise excitation abstract a cantilever with an open surface crack excited by the pedestal movement with the form of the gaussian white noise was modeled in this paper the open crack was simplified as an elastic hinge and the modal functions of the cracked cantilever were obtained the dynamical equation governing the transverse motion was gained with the kane s method after sufficient consideration of the geometrical nonlinearity and the axial inertia both quasi conservative stochastic averaging method and the improved energy envelop stochastic averaging method which is effective on strong nonlinearity were applied to analyze the responses of the cantilever model theoretically the stationary probability density function pdf of the amplitude as well as the joint stationary pdf of the displacement and velocity was investigated for the purpose of exploring the reliability of the cantilever the reliability function was established to avoid the potential fatigue or damage numerical simulations were carried out to vindicate the theoretical analysis the advantage of the improved energy envelop stochastic averaging method was explicated by the digital results of both the stationary pdf of the amplitude and the joint pdf of displacement and velocity furthermore the accuracy of the prediction on the reliability given by the theoretical analysis was also vindicated by the numerical simulation monte carlo method of the reliability

on the rate at which a homogeneous diffusion approaches a limit an application of large deviation theory to certain stochastic integrals soit x t la solution d une equation differentielle stochastique dont les coefficients sont homogenes de degre 1 sous des conditions douces on montre que des limites comme lim t 1 t log p ix t x o r existe et on donne une formule pour leur calcul

stochastic stability of coupled oscillators simple and combination resonances in coupled oscillators under a random parametric excitation are studied an efficient numerical method for obtaining the regions of mean square stability is proposed based on the method stability diagrams for different values of the excitation parameter and for four form of the coupling parameters are presented it is found that the random excitation can have a stabilizing effect under certain conditions

moment lyapunov exponents of a two dimensional system under combined harmonic and real noise excitations abstract the moment lyapunov exponents and the lyapunov exponents of a two dimensional system under combined excitations of harmonic and real noise which is modelled as an ornstein uhlenbeck process are studied the moment lyapunov exponents and the lyapunov exponents are important characteristics determining the moment and almost sure stability of a stochastic dynamical system the eigenvalue problem governing the moment lyapunov exponent is established a regular perturbation method is applied to solve the eigenvalue problem to obtain second order weak noise expansions of the moment lyapunov exponents the influence of the real noise excitation on the parametric resonance due to the harmonic excitation is investigated

the response envelope probability density function of a duffing oscillator with random narrow band excitation abstract a duffing oscillator excited by narrow band random noise can exhibit multi valued behaviour with random jumps between two pseudo stable states in this paper an approximate form of the probability density function or the envelope of the response is obtained state equations for the response envelope with the random excitation envelope as input are obtained by stochastic averaging the joint probability density function of response envelope and input envelope can be found approximately for certain ranges of the system parameters the response envelope probability density function shows two maxima in the cases where the response is multi valued numerical simulation confirms the results

moment stability for linear systems with a random parametric excitation moment stability for linear systems with a nonwhite parametric noise is considered a method of reduction of the study of this stability to the study of stability for large scale matrices is proposed mean square stability diagrams for random harmonic oscillator are presented

sample path properties of the stochastic flows we consider a stochastic flow driven by a finite dimensional brownian motion we show that almost every realization of such a flow exhibits strong statistical properties such as the exponential convergence of an initial measure to the equilibrium state and the central limit theorem the proof uses new estimates of the mixing rates of the multi point motion

the moment lyapunov exponent of a co dimension two bifurcation system driven by non gaussian colored noise in the present paper based on the concept of the pth moment lyapunov exponent the stochastic stability of a typical co dimension two bifurcation system that is on a three dimensional center manifold and possesses two pure imaginary eigenvalues and one zero eigenvalue and is excited by a non gaussian colored noise is investigated the non gaussian colored noise is treated as an ornstein uhlenbeck process by means of the path integral approach based on the perturbation approach and the green s functions method the second differential eigenvalue equation which governing the moment lyapunov exponent is established by solving the eigenvalue problem the weak noise asymptotic expansions for the finite pth moment lyapunov exponent are obtained and which matches the approximation of the numerical monte carlo simulations finally the conclusions are given

