characterizing the robustness and susceptibility of steady state dynamics in post buckled structures to stochastic perturbations abstract recent reports exemplify both the potential and the concerns encountered in implementing post buckled structures in diverse applications by numerical and experimental methods characterizations have provided useful insights on the dynamic sensitivities of such bi or multistable structures subjected to either harmonic or stochastic excitations yet the more realistic scenario of combined harmonic and stochastic loading has been not been closely examined so that such sensitivities might be fully illuminated to provide a more complete understanding on the robustness and susceptibility of multistable structures to dynamic state transitions this research establishes new analytical and experimental methods to quantify the likelihood of triggering transitions among dynamic regimes of an archetypal post buckled structure as a result of combined harmonic and stochastic loading it is discovered that persistent periodic snap through dynamics are rapidly disabled by additional noise excitation when the harmonic excitation contribution occurs at frequencies close to the linearized resonance the extra noise may also drastically compromise the integrity of small amplitude periodic responses that occur at frequencies around one half of the linearized resonance particular relative proportions of the noise standard deviation and harmonic excitation amplitude are uncovered that most readily compromise the robustness of a given steady state dynamic regime the analytical method also complements prior developments focused on stochastic resonance by uncovering the broader perspective of dynamic sensitivities of post buckled structures under arbitrary combinations of harmonic and random driving loads

axisymmetric shape of gas bubbles in a uniform flow this paper investigates axisymmetric shapes of an incompressible inviscid fluid bubble in a uniform flow approximate solutions are constructed for the governing nonlinear boundary value problem using the domain perturbation method bifurcation and turning points are located by applying a special type of hermite pade approximation

the geostatistical characteristics of the borden aquifer a complete reexamination of sudicky s 1986 field experiment for the geostatistical characterization of hydraulic conductivity at the borden aquifer in ontario canada is performed the sampled data reveal that a number of outliers low ln k values are present in the data base these low values cause difficulties in both variogram estimation and determining population statistics the analysis shows that assuming either a normal distribution or exponential distribution for log conductivity is appropriate the classical cressie hawkins and squared median of the absolute deviations smad estimators are used to compute experimental variograms none of these estimators provides completely satisfactory variograms for the borden data with the exception of the classical estimator with outliers removed from the data set theoretical exponential variogram parameters are determined from nonlinear nl estimation differences are obtained between nl fits and those of sudicky 1986 for the classical screened estimated variogram nl fits produce an ln k variance of 0 24 nugget of 0 07 and integral scales of 5 1 m horizontal and 0 21 m vertical along a a for b b these values are 0 37 0 11 8 3 and 0 34 the fitted parameter set for b b data horizontal and vertical is statistically different than the parameter set determined for a a we also evaluate a probabilistic form of dagan s 1982 1987 equations relating geostatistical parameters to a tracer cloud s spreading moments the equations are evaluated using the parameter estimates and covariances determined from line a a as input with a velocity equal to 9 0 cm day the results are compared with actual values determined from the field test but evaluated by both freyberg 1986 and rajaram and gelhar 1988 the geostatistical parameters developed from this study produce an excellent fit to both sets of calculated plume moments when combined with dagan s stochastic theory for predicting the spread of a tracer cloud

on the self averaging of dispersion for transport in quasi periodic random media in this study we present a numerical analysis for the self averaging of the longitudinal dispersion coefficient for transport in heterogeneous media this is done by investigating the mean square sample to sample fluctuations of the dispersion for finite times and finite numbers of modes for a random field using analytical arguments as well as numerical simulations we consider transport of point like injections in a quasi periodic random field with a gaussian correlation function in particular we focus on the asymptotic and pre asymptotic behaviour of the fluctuations with the aid of a probability density function for the dispersion and we verify the logarithmic growth of the sample to sample fluctuations as earlier reported in eberhard 2004 j phys a math gen 37 2549 71 we also comment on the choice of the relevant parameters to generate quasi periodic realizations with respect to the self averaging of transport in statistically homogeneous gaussian velocity fields

