on the numerical treatment of nonconvex energy problems multilevel decomposition methods for hemivariational inequalities abstract the variational formulation of mechanical problems involving nonmonotone possibly multivalued material or boundary laws leads to hemivariational inequalities since the underlying energy super potentials generally lack both convexity and smoothness these problems cannot be treated by the classical nonlinear analysis methods here we propose a method for the solution of a wide family of hemivariational inequalities it is based on a multilevel decomposition of the admissible solution space combined interactively with an appropriate structure decomposition which gives rise to a finite number of variational inequalities involving convex but nonsmooth energy functionals the latter can be solved easily using existing convex minimization algorithms concluding numerical examples illustrate the applicability of the approach

a sequential quadratically constrained quadratic programming method with an augmented lagrangian line search function based on an augmented lagrangian line search function a sequential quadratically constrained quadratic programming method is proposed for solving nonlinearly constrained optimization problems compared to quadratic programming solved in the traditional sqp methods a convex quadratically constrained quadratic programming is solved here to obtain a search direction and the maratos effect does not occur without any other corrections the active set strategy used in this subproblem can avoid recalculating the unnecessary gradients and approximate hessian matrices of the constraints under certain assumptions the proposed method is proved to be globally superlinearly and quadratically convergent as an extension general problems with inequality and equality constraints as well as nonmonotone line search are also considered

boundedness in asymmetric oscillations abstract in this paper we are concerned with the boundedness of all the solutions of the equation x ax bx f t wheref t is a smooth 2 periodic function aandbare positive constants a b

aubry mather sets for finite twist maps of a cylinder and semilinear duffing equations in this paper based on a generalized version of the aubry mather theorem about twist maps of an infinite cylinder without the usual infinite twist assumption we establish the existence of certain quasiperiodic solutions and unlinked periodic solutions in some semilinear duffing equations x g x p t where p t is a continuous periodic function of period 1 and the semilinearity means that 0 g x x k

feasible side global convergence in experimental optimization we propose a canonical form of the experimental optimization problem and review the state of the art methods to solve it as guarantees of global convergence to an optimal point via only feasible iterates are absent in these methods we present a set of sufficient conditions to enforce this behavior

nonresonance conditions for asymptotically positively homogeneous differential systems the fucik spectrum and its generalization in this article we propose how to define the fucik spectra for differential systems in rnwith jumping nonlinearities the propertyp a generalization of the non fucik spectra to positively homogeneous nonautonomous differential systems is also defined in the most general setting a quantitative estimate result concerning the parameterized completely continuous positively homogeneous perturbations of the identity in banach spaces is given in this article such a result is a generalization of one property in the spectrum theory for completely continuous linear operators using this estimate result we will give some significantly improved nonresonance results from the fucik spectra and propertypfor asymptotically positively homogeneous differential systems

boundedness for the general semilinear duffing equations via the twist theorem abstract in this paper we prove the boundedness of all solutions for the periodic equation x 2 x x g x x t p t where satisfies the diophantine condition x c 19 is bounded p t c 23 and d x i d t j g x t is bounded for 0 i j 21

periodic solutions of perturbed isochronous hamiltonian systems at resonance we look for periodic solutions of planar systems obtained by adding an asymptotically positively homogeneous nonlinear term to an isochronous hamiltonian system precise computations of the topological degree are obtained by elementary phase plane analysis c 2005 elsevier inc all rights reserved

boundedness in nonlinear asymmetric oscillations abstract in this paper the boundedness of all solutions of the nonlinear equation p x p 1 p x p x f x g x e t is discussed where e t c 7 is 2 p periodic f g are bounded c 6 functions p u u p 2 u p 2 are positive constants x max x 0 x max x 0

projected sequential quadratic programming methods in this paper we introduce and analyze a class of optimization methods called projected sequential quadratic programming sqp methods for the solution of optimization problems with nonlinear equality constraints and simple bound constraints on parts of the variables such problems frequently arise in the numerical solution of optimal control problems projected sqp methods combine the ideas of projected newton methods and sqp methods they use the simple projection onto the set defined by the bound constraints and maintain feasibility with respect to these constraints the iterates are computed using an extension of sqp methods and require only the solution of the linearized equality constraint global convergence of these methods is enforced using a constrained merit function and an armijo like line search we discuss global and local convergence properties of these methods the identification of active indices and we present numerical examples for an optimal control problem governed by a nonlinear h