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

kraichnan flow in a square an example of integrable chaos the kraichnan flow provides an example of a random dynamical system accessible to an exact analysis we study the evolution of the infinitesimal separation between two lagrangian trajectories of the flow its long time asymptotics is reflected in the large deviation regime of the statistics of stretching exponents whereas in the flow that is isotropic at small scales the distribution of such multiplicative large deviations is gaussian this does not have to be the case in the presence of an anisotropy we analyze in detail the flow in a two dimensional periodic square where the anisotropy generally persists at small scales the calculation of the large deviation rate function of the stretching exponents reduces in this case to the study of the ground state energy of an integrable periodic schrodinger operator of the lame type the underlying integrability permits to explicitly exhibit the non gaussianity of the multiplicative large deviations and to analyze the time scales at which the large deviation regime sets in in particular we indicate how the divergence of some of those time scales when the two lyapunov exponents become close allows a discontinuity of the large deviation rate function in the parameters of the flow the analysis of the two dimensional anisotropic flow permits to identify the general scenario for the appearance of multiplicative large deviations together with the restrictions on its applicability

image solution for clamped finite beams an image solution for the forced response of a clamped finite beam is developed the finite beam is replaced by an infinite beam under spatially periodic excitation and periodic support reactions the response is then found by a fourier transform approach making use of the periodicity an explicit expression is found using the poisson sum formula the aim is to describe the potential of using the image method for clamped structural acoustic and vibration problems

large deviations and stochastic flows of diffeomorphisms previous results in the theory of large deviations for additive functionals of a diffusion process on a compact manifold m are extended and then applied to the analysis of the lyapunov exponents of a stochastic flow of diffeomorphisms of m an approximation argument relates these results to the behavior near the diagonal in m2 of the associated two point motion finally it is shown under appropriate non degeneracy conditions that the two point motion is ergodic on m2 if the top lyapunov exponent is positive

moment lyapunov exponents of a two dimensional system under real noise excitation abstract the moment lyapunov exponents of a two dimensional system under real noise excitation an ornstein uhlenbeck process are studied in this paper the method of regular perturbation is applied to obtain the weak noise expansions of the moment lyapunov exponent lyapunov exponent and stability index in terms of the small fluctuation parameter

nonlinear schrodinger equations and sharp interpolation estimates a sharp sufficient condition for global existence is obtained for the nonlinear schrodinger equation begin array 20 c nls 2i phi t delta phi left phi right 2 sigma phi 0 x in mathbb r n t in mathbb r end array in the case 2 n this condition is in terms of an exact stationary solution nonlinear ground state of nls it is derived by solving a variational problem to obtain the best constant for classical interpolation estimates of nirenberg and gagliardo

on evaluating the donsker varadhan i function let x t be a feller process on a complete separable metric space a and consider the occupation measure l t omega cdot int t 0 chi cdot x s ds the i function is defined for mu in mathscr p a the set of probability measures on a by i mu inf u in mathscr d int a lu u d mu where l mathscr d is the generator of the process and mathscr d subset mathscr d consists of the strictly positive functions in mathscr d the i function determines the asymptotic rate of decay of p 1 t l t omega cdot in g for g subset mathscr p a the first difficulty encountered in evaluating i mu is that the domain mathscr d is generally not known explicitly in this paper we prove a theorem which allows us to restrict the calculation of the infimum to a nice subdomain we then apply this general result to diffusion processes with boundaries

optimal feedback control of strongly non linear systems excited by bounded noise abstract a strategy for non linear stochastic optimal control of strongly non linear systems subject to external and or parametric excitations of bounded noise is proposed a stochastic averaging procedure for strongly non linear systems under external and or parametric excitations of bounded noise is first developed then the dynamical programming equation for non linear stochastic optimal control of the system is derived from the averaged ito equations by using the stochastic dynamical programming principle and solved to yield the optimal control law the fokker planck kolmogorov equation associated with the fully completed averaged ito equations is solved to give the response of optimally controlled system the application and effectiveness of the proposed control strategy are illustrated with the control of cable vibration in cable stayed bridges and the feedback stabilization of the cable under parametric excitation of bounded noise

principal response of duffing oscillator to combined deterministic and narrow band random parametric excitation abstract the principal resonance of a duffing oscillator to combined deterministic and narrow band random parametric excitations is investigated in particular the case in which the parametric terms share close frequencies is examined the method of multiple scales is used to determine the equations of modulation of amplitude and phase the behavior stability and bifurcation of steady state response are studied by means of qualitative analyses jumps are shown to occur if the random excitation is small the effects of damping detuning and magnitudes of deterministic and narrow band parametric excitations are analyzed the theoretical analyses are verified by numerical results