impact of sampling volume on the probability density function of steady state concentration 1 in recent years statistical theory has been used to compute the ensemble mean and variance of solute concentration in aquifer formations with second order stationary velocity fields the merit of accurately estimating the mean and variance of concentration however remains unclear without knowing the shape of the probability density function pdf in a setup where a conservative solute is continuously injected into a domain the concentration is bounded between zero and the concentration value in the injected solution at small travel distances close to the fringe of the plume an observation point may fall into the plume or outside so that the statistical concentration distribution clusters at the two limiting values obviously this results in non gaussian pdf s of concentration with increasing travel distance the lateral plume boundaries are smoothed resulting in increased probability of intermediate concentrations likewise averaging the concentration in a larger sampling volume as typically done in field measurements leads to higher probabilities of intermediate concentrations we present semianalytical results of concentration pdf s for measurements with point like or larger support volumes based on stochastic theory applied to stationary media to this end we employ a reversed auxiliary transport problem in which we use analytical expressions for first and second central spatial lateral moments with an assumed gaussian pdf for the uncertainty of the first lateral moment and gauss like shapes in individual cross sections the resulting concentration pdf can be reasonably fitted by beta distributions the results are compared to monte carlo simulations of flow and steady state transport in 3 d heterogeneous domains in both methods the shape of the concentration pdf changes with distance to the contaminant source near the source the distribution is multimodal whereas it becomes a unimodal beta distribution far away from the contaminant source the semianalytical and empirical pdf s differ slightly which we contribute to the numerical artifacts in the monte carlo simulations but also to hard assumptions made in the semianalytical approach our results imply that geostatistical techniques for interpolation and other statistical inferences based on gaussian distributions such as kriging and cokriging may be feasible only far away from the contaminant source for calculations near the source the beta like distribution of concentration should be accounted for

three dimensional stochastic analysis of macrodispersion in aquifers the dispersive mixing resulting from complex flow in three dimensionally heterogeneous porous media is analyzed using stochastic continuum theory stochastic solutions of the perturbed steady flow and solute transport equations are used to construct the macroscopic dispersive flux and evaluate the resulting macrodispersivity tensor in terms of a three dimensional statistically anisotropic input covariance describing the hydraulic conductivity with a statistically isotropic input covariance the longitudinal macrodispersivity is convectively controlled but the transverse macrodispersivity is proportional to the local dispersivity and is several orders of magnitude smaller than the longitudinal term with an arbitrarily oriented anisotropic conductivity covariance all components of the macrodispersivity tensor are convectively controlled and the ratio of transverse to longitudinal dispersivity is of the order of 10 1 in this case the off diagonal components of the dispersivity tensor are significant being numerically larger than the diagonal transverse terms and the transverse dispersion process can be highly anisotropic dispersivities predicted by the stochastic theory are shown to be consistent with controlled field experiments and monte carlo simulations the theory which treats the asymptotic condition of large displacement indicates that a classical gradient transport fickian relationship is valid for large scale displacements

on response of nonlinear oscillators with random frequency of excitation abstract some types of nonlinear oscillators for which the frequency of excitation is stochastic are investigated the paper consists of two parts in the first part equations of motion are linearized with the aid of stochastic averaging differential equations for the mean and variance of the process are obtained and integrated numerically this approach is applicable for weakly nonlinear oscillators the case of strong nonlinearity is considered in the second part making use of computer simulation a number of stochastic realizations of the process are computed the stochastic process is characterized by the mean and standard deviation of these realizations calculations have been carried out for the duffing ueda and van der pol equations and for forced vibrations of a pendulum these calculations show that if attractors exist then the deterministic vibrations which may be chaotic turn regular by adding noise and the motion terminates in a stable fixed point or on a limit cycle

high order analysis of the limit cycle of the van der pol oscillator we have applied the lindstedt poincare method to study the limit cycle of the van der pol oscillator obtaining the numerical coefficients of the series for the period and for the amplitude to order 859 hermite pade approximants have been used to extract the location of the branch cut of the series with unprecedented accuracy 100 digits both series have then been resummed using an approach based on pade approximants where the exact asymptotic behaviors of the period and the amplitude are taken into account our results improve drastically all previous results obtained on this subject

