on littlewood s boundedness problem for relativistic oscillators with anharmonic potentials abstract in this paper we are concerned with the littlewood s boundedness problem and the existence of quasi periodic solutions for the nonlinear differential equation of the motions of relativistic oscillators with anharmonic potentials d d t x 1 x 2 x 1 x p t where the 1 periodic function p t is a smooth function and is a positive constant

on littlewood s boundedness problem for relativistic oscillators with singular potentials abstract in this paper we study the following nonlinear differential equations of motions of relativistic oscillators with singular potentials d d t x 1 x 2 v x p t where v is a singular potential and p is a 1 periodic function we will prove the boundedness of all solutions and the existence of infinitely many quasi periodic solutions via moser s twist theorem

limitations of robust stability of a linear delayed feedback control in this paper we consider the issue of robust stability of a linear delayed feedback control dfc mechanism in particular we consider a dfc for stabilizing fixed points of a smooth function f mathbb r m rightarrow mathbb r m of the form x k 1 f x k u k where u k is given by the formula u k sum j 1 n 1 epsilon j x k j x k j 1 we associate with each fixed point of f an explicit polynomial whose schur stability corresponds to the local asymptotic stability of the dfc at that fixed point this polynomial is the characteristic polynomial of the jacobian matrix of an auxiliary map from mathbb r mn to mathbb r mn and may be given in terms of the eigenvalues of the jacobian of f at the fixed point this enables us to evaluate the robustness of the control by considering over what possible sets of eigenvalues of the jacobian of f the associated characteristic polynomials are schur stable we will show that for a given control of the above form sta

boundedness of solutions of nonlinear differential equations abstract in this paper we are concerned with the boundedness of solutions of some nonlinear differential equations which are neither dissipative nor conservative having the form x x t x g x t 0 where f and g are odd polynomials in x with coefficients which are even and periodic in t using the kam theory for reversible systems we prove that all the solutions are bounded whenever a sharp condition on the degrees of f and g is satisfied we also obtain a boundedness result when f x t 0 i e the equation is conservative under some smoothness assumptions on g which improve the previously known ones in this case no symmetry conditions on g are needed

growth of sobolev norms in time dependent semiclassical anharmonic oscillators abstract we consider the semiclassical schrodinger equation on r d given by i t 2 2 w l x v t x where w l is an anharmonic trapping of the form w l x 1 2 l j 1 d x j 2 l l 2 is an integer and is a semiclassical small parameter we construct a smooth potential v t x bounded in time with its derivatives and an initial datum such that the sobolev norms of the solution grow at a logarithmic speed for all times of order log 1 2 1 the proof relies on two ingredients first we construct an unbounded solution to a forced mechanical anharmonic oscillator then we exploit semiclassical approximation with coherent states to obtain growth of sobolev norms for the quantum system which are valid for semiclassical time scales

boundedness of solutions for non linear quasi periodic differential equations with liouvillean frequency abstract we study the boundedness of solutions for non linear quasi periodic differential equations with liouvillean frequencies we proved that if the forcing is quasi periodic in time with two frequencies which is not super liouvillean then all solutions of the equation are bounded the proof is based on action angle variables and modified kam theory

an explicit time evolution method for acoustic wave propagation cost effective waveform modeling is the key to practical reverse time migration rtm and full waveform inversion fwi implementations we evaluated an explicit time evolution ete method to efficiently simulate wave propagation in acoustic media with high temporal accuracy we started from the constant density acoustic wave equation and obtained an analytical time marching scheme in the wavenumber domain we then formulated an ete scheme in the time space domain by introducing a cosine function approximation although the ete operator appears to be similar to the second order temporal finite difference fd operator the exact nature of the ete formula ensures high accuracy in time we further introduced a set of optimum stencils and coefficients by minimizing evolution errors in a least squares sense our numerical tests indicated that ete can achieve similar waveform accuracy as fd with four times larger time steps meanwhile the compact ete operator keeps the computation efficient the efficiency and capability to handle complex velocity field make ete an attractive engine in acoustic rtm and fwi