dynamic responses of axially moving viscoelastic beam under a randomly disordered periodic excitation abstract we investigate dynamic responses of axially moving viscoelastic beam subject to a randomly disordered periodic excitation the method of multiple scales is used to derive the analytical expression of first order uniform expansion of the solution based on the largest lyapunov exponent the almost sure stability of the trivial steady state solution is examined meanwhile we obtain the first order and the second order steady state moments for the non trivial steady state solutions specially we discuss the first mode theoretically and numerically results show that under the same conditions of the parameters as the intensity of the random excitation increases non trivial steady state solution fluctuation will become strenuous which will result in the non trivial steady state solution lose stability and the trivial steady state solution can be a possible in the case of parametric principal resonance the stochastic jump is observed for the first mode which indicates that the stationary joint probability density concentrates at the non trivial solution branch when the random excitation is small but with the increase of intensity of the random excitation the probability of the trivial steady state solution will become larger this phenomenon of stochastic jump can be defined as a stochastic bifurcation

simultaneous combination principal parametric and internal resonances in a slender beam with a lumped mass three mode interactions abstract the non linear dynamics of a parametrically base excited slender beam carrying a lumped mass with an excitation frequency nearly equal to twice the second mode natural frequency is examined the mass and the position of the attached element are so adjusted that the system exhibits 1 3 5 internal resonance by virtue of which a combination resonance of type 1 3 occurs simultaneously in the system the method of multiple scales mms is applied to reduce the second order temporal differential equations of motion of the system to a set of first order differential equations which is used to study the steady state periodic quasi periodic and chaotic responses of the system for different control parameters namely frequency and amplitude of base excitation damping internal detuning etc the results of perturbation analysis are compared with those obtained by numerically integrating the original temporal equations of motion poincare section and lyapunov exponents are used to characterize the chaotic responses

moment lyapunov exponents of a two dimensional system under both harmonic and white noise parametric excitations the moment lyapunov exponents and the lyapunov exponents of a 2d system under both harmonic and white noise excitations are studied the moment lyapunov exponents and the lyapunov exponents are important characteristics determining the moment and almost sure stability of a stochastic dynamical system the eigenvalue problem governing the moment lyapunov exponent is established a singular perturbation method is applied to solve the eigenvalue problem to obtain second order weak noise expansions of the moment lyapunov exponents the influence of the white noise excitation on the parametric resonance due to the harmonic excitation is investigated

almost sure and moment lyapunov exponents for a stochastic beam equation the transverse vibrations of an euler bernoulli beam with axial tension p and axial white noise forcing are given by mytt m yt eiyxxxx pyxx 0yxxw t 0 x l with hinged endpoints y 0 t yxx 0 t y l t yxx l t 0 the solution may be written y x t n 1 sin n xl cn t where the amplitude cn t of the nth mode of vibration is given by a two dimensional linear stochastic ordinary differential equation the main results give formulas for the almost sure and moment lyapunov exponents for the spde in terms of the almost sure and moment lyapunov exponents for each of the sdes corresponding to modes which are present in the initial condition it is shown that the almost sure and moment lyapunov exponents for the spde depend sensitively on the initial distribution of energy amongst the infinitely many modes of vibration and that they cannot be well approximated using only finitely many modes of vibration

dynamics of flexible beams undergoing overall motions abstract a modelling method for straight beams undergoing large overall motions as well as small elastic deformations is presented in this paper different from the classical linear cartesian modelling method which employs three cartesian deformation variables the present modelling method uses a non cartesian variable along with two cartesian variables to describe the elastic deformation a quadratic form of the strain energy expressed with the hybrid set of deformation variables is used to obtain the generalized active forces and a geometric constraint equation relating the non cartesian variable and cartesian variables is used to obtain the generalized inertia forces in the equations of motion the present modelling method not only provides accurate simulation results but also clarifies the limit of validity of the classical linear cartesian modelling method

t e harris s contributions to recurrent markov processes and stochastic flows this is a brief survey of t e harris s work on recurrent markov processes and on stochastic flows and of some more recent work in these fields