priors and component structures in autoregressive time series models new approaches to prior specification and structuring in autoregressive time series models are introduced and developed we focus on defining classes of prior distributions for parameters and latent variables related to latent components of an autoregressive model for an observed time series these new priors naturally permit the incorporation of both qualitative and quantitative prior information about the number and relative importance of physically meaningful components that represent low frequency trends quasi periodic subprocesses and high frequency residual noise components of observed series the class of priors also naturally incorporates uncertainty about model order and hence leads in posterior analysis to model order assessment and resulting posterior and predictive inferences that incorporate full uncertainties about model order as well as model parameters analysis also formally incorporates uncertainty and leads to inferences about unknown initial values of the time series as it does for predictions of future values posterior analysis involves easily implemented iterative simulation methods developed and described here one motivating field of application is climatology where the evaluation of latent structure especially quasi periodic structure is of critical importance in connection with issues of global climatic variability we explore the analysis of data from the southern oscillation index one of several series that has been central in recent high profile debates in the atmospheric sciences about recent apparent trends in climatic indicators

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 and unboundedness of solutions for reversible oscillators at resonance abstract in this paper we study the problem of boundedness of solutions for the perturbed asymmetric oscillator with damping x f x a x b x g x p t where a b lies on one of the fucik spectrum curves since the system in general is non hamiltonian we have to introduce reversibility assumptions to apply the twist theorem for reversible maps we then obtain a sharp result concerning the boundedness of all solutions

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

boundedness of solutions for nonlinear reversible system at resonance abstract in this paper we are concerned with the problem of boundedness of solutions for the following nonlinear p laplacian p x p 1 n p p x f x p x g x e t where p u u p 2u p 2 for some n n f g c r are odd functions e t c r is odd and 2 p periodic

efficient computation of robust weighted low rank matrix approximations using the l_1 norm the calculation of a low rank approximation to a matrix is fundamental to many algorithms in computer vision and other fields one of the primary tools used for calculating such low rank approximations is the singular value decomposition but this method is not applicable in the case where there are outliers or missing elements in the data unfortunately this is often the case in practice we present a method for low rank matrix approximation which is a generalization of the wiberg algorithm our method calculates the rank constrained factorization which minimizes the l1 norm and does so in the presence of missing data this is achieved by exploiting the differentiability of linear programs and results in an algorithm can be efficiently implemented using existing optimization software we show the results of experiments on synthetic and real data

time series decomposition a constructive result on time series decomposition is presented and illustrated developed through dynamic linear models the decomposition is useful in analysis of an observed time series through inference about underlying latent component series that may have physical interpretations particular special cases include state space autoregressive component models in which the decomposition is useful for isolating latent quasi cyclical components in particular brief summaries of analyses of some geological records related to climatic change illustrate the result

a smooth method for the finite minimax problem we consider unconstrained minimax problems where the objective function is the maximum of a finite number of smooth functions we prove that under usual assumptions it is possible to construct a continuously differentiable function whose minimizers yield the minimizers of the max function and the corresponding minimum values on this basis we can define implementable algorithms for the solution of the minimax problem which are globally convergent at a superlinear convergence rate preliminary numerical results are reported

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

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

infinity of subharmonics for asymmetric duffing equations with the lazer leach dancer condition abstract in this paper based on a generalized version of the poincare birkhoff twist theorem by franks we establish the existence of infinitely many subharmonics for the asymmetric duffing equation with the classical lazer leach dancer condition as a consequence of our result we obtain a sufficient and necessary condition for existence of arbitrarily large amplitude periodic solutions for a class of asymmetric duffing equations at resonance