boundedness of solutions for duffing s equations with semilinear potentials abstract in this paper we will prove the boundedness of all the solutions and the existence of quasiperiodic solutions for the second order scalar differential equation x arctan x ee t where e is a small parameter and the 1 periodic function e t is a smooth function

longtime closeness estimates for bounded and unbounded solutions of non recurrent duffing equations with polynomial potentials abstract in this paper we consider the littlewood s problem for the second order differential equation x x 2 n 1 q t x l p t l n where the functions p t and q t do not need to be periodic using the theory of non periodic twist mappings which is developed by kunze and ortega in 3 4 5 6 7 we obtained longtime closeness results on the bounded and unbounded solutions of the equation

invariant tori for semilinear reversible systems abstract in this paper we consider the problem of the boundedness of all solutions for the oscillator x f x x 2 x g x p t where 0 and n g x is bounded and p t p t 2 since the system in general is non hamiltonian we have to introduce reversibility assumptions to apply the twist theorem for reversible maps under some suitable conditions we then obtain the existence of invariant tori and thus the boundedness of all solutions

invariant tori of duffing type equations abstract in this paper we prove the existence of invariant tori and thus the boundedness of all solutions and the existence of quasiperiodic solutions for a class of duffing type equations d 2 x dt 2 x 2n 1 j 0 2n p j t x j 0 n 1 wherepj t j 0 1 n are 1 periodicc 1 functions and wherepj t j n 1 2n are 1 periodicc 2 functions

spectral methods in time for hyperbolic equations a pseudospectral numerical scheme for solving linear periodic hyperbolic problems is described it has infinite accuracy both in time and in space the high accuracy in time is achieved without increasing the computational work and memory space which is needed for a regular one step explicit scheme the algorithm is shown to be optimal in the sense that among all the explicit algorithms of a certain class it requires the least amount of work to achieve a certain given resolution the class of algorithms referred to consists of all explicit schemes which may be represented as a polynomial in the spatial operator

time reversal with partial information for wave refocusing and scatterer identification time reversal is a well known procedure in application fields involving wave propagation among other uses it can be applied as a computational tool for solving certain inverse problems the procedure is based on advancing the solution of the relevant wave problem backward in time one important use of numerical time reversal is that of refocusing where a reverse run is performed to recover the location of a source applied at an initial time based on measurements at a later time usually only partial noisy information is available at certain measurement locations on the field values that serve as data for the reverse run in this paper the question concerning the amount and characterization of the available data needed for a successful refocusing is studied for the scalar wave equation in particular a simple procedure is proposed which exploits multiple measurement times and is shown to be very beneficial for refocusing a tradeoff between availability of spatial and temporal information is discussed the effect of measurement noise is studied and the technique is shown to be quite robust sometimes even in the presence of very high noise levels the use of the technique as a basis for scatterer identification is also discussed a numerical study of these effects is presented employing finite elements in space and a standard explicit marching scheme in time in contrast to some previous studies the propagation medium is taken to be homogeneous

time domain multiscale full waveform inversion using the rapid expansion method and efficient step length estimation full waveform inversion fwi is rapidly becoming a standard tool for high resolution velocity estimation however the application of this method is usually limited to low frequencies due to the high computational cost of the wavefield propagation and the inversion scheme to mitigate this problem we have developed a rapid expansion method rem for numerical wavefield extrapolation inside the fwi workflow this method approximated the partial time derivative of the wave equation using chebyshev polynomials instead of the conventional finite difference fd approximation this allowed the rem to accurately propagate wavefields with bigger time steps thus improving the computational efficiency of fwi we have compared high frequency fwi results using rem and the traditional fd approximation of the time derivative to illustrate the ability of rem to remain stable and free of numerical dispersion noise even with large grid and time samplings in addition we studied a step length estimation scheme with the objective of avoiding an extra forward wavefield propagation during the line search at each fwi iteration in this scheme we estimated the step length value based on prior iterations and validated this value using the wolfe conditions if the value was accepted then the forward modeling was availed in the next iteration and no extra propagation was necessary we tried seven different step length estimation methods from which we highlighted the adaptive barzilai borwein method when combined with the steepest descent inversion scheme and the unitary step length for the l bfgs algorithm through synthetic numerical results we showed how this scheme could achieve convergence while keeping the number of extra forward modelings way below the number of fwi iterations