hausdorff dimension in stochastic dispersion we consider the evolution of a connected set in euclidean space carried by a periodic incompressible stochastic flow while for almost every realization of the random flow at time t most of the particles are at a distance of order sqrt t away from the origin there is an uncountable set of measure zero of points which escape to infinity at the linear rate in this paper we prove that this set of linear escape points has full hausdorff dimension

almost sure enhanced dissipation and uniform in diffusivity exponential mixing for advection diffusion by stochastic navier stokes we study the mixing and dissipation properties of the advection diffusion equation with diffusivity 0 kappa ll 1 and advection by a class of random velocity fields on mathbb t d d 2 3 including solutions of the 2d navier stokes equations forced by sufficiently regular in space non degenerate white in time noise we prove that the solution almost surely mixes exponentially fast uniformly in the diffusivity kappa namely that there is a deterministic exponential rate independent of kappa such that all mean zero h 1 initial data decays exponentially fast in h 1 at this rate with probability one this implies almost sure enhanced dissipation in l 2 specifically that there is a deterministic uniform in kappa exponential decay in l 2 after time t rsim left log kappa right both the mathcal o left log kappa right time scale and the uniform in kappa exponential mixing are optimal for lipschitz velocity fields this work is also a major step in our program on scalar mixing and lagrangian chaos necessary for a rigorous proof of the batchelor power spectrum of passive scalar turbulence

stochastic jump and bifurcation of a duffing oscillator under narrow band excitation abstract the variance of the displacement the probability densities of the displacement velocity and amplitude and the joint probability density of the displacement and velocity of the stationary response of a duffing oscillator to narrow band gaussian random excitation for a number of combinations of the parameters of the oscillator and excitation obtained from digital simulation are presented based on these results the stochastic jump and bifurcation of the system are examined it is shown that for each combination of the parameters all the statistics of the stationary response are unique and independent of initial conditions in a certain domain of the parameter space there are two more probable motions in the stationary response and jumps may occur the jump is a transition from one more probable motion to another or vice versa outside of this domain the stationary response is either nearly gaussian or like a diffused limit cycle as the parameters cross the boundary of the domain the qualitative behavior of the stationary response changes and it is a special kind of bifurcation it is pointed out that some previous results and explanations on this problem are incorrect

maximal lyapunov exponent and almost sure stability for stochastic mathieu duffing systems abstract the principal resonance of a mathieu duffing oscillator to random excitation is investigated the method of multiple scales is used to determine the equations of modulation of amplitude and phase the effects of damping detuning cubic term and magnitudes of random excitation are analyzed the explicit asymptotic formulas for the maximal lyapunov exponent is obtained the almost sure stability or instability of the stochastic mathieu duffing system depends on the sign of the maximal lyapunov exponent in the last part of the work the numerical results are obtained

comparison of some approaches to stability of a sdof system under random parametric excitation we give a rigorous comparison of the asymptotic stochastic averaging method and the moment lyapunov exponent approach in assessing the mean square stability through an example of a single degree of freedom system under small periodic parametric excitation with random phase modulation the comparison is based on an efficient numerical procedure to determine the regions of mean square stability

lyapunov exponents and relative entropy for a stochastic flow of diffeomorphisms the lyapunov exponents 1 2 d for a stochastic flow of diffeomorphisms of a d dimensional manifold m with a strongly recurrent one point motion describe the almost sure limiting exponential growth rates of tangent vectors under the flow this paper shows how the lyapunov exponents are related to measure preserving properties of the stochastic flow on m and of the induced stochastic flow on the projective bundle pm relative entropy is used to quantify the extent to which a measure fails to be invariant under the flow the results include the following if m is compact and if the one point motion on m is a non degenerate diffusion with stationary probability measure then 1 d 0 with equality if and only if the flow preserves almost surely if in addition the induced one point motion on pm satisfies a weak non degeneracy condition then 1 d if and only if there is a smooth riemannian structure on m with respect to which the flow is conformal almost surely

largest lyapunov exponent for second order linear systems under combined harmonic and random parametric excitations the principal resonance of a second order linear stochastic oscillator to combined harmonic and random parametric excitations is investigated the method of multiple scales is used to determine the equations of modulation of amplitude and phase the effects of damping detuning bandwidth and magnitudes of random excitation are analyzed the method of path integration is used to obtain the steady state probability density function of the system and then the largest lyapunov exponent is calculated the almost sure stability or instability of the stochastic system depends on the sign of the largest lyapunov exponent the theoretical analyses are verified by numerical results