time series decomposition into oscillation components and phase estimation many time series are naturally considered as a superposition of several oscillation components for example electroencephalogram eeg time series include oscillation components such as alpha beta and gamma we propose a method for decomposing time series into such oscillation components using state space models based on the concept of random frequency modulation gaussian linear state space models for oscillation components are developed in this model the frequency of an oscillator fluctuates by noise time series decomposition is accomplished by this model like the bayesian seasonal adjustment method since the model parameters are estimated from data by the empirical bayes method the amplitudes and the frequencies of oscillation components are determined in a data driven manner also the appropriate number of oscillation components is determined with the akaike information criterion aic in this way the proposed method provides a natural decomposition of the given time series into oscillation components in neuroscience the phase of neural time series plays an important role in neural information processing the proposed method can be used to estimate the phase of each oscillation component and has several advantages over a conventional method based on the hilbert transform thus the proposed method enables an investigation of the phase dynamics of time series numerical results show that the proposed method succeeds in extracting intermittent oscillations like ripples and detecting the phase reset phenomena we apply the proposed method to real data from various fields such as astronomy ecology tidology and neuroscience

boundedness in a piecewise linear oscillator and a variant of the small twist theorem consider the differential equation ddot x n 2 x h l x p t where n 1 2 dots is an integer p is a 2 pi periodic function and h l is the piecewise linear function h l x begin cases l text if x geq 1 lx text if x leq 1 l text if x leq 1 end cases a classical result of lazer and leach implies that this equation has a 2 pi periodic solution if and only if begin equation label ll hat p n 2l over pi end equation where hat p n 1 over 2 pi int 0 2 pi p t e int dt in this paper i prove that if p is of class c 5 then the condition ref ll is also necessary and sufficient for the boundedness of all the solutions of the equation the proof of this theorem motivates a new variant of moser s small twist theorem this variant guarantees the existence of invariant curves for certain mappings of the cylinder which have a twist that may depend on the angle 1991 mathematics subject classification 34c11 58f35

on the convergence of a new trust region algorithm in this paper we present a new trust region algorithm for general nonlinear constrained optimization problems the algorithm is based on the l infty exact penalty function under very mild conditions global convergence results for the algorithm are given local convergence properties are also studied it is shown that the penalty parameter generated by the algorithm will be eventually not less than the l 1 norm of the lagrange multipliers at the accumulation point it is proved that the method is equivalent to the sequential quadratic programming method for all large k hence superlinearly convergent results of the sqp method can be applied numerical results are also reported

roots of unity and unbounded motions of an asymmetric oscillator r 2 z the general questions that we have in mind are under what conditions ona b and f t can we say that 1 has unbounded solutions are there equationsin the class 1 where unbounded and periodic solutions can coexist equation 1 was first considered by dancer in 2 3 and 4 and byfucik in 7 they studied the periodic and dirichlet boundary valueproblems for 1 and looked at this equation as a model of the so called equations with jumping nonlinearities we sum up some of the results in 2 3 consider the family of curves in the a b parameter space 1 a 1 b 2p p 1 2 2 when a b is not in any of these curves then 1 has at least one 2 periodic solution however if a b lies in one of these curves there existfunctions f t for which 1 has no periodic solution more results on theperiodic problem for 1 have been obtained by lazer and mckenna 8 fabry 5 and fabry and fonda 6 the known results on the existenceof periodic solutions already shed some light on the boundedness problem actually the second theorem of massera implies that if 1 has no periodic

objective derivative free methods for constrained optimization we propose feasible descent methods for constrained minimization that do not make explicit use of the derivative of the objective function the methods iteratively sample the objective function value along a finite set of feasible search arcs and decrease the sampling stepsize if an improved objective function value is not sampled the search arcs are obtained by projecting search direction rays onto the feasible set and the search directions are chosen such that a subset approximately generates the cone of first order feasible variations at the current iterate we show that these methods have desirable convergence properties under certain regularity assumptions on the constraints in the case of linear constraints the projections are redundant and the regularity assumptions hold automatically numerical experience with the methods in the linearly constrained case is reported

unboundedness of the large solutions of an asymmetric oscillator abstract in this paper we consider the unboundedness problem of solutions for the following asymmetric oscillator p x p 1 p x p x f x x t where p u u p 2 u p 1 and are positive constants such that 1 p 1 p 2 with n m m n n f x x t is bounded and 2 p periodic in t where p 2 p sin p

invariant tori and unboundedness for second order differential equations with asymmetric nonlinearities depending on the derivatives abstract in this paper the boundedness and unboundedness of solutions of the oscillator x a x b x x x f x g x p t are discussed under some suitable conditions