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

numerical investigations on ergodicity of solute transport in heterogeneous aquifers 1 darcy velocities for lognormal hydraulic conductivity with small variance and finite correlation length were approximated by periodic random fields accurate simulations of two dimensional advection dispersion processes were achieved with the global random walk algorithm using 1010 particles in every transport realization reliable statistical estimations were obtained by averaging over 256 realizations the main result is a numerical evidence for the mean square convergence of the actual concentrations to the macrodispersion process predicted by a known limit theorem for small initial plumes the ergodic behavior can be expected after thousands of advection timescales when the deviation from the theoretical prediction of the cross section space averaged concentration monotonously decays and falls under 20 the increase of the transverse dimension of the initial plume slows down the approach to the quasi ergodic state and has a nonlinear effect on the variability of the actual concentrations and dispersivities

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

stratified disordered media exact solutions for transport parameters and their self averaging properties we investigate the transport of a passive tracer in a two dimensional stratified random medium with flow parallel and perpendicular to the strata assuming a gaussian random flow with a gaussian correlation function it is not only possible to derive exact expressions for the temporal behaviour of the dispersion coefficients characterizing the tracer transport but also to investigate the self averaging properties of these quantities we give explicit results for the dispersion coefficient and its mean square fluctuations as a function of time as it turns out the sample to sample fluctuations which are encoded in the latter quantity remain finite even in the asymptotic limit of infinite times which implies that the given two dimensional model is not self averaging

temporal behavior of a solute cloud in a heterogeneous porous medium 3 numerical simulations 1 the article presents systematic numerical simulations of the temporal behavior of a passive solute in a saturated three dimensional heterogeneous medium the groundwater flow is derived from the linearized solution of the darcy equation with gauss distributed log hydraulic conductivity the transport of a passive solute is studied by a random walk method which allows for a systematic study of the temporal behavior of the effective and ensemble dispersion coefficients the numerical results are compared to the second order perturbation theory expressions given in two companion papers dentz et al 2000a 2000b and to nonperturbative results which follow from corrsin s conjecture the low order perturbation theory is intrinsically based on the assumption of small heterogeneity while corrsin s conjecture does not take into account certain contributions due to higher order terms of the perturbation series the simulations yield for the first time systematic quantitative information on the validity and the limitations of these analytic approximations for increasing heterogeneities considerable deviations from the theoretically predicted transport behavior are observed

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

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

summation of the eigenvalue perturbation series by multi valued pade approximants application to resonance problems and double wells quadratic pade approximants are used to obtain energy levels both for the anharmonic oscillator x2 2 lambda x4 and for the double well x2 2 lambda x4 in the first case the complex valued energy of the resonances is reproduced by summation of the real terms of the perturbation series the second case is treated formally as an anharmonic oscillator with a purely imaginary frequency we use the expansion around the central maximum of the potential to obtain a complex perturbation series on the unphysical sheet of the energy function then we perform an analytical continuation of this solution to the neighbouring physical sheet taking into account the supplementary branch of quadratic approximants in this way we can reconstruct the real energy by summation of the complex series such an unusual approach eliminates the double degeneracy of states that makes ordinary perturbation theory around the minima of the double well potential incorrect

asymptotic dispersion in 2d heterogeneous porous media determined by parallel numerical simulations we determine the asymptotic dispersion coefficients in 2d exponentially correlated lognormally distributed permeability fields by using parallel computing fluid flow is computed by solving the flow equation discretized on a regular grid and transport triggered by advection and diffusion is simulated by a particle tracker to obtain a well defined asymptotic regime under ergodic conditions initial plume size much larger than the correlation length of the permeability field the characteristic dimension of the simulated computational domains was of the order of 103 correlation lengths with a resolution of ten cells by correlation length we determine numerically the asymptotic effective longitudinal and transverse dispersion coefficients over 100 simulations for a broad range of heterogeneities s 2 0 9 where s 2 is the lognormal permeability variance for purely advective transport the asymptotic longitudinal dispersion coefficient depends linearly on s 2 for s 2 1 and the asymptotic transverse dispersion coefficient is zero addition of homogeneous isotropic diffusion induces an increase of transverse dispersion and a decrease of longitudinal dispersion

