the recently recognized failure of predictability in newtonian dynamics modern theories of dynamical systems have very clearly demonstrated the unexpected fact that systems governed by the equations of newtonian dynamics do not necessarily exhibit the predictability property indeed very recent researches have shown that in wide classes of very simple systems satisfying those equations predictability is impossible beyond a certain definite time horizon

fractal basin boundaries abstract basin boundaries for dynamical systems can be either smooth or fractal this paper investigates fractal basin boundaries one practical consequence of such boundaries is that they can lead to great difficulty in predicting to which attractor a system eventually goes the structure of fractal basin boundaries can be classified as being either locally connected or locally disconnected examples and discussion of both types of structures are given and it appears that fractal basin boundaries should be common in typical dynamical systems lyapunov numbers and the dimension for the measure generated by inverse orbits are also discussed

expectation consistent approximate inference we propose a novel framework for approximations to intractable probabilistic models which is based on a free energy formulation the approximation can be understood as replacing an average over the original intractable distribution with a tractable one it requires two tractable probability distributions which are made consistent on a set of moments and encode different features of the original intractable distribution in this way we are able to use gaussian approximations for models with discrete or bounded variables which allow us to include non trivial correlations these are neglected in many other methods we test the framework on toy benchmark problems for binary variables on fully connected graphs and 2d grids and compare with other methods such as loopy belief propagation good performance is already achieved by using single nodes as tractable substructures significant improvements are obtained when a spanning tree is used instead

step by step community detection in volume regular graphs abstract spectral techniques have proved amongst the most effective approaches to graph clustering however in general they require explicit computation of the main eigenvectors of a suitable matrix usually the laplacian matrix of the graph recent work e g becchetti et al soda 2017 suggests that observing the temporal evolution of the power method applied to an initial random vector may at least in some cases provide enough information on the space spanned by the first two eigenvectors so as to allow recovery of a hidden partition without explicit eigenvector computations while the results of becchetti et al apply to perfectly balanced partitions and or graphs that exhibit very strong forms of regularity we extend their approach to graphs containing a hidden k partition and characterized by a milder form of volume regularity we show that the class of k volume regular graphs is the largest class of undirected possibly weighted graphs whose transition matrix admits k stepwise eigenvectors i e vectors that have constant entries over the components corresponding to the same set of the hidden partition to obtain this result we highlight a connection between volume regularity and lumpability of markov chains moreover we prove that if the stepwise eigenvectors are those associated to the first k largest eigenvalues of the transition matrix of a random walk on the graph and the gap between the k th and the k 1 th eigenvalues is sufficiently large the averaging dynamics of becchetti et al recovers the underlying community structure of the graph in logarithmic time with high probability

linear systems theory revisited this paper investigates and clarifies how different definitions of reachability observability controllability reconstructability and minimality that appear in the control literature may be equivalent or different depending on the type of linear system the differences are caused by 1 whether or not the linear system has state dimensions that vary with time 2 bounds on the time axis of the linear system 3 whether or not the initial state is non zero and 4 whether or not the system is time invariant also 5 time reversibility of systems plays a role discrete time linear strictly proper systems are considered a recently published result is used to argue that all the results carry over to continuous time out of the investigation two types of definitions emerge one type applies naturally to systems with constant dimensions while the other applies naturally to systems with variable dimensions this paper reveals that time varying state dimensions that are allowed to be zero are necessary to obtain equivalence between minimality and weak reachability together with observability at the systems level besides their theoretical significance the results of this paper are of practical importance for model reduction and control of time varying discrete time linear systems because they result in minimal realizations with smaller dimensions that are also computed more easily

production of computational portraits of bounded invariant manifolds abstract to develop an effective procedure for describing manifold tangencies an automative process for describing the invariant manifolds involved is required this paper presents such an automative process the accurate location of the saddle is achieved by a modified powell hybrid procedure which finds the local solution to a non linear system of algebraic equations the resulting eigenvalue problem based on the local jacobian matrix of the poincare map is solved and the eigenvectors evaluated these are then used to extrapolate the initial points on the invariant manifold by repeated iteration of the poincare map other points on this manifold can be located by storing the resultant data in a singly linked list and making use of a stack reference structure and negatively signed pointers ordering and inserting deletion of points can be achieved while keeping a bound on computational time and space the technique provides a basis for investigative procedures for describing global bifurcation events

periodic and chaotic behaviour in a reduction of the perturbed korteweg de vries equation the dynamical behaviour of a reduction of the forced and damped korteweg de vries equation is studied numerically chaos arising from subharmonic instability and homoclinic crossings are observed both period doubling bifurcations and the melnikov sequence of subharmonic bifurcations are found and lead to chaotic behaviour here characterised by a positive lyapunov exponent supporting theoretical analysis includes the construction of periodic solutions and homoclinic orbits and their behaviour under perturbation using melnikov functions

the bethe permanent of a nonnegative matrix it has recently been observed that the permanent of a nonnegative square matrix i e of a square matrix containing only nonnegative real entries can very well be approximated by solving a certain bethe free energy function minimization problem with the help of the sum product algorithm we call the resulting approximation of the permanent the bethe permanent in this paper we give reasons why this approach to approximating the permanent works well namely we show that the bethe free energy function is convex and that the sum product algorithm finds its minimum efficiently we then discuss the fact that the permanent is lower bounded by the bethe permanent and we comment on potential upper bounds on the permanent based on the bethe permanent we also present a combinatorial characterization of the bethe permanent in terms of permanents of so called lifted versions of the matrix under consideration moreover we comment on possibilities to modify the bethe permanent so that it approximates the permanent even better and we conclude the paper with some observations and conjectures about permanent based pseudocodewords and permanent based kernels

noise induced chaos and phase space flux abstract we study the effect of additive noise on near integrable second order dynamical systems whose unperturbed flows have homoclinic or heteroclinic orbits the noise is represented by a type of shinozuka stochastic process capable of arbitrarily closely approximating gaussian noise with any specified spectrum we derive a formula for the flux factor applicable for any asymptotic mean stationary excitation this derivation shows that to first order the effect of the external excitation on the system is mediated by a linear filter associated with the system homoclinic or heteroclinic orbit it also shows that the stationary mean distribution of the filtered excitation determines the average phase space flux this is true for both random and nonrandom excitations and indicates that for the dynamical systems considered here these two classes of excitation play substantially equivalent roles in the promotion of chaos

the mean field traveling salesman and related problems the edges of a complete graph on n vertices are assigned i i d random costs from a distribution for which the interval 0 t has probability asymptotic to t as t 0 through positive values in this so called pseudo dimension 1 mean field model we study several optimization problems of which the traveling salesman is the best known we prove that as n the cost of the minimum traveling salesman tour converges in probability to a certain number approximately 2 0415 which is characterized analytically

