stochastic mechanics of reciprocal diffusions the dynamics and kinematics of reciprocal diffusions were examined in a previous paper j math phys 34 1846 1993 where it was shown that reciprocal diffusions admit a chain of conservation laws which close after the first two laws for two disjoint subclasses of reciprocal diffusions the markov and quantum diffusions for the case of quantum diffusions the conservation laws are equivalent to schrodinger s equation the markov diffusions were employed by schrodinger sitzungsber preuss akad wiss phys math kl 144 1931 ann inst h poincare 2 269 1932 nelson dynamical theories of brownian motion princeton university princeton nj 1967 quantum fluctuations princeton university princeton nj 1985 and other researchers to develop stochastic formulations of quantum mechanics called stochastic mechanics we propose here an alternative version of stochastic mechanics based on quantum diffusions a procedure is presented for constructing the quantum diffusion associated to a given wave function it is shown that quantum diffusions satisfy the uncertainty principle and have a locality property whereby given two dynamically uncoupled but statistically correlated particles the marginal statistics of each particle depend only on the local fields to which the particle is subjected however like wigner s joint probability distribution for the position and momentum of a particle the finite joint probability densities of quantum diffusions may take negative values

study of a class of subharmonic motions using a non smooth temporal transformation nstt abstract a nonlinear boundary value problem nbvp formulation for computing strongly nonlinear subharmonic orbits of a class of harmonically forced conservative systems is presented the formulation is based on a non smooth temporal transformation nstt to replace the independent temporal variable of the problem with two new piecewise linear periodic temporal variables as a result the problem of computing the subharmonic motions is reduced to a set of nbvps with homogeneous boundary conditions some numerical and analytical solutions of the reduced nbvps are given and the asymptotic behavior in the limit of large period of the derived analytical approximations is discussed in particular the formulation shows a definite connection between vibro impact oscillations considered in mechanics and strongly nonlinear oscillations close to a heteroclinic orbit an interesting feature of a family of subharmonic solutions analyzed in this work is that in the limit of large periods it degenerates to the perturbed stable and unstable manifolds of an unstable periodic orbit of the system

correspondence a note on efficient smoothing for boundary value models abstract the smoothing problem for two point boundary value models is investigated it is shown that by triangularizing the relevant hamiltonian system a new smoothing algorithm can be derived which is twice as fast as that obtained by diagonalizing the hamiltonian

existence of exponentially small separatrix splittings and homoclinic connections between whiskered tori in weakly hyperbolic near integrable hamiltonian systems abstract in this paper we study a generalization of arnold s original example in which he discussed the existence of a mechanism for instability caused by the splitting of the homoclinic manifolds of the weakly hyperbolic tori that has subsequently been referred to as arnold diffusion in case when the number of degrees of freedom n 3 namely we consider a widely studied model of a pendulum weakly coupled with n 1 rotors with the degeneracies in the unperturbed hamiltonian corresponding to different timescales existing in the problem using an alloy of the iterative and direct methods developed within the last years we give exponentially small upper bounds for the splitting measure of transversality for the case of an even analytic perturbation thus improving the estimate of gallavotti 1994 which he calls quasiflat and generalizing the analogous recent estimate of delshams et al 1996 for the rapidly quasiperiodically forced pendulum to a much larger class of hamiltonian systems in particular the exponentially small upper bound for the transversality measure of the splitting applies when the hamiltonian has extra degeneracies namely when the frequencies on a torus become near resonant in fact we show that in such a case the quantity in question becomes smaller which is the incarnation of the general fact that resonant regions in the action space are more stable in the sense that they have larger nekhoroshev exponent nekhoroshev exponent nevertheless we emphasize that getting uniform estimates for an arbitrary n 3 is very hard unless one makes some additional assumptions on the approximation properties of the frequency vector although recent developments show that the first order of canonical perturbation theory given by melnikov integrals generally cannot be accepted as the leading order answer for the splitting distance for the case of more than two degrees of freedom including the rapidly quasiperiodically forced pendulum problem we suggest an analytic perturbation the majority of whose fourier components are strictly nondashzero for which melnikov integrals can be vindicated as the leading order approximation for the components of the splitting distance in different directions if the frequencies on the invariant tori satisfy certain arithmetic conditions this allows us to bound the splitting distance from below furthermore having such a perturbation for the case of three degrees of freedom we use a simple number theoretical argument to find the asymptotics of the fourier series with exponentially small coefficients involved this enables us to compute the numerous homoclinic orbits for the whiskered tori of asymptotically full measure and by proving the domineering contribution of the first order of perturbation theory for the transversality measure to suggest a leading order answer for this quantity which implies the existence of an infinite number of heteroclinic connections between tori with close diophantine frequencies we elucidate the numerous arithmetic issues that obstruct getting a compact leading order approximation for the splitting size most of which can be overcome in the case of three degrees of freedom as our example shows these obstacles can be also possibly avoided in the same fashion for an arbitrary n 3 if one treats the case when the frequencies of the rotors are near a resonance of multiplicity n 3 or n 2

