two parameter bifurcations in a network of two neurons with multiple delays abstract we consider a network of two coupled neurons with delayed feedback we show that the connection topology of the network plays a fundamental role in classifying the rich dynamics and bifurcation phenomena regarding eigenvalues of the connection matrix as bifurcation parameters we obtain codimension 1 bifurcations including a fold bifurcation and a hopf bifurcation and codimension 2 bifurcations including fold hopf bifurcations and hopf hopf bifurcations we also give concrete formulae for the normal form coefficients derived via the center manifold reduction that give detailed information about the bifurcation and stability of various bifurcated solutions in particular we obtain stable or unstable equilibria periodic solutions quasi periodic solutions and sphere like surfaces of solutions we also show how to evaluate critical normal form coefficients from the original system of delay differential equations without computing the corresponding center manifolds

stochastic dynamics of a nonlinear aeroelastic system we investigate the asymptotic behaviors of a thin panel in high supersonic flow with a turbulent boundary layer the objective of this investigation is achieved by formulating methods to analyze complex interactions among aerodynamic structural dynamic nonlinearities turbulence and stability we reduce an infinite dimensional model to a finite dimensional system in the galerkin s sense the normal form technique is employed not only to capture the essential dynamics of the system in which significant nonlinearities are considered but also to reduce the dimensionality of the dynamical system because of the random nature of forcing characteristics associated with the turbulence the theory of stochastic processes is used to explore panel responses in the presence of turbulent boundary layer after adequate scaling of parameters a nonstandard reduction through stochastic averaging is achieved it turns out that the solution of the reduced model is approximated by a markov process that takes its value on a graph with gluing conditions that furnish the complete specification of the dynamics of the reduced model with the aid of the infinitesimal generator of the reduced markov process on the graph we examine stochastic analyses such as mean exit time probability density and stochastic bifurcation in the phenomenological sense

an mrf model based approach to simultaneous recovery of depth and restoration from defocused images in this paper we propose a map markov random field mrf based scheme for recovering the depth and the focused image of a scene from two defocused images the space variant blur parameter and the focused image of the scene are both modeled as mrfs and their map estimates are obtained using simulated annealing the scheme is amenable to the incorporation of smoothness constraints on the spatial variations of the blur parameter as well as the scene intensity it also allows for inclusion of line fields to preserve discontinuities the performance of the proposed scheme is tested on synthetic as well as real data and the estimates of the depth are found to be better than that of the existing window based depth from defocus technique the quality of the space variant restored image of the scene is quite good even under severe space varying blurring conditions

performance analysis of maximum likelihood estimator for recovery of depth from defocused images and optimal selection ofcamera parameters the recovery of depth from defocused images involves calculating the depth of various points in a scene by modeling the effect that the focal parameters of the camera have on images acquired with a small depth of field in the existing methods on depth from defocus dfd two defocused images of a scene are obtained by capturing the scene with different sets of camera parameters although the dfd technique is computationally simple the accuracy is somewhat limited compared to the stereo algorithms further an arbitrary selection of the camera settings can result in observed images whose relative blurring is insufficient to yield a good estimate of the depth in this paper we address the dfd problem as a maximum likelihood ml based blur identification problem we carry out performance analysis of the ml estimator and study the effect of the degree of relative blurring on the accuracy of the estimate of the depth we propose a criterion for optimal selection of camera parameters to obtain an improved estimate of the depth the optimality criterion is based on the cramer rao bound of the variance of the error in the estimate of blur a number of simulations as well as experimental results on real images are presented to substantiate our claims

rotating chemical waves in the gray scott model a set of reaction diffusion equations is considered known as the gray scott model defined on a circle and the stability of rotating wave solutions formed via hopf bifurcations that break the circular o 2 symmetry is investigated using a hybrid numerical analytical technique center manifold normal form reductions are performed to analyze symmetry breaking hopf bifurcations degenerate hopf bifurcations and hopf hopf mode interactions it is found that stable rotating waves exist over broad ranges of parameter values and that the bifurcation behavior of this relatively simple model can be quite complex e g two and three frequency motions exist