stochastic nestedness and the belief sharing information pattern solutions to decentralized stochastic optimization problems lead to recursions in which the state space enlarges with the time horizon thus leading to non tractability of classical dynamic programming a common joint information state supplied to each of the agents leads to a tractable recursion as is evident in the one step delayed information sharing structure case or when deterministic nestedness in information holds when there is a causality relationship as in the case of partially nested information structure however communication requirements for such conditions require exchange of very large data noiselessly hence these assumptions are generally impractical in this paper we present a weaker notion of nestedness which we term as stochastic nestedness which is characterized by a sequence of markov chain conditions it is shown that if the information structure is stochastically nested then an optimization problem is tractable and in particular for lqg problems the team optimal solution is linear despite the lack of deterministic nestedness or partial nestedness one other contribution of this paper is that by regarding the multiple decision makers as a single decision maker and using witsenhausen s equivalent model for discrete stochastic control it is shown that the common state required need not consist of observations and it suffices to share beliefs on the state and control actions a pattern we refer to as k stage belief sharing pattern we discuss the minimum amount of information exchange required to achieve such an information pattern for k 1 the information exchange needed is generally strictly less than what is needed for deterministic nestedness and is zero whenever stochastic nestedness applies in view of nestedness we present a discussion on the monotone values of information channels

transient periodic behaviour related to a saddle node bifurcation the authors investigate transient periodic orbits of dissipative invertible maps of r2 such orbits exist just before in parameter space a saddle node pair is formed they obtain numerically and analytically simple scaling laws for the duration of the transient and for the region of initial conditions which evolve into transient periodic orbits an estimate of this region is then obtained by the construction after extension of the map to c2 of the stable manifolds of the two complex saddles in c2 that bifurcate ino the real saddle node pair

a unified control framework of the non regular dynamics of mechanical oscillators a method for controlling non linear dynamics and chaos based on avoiding homo heteroclinic bifurcations is considered in a unified framework various non linear oscillators both hardening and softening symmetric and asymmetric considered as archetypes of a more general class of single d o f systems are analyzed the melnikov s method is applied to analytically detect the homo heteroclinic bifurcations and the results are used to select the optimal shape of the excitation permitting the maximum shift of the undesired bifurcations in parameter space the generic character of the optimization problem is highlighted and the problem itself is discussed in detail various control strategies are proposed based on the elimination either of a single bifurcation one side controls or of all bifurcations global control the optimization problems are solved analytically and numerically under various forms by taking into account the physical admissibility of the related optimal excitation and the easiness of implementation in practical applications it is shown that the solutions of one side control problems are always system independent in turn the solutions of global control problems are system independent for the large class of symmetric systems considered in this work although there are other systems whose solutions are system dependent these considerations support the very generic nature of the optimal control

an examination of initial condition maps for the sinusoidally excited buckled beam modeled by the duffing s equation abstract the initial condition problem is studied for the dynamics of the buckled beam by using the forced holmes duffing s equation in an attempt to understand the route to chaos earlier results generated by moon 1 for different parameter values are reconsidered and new techniques are used in order to understand more of the global behavior of the system in particular fractal basin boundaries are determined for two types of basins of attraction these are correlated with previous results to help provide an explanation for observed chaotic system behavior

crises sudden changes in chaotic attractors and transient chaos abstract the occurrence of sudden qualitative changes of chaotic dynamics as a parameter is varied is discussed and illustrated it is shown that such changes may result from the collision of an unstable periodic orbit and a coexisting chaotic attractor we call such collisions crises phenomena associated with crises include sudden changes in the size of chaotic attractors sudden appearances of chaotic attractors a possible route to chaos and sudden destructions of chaotic attractors and their basins this paper presents examples illustrating that crisis events are prevalent in many circumstances and systems and that just past a crisis certain characteristic statistical behavior whose type depends on the type of crisis occurs in particular the phenomenon of chaotic transients is investigated the examples discussed illustrate crises in progressively higher dimension and include the one dimensional quadratic map the two dimensional henon map systems of ordinary differential equations in three dimensions and a three dimensional map in the case of our study of the three dimensional map a new route to chaos is proposed which is possible only in invertible maps or flows of dimension at least three or four respectively based on the examples presented the following conjecture is proposed almost all sudden changes in the size of chaotic attractors and almost all sudden destruction or creations of chaotic attractors and their basins are due to crises

integrity measures quantifying the erosion of smooth and fractal basins of attraction abstract the phase portraits of a dissipative dynamical system are characterized by the existence of one or more stable attractors which typically include point equilibria periodic oscillations harmonic and subharmonic quasi periodic solutions and chaotic attractors each attractor is embedded in its own domain or basin of attraction bounded by a separatrix associated with an unstable saddle solution under the variation of a control parameter as the attractors move and bifurcate the basins also undergo corresponding changes and metamorphoses these changes in size and shape are usually continuous but can be discontinuous as when an attractor vanishes along with its basin at a saddle node bifurcation associated with the homoclinic tangling of the invariant manifolds of the saddle solution basin boundaries can also change in nature from smooth to fractal in this paper the escape of a driven oscillator from a cubic potential well as an archetypal example is used to explore the engineering significance of the basin erosions that occur under increased forcing various measures of engineering integrity of the constrained attractor are introduced a global measure assesses the overall basin area a local measure assesses the distance from the attractor to the basin boundary and a velocity measure is related to the size of impulse that could be sustained without failure since engineering systems may be subjected to pulse loads of finite duration attention is given to both the absolute and transient basin boudaries the significant erosion of these at homoclinic tangencies is particularly highlighted in the present study the fractal basins having a severely reduced integrity under all three criteria

l_ p consonant approximations of belief functions in this paper we solve the problem of approximating a belief measure with a necessity measure or consonant belief function in a geometric framework consonant belief functions form a simplicial complex in both the space of all belief functions and the space of all mass vectors partial approximations are first sought in each component of the complex while global solutions are selected among them as a first step in this line of study we seek here approximations that minimize lp norms approximations in the mass space can be interpreted in terms of mass redistribution while approximations in the belief space generalize the maximal outer consonant approximation we compare them with each other and with other classical approximations and illustrate them with the help of a running example

high dimensional gaussian graphical model selection walk summability and local separation criterion we consider the problem of high dimensional gaussian graphical model selection we identify a set of graphs for which an efficient estimation algorithm exists and this algorithm is based on thresholding of empirical conditional covariances under a set of transparent conditions we establish structural consistency or sparsistency for the proposed algorithm when the number of samples n jmin 2 log p where p is the number of variables and jmin is the minimum absolute edge potential of the graphical model the sufficient conditions for sparsistency are based on the notion of walk summability of the model and the presence of sparse local vertex separators in the underlying graph we also derive novel non asymptotic necessary conditions on the number of samples required for sparsistency