exponentially small splitting of separatrices in a weakly hyperbolic case abstract we validate the poincare melnikov method in the singular case of high frequency periodic perturbations of the hamiltonian h 0 x y 1 2 y 2 x 3 x 4 under appropriate conditions which among other things imply that we are considering the bifurcation case when the character of the fixed point changes from parabolic in the unperturbed case to hyperbolic in the perturbed one the splitting is exponentially small

geometry and chaos near resonant equilibria of 3 dof hamiltonian systems in this paper we study the dynamics near resonant elliptic equilibria in three degree of freedom hamiltonian systems the resonances we consider have multiplicity two and the corresponding local normal form for the equilibrium is integrable at cubic order we prove the existence of families of 3 tori and whiskered 2 tori with nearby chaotic dynamics in the quartic normal form the whiskers of the 2 tori intersect in a non trivial way giving rise to multi pulse homoclinic and heteroclinic connections these connections survive in the full system as orbits homoclinic to invariant 3 spheres

linear estimation of boundary value stochastic processes part i the role and construction of complementary models this paper presents a substantial extension of the method of complementary models for minimum variance linear estimation introduced by weinert and desai in their important paper 1 specifically the method of complementary models is extended to solve estimation problems for both discrete and continuous parameter linear boundary value stochastic processes in one and higher dimensions a major contribution of this paper is an application of green s identity in deriving a differential operator representation of the estimator to clarify the development and to illustrate the range of applications of our approach two brief examples are provided one is a 1 d discrete two point boundary value process and the other is a 2 d process governed by poisson s equation on the unit disk

non existence of truly solitary waves in water with small surface tension this paper considers permanent capillary gravity waves on the free surface of a two dimensional incompressible inviscid fluid of finite depth it is shown that there are no solitary wave solutions of the exact governing equations of the flow that decay to zero exponentially at infinity if the surface tension coefficient is less than its critical value and lies in some intervals the proof is based upon an estimate of a constant that is related to the approximation of the solution if it exists near its singularity the approximation satisfies a fourth order nonlinear ordinary differential equation when the solution is extended to the complex plane then the non existence of truly solitary waves is obtained by using a contradiction on this constant

interaction of sine gordon kinks with defects the two bounce resonance a model of soliton defect interactions in the sine gordon equations is studied using singular perturbation theory melnikov theory is used to derive a critical velocity for strong interactions which is shown to be exponentially small for weak defects matched asymptotic expansions for nearly heteroclinic orbits are constructed for the initial value problem which are then used to derive analytical formulas for the locations of the well known two and three bounce resonance windows as well as several other phenomena seen in numerical simulations

some comments on conditionally markov and reciprocal gaussian processes corresp mehr and mcfadden s conditionally markov processes and jamison s reciprocal processes are two independent extensions of slepian s work on a particular stationary gaussian process the relationship between them and their relationship with gauss markov processes are discussed

bayesian estimation and the kalman filter in this tutorial article we give a bayesian derivation of a basic state estimation result for discrete time markov process models with independent process and measurement noise and measurements not affecting the state we then list some properties of gaussian random vectors and show how the kalman filtering algorithm follows from the general state estimation result and a linear gaussian model definition we give some illustrative examples including a probabilistic turing machine dynamic classification and tracking a moving object

the discrete frenkel kontorova model and its extensions i exact results for the ground states abstract we present a rigorous study of the classical ground states under boundary conditions of a class of one dimensional models generalizing the discrete frenkel kontorova model the extremalization equations of the energy of these models turn out to define area preserving twist maps which exhibits periodic quasi periodic and chaotic orbits for all boundary conditions we select among all the extremum solutions of the energy of the model those which correspond to the ground states of the infinite system we prove that these ground states are either periodic commensurate or quasi periodic incommensurate but are never chaotic we also prove the existence of elementary discommensurations which are minimum energy configuration of the model for certain special boundary conditions the topological structure of the whole set of ground states is described in details in addition to physical applications consequences for twist map homeomorphisms are mentioned part ii s aubry p y ledaeron and g andre will be mostly devoted to exact results on the transition by breaking of analyticity which occurs on the incommensurate ground states when the model parameters vary and on its connection with the stochasticity threshold in the corresponding twist map