a nonreflecting boundary condition for discrete acoustic and elastic wave equations one of the nagging problems which arises in application of discrete solution methods for wave propagation calculations is the presence of reflections or wraparound from the boundaries of the numerical mesh these undesired events eventually override the actual seismic signals which propagate in the modeled region the solution to avoiding boundary effects used to be to enlarge the numerical mesh thus delaying the side reflections and wraparound longer than the range of times involved in the modeling obviously this solution considerably increases the expense of computation more recently nonreflecting boundary conditions were introduced for the finite difference method clayton and enquist 1977 reynolds 1978 these boundary conditions are based on replacing the wave equation in the boundary region by one way wave equations which do not permit energy to propagate from the boundaries into the numerical mesh this approach has been relatively successful except that its effectiveness degrades for event

the existence of quasiperiodic solutions for coupled duffing type equations in this paper we prove the existence of infinitely many quasiperiodic solutions for a class of coupled duffing type equations via kam theorem moreover the set of quasiperiodic solutions is of infinitely lebesgue measure in the phase space

finite difference time dispersion transforms for wave propagation the finite difference fd wave equation is widely implemented in seismic imaging for oil exploration but the numerical dispersion due to discretization of time and space derivatives can introduce severe numerical errors we have investigated time dispersion which might distort the phase and introduce severe artifacts to the data and images especially for long time propagation we first studied precisely how the time dispersion was produced and then we developed an efficient approach the time dispersion transforms to cope with time dispersion errors in wave propagation the proposed forward time dispersion transform can predict or simulate the time dispersion errors and it can be applied to remove time dispersion errors in reverse time migration the inverse time dispersion transform can be used to eliminate time dispersion errors from synthetic data after modeling the proposed method is general and works for low and high order time fd schemes by using this method large time steps were allowed in wave propagation thus we can remove the time dispersion errors at a negligible numerical cost and the efficiency is not affected to test the effectiveness and robustness of the proposed time dispersion transforms we used isotropic and anisotropic examples as well as modeling and imaging examples

a k space operator based least squares staggered grid finite difference method for modeling scalar wave propagation abstracttwo staggered grid finite difference sgfd schemes with fourth and sixth order accuracies in time have been developed recently based on new sgfd stencils the sgfd coefficients of the two schemes are determined by a taylor series expansion te approach which is accurate only near a zero wavenumber we have adopted the same sgfd stencils and determined the sgfd coefficients by minimizing the errors between the wavenumber responses of the sgfd operators and the first order k wavenumber space operator in a least squares ls sense we solved the ls problems by performing a weighted pseudoinverse of nonsquare matrices to obtain the sgfd coefficients and to yield ls based sgfd methods we have developed an efficient ls implementation by using a small number of representative wavenumbers which makes the computational time for estimating the coefficients negligible dispersion analysis and numerical examples demonstrate that our ls based sgfd methods can preserve the original temporal accuracy and

time evolution of the wave equation using rapid expansion method forward modeling of seismic data and reverse time migration are based on the time evolution of wavefields for the case of spatially varying velocity we have worked on two approaches to evaluate the time evolution of seismic wavefields an exact solution for the constant velocity acoustic wave equation can be used to simulate the pressure response at any time for a spatially varying velocity a one step method can be developed where no intermediate time responses are required using this approach we have solved for the pressure response at intermediate times and have developed a recursive solution the solution has a very high degree of accuracy and can be reduced to various finite difference time derivative methods depending on the approximations used although the two approaches are closely related each has advantages depending on the problem being solved