an improved theory of asymptotic unfoldings abstract an asymptotic unfolding of a dynamical system near a rest point is a system with additional parameters such that every one parameter deformation of the original system can be embedded in the unfolding preserving all properties that can be detected by asymptotic methods asymptotic unfoldings are computed using normal and hypernormal form methods we present a simplified and improved method of computing such unfoldings that can be used in any normal form style

hopf bifurcation for equivariant conservative and time reversible systems we study the bifurcation of small periodic solutions at a non semi simple 1 1 resonance in equivariant conservative or equivariant time reversible systems by using an equivariant liapunov schmidt method and restricting to solutions with an appropriate isotropy we reduce the problem to a scalar bifurcation equation the analysis of this equation shows a bifurcation behaviour similar to that found for the hamiltonian hopf bifurcation

absolute depth estimation from a single defocused image shape from defocus sfd is one of the most popular techniques in monocular 3d vision while most sfd approaches require two or more images of the same scene captured at a fixed view point this paper presents an efficient approach to estimate absolute depth from a single defocused image instead of directly measuring defocus level of each pixel we propose to design a sequence of aperture shape filters to segment a defocused image by defocus level a boundary weighted belief propagation algorithm is employed to obtain a smooth depth map we also give an estimation of depth error extensive experiments show that our approach outperforms the state of the art single image sfd approaches both in precision of the estimated absolute depth and running time

global characterization of the normal form for a vector field near a closed orbit on deduit une generalisation naturelle de la caracterisation globale de elphick pour des champs de vecteurs pres d une orbite fermee

codimension 2 hopf bifurcation of a two degree of freedom vibro impact system abstract codimension 2 hopf bifurcation problem of a two degree of freedom system vibrating against a rigid surface is investigated in this paper the four dimensional poincare map of the vibro impact system is reduced to a two dimensional normal form by virtue of a center manifold reduction and a normal form technique then the theory of hopf bifurcation of maps in r 2 is applied to conclude the existence of codimension 2 hopf bifurcation of the vibro impact system the quasi periodic response of the system by theoretical analysis is well supported by numerical simulations it is shown that there exists codimension 2 hopf bifurcation in multi degree of freedom vibro impact systems the codimension 2 tori doubling phenomenon and the routes of quasi periodic impacts to chaos are reported briefly

shape from focus the shape from focus method presented here uses different focus levels to obtain a sequence of object images the sum modified laplacian sml operator is developed to provide local measures of the quality of image focus the operator is applied to the image sequence to determine a set of focus measures at each image point a depth estimation algorithm interpolates a small number of focus measure values to obtain accurate depth estimates a fully automated shape from focus system has been implemented using an optical microscope and tested on a variety of industrial samples experimental results are presented that demonstrate the accuracy and robustness of the proposed method these results suggest shape from focus to be an effective approach for a variety of challenging visual inspection tasks

bifurcation and stability analyses for a coupled brusselator model abstract this paper addresses the dynamic behaviour of a chemical oscillator arising from the series coupling of two brusselators of particular interest is the study of the associated hopf bifurcation and double hopf bifurcations the motion of the oscillator may either be periodic bifurcating from a hopf type critical point or quasi periodic bifurcating from a compound critical point furthermore bifurcation analysis reveals that the limit cycles associated with the first brusselator are always stable while that generated by the second brusselator may be unstable if the parameter values are chosen far from the stability boundary it is interesting to note that in the vicinity of the double hopf compound critical point there exist periodic as well as quasi periodic solutions the quasi periodic motion is stable for a small parameter region a robust gauss seidel like implicit finite difference method gs1 has been developed and used for the solution of the resulting initial value problem ivp in addition to being of comparable accuracy judging by the similarity of the profiles generated with the fourth order runge kutta method rk4 the gs1 method will be seen to have better numerical stability property than rk4 unlike the rk4 which fails when large time steps are used to integrate the ivp extensive numerical simulations with appropriate initial data suggest that the gs1 method is unconditionally convergent moreover it is more economical computationally