almost periodic parameters for the second moment stability of linear stochastic systems the technical note presents conditions to assure the uniform second moment stability for a class of linear time varying stochastic systems the system parameters are assumed to be almost periodic a concept that is weaker than the periodic one under the existence of the long run average cost associated with the stochastic system we apply the almost periodicity to prove the desired stability result an application illustrates the usefulness of the approach by implementing an almost periodic state feedback strategy to control the velocity of a dc motor device

control sets and their boundaries under parameter variation abstract control sets i e maximal sets of approximate controllability are described for systems with parameter dependent control range if the so called inner pair condition is satisfied it is known that generically they change continuously under parameter variation while mergers of control sets produce discontinuous transitions the inner pair condition is proved for a class of control affine systems on r n furthermore continuity results for exit and entrance boundaries of control sets are given both for one control set that changes continuously and for two merging control sets the results are illustrated by means of the controlled escape equation

a chaotic blue sky catastrophe in forced relaxation oscillations abstract the chaotic attractor of a periodically forced van der pol oscillator shaw variant is observed in digital simulation and is made to vanish in a blue sky catastrophe by increasing a constant bias term in the force the detailed bifurcation diagram based on extensive simulations reveals the involvement of the homoclinic outset of a nearby limit cycle of saddle type

quadratic stochastic euclidean bipartite matching problem we propose a new approach for the study of the quadratic stochastic euclidean bipartite matching problem between two sets of n points each n 1 the points are supposed independently randomly generated on a domain r d with a given distribution x on in particular we derive a general expression for the correlation function and for the average optimal cost of the optimal matching a previous ansatz for the matching problem on the flat hypertorus is obtained as a particular case

constructing free energy approximations and generalized belief propagation algorithms important inference problems in statistical physics computer vision error correcting coding theory and artificial intelligence can all be reformulated as the computation of marginal probabilities on factor graphs the belief propagation bp algorithm is an efficient way to solve these problems that is exact when the factor graph is a tree but only approximate when the factor graph has cycles we show that bp fixed points correspond to the stationary points of the bethe approximation of the free energy for a factor graph we explain how to obtain region based free energy approximations that improve the bethe approximation and corresponding generalized belief propagation gbp algorithms we emphasize the conditions a free energy approximation must satisfy in order to be a valid or maxent normal approximation we describe the relationship between four different methods that can be used to generate valid approximations the bethe method the junction graph method the cluster variation method and the region graph method finally we explain how to tell whether a region based approximation and its corresponding gbp algorithm is likely to be accurate and describe empirical results showing that gbp can significantly outperform bp

geometrical concepts and computational techniques of nonlinear dynamics abstract this paper presents a brief overview of some of the geometrical concepts of nonlinear dissipative dynamics computational techniques are described for locating fixed points of poincare maps local bifurcations and homoclinic tangencies of invariant manifolds and following them as parameters vary the cell to cell mapping technique for locating basins of attraction in a computationally efficient way is outlined

cauchy matrix for linear almost periodic systems and some consequences abstract a novel method to construct the fundamental matrix for a linear almost periodic system is proposed provided that the diagonal terms satisfy an average separation condition and the off diagonal coefficients are l small the idea is to transform the system in a set of riccati type equations and use exponential dichotomy and its consequences it is shown that the method yields easy computation procedures with simple and direct conditions depending on the coefficients finally our result enables us to obtain i explicit almost periodic matrices q t q 1 t and q t which diagonalize the original system and ii sufficient conditions for the stability two illustrative examples are shown

one dimensional euclidean matching problem exact solutions correlation functions and universality we discuss the equivalence relation between the euclidean bipartite matching problem on the line and on the circumference and the brownian bridge process on the same domains the equivalence allows us to compute the correlation function and the optimal cost of the original combinatorial problem in the thermodynamic limit moreover we solve also the minimax problem on the line and on the circumference the properties of the average cost and correlation functions are discussed

dynamics of an eccentric disk on a curved surface abstract this paper investigates the dynamic behavior of an eccentric disk rolling on a curve of arbitrary shape and then on a curve defined as a cubic function comparisons are made to a disk without eccentricity and the related point mass approximation the curve is subject to sinusoidal base excitation and the system is considered from the perspective of a potential well problem where escape is possible on one side the equations of motion are derived using a roll without slip constraint and the behavior is investigated by means of approximate analytical solutions numerically simulated frequency and amplitude sweeps and by considering the basins of attraction when initial conditions or forcing parameters are varied

exact value for the average optimal cost of the bipartite traveling salesman and two factor problems in two dimensions we show that the average cost for the traveling salesman problem in two dimensions which is the archetypal problem in combinatorial optimization in the bipartite case is simply related to the average cost of the assignment problem with the same euclidean increasing convex weights in this way we extend a result already known in one dimension where exact solutions are avalaible the recently determined average cost for the assignment when the cost function is the square of the distance between the points provides therefore an exact prediction overline e n frac 1 pi log n for large number of points 2n as a byproduct of our analysis also the loop covering problem has the same optimal average cost we also explain why this result cannot be extended at higher dimensions we numerically check the exact predictions

polynomially solvable cases of the bipartite traveling salesman problem abstract given two sets r and b consisting of n cities each in the bipartite traveling salesman problem one looks for the shortest way of visiting alternately the cities of r and b returning to the city of origin this problem is known to be np hard for arbitrary sets r and b in this paper we provide an o n 6 algorithm to solve the bipartite traveling salesman problem if the quadrangle property holds in particular this algorithm can be applied to solve in o n 6 time the bipartite traveling salesman problem in the following cases s r b is a convex point set in the plane s r b is the set of vertices of a simple polygon and v r b is the set of vertices of a circular graph for this last case we also describe another algorithm which runs in o n 2 time

exact solutions to the transportation problem on the line given distributions w of production and v of consumption on the line the monge kantorovich problem is to decide which producer should supply each consumer in order to minimize the total transportation costs here cost will be assumed to be a strictly concave function of the distance which translates into an economy of scale for longer trips and may encourage cross hauling the resulting solutions display a hierarchical structure that reflects a striking separation into local and global scales also found in the real world moreover this structure can be exploited to reduce the infinite dimensional linear problem to a convex minimization in m variables where 2 m 2 counts the number of times that w v changes sign a combinatorial algorithm is then derived which yields exact solutions by optimizing a certain finite sequence of convex separable network flows