on combining feasibility descent and superlinear convergence in inequality constrained optimization extension of quasi newton techniques from unconstrained to constrained optimization via sequential quadratic programming sqp presents several difficulties among these are the possible inconsistency away from the solution of first order approximations to the constraints resulting in infeasibility of the quadratic programs and the task of selecting a suitable merit function to induce global convergence in ths case of inequality constrained optimization both of these difficulties disappear if the algorithm is forced to generate iterates that all satisfy the constraints and that yield monotonically decreasing objective function values feasibility of the successive iterates is in fact required in many contexts such as in real time applications or when the objective function is not well defined outside the feasible set it has been recently shown that this can be achieved while preserving local two step superlinear convergence in this note the essential ingredients for an sqp based method exhibiting the desired properties are highlighted correspondingly a class of such algorithms is described and analyzed tests performed with an efficient implementation are discussed

boundedness problem for solutions of semilinear asymmetric equations in this paper we consider the boundedness problem for the solutions of the following semilinear differential equation f p x p 1 email protected p x email protected f p x g x e t where f p u u p 2u p 2 a b are positive constants satisfying a 1 p email protected 1 p email protected 1 g x c r is semilinear and e t c r z

nonlinear resonance in asymmetric oscillators we study the periodic solutions of equations with asymmetric nonlinearities at resonance with the fucik spectrum we compute the associated topological degree and prove existence multiplicity and stability of large amplitude oscillations for equations with a small friction term such equations can be viewed e g as simple models for investigating vertical oscillations of long span suspension bridges the results are typically of a nonlinear nature as some of the situations observed cannot occur with a linear equation c 1998 academic press

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

periodic and unbounded motions in asymmetric oscillators at resonance abstract we consider the existence of periodic and unbounded solutions for the asymmetric oscillator x a x b x g x p t where x max x 0 x max x 0 a and b are two positive constants p t is a 2 periodic smooth function and g x satisfies lim x x 1 g x 0 we have proved previously that the boundedness of all the solutions and the existence of unbounded solutions have a close relation to the interaction of some well defined functions p and h in this paper we consider some critical cases and obtain some new sufficient conditions for the existence of 2 periodic and unbounded solutions in particular the obtained periodic solutions are new and cannot be deduced from landesman lazer s and fabry fonda s existence conditions in contrast with many existing results the function g x may be unbounded or oscillatory without asymptotic limits

nonlinear resonance in asymmetric oscillations in this paper we study the periodic problem f p x email protected f p x email protected f p x f x t where f p u u p 2u p 1 a 1 email protected 1 p 2n 1 email protected n by computing the associated topological degree we prove existence of periodic solutions and stability of large amplitude oscillations for equations with a small damping term

nonlinear programming and nonsmooth optimization by successive linear programming methods are considered for solving nonlinear programming problems using an exactl1 penalty function lp like subproblems incorporating a trust region constraint are solved successively both to estimate the active set and to provide a foundation for proving global convergence in one particular method second order information is represented by approximating the reduced hessian matrix and coleman conn steps are taken a criterion for accepting these steps is given which enables the superlinear convergence properties of the coleman conn method to be retained whilst preserving global convergence and avoiding the maratos effect the methods generalize to solve a wide range of composite nonsmooth optimization problems and the theory is presented in this general setting a range of numerical experiments on small test problems is described

boundedness in forced isochronous oscillators abstract in this paper we are concerned with the boundedness of all solutions for the forced isochronous oscillator x v x g x f t where v is a so called t isochronous potential the perturbation g is assumed to be bounded and the 2 periodic function f t is smooth using the resonant small twist theorem and averaged small twist theorem established by ortega we will prove the boundedness of all solutions for the above forced isochronous oscillator in the resonant and non resonant cases under some reasonable assumptions respectively

boundedness of solutions for second order differential equations with asymmetric nonlinearity abstract in this paper we are concerned with the boundedness of all solutions of the second order differential equations x f x a x b x g x p t where a b does not lie on one of the fucik spectrum curves the functions f and g are bounded and p is 2 periodic

a feasible trust region sequential quadratic programming algorithm an algorithm for smooth nonlinear constrained optimization problems is described in which a sequence of feasible iterates is generated by solving a trust region sequential quadratic programming sqp subproblem at each iteration and by perturbing the resulting step to retain feasibility of each iterate by retaining feasibility the algorithm avoids several complications of other trust region sqp approaches the objective function can be used as a merit function and the sqp subproblems are feasible for all choices of the trust region radius global convergence properties are analyzed under various assumptions on the approximate hessian under additional assumptions superlinear convergence to points satisfying second order sufficient conditions is proved