on the c8 3 regularisation of simultaneous binary collisions in the collinear 4 body problem abstract a conjecture on the nature of the simultaneous binary collision is explored in the collinear 4 body problem it is known that any attempt to remove the singularity via block regularisation will result in a regularised flow that is no more than c 8 3 differentiable with respect to initial conditions through a blow up of the singularity this loss of differentiability is investigated and a new proof of the c 8 3 regularity is provided in the process it is revealed that the collision manifold consists of two manifolds of normally hyperbolic saddle singularities which are connected by a manifold of heteroclinics by utilising recent work on transitions near such objects and their normal forms an asymptotic series of the transition past the singularity is explicitly computed it becomes remarkably apparent that the finite differentiability at 8 3 is due to the inability to construct a set of integrals local to the simultaneous binary collision the finite differentiability is shown to be independent of the value of the masses

heteroclinic tangles and homoclinic snaking in the unfolding of a degenerate reversible hamiltonian hopft bifurcation this paper considers an unfolding of a degenerate reversible 1 1 resonance or hamiltonian hopf bifurcation for four dimensional systems of time reversible ordinary differential equations odes this bifurcation occurs when a complex quadruple of eigenvalues of an equilibrium coalesce on the imaginary axis to become imaginary pairs the degeneracy occurs via the vanishing of a normal form coefficient q2 0 that determines whether the bifurcation is super or subcritical of particular concern is the behaviour of homoclinic and heteroclinic connections between the trivial equilibrium and simple periodic orbits a partial unfolding of such solutions already occurs in the work of dias and iooss eur j mech b fluids 15 1996 367 393 given a sign of the coefficient of a higher order term q4 0 and 1 q2 0 the normal form then shows a small amplitude bifurcating branch of homoclinic solutions terminating at a heteroclinic connection to a simple periodic orbit a genericity argument shows that this connection is not structurally stable and should break into a pair of heteroclinic tangencies this is confirmed by numerics which shows that branches of the simplest homoclinic orbits undergo an infinite snaking sequence of limit points accumulating on the parameter values of the two tangencies at each limit point the homoclinic orbit generates another bump close to the periodic orbit as q2 is further decreased from zero the snake widens until at a critical value a branch is formed of heteroclinic connections from the origin to a nontrivial equilibrium a kink for q2 less than this value the kinks heteroclinic tangencies and snake like curves no longer occur

the limited effectiveness of normal forms a critical review and extension of local bifurcation studies of the brusselator pde abstract the detection and unfolding of degenerate local bifuractions provides one of very few generally applicable analytical tools for studying complex dynamics in systems of arbitrarily high dimension using the brusselator partial differential equations pdes prigogine and lefever 1968 as motivation and main example we critically review this method we extend and correct previous calculations presenting explicit formulae from which normal forms accurate to third order may be computed and for the first time we carefully compare bifurcations and dynamics of these normal forms with those of the untransformed systems restricted to a center manifold and with galerkin and finite difference approximations of the original pde while judicious use of symbolic manipulations makes feasible such high order center manifold and normal form calculations we show that the conclusions drawn from them are of limited use in understanding spatio temporal complexity and chaos as guckenheimer 1981 argued the method permits proof of existence of quasi periodic motions and under mild genericity assumptions sil nikov chaos sub shifts of finite type but the parameter and phase space ranges in which these results may be applied are extremely small

normal forms symmetry and linearization of dynamical systems we discuss how the presence of a suitable symmetry can guarantee the perturbative linearizability of a dynamical system or a parameter dependent family via the poincare normal form approach we discuss this formally at first and later pay attention to the convergence of the linearizing procedure we also discuss some generalizations of our main result

pattern formation and bistability in flow between counterrotating cylinders abstract in the taykor coutte experiment on fluid flow counterrotating cylinders there is a bicritical point where the onset of instabilities to taylor vortex flow a steady state bifurcation and spiral vortex flow a hopf bifurcation meet the nonlinear mode interactions near this bicritical point are analyzed exploiting the role of symmetry in the bifurcation theory and with emphasis of the relevance to experiments for a range of raduis ratios 0 43 0 98 the mechanism of the pattern formation is elucidated and several new flow patterns and transitions are predicted including wavy vortices bistability hysteresis and up to 7 quasiperiodic flows

the structure of wtc expansions and applications we construct generalized painleve expansions with logarithmic terms for a general class of non integrable scalar equations and describe their structure in detail these expansions were introduced without logarithms by weiss tabor carnevale wtc 1983 the construction of the formal solutions is shown to involve semi invariants of binary forms and tools from invariant theory are applied to the determination of the type of logarithmic terms that are required for the most general singular series the structure of the series depends strongly on whether 1 is or is not a resonance the convergence of these series is obtained as a consequence of the general results of littman and kichenassamy 1993 the results are illustrated on a family of fifth order models for water waves and other examples we also give necessary and sufficient conditions for 1 to be a resonance