closure method and asymptotic expansions for linear stochastic systems a closure procedure for the hierarchy of moment equations related to linear systems of ordinary differential equations with a random parametric excitation is introduced a generalization of pringsheim s theorem for continued fractions is used in a proof of the procedure convergence the boundary function method for singular perturbation problems is applied to obtain asymptotic expansions for the moments of the solutions of such systems

study on connections of the mfs trefftz method indirect biem and invariant mfs in the three dimensional laplace problems containing spherical boundaries following the success of a study on the method of fundamental solutions using an image concept 13 we extend to solve the three dimensional laplace problems containing spherical boundaries by using the three approaches the case of eccentric sphere for the laplace problem is considered the optimal locations for the source distribution to include the foci in the mfs are also examined by using the image concept in the 3d problems whether a free constant is required or not in the mfs is also studied the error distribution is discussed after comparing with the analytical solution derived by using the bispherical coordinates besides the relationship between the trefftz bases and the singularity in the mfs for the three dimensional laplace problems is also addressed it is found that one source of the mfs contains several interior and exterior trefftz sets through a degenerate kernel on the contrary one single trefftz base can be superimposed by some lumped sources in the mfs through an indirect biem based on this finding the relationship between the fictitious boundary densities of the indirect biem and the singularity strength in the mfs can be constructed due to the fact that the mfs is a lumped version of an indirect biem

moment lyapunov exponents and stochastic stability of a double beam system under compressive axial loading the lyapunov exponent and moment lyapunov exponents of two degrees of freedom linear systems subjected to white noise parametric excitation are investigated the method of regular perturbation is used to determine the explicit asymptotic expressions for these exponents in the presence of small intensity noises the lyapunov exponent and moment lyapunov exponents are important characteristics for determining the almost sure and moment stability of a stochastic dynamical system as an example we study the almost sure and moment stability of the double beam system under stochastic compressive axial loading the validity of the approximate results for moment lyapunov exponents is checked by the numerical monte carlo simulation method for this stochastic system

a stochastic hopf bifurcation let x t t 0 be the solution of a stochastic differential equation sde in d which fixes 0 and let denote the lyapunov exponent for the linear sde obtained by linearizing the original sde at 0 it is known that under appropriate conditions the sign of controls the stability instability of 0 and the transience recurrence of x t t 0 on d 0 in particular if the coefficients in the sde depend on some parameterz which is varied in such a way that the corresponding lyapunov exponent z changes sign from negative to positive the almost surely stable fixed point at 0 is replaced by an almost surely unstable fixed point at 0 together with an attracting invariant probability measure z on d 0 in this paper we investigate the limiting behavior of z as z converges to 0 from above the main result is that the rescaled measures 1 z z converge in an appropriate weak sense to a non trivial finite measure on d 0

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

ground state of the mass critical inhomogeneous nonlinear schrodinger functional abstract we study the ground state problem of the nonlinear schrodinger functional with a mass critical inhomogeneous nonlinear term we provide the optimal condition for the existence of ground states and show that in the critical focusing regime there is a universal blow up profile given by the unique optimizer of a gagliardo nirenberg interpolation inequality

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

a function space large deviation principle for certain stochastic integrals consider a process y t of the form y t sum limits 1 d mathop smallint limits 0 t y k x s circ d beta kappa s mathop smallint limits 0 t y 0 x s ds where x t is a diffusion process on a compact manifold and k t k 1 d are independent wiener processes we derive a schilder type large deviation principle for the two families left frac 1 t cdot sum limits 0 n 1 y k t y k t in 0 1 right n 1 2 3 and left frac 1 t cdot y t cdot t t in 0 1 right text t geqq 0 as n respectively t and identify the corresponding rate functions on c 0 1 we give a few applications to the asymptotic distribution of the liapunov exponent of a homogeneous system

lyapunov exponent and rotation number of two dimensional linear stochastic systems with small diffusion by means of a singular perturbation scheme and large deviation estimates we expand the lyapunov exponent and rotation number in terms of the small noise intensity varepsilon for all 2 times 2 systems dx t varepsilon ax t varepsilon dt sqrt varepsilon sum nolimits k 1 r b k x t varepsilon circ dw t k for which a is complex diagonalizable examples are given