symplectic scheme and the poynting vector in reverse time migration abstractwe developed a new numerical solution for the wave equation that combines symplectic integrators and the rapid expansion method rem this solution can be used for seismic modeling and reverse time migration rtm in seismic modeling and rtm spatial derivatives are usually calculated by finite differences fds or by the fourier method and the time evolution is normally obtained by a second order fd approach if the spatial derivatives are computed by higher order fd schemes then the time step needs to be small enough to avoid numerical dispersion therefore increasing the computational time however by using rem with the fourier method for the spatial derivatives we can apply the proposed method to propagate the wavefield for larger time steps moreover if the appropriate number of expansion terms is chosen this method is unconditionally stable and propagates seismic waves free of numerical dispersion the use of a symplectic numerical scheme provides the solution of the wave equation an

time step n tupling for wave equations abstractwe have constructed novel temporal discretizations for wave equations we first select an explicit time integrator that is of second order leading to classic time marching schemes in which the next value of the wavefield at the discrete time t t is computed from current values known at time t and the previous time t t then we determine how the time step can be doubled tripled or generally n tupled producing a new time stepping method in which the next value of the wavefield at the discrete time t n t is computed from current values known at time t and the previous time t n t in between time values of the wavefield are eliminated using the fourier method to calculate space derivatives the new time integrators allow larger stable time steps than traditional time integrators however like the lax wendroff procedure they require more computational effort per time step because the new schemes are developed from the classic second order time stepping scheme they will have the same proper

forward modeling by a fourier method a fourier or pseudospectral forward modeling algorithm for solving the two dimensional acoustic wave equation is presented the method utilizes a spatial numerical grid to calculate spatial derivatives by the fast fourier transform time derivatives which appear in the wave equation are calculated by second order differcncing the scheme requires fewer grid points than finite diffcrcnce methods to achieve the same accuracy it is therefore believed that the fourier method will prove more efficient than finitedifference methods especially when dealing with threedimensional models the fourier forward modeling method was tested against two problems a single layer problem with a known analytic solution and a wedge problem which was also tested by physical modeling the numerical results agreed with both the analytic and physical model results furthermore the numerical model facilitates the explanation of certain events on the time section of the physical model which otherwise could not easily be taken into account

controlling chaos using nonlinear feedback with delay we generalize a method of control of chaos that uses delayed feedback at the period of an unstable orbit to stabilize that orbit the generalization consists of substituting some portion of the nonlinear dynamical system with a delayed dynamics rather than using a linear delay function for control a further generalization in which the control function retains memory of all previous periods allows the region of the parameter space over which control can be achieved to be extended but at the price of losing the ability to achieve superstability nonlinear feedback results in a larger basin of attraction to the stabilized orbit than linear feedback for the simple test mapping studied the logistic map the dimension of the system increases from 1 to 2 by introducing control we show in the case involving memory for a particular choice of the relationship between the control parameters that the superstable orbit can be recovered without reducing the parameter space that can be controlled this particular solution in addition to having the largest basin of attraction of the methods considered retains the dimension of the uncontrolled system textcopyright 1996 the american physical society

the coexistence of quasi periodic and blow up solutions in a class of hamiltonian systems abstract the coexistence of quasi periodic solutions and blow up phenomena in a class of higher dimensional duffing type equations is proved in this paper moreover we show that the initial point sets for both kinds of solutions are of infinite lebesgue measure in the phase space for the part of quasi periodic solutions the tool we used is the small twist theorem for higher dimensional cases

using the rapid expansion method for accurate time stepping in modeling and reverse time migration reverse time migration is based on seismic forward modeling algorithms where spatial derivatives usually are calculated by finite differences or by the fourier method time integration in general is done by finite difference time stepping of low orders if the spatial derivatives are calculated by high order methods and time stepping is based on low order methods there is an imbalance that might require that the time step size needs to be very small to avoid numerical dispersion as a result computing times increase using the rapid expansion method rem avoids numerical dispersion if the number of expansion terms is chosen properly comparisons with analytical solutions show that the rem is preferable especially at larger propagation times for reverse time migration the rem needs to be applied in a time stepping manner this is necessary because the original implementation based on very large time spans requires that the source term is separable in space and time this is not appropriate for reverse time migration where the sources have different time histories in reverse time migration it might be desirable to use the poynting vector information to estimate opening angles to improve the quality of the image in the solution of the wave equation this requires that one calculates not only the pressure wavefield but also its time derivative the rapid expansion method can be extended easily to provide this time derivative with negligible extra cost