evaluation of the first order approximations for transport in heterogeneous media 1 longitudinal dispersion coefficients in given realizations of the transport computed by two currently used approximations of the first order in velocity variance are compared with accurate global random walk simulations the comparisons are performed for the same ensemble of realizations of the darcy velocity field approximated by a quasiperiodic random field for lognormal hydraulic conductivity with small variance and finite correlation lengths the results show that at finite times of about one dispersion timescale the mean coefficient is underestimated by 20 and the fluctuations are overestimated by 80 at larger times the errors decrease monotonously and the first order approximations yield fairly good predictions for the mean and the fluctuations of the dispersion coefficient

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

a hermite pade perspective on the renormalization group with an application to the correlation function of lieb liniger gas renormalization group rg methods are based on symmetry principles that are peculiar to quantum field theory and certain related settings however in cases when the function depends only on the coupling the rg technology does include a part which is related to the equation for the running coupling and which we will call the single variable rg svrg that turns out to be a purely formal method for improving the convergence properties of series expansions in fact we show that svrg is an integral hermite pade approximation preceded by a series reversion a complementary viewpoint on what svrg does to an expansion is that it performs a minimal upgrade to a self similar form the minimality being crucial in applications svrg is especially useful in problems in which one knows not only the first few terms of an expansion valid for small values of a variable but also the exponent but not the prefactor of the scaling law for large values of the variable as a practical example we extract the scaling law prefactor for the one body density matrix of the lieb liniger gas using a new result for the fourth order term in the short distance expansion we find remarkable agreement with known quantum diffusion monte carlo numerical results

subharmonic response of a single degree of freedom nonlinear vibroimpact system to a randomly disordered periodic excitation the subharmonic resonant response of a single degree of freedom nonlinear vibroimpact oscillator with a one sided barrier to narrow band random excitation is investigated the analysis is based on a special zhuravlev transformation which reduces the system to one without impacts or velocity jumps thereby permitting the applications of asymptotic averaging over the period for slowly varying random processes the averaged equations are solved exactly and algebraic equation of the amplitude of the response is obtained in the case without random disorder a perturbation based moment closure scheme is proposed and an iterative calculation equation for the mean square response amplitude is derived in the case with random disorder the effects of damping nonlinear intensity detuning and magnitudes of random excitations are analyzed the theoretical analyses are verified by numerical results theoretical analyses and numerical simulations show that the peak amplitudes may be strongly reduced at large detuning or large nonlinear stiffness and when intensity of the random disorder increase the steady state solution may change from a limit cycle to a diffused limit cycle and even change to a chaos one

an exact nonlinear correction to transverse macrodispersivity for transport in heterogeneous formations the random two dimensional and steady in time eulerian velocity field is assumed to be given it pertains to flow through heterogeneous porous media of spatially variable permeability advective transport of a passive scalar is modeled by the lagrangian approach by seeking the statistical moments of the trajectory of a tagged particle the trajectories variances and the associated dispersion coefficients are evaluated by a perturbation expansion in the velocity variances and by corrsin s conjecture first order results obtained in the past show that for flow through porous formations of log permeability two point covariance of finite integral scale the transverse dispersion coefficient tends asymptotically to zero in contrast corrsin s conjecture leads to a finite dispersion coefficient quadratic in the velocity variance to investigate this effect an exact nonlinear correction to the transverse dispersion coefficient is derived for a normal velocity field it is shown that this term tends to zero asymptotically in contrast with the result based on corrsin s conjecture furthermore an illustrative computation shows the transport related nonlinear correction to be small at any time