lyapunov exponents for small random perturbations of hamiltonian systems consider the stochastic nonlinear oscillator equation ddot x x x 3 varepsilon 2 beta dot x varepsilon sigma x dot w t with beta 0 and sigma neq 0 if 4 beta sigma 2 0 then for small enough varepsilon 0 the system x dot x is positive recurrent in bf r 2 setminus 0 0 now let overline lambda varepsilon denote the top lyapunov exponent for the linearization of this equation along trajectories the main result asserts that overline lambda varepsilon varepsilon 2 3 overline lambda o varepsilon 4 3 qquad mbox as varepsilon to 0 with overline lambda 0 this result depends crucially on the fact that the system above is a small perturbation of a hamiltonian system the method of proof can be applied to a more general class of small perturbations of two dimensional hamiltonian systems the techniques used include i an extension of results of pinsky and wihstutz for perturbations of nilpotent linear systems and ii a stochastic averaging argument involving

stability analysis of parametrically excited systems using spectral collocation abstract the spectral collocation method is used to determine the stability of parametrically excited systems and compared with the traditional transition matrix approach results from a series of test problems demonstrate that spectral collocation converges rapidly in addition the spectral collocation method preserves the sparsity of the underlying system matrices a property not shared by the transition matrix approach as a result spectral collocation can be used for very large systems and can utilize sparse eigensolvers to reduce computational memory and time for the large scale system studied up to 40 degrees of freedom the spectral collocation method was on average an order of magnitude faster than the transition matrix approach using matlab this computational advantage is implementation specific in a c implementation of the algorithm the transition matrix method is faster than the spectral collocation overall the method proves to be simple efficient reliable and generally competitive with the transition matrix method

dynamic modelling and stability analysis of axially oscillating cantilever beams dynamic stability of an axially oscillating cantilever beam is investigated in this paper equations of motion for the axially oscillating beam are derived and transformed into dimensionless forms the equations include harmonically oscillating parameters which are related to the motion induced stiffness variation stability diagrams of the first and the second order approximate solutions are obtained by using the multiple scale perturbation method the stability diagrams show that there exist significant difference between the first and the second order approximate solutions it is also found that relatively large unstable regions exist around the first bending natural frequency twice the first bending natural frequency and twice the second being natural frequency the validity of the stability diagram is verified by direct numerical integrations of the equations of motion of the system

the rate function of hypoelliptic diffusions let be a hypoelliptic diffusion operator on a compact manifold m given an a priori smooth reference measure on m we can then rewrite l as the sum of a symmetric part l0 and a first order drift part y the paper investigates the effect of the drift y on the donsker varadhan rate function corresponding to the large deviations of the empirical measure of the diffusion when y is in the linear span of the first and second order lie brackets of the xi s we derive an affine bound relating the rate functions associated with l and l0 as soon as one point exists where y is not in the linear span of the first and second order lie brackets of the xi s we show that such an affine bound is impossible 1994 john wiley sons inc

phase diagram of the random frequency oscillator the case of ornstein uhlenbeck noise abstract we study the stability of a stochastic oscillator whose frequency is a random process with finite time memory represented by an ornstein uhlenbeck noise this system undergoes a noise induced bifurcation when the amplitude of the noise grows the critical curve that separates the absorbing phase from an extended non equilibrium steady state corresponds to the vanishing of the lyapunov exponent that measures the asymptotic logarithmic growth rate of the energy we derive various expressions for this lyapunov exponent by using different approximation schemes this allows us to study quantitatively the phase diagram of the random parametric oscillator

dynamics of a slender beam with an attached mass under combination parametric and internal resonances part i steady state response the non linear behaviour of a slender beam with an attached mass at an arbitrary position under vertical base excitation is investigated with combination parametric and internal resonances the governing equation which retains the cubic non linearities of geometric and inertial type is discretized by using galerkin s method and the resulting second order temporal differential equation is then reduced by the method of multiple scales to a set of first order non linear differential equations steady state response and its stability are obtained numerically from these reduced equations super and sub critical hopf bifurcations in the trivial as well as non trivial branches and the saddle node or fold type bifurcations in the non trivial branches of the response curves are found the effect of damping amplitude as well as frequency of base excitation the mass ratio and the location of the concentrated mass on the non linear response of the system having internal resonance of 3 1 is studied at length hysteresis saturation and blue sky catastrophe phenomena with bistability interval in the response curves are observed for a wide range of bifurcating parameters