the application of high order differencing to the scalar wave equation the second order central difference is often used to approximate the derivatives of the wave equation it is demonstrated that gains in computational efficiency can be made by using high order approximations for these derivatives a one dimensional model is used to illustrate the relative accuracy of o t2 x2 o t2 x4 o t4 x4 ando t4 x10 central difference schemes for comparison o t2 ando t4 pseudospectral schemes are used as an additional measure of performance the results indicate that o t4 x10 differencing can achieve similar accuracy as the o t4 spectral scheme for practical illustration a two dimensional form of the o t4 x10 algorithm is used to compute the exploding reflector response of a salt dome model and compared with a fine grid o t2 x4 result transmissive sponge like boundary conditions are also examined and shown to be effective

lagrange stability for duffing type equations abstract in this paper the duffing type equation d 2 x dt 2 x 2 n 1 p 2 n t x 2 n p 1 t x p 0 t 0 is studied where the p j s are 1 periodic it is shown that all solutions of this equation are bounded provided that the p j t 0 j n are of bounded variation in 0 1 and that the derivatives of p j t n j 2 n are lipschitzian it is also shown that there exist p j s being discontinuous everywhere such that all solutions of the equation are bounded this implies that the continuity of p j s is not necessary for the boundedness of solutions of the equation

some discrete time counterparts to kharitonov s stability criterion for uncertain systems in 1 kharitonov gave an elegant and simple stability criterion for continuous time systems this note reports on similar results for discrete time systems

delayed feedback control of collective synchrony an approach to suppression of pathological brain rhythms we suggest a method for suppression of synchrony in a globally coupled oscillator network based on the time delayed feedback via the mean field having in mind possible applications for suppression of pathological rhythms in neural ensembles we present numerical results for different models of coupled bursting neurons a theory is developed based on the consideration of the synchronization transition as a hopf bifurcation

lagrange stability for impulsive duffing equations abstract this work discusses the boundedness of solutions for impulsive duffing equation with time dependent polynomial potentials by kam theorem we prove that all solutions of the duffing equation with low regularity in time undergoing suitable impulses are bounded for all time and that there are many positive lebesgue measure quasi periodic solutions clustering at infinity this result extends some well known results on duffing equations to impulsive duffing equations

efficient 3d frequency response modeling with spectral accuracy by the rapid expansion method frequency responses of seismic wave propagation can be obtained either by directly solving the frequency domain wave equations or by transforming the time domain wavefields using the fourier transform the former approach requires solving systems of linear equations which becomes progressively difficult to tackle for larger scale models and for higher frequency components on the contrary the latter approach can be efficiently implemented using explicit time integration methods in conjunction with running summations as the computation progresses commonly used explicit time integration methods correspond to the truncated taylor series approximations that can cause significant errors for large time steps the rapid expansion method rem uses the chebyshev expansion and offers an optimal solution to the second order in time wave equations when applying the fourier transform to the time domain wavefield solution computed by the rem we can derive a frequency response modeling formula that has the same form as the original time domain rem equation but with different summation coefficients in particular the summation coefficients for the frequency response modeling formula corresponds to the fourier transform of those for the time domain modeling equation as a result we can directly compute frequency responses from the chebyshev expansion polynomials rather than the time domain wavefield snapshots as do other time domain frequency response modeling methods when combined with the pseudospectral method in space this new frequency response modeling method can produce spectrally accurate results with high efficiency