the break up of a heteroclinic connection in a volume preserving mapping abstract we consider a family of three dimensional volume preserving maps depending on a small parameter e as e 0 these maps asymptote to flows which attain a heteroclinic connection we show that for small e the heteroclinic connection breaks up and that the splitting between its components scales with e like e exp e we estimate using the singularities of the e 0 heteroclinic orbit in the complex plane we then estimate using linearization about orbits in the complex plane while these estimates are not proven they are well supported by our numerical calculations the work described here is a special case of the theory derived by amick et al which applies to q dimensional volume preserving mappings

polynomial identities for the jordan algebra of a symmetric bilinear form 1 the main purpose of this paper is to obtain a description of the relatively free algebras fj b and fj of the varieties of unitary jordan algebras 2b var g and 2 var gk the g module structure of these algebras is established and their hilbert or poincare series are computed a formula for the codimensions of 22 is given and the codimensions of i and 2bk are estimated asymptotically the varieties 2bk are described as subvarieties of 93 and lbk the proofs of these results use the representation theory of the symmetric and general linear groups the essential results are contained in sections 5 7 while the first four sections give the necessary preliminaries section 8 lists some open problems on varieties of jordan algebras motivated by results and problems from the associative theory 1

gaussian conditionally markov sequences dynamic models and representations of reciprocal and other classes conditionally markov cm sequences are powerful mathematical tools for modeling problems one class of cm sequences is the reciprocal sequence in application we need not only cm dynamic models but also know how to design model parameters models of two important classes of nonsingular gaussian ng cm sequences called cm l and cm f models and a model of the ng reciprocal sequence called reciprocal cm l model were presented in our previous works and their applications were discussed in this paper these models are studied in more detail in particular their parameter design it is shown that every reciprocal cm l model can be induced by a markov model then parameters of each reciprocal cm l model can be obtained from those of the markov model also it is shown that an ng cm l cm f sequence can be represented by a sum of an ng markov sequence and an uncorrelated ng vector this necessary and sufficient representation provides a basis for designing parameters of a cm l cm f model from the cm viewpoint a representation is also obtained for ng reciprocal sequences this representation is simple and reveals an important property of reciprocal sequences as a result the significance of studying reciprocal sequences from the cm viewpoint is demonstrated a full spectrum of dynamic models from a cm l model to a reciprocal cm l model is also presented some examples are presented for illustration

gaussian conditionally markov sequences singular nonsingular most existing results about modeling and characterizing gaussian markov reciprocal and conditionally markov cm processes assume nonsingularity of the processes this assumption makes the analysis easier but restricts application of these processes this paper studies models and characterizes the general singular nonsingular gaussian cm including reciprocal and markov sequence for example to our knowledge there is no dynamic model for the general singular nonsingular gaussian reciprocal sequence in the literature we obtain two such models from the cm viewpoint as a result the significance of studying reciprocal sequences from the cm viewpoint is demonstrated the results of this paper unify singular and nonsingular gaussian cm including reciprocal and markov sequences and provide tools for their application an application of cm sequences in trajectory modeling with a destination is discussed and illustrative examples are presented

exponentially small splitting of separatrices under fast quasiperiodic forcing we consider fast quasiperiodic perturbations with two frequencies 1 epsiv of a pendulum where is the golden mean number the complete system has a two dimensional invariant torus in a neighbourhood of the saddle point we study the splitting of the three dimensional invariant manifolds associated to this torus provided that the perturbation amplitude is small enough with respect to and some of its fourier coefficients the ones associated to fibonacci numbers are separated from zero it is proved that the invariant manifolds split and that the value of the splitting which turns out to be exponentially small with respect to is correctly predicted by the melnikov function

discrete time complementary models and smoothing abstract the fact that the smoothing error is a wide sense markov process is somehow surprising since smoothed estimates depend upon both past and future data in this paper we first give a simple and general proof of this fact then we use the so called complementary models introduced by weinert and desai to derive forwards and backwards markovian models for the smoothing error in state space models by exploring the structure of the complementary models we show that under certain restrictions only two simple structured models exist one that runs forwards in time and the other that runs backwards in time the forwards complementary model leads to the forward rauch tung striebel rts smoothing formula and to a backwards markovian model for the error whereas the backwards model leads to the backward rts formula and to a forwards error model the two models for the smoothing error can be derived one from the other by a forward backward transformation that preserves the sample paths finally by using