a numerically stable dual method for solving strictly convex quadratic programs an efficient and numerically stable dual algorithm for positive definite quadratic programming is described which takes advantage of the fact that the unconstrained minimum of the objective function can be used as a starting point its implementation utilizes the cholesky and qr factorizations and procedures for updating them the performance of the dual algorithm is compared against that of primal algorithms when used to solve randomly generated test problems and quadratic programs generated in the course of solving nonlinear programming problems by a successive quadratic programming code the principal motivation for the development of the algorithm these computational results indicate that the dual algorithm is superior to primal algorithms when a primal feasible point is not readily available the algorithm is also compared theoretically to the modified simplex type dual methods of lemke and van de panne and whinston and it is illustrated by a numerical example

on the convergence of successive linear quadratic programming algorithms the global convergence properties of a class of penalty methods for nonlinear programming are analyzed these methods include successive linear programming approaches and more specifically the successive linear quadratic programming approach presented by byrd et al math program 100 2004 pp 27 48 every iteration requires the solution of two trust region subproblems involving piecewise linear and quadratic models respectively it is shown that for a fixed penalty parameter the sequence of iterates approaches stationarity of the penalty function a procedure for dynamically adjusting the penalty parameter is described and global convergence results for it are established

inexact newton kantorovich methods for constrained nonlinear model predictive control in this paper we consider newton kantorovich type methods for solving control constrained optimal control problems that appear in model predictive control conditions for convergence are established for an inexact version of the newton kantorovich method applied to variational inequalities based on these results two groups of algorithms are proposed to solve the optimality system the first group includes exact and inexact newton and newton kantorovich implementations of the sequential quadratic programming in the second group exact and inexact newton and newton kantorovich methods are developed for solving a nonsmooth normal map equation equivalent to the optimality system numerical simulations featuring examples from the aerospace and automotive domain are presented which show that inexact newton kantorovich type methods can achieve significant reduction of the computational time

unbounded solutions of a class of planar systems abstract in this paper the unbounded solutions for the following nonlinear planar system x a y a y f t y b x b x g t is discussed where a b are positive constants satisfying 1 a b 1 a b 1 a b 1 a b 4 x max x 0 y max y 0 r q f t g t l 0 2 are 2 periodic functions

a feasible trust region algorithm for inequality constrained optimization the paper presents an algorithm for smooth nonlinearly inequality constrained optimization problems in which a sequence of feasible iterates is generated by a trust region sequential quadratic programming subproblem at each iteration because of retaining feasibility the objective function can be used as a merit function and the subproblems are feasible under common assumptions the algorithm is globally convergent the numerical results show it is promising

positively homogeneous hamiltonian systems in the plane i try to give a general description of the dynamics of the solutions for a planar hamiltonian system with positively homogeneous hamiltonian function and periodic forcing term most of the results obtained are already known in the special case of a scalar second order differential equation with asymmetric nonlinearity

on the superlinear convergence of a trust region algorithm for nonsmooth optimization it is proved that the second order correction trust region algorithm of fletcher 5 ensures superlinear convergence if some mild conditions are satisfied

projected newton methods for optimization problems with simple constraints we consider the problem min f x x geqq 0 and propose algorithms of the form x k 1 x k alpha k d k nabla f x k where cdot denotes projection on the positive orthant alpha k is a stepsize chosen by an armijo like rule and d k is a positive definite symmetric matrix which is partly diagonal we show that d k can be calculated simply on the basis of second derivatives of f so that the resulting newton like algorithm has a typically superlinear rate of convergence with other choices of d k convergence at a typically linear rate is obtained the algorithms are almost as simple as their unconstrained counterparts they are well suited for problems of large dimension such as those arising in optimal control while being competitive with existing methods for low dimensional problems the effectiveness of the newton like algorithm is demonstrated via computational examples involving as many as 10 000 variables extensions to general linearly constrained pr