third order symplectic integration method with inverse time dispersion transform for long term simulation the symplectic integration method is popular in high accuracy numerical simulations when discretizing temporal derivatives however it still suffers from time dispersion error when the temporal interval is coarse especially for long term simulations and large scale models we employ the inverse time dispersion transform itdt to the third order symplectic integration method to reduce the time dispersion error first we adopt the pseudospectral algorithm for the spatial discretization and the third order symplectic integration method for the temporal discretization then we apply the itdt to eliminate time dispersion error from the synthetic data as a post processing method the itdt can be easily cascaded in traditional numerical simulations we implement the itdt in one typical exiting third order symplectic scheme and compare its performances with the performances of the conventional second order scheme and the rapid expansion method theoretical analyses and numerical experiments show that the itdt can significantly reduce the time dispersion error especially for long travel times the implementation of the itdt requires some additional computations on correcting the time dispersion error but it allows us to use the maximum temporal interval under stability conditions thus its final computational efficiency would be higher than that of the traditional symplectic integration method for long term simulations with the aid of the itdt we can obtain much more accurate simulation results but with a lower computational cost we use the inverse time dispersion transform to the 3rd order symplectic integration it can greatly reduce the time dispersion error especially for long travel times it is superior to the conventional 2nd and high order finite difference methods it allows the maximum temporal interval under stability conditions thus is efficient we can obtain more accurate simulation results but with a lower computational cost

boundedness for solutions of nonlinear hill s equations with periodic forcing terms via moser s twist theorem we consider the nonlinear hill s equation with periodic forcing term x x2n 1 a1 ea t x p t n 1 where a t and p t are continuous and 1 periodic a1 and are positive constants and e is a small parameter the purpose of this paper is to prove that all solutions of the above mentioned equation are bounded for t r and that there are infinitely many quasi periodic solutions and an infinity of periodic solutions of minimal period m for each positive integer m

quasiperiodic motions in superquadratic time periodic potentials it is shown that for a large class of potentials on the line with superquadratic growth at infinity and with the additional time periodic dependence all possible motions under the influence of such potentials are bounded for all time and that most in a precise sense motions are in fact quasiperiodic the class of potentials includes as very particular examples the exponential polynomial and much more this extends earlier results and gives an answer to a problem posed by littlewood in the mid 1960 s along the way machinery is developed for estimating the action angle transformation directly in terms of the potential and also some apparently new identities involving singular integrals are derived

pseudo analytical finite difference elastic wave extrapolation based on the k space method abstractcost effective elastic wave modeling is the key to practical elastic reverse time migration and full waveform inversion implementations we have developed an efficient elastic pseudo analytical finite difference pafd scheme for elastic wave extrapolation the elastic pafd scheme is based on a modified pseudo spectral method k space method in which a pseudo analytical operator is used to ensure the high accuracy of elastic wave extrapolation however the k space method is motivated for a pure wave mode and thus its application in coupled first order elastic wave equations may cause the elastic pseudo analytical operators to suffer from crosstalk between the p and s wavefields the approaches presented attempt to overcome these shortcomings by introducing two improvements to achieve the goal this is done first by performing a predictor corrector strategy in first order elastic wave equations to eliminate those errors during wave extrapolation considering the massive computational cost in

acoustic and elastic modeling by optimal time space domain staggered grid finite difference schemes abstractstaggered grid finite difference sfd methods are widely used in modeling seismic wave propagation and the coefficients of finite difference fd operators can be estimated by minimizing dispersion errors using taylor series expansion te or optimization we developed novel optimal time space domain sfd schemes for acoustic and elastic wave equation modeling in our schemes a fourth order multiextreme value objective function with respect to fd coefficients was involved to yield the globally optimal solution with low computational cost we first used variable substitution to turn our optimization problem into a quadratic convex one and then used least squares ls to derive the optimal sfd coefficients by minimizing the relative error of time space domain dispersion relations over a given frequency range to ensure the robustness of our schemes a constraint condition was imposed that the dispersion error at each frequency point did not exceed a given threshold moreover the hybrid absorbin

boundedness of solutions for reversible system via moser s twist theorem abstract in this paper we consider the problem of the boundedness of all solutions for the reversible system x j 0 l b j t x 2 j 1 x x 2 n 1 i 0 n 1 a i t x 2 i 1 0 it is shown that all the solutions are bounded provided that the a i t 0 i n 1 2 are of bounded variation in 0 1 and the derivatives of b j t and a i t n 1 2 1 i n 1 0 j l are lipschitzian it is also shown that there exist a i s being discontinuous everywhere such that all solutions of the equation are bounded this implies that the continuity of a i s is not necessary for the boundedness of solutions of the equation