on the role of high order resonances in normal forms and in separatrix splitting we discuss the role of high order resonances in the construction of normal forms for e close to integrable hamiltonian systems by heuristic considerations based on standard estimates we show that the remainder of normal forms is dominated by the terms corresponding to the main high order resonances and we provide a general argument to show that the size of such leading terms is exponentially small in 1e we apply this method to the problem of estimating the splitting of separatrices in resonant perturbed systems

exponentially small splitting of separatrices beyond melnikov analysis rigorous results in this paper we study the problem of exponentially small splitting of separatrices of one degree of freedom classical hamiltonian systems with a non autonomous perturbation which is fast and periodic in time we provide a result valid for general systems which are polynomials or trigonometric polynomials in the state variables our result consists in obtaining a rigorous proof of the asymptotic formula for the measure of the splitting we have obtained that the splitting has the asymptotic behavior k e a identifying the constants k and a in terms of the features of the system the study of our problem leads us to consider several cases in some cases assuming the per turbation is small enough it turns out that the values of k coincides with the classical melnikov approach we have identified the limit size of the perturbation for which this classical theory holds true however for the limit cases which appear naturally both in averaging theory and bifurcation theory we encounter that generically neither k nor are actually well predicted by melnikov theory

oscillatory motions for the restricted planar circular three body problem the restricted three body problem models the motion of a massless body under the influence of the newtonian gravitational force caused by two other bodies called the primaries when they move along circular keplerian orbits and the third body moves in the same plane one has the restricted planar circular three body problem rpc3bp in suitable coordinates it is a hamiltonian system of two degrees of freedom the conserved energy is usually called the jacobi constant llibre and simo math ann 248 2 153 184 1980 proved the existence of oscillatory motions for this system that is orbits which leave every bounded region but which return infinitely often to some fixed bounded region to prove their existence they had to assume the ratio between the masses of the primaries to be small enough in this paper we prove the existence of such motions for any value of the mass ratio mu closing the problem of existence of oscillatory motions in the rpc3bp to obtain such motions we restrict ourselves to the level sets of the jacobi constant we show that for any value of the mass ratio and for large values of the jacobi constant there exist transversal intersections between the stable and unstable manifolds of infinity in these level sets these transversal intersections guarantee the existence of a symbolic dynamics that creates the oscillatory orbits the main achievement is to prove the existence of these orbits without assuming the mass ratio mu small when mu is not small this transversality can not be checked by means of classical perturbation theory since our method is valid for all values of mu we are able to detect a curve in the parameter space formed by mu and the jacobi constant where cubic homoclinic tangencies between the invariant manifolds of infinity appear

nonpersistence of breather families for the perturbed sine gordon equation we show that up to one exception and as a consequence of first order perturbation theory only it is impossible that a large portion of the well known family of breather solutions to the sine gordon equation could persist under any nontrivial perturbation of the form u tt u xx sin u varepsilon delta left u right o left varepsilon 2 right where is an analytic function in anarbitrarily small neighbourhood ofu 0 improving known results we analyze and overcome the particular difficulties that arise when one allows the domain of analyticity of to be small the single exception is a one dimensional linear space of perturbation functions under which the full family of breathers does persist up to first order in e

melnikov method and exponentially small splitting of separatrices abstract we study the splitting of separatrices for high frequency perturbations of a planar hamiltonian system it is well known that in the analytic case the splitting of separatrices is exponentially small with respect to the perturbation period 2 e and that the constant in the exponent is related to the position of singularities of the unperturbed separatrix in the complex time plane on the other hand in many examples the melnikov function can be calculated explicitly and is exponentially small with an exponent very close but larger than the exponent from the upper estimates so the melnikov function appears to be smaller than the upper estimate as well as the standard estimate of the approximation error we found a class of hamiltonians such that the melnikov function gives actual asymptotics for the splitting provided the perturbation amplitude is less than 0ep for some p 0