dynamics of a slender beam with an attached mass under combination parametric and internal resonances part ii periodic and chaotic responses abstract the governing second order temporal differential equation of a slender beam with an attached mass at an arbitrary position under vertical base excitation which retains the cubic non linearities of geometric and inertial type is reduced to a set of first order differential equations by the method of normal forms for combination parametric and internal resonances of 3 1 these equations are used to find the periodic quasi periodic and chaotic responses of the system for various bifurcating parameters namely damping amplitude and frequency of base motion attached mass and its location bifurcation set mixed mode oscillation period doubling quasi periodic orbits and different routes to chaos namely alternate periodic chaotic transition torus breakdown and intermittency have been studied for the above mentioned bifurcating parameters using phase portrait poincare section time and power spectra

saturation and resonance of nonlinear system under bounded noise excitation the principal resonance of a 2dof nonlinear oscillator due to bounded random excitations is investigated equations of modulation of response amplitude and phase are derived by the method of multiple scales steady state moments for the response amplitude of the system are determined through the linearized ito differential equation the results of theoretical analyses are verified by numerical simulations saturation phenomena are found in the random counterpart some recommendations for potential applications of this random saturation phenomenon to vibration control problems are given at the end of the paper

explanation on the importance of narrow band random excitation characters in the response of a cantilever beam a detailed theoretical investigation into the first and second mode response of a parametrically excited slender cantilever beam where the narrow band random excitation characters are taken into consideration is presented the method of multiple scales is used to determine a uniform first order expansion of the solution of the nonlinear integro differential equations the first order moment frequency and force response data curves of a specimen beam tested by other investigators are obtained further comparisons have been made and results show that whether the first order moment frequency response data curves or the first order moment force response data curves of the first two modes are all in agreement with other investigators experimental results furthermore the stochastic jump and bifurcation have been investigated for the first modal parametric principal resonance by using the stationary joint probability of amplitude and phase results show that stochastic jump occurs mainly in the region of triple valued solution for the frequency response domain if the bandwidth is a variable and others keep constant the basic phenomena indicate that the most probable motion is around the higher branch when the bandwidth is smaller whereas the most probable motion gradually approaches the lower one when the bandwidth becomes higher if the excitation central frequency f is a variable and others keep constant the basic phenomena imply that the higher is f the more probable is the jump from the higher branch to the lower one once f exceeds an certain value for the force response domain there is a region of excitation acceleration a within which the joint probability density has two peaks an upper peak and a lower peak results show that the upper peak decreases while the lower peak increases as the value of a decreases

instability of the harmonic oscillator with small noise we construct asymptotic expansions for the exponential growth rate lyapunov exponent and rotation number of the random oscillator when the noise is small and is defined by a temporally homogeneous markov process with a finite number of states in the case of two states the telegraph process we obtain additional terms in the expansions affording comparison between the exact values and the asymptotic formulas

largest lyapunov exponent and almost certain stability analysis of slender beams under a large linear motion of basement subject to narrowband parametric excitation the first order approximate solutions of a set of non liner differential equations which is established by using kane s method and governs the planar motion of beams under a large linear motion of basement are systematically derived via the method of multiple scales the non linear dynamic behaviors of a simply supported beam subject to narrowband random parametric excitation in which either the principal parametric resonance of its first mode or a combination parametric resonance of the additive type of its first two modes with or without 3 1 internal resonance between the first two modes is taken into consideration are analyzed in detail the largest lyapunov exponent is numerically obtained to determine the almost certain stability or instability of the trivial response of the system and the validity of the stability is verified by direct numerical integration of the equation of motion of the system