a successive integration technique for the solution of the duffing oscillator equation with damping and excitation abstract in this paper a successive integration technique is suggested for solving the duffing oscillator equation with damping and excitation in the technique one performs integrations from some initial values over a wider range some of them may even be divergent however some of them will give a convergent result such that the periodic condition of motion is satisfied in fact the convergent result represents a stable periodic solution for the motion if a convergent result or a stable periodic solution is obtained the stability test is passed a harmonic balance method in conjunction with the successive integration technique abbreviated as hbmsit is also suggested in the method the initial values are obtained from the harmonic balance method therefore the hbmsit belongs to the successive integration techniques many examples with calculated results are presented

hermite pade approximation approach to mhd jeffery hamel flows in this paper we studied the problem of nonlinear flow of an incompressible conducting viscous fluid in convergent divergent channels under the influence of an externally applied homogeneous magnetic field the governing equations is obtained and solved using a special type of hermite pade approximation semi numerical approach this technique offers some advantages over solutions obtained by using traditional methods such as finite differences spectral method shooting method etc it reveals the analytical structure of the solution function and the important properties of overall flow structure including velocity field flow reversal control and bifurcations are discussed

vector orthogonal polynomials abstract the following well known framework will be applied to hermite pade approximants first the hermite pade type approximants will be defined from the error of approximation some supplementary conditions will be fixed in order to obtain hermite pade forms vector orthogonal polynomials will be deduced from these conditions their existence and uniqueness are studied in the normal case these results are ensured then some properties of vector orthogonal polynomials will be proved

approximants de pade hermite 2eme partie programmation in this second paper we propose an algorithm for the construction of the tables and of the pade hermite tables 3 then we present some numerical applications

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

numerical study of bifurcations by analytic continuation of a function defined by a power series a novel computational approach to the investigation of bifurcations relying on the use of power series in the bifurcation parameter for a particular solution branch is presented the first part of the paper is devoted to the description of a series summation technique based on the assumption that the given series is the local representation of a function algebraic in the independent variable the procedure leads to a special type of hermite pade approximant although no mathematical analysis is presented the numerical evidence suggests that the error decays faster than exponentially with the number of terms of the series used the procedure s chief merit is its ability to reveal solution branches of the underlying problem in addition to the one represented by the original series in the final part of the paper an algorithm is described for numerically generating the required power series where standard perturbation methods are inadequate thus it is shown how path following techniques may be combined

summation of asymptotic expansions of multiple valued functions using algebraic approximants application to anharmonic oscillators the divergent rayleigh schrodinger perturbation expansions for energy eigenvalues of cubic quartic sextic and octic oscillators are summed using algebraic approximants these approximants are generalized pade approximants that are obtained from an algebraic equation of arbitrary degree numerical results indicate that given enough terms in the asymptotic expansion the rate of convergence of the diagonal staircase approximant sequence increases with the degree different branches of the approximants converge to different branches of the function the success of the high degree approximants is attributed to their ability to model the function on multiple sheets of the riemann surface and to reproduce the correct singularity structure in the limit of large perturbation parameter an efficient recursive algorithm for computing the diagonal approximant sequence is presented

approximants de pade hermite 1ere partie theorie we present in this first paper a generalization of pade approximants which gives us as particular cases shafer s and baker sd log approximants first we define these approximants following an old idea of hermite then we prove some fundamental properties for their constructions

applications of hermite pade approximants to water waves and the harmonic oscillator on a lattice three types of hermite pade approximants are considered known respectively as quadratic integral and differential pade approximants the singularity structure of each type of approximant is described it is more complicated than that of the standard pade approximant and this property may often be used to estimate the types of singularity of a function from its power series expansion as well as evaluate it on its branch cuts two applications in different fields are described which illustrate these properties of the above types of hermite pade approximants the first concerns the characteristic values of mathieu s equation which are related to the energy eigenvalues of the harmonic oscillator on a lattice the second concerns the investigation of the singularity structure and values of various physical quantities associated with periodic and solitary water waves