2 graded polynomial identities for the jordan algebra of the symmetric matrices of order two abstract the jordan algebra of the symmetric matrices of order two over a field k has two natural gradings by z 2 the cyclic group of order 2 we describe the graded polynomial identities for these two gradings when the base field is infinite and of characteristic different from 2 we exhibit bases for these identities in each of the two cases in one of the cases we perform a series of computations in order to reduce the problem to dealing with associators while in the other case one employs methods and results from invariant theory moreover we extend the latter grading to a z 2 grading on b n the jordan algebra of a symmetric bilinear form in a vector space of dimension n n 1 2 we call this grading the scalar one since its even part consists only of the scalars as a by product we obtain finite bases of the z 2 graded identities for b n in fact the last result describes the weak jordan polynomial identities for the pair b n v n

stationarity and reciprocity in stochastic multipoint boundary value systems investigates certain statistical properties of the state vector in a stochastic multipoint boundary value system the authors give necessary and sufficient conditions for the state vector to be stationary and prove that the state vector is locally reciprocal

non integrability and continuation of fixed points of 2 n dimensional perturbed twist maps abstract in this paper a simple criterion to prove non integrability of symplectic perturbed twist mappings in 2 n dimensions is developed for sufficiently small perturbations in addition an upper bound for the number of isolating integrals the system can possess is provided a criterion for the analytic continuation of isolated periodic orbits in case of a small nonzero perturbation of twist maps is found the evaluation of their linear stability character by obtaining a simplified expression of the eigenvalues of the jacobian matrix concludes the theoretical part the theory is finally applied to a twist map perturbed by the suris potential and also a four dimensional map with physical interest

width of stochastic layers in near integrable two dimensional symplectic maps abstract width of stochastic layers around separatrix eight is studied for two dimensional symplectic maps close to integrable we prove that under rather general assumptions the relation w d 1 log holds where w is the width of the stochastic layer d is the width of a lobe domain and is a multiplier at the corresponding hyperbolic point

exponentially small estimate for the amplitude of capillary ripples of a generalized solitary wave abstract a recently discovered phenomenon of capillary gravity waves on water is a solitary wave of elevation plus small amplitude ripples at infinity it has been conjectured by beale comm pure appl math 44 1991 211 257 that the amplitude of the ripples should be exponentially small beyond all orders in this paper we prove this conjecture

separatrix splitting for systems with three time scales an exact expression for the determinant of the splitting matrix is derived for three degrees of freedom systems with three time scales it allows us to analyze the asymptotic behaviour needed to amend the large angles theorem proposed in ann inst h poincare b 60 1 1994 the asymptotic validity of mel nikov s integrals is proved for the class of models considered which are polynomial perturbations the technique for exhibiting cancellations is inspired by renormalization theory in quantum electrodynamics and uses an analogue of dyson s equations to prove an infinite family of identities due to symmetries that remind us of ward s identities

a refined formula for the separatrix splitting for the standard map abstract we study the standard map for small values of a parameter e an asymptotic formula is found which describes the size of the splitting of the stable and unstable separatrices of the origin considered as a hyperbolic fixed point the formula contains an exponentially decreasing expression times an asymptotic series on powers of a small parameter the coefficients in the asymptotic expansion are to be found in the process of solving numerically a nonlinear recurrent system of difference equations in the complex domain

on the existence of small amplitude solitary waves with strong surface tension an existence theory for small amplitude solitary waves with surface tension effects included is developed for large values of the surface tension parameter 13 using ideas of beale the nash moser implicit function theorem is applied to justify the well known approximation of korteweg and devries some of the recent results of amick and kirchgassner are thereby achieved more directly and additional insight obtained for the open case 13

representations for multivariate reciprocal gaussian processes multivariate reciprocal gaussian processes are represented as a sum of two independent processes a piecewise markov process which is also represented in terms of a wiener type process and a time dependent linear transformation of a normally distributed random vector this result is then applied to the first passage time problem

a maximum entropy solution of the covariance extension problem for reciprocal processes stationary reciprocal processes defined on a finite interval of the integer line can be seen as a special class of markov random fields restricted to one dimension nonstationary reciprocal processes have been extensively studied in the past especially by jamison et al the specialization of the nonstationary theory to the stationary case however does not seem to have been pursued in sufficient depth in the literature stationary reciprocal processes and reciprocal stochastic models are potentially useful for describing signals which naturally live in a finite region of the time or space line estimation or identification of these models starting from observed data seems still to be an open problem which can lead to many interesting applications in signal and image processing in this paper we discuss a class of reciprocal processes which is the acausal analog of auto regressive ar processes familiar in control and signal processing we show that maximum likelihood identification of these processes leads to a covariance extension problem for block circulant covariance matrices this generalizes the famous covariance band extension problem for stationary processes on the integer line as in the usual stationary setting on the integer line the covariance extension problem turns out to be a basic conceptual and practical step in solving the identification problem we show that the maximum entropy principle leads to a complete solution of the problem