stochastic belief propagation a low complexity alternative to the sum product algorithm the belief propagation bp or sum product algorithm is a widely used message passing method for computing marginal distributions in graphical models at the core of the bp message updates when applied to a graphical model involving discrete variables with pairwise interactions lies a matrix vector product with complexity that is quadratic in the state dimension d and requires transmission of a d 1 dimensional vector of real numbers messages to its neighbors since various applications involve very large state dimensions such computation and communication complexities can be prohibitively complex in this paper we propose a low complexity variant of bp referred to as stochastic belief propagation sbp as suggested by the name it is an adaptively randomized version of the bp message updates in which each node passes randomly chosen information to each of its neighbors the sbp message updates reduce the computational complexity per iteration from quadratic to linear in d without assuming any particular structure of the potentials and also reduce the communication complexity significantly requiring only log2d bits transmission per edge moreover we establish a number of theoretical guarantees for the performance of sbp showing that it converges almost surely to the bp fixed point for any tree structured graph and for any graph with cycles satisfying a contractivity condition in addition for these graphical models we provide nonasymptotic upper bounds on the convergence rate showing that the l norm of the error vector decays no slower than o 1 t with the number of iterations t on trees and the normalized mean squared error decays as o 1 t for general graphs this analysis also supported by experimental results shows that sbp can provably yield reductions in computational and communication complexities for various classes of graphical models

safe basins for a nonlinear oscillator with ramped forcing a nonlinear oscillator is studied in the presence of external forcing for which the amplitude initially depends on time the focus is on the sizes of the basins of attraction which do not lead to unbounded motions collectively termed the safe basin direct comparisons are drawn between the regime of constant forcing amplitude and that where the forcing amplitude initially depends on time in the process questions from previous literature are answered and previously unexplained phenomena are understood furthermore we witness a new phenomenon not previously observed for the system studied

a polynomial counterexample to the markus yamabe conjecture abstract we give a polynomial counterexample to both the markus yamabe conjecture and the discrete markus yamabe problem for all dimensions 3

chaotic behavior resulting in transient and steady state instabilities of pressure loaded shallow spherical shells abstract in this paper the axisymmetric dynamic behavior and snap through buckling of thin elastic shallow spherical shells under harmonic excitation is investigated based on marguerre kinematical assumptions the governing partial differential equations of motion for a pre loaded cap are presented in the form of a compatibility equation and a transverse motion equation the continuous model is reduced to a finite degree of freedom system using the galerkin method and a fourier bessel approach results show that pre loaded shells may exhibit co existing stable equilibrium states and that with the application of sufficiently large dynamic loads the structure escapes from the well corresponding to pre buckling configurations to another this escape load may be much lower than the corresponding quasi static buckling load indeed complex resonances can occur until the system snaps through often signalling the loss of stability as parameters are slowly varied steady state instabilities may occur these can include jumps to resonance subharmonic period doubling bifurcations cascades to chaos etc moreover a sudden pulse of excitation may lead to a transient failure of the system in this paper we examine how spherical caps under harmonic loading may be assessed in an engineering context with a view to design against steady state instabilities as well as the various modes of transient failure steady state and transient stability boundaries are presented in which special attention is devoted to the determination of the critical load conditions from this theoretical analysis dynamic buckling criteria can be properly established which may constitute a consistent and rational basis for design of these shell structures under harmonic loading

average cost and stability of time varying linear systems in this technical note the stability for time varying discrete time stochastic linear systems with possibly unbounded trajectories is associated to the existence of the long run average cost criteria under controllability and observability the stochastic system is stable in the sense that its state tends asymptotically to the origin in the mean further conditions related to a lower bound on the long run average cost or with periodic time varying systems provides uniform second moment stability

indeterminate jumps to resonance from a tangled saddle node bifurcation we identify in this paper a new type of bifurcation which yields an unpredictable outcome and therefore seems to be an important new ingredient of nonlinear dissipative dynamics it arises when the unstable manifold of the saddle of a saddle node bifurcation is heteroclinically tangled with the inset of a distant saddle which is itself homoclinically tangled so that it forms a fractal basin boundary between two remote attractors at the saddle node fold a slowly evolving system will thus find itself sitting precisely on a fractal basin boundary and in the presence of even infinitesimal noise we cannot predict to which of the two remote attractors the system will jump we show here that such an indeterminate tangled saddle node bifurcation is a common ingredient in the resonance of softening systems

on stability of linear time varying infinite dimensional discrete time systems the asymptotic behaviour of linear time varying infinite dimensional discrete lime systems is considered the introduced notions are weak power equistability power equistability uniform power equistability l p equistability uniform l p equistability and l p x equistability it is shown that they are identical a generalization of the concept of spectral radius of a single operator is also proposed it is proven that any time varying system is uniformly power equistable if and only if the generalized spectral radius of the sequence of the operators which define the system considered is less than one

solution for a bipartite euclidean traveling salesman problem in one dimension the traveling salesman problem is one of the most studied combinatorial optimization problems because of the simplicity in its statement and the difficulty in its solution we characterize the optimal cycle for every convex and increasing cost function when the points are thrown independently and with an identical probability distribution in a compact interval we compute the average optimal cost for every number of points when the distance function is the square of the euclidean distance we also show that the average optimal cost is not a self averaging quantity by explicitly computing the variance of its distribution in the thermodynamic limit moreover we prove that the cost of the optimal cycle is not smaller than twice the cost of the optimal assignment of the same set of points interestingly this bound is saturated in the thermodynamic limit

statistical mechanics of image restoration and error correcting codes we develop a statistical mechanical formulation for image restoration and error correcting codes these problems are shown to be equivalent to the ising spin glass with ferromagnetic bias under random external fields we prove that the quality of restoration decoding is maximized at a specific set of parameter values determined by the source and channel properties for image restoration in a mean field system a line of optimal performance is shown to exist in the parameter space these results are illustrated by solving exactly the infinite range model the solutions enable us to determine how precisely one should estimate unknown parameters monte carlo simulations are carried out to see how far the conclusions from the infinite range model are applicable to the more realistic two dimensional case in image restoration

statistical mechanical approach to image processing the basic frameworks and techniques of the bayesian approach to image restoration are reviewed from the statistical mechanical point of view first a few basic notions in digital image processing are explained to convince the reader that statistical mechanics has a close formal similarity to this problem second the basic formulation of the statistical estimation from the observed degraded image by using the bayes formula is demonstrated the relationship between bayesian statistics and statistical mechanics is also listed particularly it is explained that some correlation inequalities on the nishimori line of the random spin model also play an important role in bayesian image restoration third the framework of bayesian image restoration for binary images by means of the ising model is reviewed some practical algorithms for binary image restoration are given by employing the mean field and the bethe approximations finally bayesian image restoration for a grey level image using the gaussian model is reviewed and the gaussian model is extended to a more practical probabilistic model by introducing the line state to treat the effects of edges the line state is also extended to quantized values

exploring the dynamics of a class of post tensioned moment resisting frames in this paper we present an equivalent low order nonlinear system that describes the dynamics of a generic class of post tensioned frames the proposed nonlinear single degree of freedom system is derived from energy considerations we demonstrate that the equation of motion for the entire planar post tensioned frame is equivalent to the dynamics of a single tied rocking block on an elastic foundation as validation for this analytical model we present physical tests 1 4 scale undertaken at bristol quasi static push pull over tests and dynamic frequency sine sweep shake table tests are conducted on the physical model comparison of results indicate that the analytical model predicts both quasi static nonlinear push over and nonlinear dynamic resonant behaviour very well further numerical simulations on the analytical model identify the nonlinear resonant frequency backbone curves for a range of system parameters we explore catchment basins of both poincare phase and system parameter spaces in addition we describe failure boundaries and system integrity surfaces giving an indication as to likely bounds on forcing amplitudes