on the stability properties of a van der pol duffing oscillator that is driven by a real noise abstract in this paper we consider a van der pol duffing oscillator that is excited parametrically by a small intensity real noise which is assumed to be an integrable function of an n dimensional ornstein uhlenbeck vector process that is an output of a linear filter system the stability properties include the moment lyapunov exponent g p x 0 and the maximal lyapunov exponent and the stability in probability are examined to study a model of enhanced generality we remove both the detailed balance condition and the strong mixing condition in the case of an arbitrary finite real number p we employ the perturbation method and a spectrum representation of the fokker planck operator of the linear filter system to construct asymptotic expansions of the pth moment lyapunov exponent and the top lyapunov exponent the same methods are also used for a nonlinear stochastic system to obtain the fpk fokker planck kolmogonov equation for the amplitude process which is identical to the one that is derived from the stochastic averaging method in the case of a broadband noise excitation on the basis of this fpk equation we also examine the almost sure stability condition of the ito stochastic differential equation for the amplitude process which matches the result that is derived from the maximal lyapunov exponent finally the method proposed by lin and cai probabilistic structural dynamics advanced theory and application mcgraw hill new york 1995 is adopted to examine the stability in probability of the amplitude process for the nonlinear ito differential equation

mean square stability of a second order parametric linear system excited by a colored gaussian noise abstract the stochastic stability of a second order linear parametric oscillator whose stiffness is perturbed by a stationary zero mean colored gaussian process is investigated in this paper referring to the stability in the second response moments mean square stability the stochastic perturbation is the output of a linear filter excited by a stationary gaussian white noise stochastic process and the hypothesis of weak excitation is not formulated the first assumption allows using ito s stochastic differential calculus the moment equations of the response are written till the second order by means of ito s differential rule they form an infinite hierarchy that is the equations for the moments of order r contain moments of order larger than r in order to close the hierarchy the cumulant neglect closure method is applied herein the moment equation set is closed by neglecting the cumulants of third order nonlinear moment equations result they are linearized and the limit of stability is searched by studying the eigenvalues of the matrix of the coefficients of the linearized moment equations several numerical analyses are performed to evaluate the critical mean square value of the excitation when a system parameter is varied for the cases of both first and second order parametric excitation a first order excitation an ornstein uhlenbeck process does not cause stochastic resonance while a second order excitation does the first result was not known previously

determining lyapunov exponents from a time series we present the first algorithms that allow the estimation of non negative lyapunov exponents from an experimental time series lyapunov exponents which provide a qualitative and quantitative characterization of dynamical behavior are related to the exponentially fast divergence or convergence of nearby orbits in phase space a system with one or more positive lyapunov exponents is defined to be chaotic our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems we monitor the long term growth rate of small volume elements in an attractor the method is tested on model systems with known lyapunov spectra and applied to data for the belousov zhabotinskii reaction and couette taylor flow

stochastic averaging of strongly non linear oscillators under bounded noise excitation abstract a stochastic averaging method for strongly non linear oscillators under external and or parametric excitation of bounded noise is proposed by using the so called generalized harmonics functions the method is then applied to study the primary resonance of duffing oscillator with hardening spring under external excitation of bounded noise the stochastic jump and its bifurcation of the system are observed and explained by using the stationary probability density of amplitude and phase subsequently the method is applied to study the dynamical instability and parametric resonance of duffing oscillator with hardening spring under parametric excitation of bounded noise the primary unstable region is delineated by evaluating the lyapunov exponent of linearized system and the response and jump of non linear system around the unstable region are examined by using the sample functions and stationary probability density of amplitude and phase

image solutions for boundary value problems without sources in this paper we employ the image method to solve boundary value problems in domains containing circular or spherical shaped boundaries free of sources two and threed problems as well as symmetric and anti symmetric cases are considered by treating the image method as a special case of method of fundamental solutions only at most four unknown strengths distributed at the center two locations of frozen images and one free constant need to be determined besides the optimal locations of sources are determined for the symmetric and anti symmetric cases only two coefficients are required to match the two boundary conditions the convergence rate versus number of image group is numerically performed the differences of the image solutions between 2d and 3d problems are addressed it is found that the 2d solution in terms of the bipolar coordinates is mathematically equivalent to that of the simplest mfs with only two sources and one free constant finally several examples are demonstrated to see the validity of the image method for boundary value problems

intermittency type estimates for some nondegenerate spde s in this paper we prove some intermittency type estimates for the stochastic partial differential equation du mathscr l u dt mathscr m lu circ dw l t where mathscr l is a strongly elliptic second order partial differential operator and the mathscr m l s are first order partial differential operators here the w l s are standard wiener processes and circ denotes stratonovich integration we assume for simplicity that u 0 cdot equiv 1 our interest here is the behavior of mathbb e lbrack u t x p rbrack for large time and large p more specifically our interest is the growth of p 2t 1 log mathbb e lbrack u t x p rbrack as t then p become large