on nonlinear boundary value problems ghosts parasites and discretizations ghosts are a new class of approximate solutions to nonlinear boundary value problems bvps they are bounded away from true solutions but solve an initial value problem ivp for the differential equations exactly and approximate boundary and internal matching conditions with exponentially decaying error as a control parameter approaches infinity ghosts are characteristic of singularly perturbed differential equations in which solutions are localized and the ivp depends sensitively on initial data they correspond to shifted localized solutions and they can be observed in imperfect physical experiments and be approximated by convergent discretization schemes the latter property suggests that the concept of the numerically irrelevant solution introduced for discretized second order bvps should be re examined

hamiltonian maps in the complex plane abstract the standard map is a nonintegrable discrete time analog of the vertical pendulum detailed calculations are presented and illustrated graphically for the standard map at the golden mean frequency the functional dependence of the coordinate q on the canonical angle variable is analtyically continued into the complex plane where natural boundaries are found at constant absolute values of im the boundaries represent the appearance of chaotic motion in the complex plane when the domain of analyticity shrinks to zero the kam invariant curve is destroyed two independent numerical methods with fourier analysis in the angle variable were used one based on a variation annihilation method and the other on a double expansion the results were further checked by direct solution of the complex equations of motion the numerically simpler but intrinsically complex semipendulum and semistandard map are also studied we conjecture that natural boundaries appear in the analogous analytic continuation of the invariant tori or kam surfaces of general nonintegrable systems with analytic hamiltonians

optimal smoothing for finite state hidden reciprocal processes this technical note addresses modelling and estimation of a class of finite state random processes called hidden reciprocal chains hrc a hidden reciprocal chain consists of a finite state reciprocal process together with an observation process conditioned on the reciprocal process much as in the case of a hidden markov model hmm the key difference between markov models and reciprocal models is that reciprocal models are non causal the technical note presents a characterization of a hrc by a finite set of hidden markov bridges which are hmms with the final state fixed the technical note then uses this characterization to derive the optimal fixed interval smoother for a hrc performance of linear and optimal smoothers derived for both hmm and hrc are compared using simulations for a class of hrc derived from underlying markov transitions these experiments suggest that not surprisingly the performance of the optimal hmm and hrc smoothers are signifcantly better than their linear counterparts and that some performance improvement is obtained using the hrc smoothers compared to the hmm smoothers the technical note concludes by mentioning some ongoing and future work which exploits this new markov bridge characterization of a hrc

modeling and estimation of discrete time gaussian reciprocal processes discrete time gaussian reciprocal processes are characterized in terms of a second order two point boundary value nearest neighbor model driven by a locally correlated noise whose correlation is specified by the model dynamics this second order model is the analog for reciprocal processes of the standard first order state space models for markov processes the model is used to obtain a solution to the smoothing problem for reciprocal processes the resulting smoother obeys second order equations whose structure is similar to that of the kalman filter for gauss markov processes it is shown that the smoothing error is itself a reciprocal process

modelling and estimation for finite state reciprocal processes a reciprocal equation is a kind of descriptor linear discrete index stochastic system which is well known be satisfied pathwise by all gaussian reciprocal processes from a system theoretic point of view it is a kind of noncausal linear system in the sense that the solution of it cannot be determined by only an initial condition indeed requiring the terminal state as well besides all the input function between initial and terminal states also nice properties are known of a reciprocal equation such as the equivalence of it with a couple of ordinary causal dynamic systems running in opposite directions for these reasons here we assume a reciprocal equation as the target of stochastic realization for the class of finite state reciprocal processes also named reciprocal chains the central result of the present paper is showing that any canonical reciprocal chain i e valued in the canonical base of realrn n being the cardinality of the set of chain s states satisfies pathwise a reciprocal equation in a n2 dimensional canonical variable or in other word a quadratic reciprocal equation named augmented state reciprocal model asrm also for a partially observed reciprocal chain a linear optimal smoother is derived all the results here presented are based upon the idea that a reciprocal chain is a combination of markov bridges to this purpose other forms besides the asrm are presented in order to make clear the meaning of this combination as well as to prove that the linear smoother can be actually implemented as n smoothers all operating independently on each markov bridge component