catastrophes with indeterminate outcome a catastrophe in a dissipative dynamical system which causes an attractor to completely lose stability will result in a transient trajectory making a rapid jump in phase space to some other attractor in systems where more than one other attractor is available the attractor chosen may depend very sensitively on how the catastrophe is realized two examples in forced oscillators of duffing type illustrate how the probabilities of different outcomes can be estimated using the phase space geometry of invariant manifolds

on the optimality of solutions of the max product belief propagation algorithm in arbitrary graphs graphical models such as bayesian networks and markov random fields mrfs represent statistical dependencies of variables by a graph the max product belief propagation algorithm is a local message passing algorithm on this graph that is known to converge to a unique fixed point when the graph is a tree furthermore when the graph is a tree the assignment based on the fixed point yields the most probable values of the unobserved variables given the observed ones good empirical performance has been obtained by running the max product algorithm or the equivalent min sum algorithm on graphs with loops for applications including the decoding of turbo codes except for two simple graphs cycle codes and single loop graphs there has been little theoretical understanding of the max product algorithm on graphs with loops here we prove a result on the fixed points of max product on a graph with arbitrary topology and with arbitrary probability distributions discrete or continuous valued nodes we show that the assignment based on a fixed point is a neighborhood maximum of the posterior probability the posterior probability of the max product assignment is guaranteed to be greater than all other assignments in a particular large region around that assignment the region includes all assignments that differ from the max product assignment in any subset of nodes that form no more than a single loop in the graph in some graphs this neighborhood is exponentially large we illustrate the analysis with examples

lobe dynamics and the escape from a potential well periodically forced nonlinear oscillators that permit escape from a potential well frequently possess unstable periodic orbits whose invariant manifolds are homoclinically tangled the trellises formed by such overlapping manifolds separate the poincare plane into regions called lobes whose fates at successive time periods are amenable to analysis the lobe configurations observed in simple escape systems are discussed together with their interrelation with the location of periodic orbits and other invariant sets three applications of this procedure are illustrated i a numerical technique for localizing the non wandering set ii a demonstration that the existence of a saddle possessing a bounded homoclinically tangled unstable manifold branch implies the existence of safe open regions in the global basin of attraction iii the proposal of definitions of global integrity and escape times which may be used to relate safe basin erosion to the evolution of a homoclinic tangle

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

correctness of local probability propagation in graphical models with loops graphical models such as bayesian networks and markov networks represent joint distributions over a set of variables by means of a graph when the graph is singly connected local propagation rules of the sort proposed by pearl 1988 are guaranteed to converge to the correct posterior probabilities recently a number of researchers have empirically demonstrated good performance of these same local propagation schemes on graphs with loops but a theoretical understanding of this performance has yet to be achieved for graphical models with a single loop we derive an analytical relationship between the probabilities computed using local propagation and the correct marginals using this relationship we show a category of graphical models with loops for which local propagation gives rise to provably optimal maximum a posteriori assignments although the computed marginals will be incorrect we also show how nodes can use local information in the messages they receive in order to correct their computed marginals we discuss how these results can be extended to graphical models with multiple loops and show simulation results suggesting that some properties of propagation on single loop graphs may hold for a larger class of graphs specifically we discuss the implication of our results for understanding a class of recently proposed error correcting codes known as turbo codes

chaotic phenomena triggering the escape from a potential well this paper explores the manner in which a driven mechanical oscillator escapes from the cubic potential well typical of a metastable system close to a fold the aim is to show how the well known atoms of dissipative dynamics saddle node folds period doubling flips cascades to chaos boundary crises etc assemble to form molecules of overall response hierarchies of cusps incomplete feigenbaum trees etc particular attention is given to the basin of attraction and the loss of engineering integrity that is triggered by a homoclinic tangle the latter being accurately predicted by a melnikov analysis after escape chaotic transients are shown to conform to recent scaling laws analytical constraints on the mapping eigenvalues are used to demonstrate that sequences of flips and folds commonly predicted by harmonic balance analysis are in fact physically inadmissible

invariant sets of planar diffeomorphisms in nonlinear vibrations using the standard technique of poincare sampling many simple periodically forced nonlinear oscillators can be represented by area contracting planar diffeomorphisms a class of such diffeomorphisms referred to as single sided systems are considered these permit escape of orbits to arbitrarily large displacement six closure relations which relate the invariant manifolds of the hilltop saddle cycle to other invariant sets are derived these relations embody much of the underlying organization that is observed in the dynamics of these systems

stereo matching using belief propagation in this paper we formulate the stereo matching problem as a markov network and solve it using bayesian belief propagation the stereo markov network consists of three coupled markov random fields that model the following a smooth field for depth disparity a line process for depth discontinuity and a binary process for occlusion after eliminating the line process and the binary process by introducing two robust functions we apply the belief propagation algorithm to obtain the maximum a posteriori map estimation in the markov network other low level visual cues e g image segmentation can also be easily incorporated in our stereo model to obtain better stereo results experiments demonstrate that our methods are comparable to the state of the art stereo algorithms for many test cases

scaling hypothesis for the euclidean bipartite matching problem ii correlation functions we analyze the random euclidean bipartite matching problem on the hypertorus in d dimensions with quadratic cost and we derive the two point correlation function for the optimal matching using a proper ansatz introduced by caracciolo et al phys rev e 90 012118 2014 to evaluate the average optimal matching cost we consider both the grid poisson matching problem and the poisson poisson matching problem we also show that the correlation function is strictly related to the green s function of the laplace operator on the hypertorus

the bethe free energy allows to compute the conditional entropy of graphical code instances a proof from the polymer expansion the main objective of this paper is to explore the precise relationship between the bethe free energy or entropy and the shannon conditional entropy of graphical error correcting codes the main result shows that the bethe free energy associated with a low density parity check code used over a binary symmetric channel in a large noise regime is with high probability asymptotically exact as the block length grows to arrive at this result we develop new techniques for rather general graphical models based on the loop sum as a starting point and the polymer expansion from statistical mechanics the true free energy is computed as a series expansion containing the bethe free energy as its zeroth order term plus a series of corrections it is easily seen that convergence criteria for such expansions are satisfied for general high temperature models we apply these general results to the ensembles of low density generator matrix and parity check codes while the application to generator matrix codes follows standard high temperature methods the case of parity check codes requires non trivial new ideas because the hard constraints correspond to a zero temperature regime nevertheless one can combine the polymer expansion with expander and counting arguments to show that the difference between the true and bethe free energies vanishes with high probability in the large block length limit