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

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

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

approximations for transport parameters and self averaging properties for point like injections in heterogeneous media we focus on transport parameters in heterogeneous media with a flow modelled by an ensemble of periodic and gaussian random fields the parameters are determined by ensemble averages we study to what extent these averages represent the behaviour in a single realization we calculate the centre of mass velocity and the dispersion coefficient using approximations based on a perturbative expansion for the transport equation and on the iterative solution of the langevin equation compared with simulations the perturbation theory reproduces the numerical results only poorly whereas the iterative solution yields good results using these approximations we investigate the self averaging properties the ensemble average of the velocity characterizes the behaviour of a realization for large times in both ensembles the dispersion coefficient is not self averaging in the ensemble of periodic fields for the gaussian ensemble the asymptotic dispersion coefficient is self averaging for finite times however the fluctuations are so large that the average does not represent the behaviour in a single realization

dispersion variance for transport in heterogeneous porous media 1 we study dispersion in heterogeneous porous media for solutes evolving from point like and extended source distributions in d 2 and d 3 spatial dimensions the impact of heterogeneity on the dispersion behavior is captured by a stochastic modeling approach that represents the spatially fluctuating flow velocity as a spatial random field we focus here on the sample to sample fluctuations of the dispersion coefficients about their ensemble mean for finite source sizes the definition of dispersion coefficients in single realizations is not unique we consider dispersion measures that describe the extension of the solute distribution as well as dispersion coefficients that quantify the solute spreading relative to injection points of the partial plumes that constitute the solute distribution while the ensemble averages of these dispersion quantities may be identical their fluctuation behavior is found to be different using a perturbation approach in the fluctuations of the random flow field we derive explicit expressions for the temporal evolution of the variances of the dispersion coefficients between realizations their evolution is governed by the typical dispersion time over the characteristic heterogeneity scale and the dimensions of the source distribution we find that the dispersion variance decreases toward zero with time in d 3 spatial dimensions while in d 2 it converges toward a finite long time value that is independent of the source dimensions

a split frequency harmonic balance method for nonlinear oscillators with multi harmonic forcing abstract a new harmonic balance method hbm is presented for accurately computing the periodic responses of a nonlinear sdof oscillator with multi harmonic forcing and non expansible nonlinearities the presence of multi harmonic forcing requires a large number of solution harmonics with a substantial increase in computational demand for either the conventional or the incremental hbm in this method the oscillator equation error is first defined in terms of two functions originally proposed for obtaining free vibration periods in r e mickens iteration procedure for determining approximate solutions to nonlinear oscillator equations journal of sound and vibration 116 1987 185 187 and more recently r e mickens a generalised iteration procedure for calculating approximations to periodic solutions of truly nonlinear oscillations journal of sound and vibration 287 2005 1045 1051 a fourier series solution is assumed in which the total number of harmonics is fixed by the chosen discrete time interval this series is split into two partial sums nominally associated with either low frequency or high frequency harmonics by exploiting a convergence property of the equation error functions the total solution is obtained in a new iterative scheme in which the low frequency components are computed via a conventional hbm using a small number of algebraic equations whereas the high frequency components are obtained in a separate step by updating by gradually increasing the number of harmonics in the low frequency group the equation error can be progressively reduced efficient use is made of fft based algebraic equation generation which allows an important class of non expansible nonlinearities to be handled the proposed method is tested on a duffing type oscillator and an oscillator with a non expansible 7th power stiffness term where in both cases up to 24 component multi harmonic forcing is applied as a comparison a conventional hbm is also used on the duffing model in which the algebraic equations are generated in symbolic form to totally avoid errors from entering the formulation through complicated expansion of the cubic stiffness term as in i senjanovic harmonic analysis of nonlinear oscillations of cubic dynamical systems journal of ship research 38 3 1994 225 238 and in a raghothama s narayanan periodic response and chaos in nonlinear systems with parametric excitation and time delay nonlinear dynamics 27 2002 341 365 the paper shows that in obtaining period 1 solutions the computational accuracy and efficiency of the proposed method is very good

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