the rotation number approach to the periodic fuc caron ik spectrum abstract in this paper we study the fucik spectrum of the problem q t x q t x 0 with the 2 periodic boundary condition where q t are 2 periodic after introducing a rotation number function for we prove using the hamiltonian structure and the positive homogeneity of that for any positive integer n the two boundary curves of the domain 1 n 2 in the plane are fucik curves of the result obtained in this paper shows that such a spectrum problem is much like that of the higher dimensional fucik spectrum with the dirichlet condition in particular it remains open if the fucik spectrum of is composed of only these curves

on the accurate identification of active constraints we consider nonlinear programs with inequality constraints and we focus on the problem of identifying those constraints which will be active at an isolated local solution the correct identification of active constraints is important from both a theoretical and a practical point of view such an identification removes the combinatorial aspect of the problem and locally reduces the inequality constrained minimization problem to an equality constrained problem which can be more easily dealt with we present a new technique which identifies active constraints in a neighborhood of a solution and which requires neither complementary slackness nor uniqueness of the multipliers we also present extensions to variational inequalities and numerical examples illustrating the identification technique

invariant tori for a reversible oscillator with a nonlinear damping and periodic forcing term abstract in this paper the boundedness of all solutions of the oscillator x f x x 2 x x p t is discussed where is a positive constant f x y x and p t are smooth functions and p t p t 2

bayesian non parametric signal extraction for gaussian time series abstract we consider the problem of unobserved components in time series from a bayesian non parametric perspective the identification conditions are treated as unknown and analyzed in a probabilistic framework in particular informative prior distributions force the spectral decomposition to be in an identifiable region then the likelihood function adapts the prior decompositions to the data a full bayesian analysis of unobserved components will be presented for financial high frequency data particularly a three component model long term intra daily and short term will be analyzed to emphasize the importance and the potential of this work when dealing with the value at risk analysis a second astronomical application will show how to deal with multiple periodicities

boundedness of solutions for equations with p laplacian and an asymmetric nonlinear term abstract for p 1 and p s s p 2 s we are concerned with the boundedness of solutions for the equation p x p x p x f t x where x max x 0 x max x 0 and f t x is 2 periodic in t when p 1 p p 1 p 2 n the resonant situation and f has limits f t as x there is a function z plays a central role for the boundedness of solutions more precisely if z is of constant sign then all solutions are bounded moreover such condition also guarantees the boundedness when near a fucik curve

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

an exact multivariate model based structural decomposition we propose a simple and structured procedure for decomposing a vector of time series into trend cycle seasonal and irregular components contrary to common practice we do not assume these components to be orthogonal conditional on their past however the state space representation employed ensures that their estimates converge to values with null variances and covariances null variances are very important as they ensure that the components do not change when the sample increases this lack of revisions is the most important feature of our method in comparison with most alternative procedures on the other hand null covariances provide a solid statistical foundation for the decomposition as it ensures that a given component can be analyzed and interpreted independently of any other component s other convenient properties of our method derive from the use of a state space approach first defining the problem in state space avoids dependence on particular model specifications so the same proc

almost periodic solutions for an asymmetric oscillation abstract in this paper we study the dynamical behavior of the differential equation x a x b x f t where x max x 0 x max x 0 a and b are two different positive constants f t is a real analytic almost periodic function for this purpose firstly we have to establish some variants of the invariant curve theorem of planar almost periodic mappings which was proved recently by the authors see 11 then we will discuss the existence of almost periodic solutions and the boundedness of all solutions for the above asymmetric oscillation

boundedness of solutions for semilinear duffing equations in this paper we are concerned with the boundedness of all the solutions and the existence of quasi periodic solutions for some semilinear duffing equationx g x e t wheree t is a smooth 2 periodic function and0 g x x k

boundedness of semilinear duffing equations at resonance in a critical situation abstract in this paper we propose a sufficient and necessary condition for the boundedness of all the solutions for the equation x n 2 x g x p t with the critical situation that 0 2 p t e i n t d t 2 g g on g and p where n n p t is periodic and g x is bounded

quasi periodic solutions of a forced asymmetric oscillator at resonance abstract it is proved the existence of aubry mather sets and infinitely many subharmonic solutions to an equation of the form u au bu u p t where u max u 0 u max u 0 r r is continuous p 0 2 r is continuous and 2 periodic we deal with the situation when a b are two positive constants satisfying 1 a 1 b 2 n n n