counting in graph covers a combinatorial characterization of the bethe entropy function we present a combinatorial characterization of the bethe entropy function of a factor graph such a characterization being in contrast to the original analytical definition of this function we achieve this combinatorial characterization by counting valid configurations in finite graph covers of the factor graph analogously we give a combinatorial characterization of the bethe partition function whose original definition was also of an analytical nature as we point out our approach has similarities to the replica method but also stark differences the above findings are a natural backdrop for introducing a decoder for graph based codes that we will call symbolwise graph cover decoding a decoder that extends our earlier work on blockwise graph cover decoding both graph cover decoders are theoretical tools that help toward a better understanding of message passing iterative decoding namely blockwise graph cover decoding links max product min sum algorithm decoding with linear programming decoding and symbolwise graph cover decoding links sum product algorithm decoding with bethe free energy function minimization at temperature one in contrast to the gibbs entropy function which is a concave function the bethe entropy function is in general not concave everywhere in particular we show that every code picked from an ensemble of regular low density parity check codes with minimum hamming distance growing with high probability linearly with the block length has a bethe entropy function that is convex in certain regions of its domain

optimal escape from potential wells patterns of regular and chaotic bifurcation abstract the patterns of bifurcation governing the escape of periodically forced oscillations from a potential well over a smooth potential barrier are studied by numerical simulation both the generic asymmetric single well cubic potential and the symmetric twin well potential duffing oscillator are surveyed by varying three parameters forcing frequency forcing amplitude and damping coefficient the close relationship between optimal escape and nonlinear resonance within the well is confirmed over a wide range of damping subtle but significant differences are observed at higher damping ratios the possibility of indeterminate outcomes of jumps to and from resonance near optimal escape is cmppletely suppressed above a critical level of the damping ratio about 0 12 for the asymmetric single well oscillator coincidentally at almost the same level of damping the optimal escape condition becomes distinct from the apex in the f plane of the bistable regime this corresponds to the appearance of chaotic attractors which subsume both resonant and non resonant motions within one well at higher damping levels further changes occur involving conversions from chaotic saddle to regular saddle bifurcations these changes in optimal escape phenomena correspond to codimension three bifurcations at exceptional points in the space of three parameters these bifurcations are described in terms of homoclinic and heteroclinic structures of invariant manifolds and changes in accessible boundary orbits the same sequence of codimension three bifurcations is observed in both the twin well duffing oscillator and the asymmetric single well escape equation within the codimension three bifurcation patterns governing escape one particular codimension two global bifurcation involves a chaotic attractor explosion or interior crisis compounded with a blue sky catastrophe or boundary crisis of the exploded attractor this codimension two bifurcation has structure containing a form of predictive power knowledge of attractor bifurcations in part of the codimension two pattern permits inference of the attractor and basin bifurcations in the remainder this predictive power is applicable beyond the context of escape from potential wells quantitative correlation of bifurcation patterns between the two equations according to simple scaling laws is tested the unstable periodic orbits which figure most prominently in the major attractor basin bifurcations are of periods one and three their linking is conveniently interpreted by a three layer spiral horseshoe structure for the folding action in phase space within a well the structure of this 3 shoe implies a partial ordering among order three subharmonic saddle node bifurcations this helps explain the sequence of codimension three bifurcations near optimal escape some bifurcational precedence relations are known to follow from the linking of periodic orbits in a braid on a 3 shoe additional bifurcational precedence relations follow from a quantitative property of generic potential wells the dynamic hilltop saddle has a very large expanding multiplier over one cycle of forcing near fundamental resonance this quantitative property explains the close coincidence of codimension three bifurcations near the suppression of indeterminate outcomes an experimentalist s approach to identifying the three layer template structure from time series data is discussed including a consistency check involving poincare indices the bifurcation patterns emerging at higher damling values create favorable conditions for realizing experimental strategies to recognize optimal escape and locate it in parameter space strategies based solely on observations of quasi steady behavior while remaining always within one well are discussed

belief propagation algorithms on noisy hardware the wide recognition that emerging nano devices will be inherently unreliable motivates the evaluation of information processing algorithms running on noisy hardware as well as the design of robust schemes for reliable performance against hardware errors of varied characteristics in this paper we investigate the performance of a popular statistical inference algorithm belief propagation bp on probabilistic graphical models implemented on noisy hardware and we propose two robust implementations of the bp algorithm targeting different computation noise distributions we assume that the bp messages are subject to zero mean transient additive computation noise we focus on graphical models satisfying the contraction mapping condition that guarantees the convergence of the noise free bp we first upper bound the distances between the noisy bp messages and the fixed point of noise free bp as a function of the iteration number next we propose two implementations of bp namely censoring bp and averaging bp that are robust to computation noise censoring bp rejects incorrect computations to keep the algorithm on the right track to convergence while averaging bp takes the average of the messages in all iterations up to date to mitigate the effects of computation noise censoring bp works effectively when with high probability the computation noise is exactly zero and averaging bp although having a slightly larger overhead works effectively for general zero mean computation noise distributions sufficient conditions on the convergence of censoring bp and averaging bp are derived simulations on the ising model demonstrate that the two proposed implementations successfully converge to the fixed point achieved by noise free bp additionally we apply averaging bp to a bp based image denoising algorithm and as a bp decoder for ldpc codes in the image denoising application averaging bp successfully denoises an image even when nominal bp fails to do so in the presence of computation noise in the bp ldpc decoder application the power of averaging bp is manifested by the reduction in the residual error rates compared with the nominal bp decoder

inference algorithm for finite dimensional spin glasses belief propagation on the dual lattice starting from a cluster variational method and inspired by the correctness of the paramagnetic ansatz at high temperatures in general and at any temperature in the two dimensional 2d edwards anderson ea model we propose a message passing algorithm the dual algorithm to estimate the marginal probabilities of spin glasses on finite dimensional lattices we use the ea models in 2d and 3d as benchmarks the dual algorithm improves the bethe approximation and we show that in a wide range of temperatures compared to the bethe critical temperature our algorithm compares very well with monte carlo simulations with the double loop algorithm and with exact calculation of the ground state of 2d systems with bimodal and gaussian interactions moreover it is usually 100 times faster than other provably convergent methods as the double loop algorithm in 2d and 3d the quality of the inference deteriorates only where the correlation length becomes very large i e at low temperatures in 2d and close to the critical temperature in 3d

control systems with almost periodic excitations for control systems described by ordinary differential equations subject to almost periodic excitations the controllability properties depend on the specific excitation here these properties and in particular control sets and chain control sets are discussed for all excitations in the closure of all time shifts of a given almost periodic function then relations between heteroclinic orbits of an uncontrolled and unperturbed system and controllability for small control ranges and small perturbations are studied using melnikov s method finally a system with two well potential is studied in detail