a proof of the exponentially small transversality of the separatrices for the standard map in 1984 v f lazutkin laz84 lst89 obtained an asymptotic formula for the separatrix splitting angle for the standard map the difficulty of this problem is related to the exponential smallness of the splitting with respect to a perturbation parameter lazutkin s proof was based on two conjectures probably the original form of those conjectures was incorrect but lazutkin s method was very efficient and inspired a large number of studies on the exponentially small splitting of separatrices the consequent works laz91 laz92 gls94 and gel96 prepared the base to fill all the gaps of the original proof the present paper contains a complete and self contained proof of a refined version of the original formula formula 1 7 of the present paper in this form the formula was obtained in gls94 the proof is inspired by the ideas of lazutkin s original paper laz84

melnikov potential for exact symplectic maps the splitting of separatrices of hyperbolic fixed points for exact symplectic maps of n degrees of freedom is considered the non degenerate critical points of a real valued function called the melnikov potential are associated to transverse homo clinic orbits and an asymptotic expression for the symplectic area between homoclinic orbits is given moreover if the unperturbed invariant manifolds are completely dou bled it is shown that there exist in general at least 4 primary homoclinic orbits 4 n in antisymmetric maps both lower bounds are optimal two examples are presented a 2n dimensional central standard like map and the hamiltonian map associated to a magnetized spherical pendulum several topics are studied about these examples existence of splitting explicit computations of melnikov potentials transverse homoclinic orbits exponentially small splitting etc

smoothing and likelihood ratio for gaussian boundary value processes a new derivation which does not need the invertibility assumption of the covariance matrix of the boundary data is given for the smoothing of gaussian two point boundary value processes tpbvp the likelihood ratio for tpbv processes is then derived in terms of the system parameters by using the krein factorization the likelihood ratio involves the smoother of the process an alternate expression for the likelihood ratio based on the filtered estimate of the state is also given

splitting of separatrices for standard and semistandard mappings abstract we investigate chirikov s standard mapping which is a model for the onset of nonintegrability in nonlinear hamiltonian systems the asymptotic formula when the deviation from an integrable system is small for the angle of intersection of separatrices at a homoclinic point is derived on the basis of approximation by the green percival semistandard mapping in the complex domain computed data on the splitting of separatrices for the semi standard map are presented from which we derive the pre exponential coefficient in the formula for splitting of separatrices in the standard mapping

an asymptotic expression for the splitting of separatrices of the rapidly forced pendulum the measure of the splitting of the separatrices of the rapidly forced pendulum ddot x sin x mu sin frac t varepsilon is considered as a model problem that has been studied by different authors here e are small parameters e 0 but otherwise independent the following formula for the angle between separatrices is established alpha frac pi 2 varepsilon frac mu cosh frac pi 2 varepsilon left 1 o mu varepsilon 2 right

splitting of separatrices and symbolic dynamics for some degenerate standard like maps we consider standard like maps of class c1 with parabolic fixed points and such that the second derivative has a jump discontinuity at the fixed points an asymptotic formula for the splitting of the separatrices is obtained the structure of certain chaotic set in the stochastic layer around the separatrices is described in terms of symbolic dynamics it turns out in particular that swinging and rotating regimes of motion may alternate in time arbitrarily analogous results are stated for the so called separatrix mapping and for the more smooth standard like maps

on the numerical solution of the sine gordon equation in this the first of two papers on the numerical solution of the sine gordon equation we investigate the numerical behavior of a double discrete completely integrable discretization of the sine gordon equation for certain initial values in the vicinity of homoclinic manifolds this discretization admits an instability in the form of grid scale oscillations we clarify the nature of the instability through an analytical investigation supported by numerical experiments in particular a perturbation analysis of the associated linear spectral problem shows that the initial values used for the numerical experiments lie exponentially close to a homoclinic manifold this paves the way for the second paper where we use the non linear spectrum as a basis for comparing different numerical schemes

telescoping recursive representations and estimation of gauss markov random fields we present telescoping recursive representations for both continuous and discrete indexed noncausal gauss markov random fields our recursions start at the boundary a hypersurface in and telescope inwards for example for images the telescoping representation reduce recursions from to i e to recursions on a single dimension under appropriate conditions the recursions for the random field are linear stochastic differential difference equations driven by white noise for which we derive recursive estimation algorithms that extend standard algorithms like the kalman bucy filter and the rauch tung striebel smoother to noncausal markov random fields

splitting of a small separatrix loop near the saddle center bifurcation in area preserving maps abstract when the saddle center bifurcation occurs in an analytic family of area preserving maps a parabolic fixed point first appears at the origin and then this point bifurcates creating an elliptic fixed point and a hyperbolic one separatrices of the hyperbolic fixed point form a small loop around the elliptic point in general the separatrices intersect transversely and the splitting is exponentially small with respect to the perturbation parameter we derive an asymptotic formula which describes the splitting and study the properties of the pre exponential factor