cooperative spectrum sensing in the presence of correlated and malicious cognitive radios while cooperative spectrum sensing improves sensing reliability when secondary users sus have independent measurements these gains are limited in the presence of correlated or malicious sus in this paper we propose algorithms to improve spectrum sensing performance when the system contains both honest and malicious sus all of whom may be experiencing correlated fading channels we show that when the sus reports at each time slot are independent and identically distributed it is impossible to distinguish between malicious collaboration i e collusion and correlations caused by the environment thus the optimal test statistic for cooperative spectrum sensing depends on the correlations in the su reports but not on the source of these correlations we propose two algorithms one to identify sus whose reports are correlated and another to infer the spectrum occupancy by using the learned structure of correlations groups of sus whose reports are correlated are identified by learning the structure of the underlying bayesian network model this structure is implemented as a factor graph to infer the spectrum occupancy using a loopy belief propagation algorithm we derive an upper bound of the error probability of the proposed structure learning algorithm and prove convergence of our loopy belief propagation algorithm

one loop diagrams in the random euclidean matching problem the matching problem is a notorious combinatorial optimization problem that has attracted for many years the attention of the statistical physics community here we analyze the euclidean version of the problem i e the optimal matching problem between points randomly distributed on a d dimensional euclidean space where the cost to minimize depends on the points pairwise distances using mayer s cluster expansion we write a formal expression for the replicated action that is suitable for a saddle point computation we give the diagrammatic rules for each term of the expansion and we analyze in detail the one loop diagrams a characteristic feature of the theory when diagrams are perturbatively computed around the mean field part of the action is the vanishing of the mass at zero momentum in the non euclidean case of uncorrelated costs instead we predict and numerically verify an anomalous scaling for the sub sub leading correction to the asymptotic average cost

on the uniqueness of loopy belief propagation fixed points we derive sufficient conditions for the uniqueness of loopy belief propagation fixed points these conditions depend on both the structure of the graph and the strength of the potentials and naturally extend those for convexity of the bethe free energy we compare them with a strengthened version of conditions derived elsewhere for pairwise potentials we discuss possible implications for convergent algorithms as well as for other approximate free energies

learning multiple belief propagation fixed points for real time inference in the context of inference with expectation constraints we propose an approach based on the loopy belief propagation algorithm lbp as a surrogate to an exact markov random field mrf modelling a prior information composed of correlations among a large set of n variables is encoded into a graphical model this encoding is optimized with respect to an approximate decoding procedure lbp which is used to infer hidden variables from an observed subset we focus on the situation where the underlying data have many different statistical components representing a variety of independent patterns considering a single parameter family of models we show how lbp may be used to encode and decode efficiently such information without solving the np hard inverse problem yielding the optimal mrf contrary to usual practice we work in the non convex bethe free energy minimization framework and manage to associate a belief propagation fixed point to each component of the underlying probabilistic mixture the mean field limit is considered and yields an exact connection with the hopfield model at finite temperature and steady state when the number of mixture components is proportional to the number of variables in addition we provide an enhanced learning procedure based on a straightforward multi parameter extension of the model in conjunction with an effective continuous optimization procedure this is performed using the stochastic search heuristic cmaes and yields a significant improvement with respect to the single parameter basic model

a nonlinear probabilistic method for predicting vessel capsizing in random beam seas the capsizing of vessels in random beam seas is investigated using a single degree of freedom model which is limited to the roll motion several factors including sea wave spectrum nonlinear righting moment characteristics and nonlinear damping are taken into account in the analysis a nonlinear probabilistic method is developed by combining ideas from modern nonlinear dynamics specifically the melnikov function and the phase space area flux and random vibrations conditions for the onset of vessel capsizing are obtained in terms of the sea state characteristics significant wave height and characteristic wave period and the vessel parameters damping and stiffness coefficients extensive numerical simulations are carried out to demonstrate the validity of the analytical results it is found that there exists an excellent correlation between the rate of phase space flux and the probability of capsizing

a wiener filter state space flux optimal control against escape from a potential well escape from a potential well often identified with system failure is reformulated for a class of two dimensional dynamical systems as exiting the safe region of the system state space bounded by the separatrix derived from the potential well a control against system failure based on minimizing the state space flux out of this safe region is considered for systems subject to weak external forcing with available control power exceeding a certain threshold the finite lag state space flux optimal control against failure is shown to involve a wiener filter a non wiener solution is proposed for available power below this threshold these results hold for wide sense stationary forcing with any statistics and any spectrum

anomalous scaling of the optimal cost in the one dimensional random assignment problem we consider the random euclidean assignment problem on the line between two sets of n random points independently generated with the same probability density function varrho the cost of the matching is supposed to be dependent on a power p 1 of the euclidean distance of the matched pairs we discuss an integral expression for the average optimal cost for n gg 1 that generalizes a previous result obtained for p 2 we also study the possible divergence of the given expression due to the vanishing of the probability density function the provided regularization recipe allows us to recover the proper scaling law for the cost in the divergent cases and possibly some of the involved coefficients the possibility that the support of varrho is a disconnected interval is also analysed we exemplify the proposed procedure and we compare our predictions with the results of numerical simulations

convergence of weighted min sum decoding via dynamic programming on trees applying the max product and sum product algorithms to loopy graphs is now quite popular for best assignment problems this is largely due to their low computational complexity and impressive performance in practice still there is no general understanding of the conditions required for convergence or optimality of converged solutions or both this paper presents an analysis of both attenuated max product decoding and weighted min sum decoding for low density parity check ldpc codes which guarantees convergence to a fixed point when a weight factor is sufficiently small it also shows that if the fixed point satisfies some consistency conditions then it must be both a linear programming lp and maximum likelihood ml decoding solution for dv dc regular ldpc codes the weight factor must satisfy dv 1 1 are given finally connections are explored with recent work on the threshold of lp decoding

dynamics of the maximum marginal likelihood hyperparameter estimation in image restoration gradient descent versus expectation and maximization algorithm dynamical properties of image restoration and hyperparameter estimation are investigated by means of statistical mechanics we introduce an exactly solvable model for image restoration and derive differential equations with respect to macroscopic quantities from these equations we evaluate relaxation processes of the system to the equilibrium state our statistical mechanical approach also enables us to investigate the hyperparameter estimation by means of maximization of the marginal likelihood by using gradient descent and the expectation and maximization algorithm from the dynamical point of view

stability and stabilizability of switched linear systems a survey of recent results during the past several years there have been increasing research activities in the field of stability analysis and switching stabilization for switched systems this paper aims to briefly survey recent results in this field first the stability analysis for switched systems is reviewed we focus on the stability analysis for switched linear systems under arbitrary switching and we highlight necessary and sufficient conditions for asymptotic stability after a brief review of the stability analysis under restricted switching and the multiple lyapunov function theory the switching stabilization problem is studied and a variety of switching stabilization methods found in the literature are outlined then the switching stabilizability problem is investigated that is under what condition it is possible to stabilize a switched system by properly designing switching control laws note that the switching stabilizability problem has been one of the most elusive problems in the switched systems literature a necessary and sufficient condition for asymptotic stabilizability of switched linear systems is described here

random euclidean matching problems in one dimension we discuss the optimal matching solution for both the assignment problem and the matching problem in one dimension for a large class of convex cost functions we consider the problem in a compact set with the topology both of the interval and of the circumference afterwards we assume the points positions to be random variables identically and independently distributed on the considered domain we analytically obtain the average optimal cost in the asymptotic regime of very large number of points n and some correlation functions for a power law type cost function in the form c z z p both in the p 1 case and in the p 1 whereas in both cases it is a constant when p 0 finally our predictions are compared with the results of numerical simulations

on the harmonic response of an oscillator with unsymmetric restoring force abstract the approximate fundamental response of a driven mechanical oscillator with a quadratic stiffness characteristic is studied by using the method of harmonic balance the accuracy of the periodic solutions based on first approximation theory and their stability are compared with exact results obtained from direct time integration

sufficient conditions for convergence of the sum product algorithm novel conditions are derived that guarantee convergence of the sum product algorithm also known as loopy belief propagation or simply belief propagation bp to a unique fixed point irrespective of the initial messages for parallel synchronous updates the computational complexity of the conditions is polynomial in the number of variables in contrast with previously existing conditions our results are directly applicable to arbitrary factor graphs with discrete variables and are shown to be valid also in the case of factors containing zeros under some additional conditions the conditions are compared with existing ones numerically and if possible analytically for binary variables with pairwise interactions sufficient conditions are derived that take into account local evidence i e single variable factors and the type of pair interactions attractive or repulsive it is shown empirically that this bound outperforms existing bounds

estimation for almost periodic processes processes with almost periodic covariance functions have spectral mass on lines parallel to the diagonal in the two dimensional spectral plane methods have been given for estimation of spectral mass on the lines of spectral concentration if the locations of the lines are known here methods for estimating the intercepts of the lines of spectral concentration in the gaussian case are given under appropriate conditions the methods determine rates of convergence sufficiently fast as the sample size n so that the spectral estimation on the estimated lines can then proceed effectively this task involves bounding the maximum of an interesting class of non gaussian possibly nonstationary processes

belief propagation for continuous state spaces stochastic message passing with quantitative guarantees the sum product or belief propagation bp algorithm is a widely used message passing technique for computing approximate marginals in graphical models we introduce a new technique called stochastic orthogonal series message passing sosmp for computing the bp fixed point in models with continuous random variables it is based on a deterministic approximation of the messages via orthogonal series basis expansion and a stochastic estimation of the basis coefficients via monte carlo techniques and damped updates we prove that the sosmp iterates converge to a neighborhood of the unique bp fixed point for any tree structured graph and for any graphs with cycles in which the bp updates satisfy a contractivity condition in addition we demonstrate how to choose the number of basis coefficients as a function of the desired approximation accuracy and smoothness of the compatibility functions we illustrate our theory with both simulated examples and in application to optical flow estimation

basin boundary metamorphoses changes in accessible boundary orbits abstract basin boundaries sometimes undergo sudden metamophoses these metamorphoses can lead to the conversion of a smooth basin boundary to one which is fractal or else can cause a fractal or else can basin boundary to suddenly jump in size and change its character although remaining fractal for an invertible map in the plane there may be an infinite number of saddle periodic orbits in a basin boundary that is fractal nonetheless we have found that typically only one of them can be reached or accessed directly from a given basin the other periodic orbits are buried beneath infinitely many layers of the fractal structure of the boundary the boundary metamorphoses which we investigate are characterized by a sudden replacement of the basin boundary s accessible orbit

detectability and stabilizability of time varying discrete time linear systems the concepts of detectability and stabilizability are explored for time varying systems we study duality invariance under feedback an extended version of the lemma of lyapunov existence of stabilizing feedback laws linear quadratic filtering and control and the existence of approximate canonical forms

fractal control boundaries of driven oscillators and their relevance to safe engineering design when a metastable damped oscillator is driven by strong periodic forcing the catchment basin of constrained motions in the space of the starting conditions x 0 0 develops a fractal boundary associated with a homoclinic tangling of the governing invariant manifolds the four dimensional basin in the phase control space spanned by x 0 0 f where f is the magnitude and the frequency of the excitation will likewise acquire a fractal boundary and we here explore the engineering significance of the control cross section corresponding for example to x 0 0 0 the fractal boundary in this section is a failure locus for a mechanical or electrical system subjected while resting in its ambient equilibrium state to a sudden pulse of excitation we assess here the relative magnitude of the uncertainties implied by this fractal structure for the optimal escape from a universal cubic potential well absolute and transient basins are examined giving control space maps analogous to familiar pictures of the mandelbrot set generalizing from this prototype study it is argued that in engineering design against boat capsize or earthquake damage for example a study of safe basins should augment and perhaps entirely replace conventional analysis of the steady state attracting solutions

periodic and chaotic behaviour in a reduced form of the perturbed generalized korteweg de vries and kadomtsev petviashvili equations abstract the dynamical behaviour of a reduced form of the perturbed generalized korteweg de vries and kadomtsev petviashvili equations extension of the korteweg de vries equation to two space variables are studied in this paper harmonic solutions of non resonance and primary resonance are obtained using the perturbation method chaotic motion under harmonic excitations is studied using the melnikov method a wide range of solutions for the reduced perturbed generalized korteweg de vries equations in which non linear phenomena appearing within transition from regular harmonic response periodic solutions to chaotic motion are obtained using the time integration runge kutta method when chaos is found it is detected by examining the phase plane the poincare map the sensitivity solution of the solution to initial conditions and by calculating the largest lyapunov exponent

first passage time of nonlinear ship rolling in narrow band non stationary random seas the generalized extended stochastic central difference gescd method is applied to study the response statistics and first passage time of nonlinear ship rolling in narrow band stationary and non stationary random seas the gescd method is based on a combination of the extended stochastic central difference method with a statistical linearization technique modified adaptive time scheme and time coordinate transformation the extended stochastic central difference method is however an extension of the stochastic central difference method for the determination of the recursive mean square or covariance of responses of systems under narrow band stationary and non stationary random disturbances approximate first passage probabilities of nonlinear systems based on the modified mean rate of various crossings proposed earlier by the first author were determined it is concluded that the gescd method is very accurate simple and efficient to apply compared with monte carlo simulation the proposed method is applicable to cases with large nonlinearities and intensive random excitations the approximate first passage probabilities of the nonlinear system determined by the proposed approach are very accurate as they are in excellent agreement with those evaluated by the monte carlo simulation it is believed that the model considered in this paper is a closer representation to reality than those reported earlier in the literature