a new characterization for multivariate gaussian reciprocal processes established a new necessary and sufficient condition for a multivariate gaussian random process to be reciprocal this result is then used to prove that the state vector of a two point boundary value system is reciprocal the authors do not impose any nonsingularity conditions on the covariance function of the state vector

asymptotic values for degrees associated with strips of young diagrams abstract the sums s l n occur in the representations of the symmetric and the general linear groups in combinatorics and in pi algebras we give asymptotic values for these sums

maximum likelihood sequence estimation for hidden reciprocal processes this paper addresses the problem of maximum likelihood sequence estimation mlse based on a hidden reciprocal chain hrc as the underlying target model hrcs are non causal discrete time finite state stochastic processes which can be regarded as the one dimensional version of a markov random field although they are not in general markov processes this paper describes a procedure for evaluating the mlse for hrc and compares the resultant estimator with its markov model equivalent the viterbi algorithm in addition the performance of the newly proposed reciprocal mlse is compared to a hrc based optimal smoother

weakly nonlocal solitary waves in a singularly perturbed korteweg de vries equation a fifth order korteweg de vries equation is considered where the fifth order derivative term is multiplied by a small parameter it is known that solitary wave solutions of this model equation are nonlocal in that the central core of the wave is accompanied by copropagating trailing oscillations here using the techniques of exponential asymptotics these solutions are reexamined and it is established that they form a one parameter family characterized by the phase shift of the trailing oscillations explicit asymptotic formula relating the oscillation amplitude to the phase shift are obtained

asymptotic expansion of homoclinic structures in a symplectic mapping the phenomenon of separatrix splitting and complicated homoclinic structures constitute a symptom of chaos as pointed out by poincare more than a century ago taking a time discrete dynamical system with a double well potential as an example this interesting feature is demonstrated analytically on the basis of asymptotic expansions beyond all orders violent undulations of the unstable manifold in the extreme vicinity of hyperbolic fixed points are recovered excellently comparison with results for standard and henon maps is also made

a practical use of the melnikov homoclinic method using cutoff functions and periodic extensions we prove that the melnikov homoclinic method gives a criterium to show that for a finite time interval t t with t arbitrarily large the perturbed system is conjugated to a chaotic one for quite general classes of perturbation functions the method is applied to specific perturbations of the pendulum and of the gylden s problem

chaotic boundary of a hamiltonial map abstract the half plane maps are a class of complex hamiltonial maps whose invariant curves at most fixed irrational frequencies can be obtained as a convergent taylor series expansion for these maps the boundary between regular ordered motion on invariant curves and irregular chaotic motion is given by the radius of convergence of the series the successive terms of the series oscillate wildly due to the presence of small divisors methods are presented for taming the series based on the conversion of the convergenceexponentc ln c into the integral of a continuous but nondifferentiable lambda function whose graph whows a similarity structure on small scales self similarity properties are illustrated for the chaotic boundary function c where v is the frequency

dynamics in chaotic zones of area preserving maps close to separatrix and global instability zones abstract the purpose of this paper is to study phenomena in chaotic zones of area preserving maps using simpler models which are easier to analyse theoretically and numerically first of all the study of the dynamics in a neighbourhood of the separatrices of a resonant zone is carried out the well known separatrix map defined on a figure eight when needed is used to determine the location of rotational invariant curves r i c inside and outside the resonance the interest in this part is on a quantitative description of the dynamics in a neighbourhood of the separatrices to produce theoretical estimates of the width of the stochastic zone distance to the r i c existence of tiny islands close to the separatrices in every one of the studied items one has tried to complement the limit analytic study with realistic numerical simulations describing the analogy when possible after this study we focus on the formation of larger domains without r i c e g birkhoff domains to this end we introduce the biseparatrix map model although this is a qualitative model the mechanism of destruction of the last r i c and hence the process of creation of zones without r i c is clarified by means of this simple model several numerical examples illustrate the results obtained and are used as a test of the theoretical quantitative predictions

recursive filtering and smoothing for reciprocal gaussian processes pinned boundary case the least square estimation problem for pinned to zero discrete index reciprocal gaussian processes in additive white noise is solved thus completing and extending some previous results available in the literature in particular following the innovations approach a finite set of recursive equations is obtained for the filter and for the three standard classes of smoothers fixed point fixed interval fixed lag recursive expressions for the mean square performance of the proposed estimators are also given