heteroclinic bifurcations near hopf zero bifurcation in reversible vector fields in r 3 abstract we study the dynamics near a symmetric hopf zero also known as saddle node hopf or fold hopf bifurcation in a reversible vector field in r 3 with involutory an reversing symmetry whose fixed point subspace is one dimensional we focus on the case in which the normal form for this bifurcation displays a degenerate family of heteroclinics between two asymmetric saddle foci we study local perturbations of this degenerate family of heteroclinics within the class of reversible vector fields and establish the generic existence of hyperbolic basic sets horseshoes independent of the eigenvalues of the saddle foci as well as cascades of bifurcations of periodic heteroclinic and homoclinic orbits finally we discuss the application of our results to the michelson system describing stationary states and travelling waves of the kuramoto sivashinsky pde

three dimensional gravity capillary waves on water small surface tension case abstract this paper considers three dimensional gravity capillary waves on water of finite depth which are uniformly translating in a horizontal propagating direction and periodic in a transverse direction the exact euler equations are formulated as a spatial dynamic system in which the variable used for the propagating direction is a time like variable the existence of the solutions of the system is determined by two non dimensional constants the bond number b and the froude number f which in turn give the number of eigenvalues on the imaginary axis of the complex plane for the corresponding linearized operator around a uniform flow assume that f 2 c 1 is the curve in the b plane on which the first two eigenvalues for three dimensional waves collide at the imaginary axis and the intersection point of c 1 with 1 is b 1 0 in this paper the case for 0 b b 1 and b near c 1 is considered a center manifold reduction technique and a normal form analysis are applied to show that the dynamical system can be reduced to a system of ordinary differential equations using the existence of a homoclinic orbit connecting to a two dimensional periodic called generalized solitary wave thereafter solution for the reduced system it is shown that such a generalized solitary wave solution persists for the original system by applying a perturbation method and adjusting some appropriate constants

quasi genericity of bifurcations to high dimensional invariant tori for maps we consider a family of maps in a banach spacee near the situation when the derivative at the fixed point has two pairs of complex eigenvalues lying on the unit circle the other part of the spectrum being strictly inside the unit disc we focus our attention on the region of the parameter space where the truncated normal form of the maps shows a bifurcation of a family of invariantt 1 circles into a family of invariantt 2 tori we show that this problem needs a 3 dimensional parameter unfolding and that for the complete maps bifurcation occurs at points where is the rotation number on the non normally hyperbolict 1 circle ande 2i are the eigenvalues of the constant matrix conjugated to the non contracting part of the linearization on the normal fiber bundle overt 1 making some non resonance and diophantine assumptions on leading to a positive measure cantor set int 2 we show that in paraboloidal regions of the 3 dim parameter space we have clean bifurcations as for the truncated normal form the complement of these regions forms a set of bubbles such as the ones obtained by chenciner in chen for a codimension 2 problem for maps in 2 the main tool here is a generalization for a matrix function ont 1 close to a constant of the quasi conjugacy to a constant modulo a minimum of additional parameters moved quasi conjugacy for the infinite dimensional case we use ac decoupling result on the angular dependent linear parts into a contraction still angular dependent and another part quasi conjugated to a constant matrix this type of analysis applies for a wide range of problems where truncated normal forms of the maps give bifurcations fromt n tot n 1 tori and this needs a n 1 dimensional parameter unfolding

numerical periodic normalization for codim 1 bifurcations of limit cycles explicit computational formulas for the coefficients of the periodic normal forms for all codim 1 bifurcations of limit cycles in generic autonomous odes are derived they include second order coefficients for the fold limit point bifurcation as well as third order coefficients for the flip period doubling and neimark sacker torus bifurcations the formulas are independent of the dimension of the phase space and involve solutions of certain boundary value problems on the interval 0 t where t is the period of the critical cycle as well as multilinear functions from the taylor expansion of the right hand sides near the cycle the formulas allow us to distinguish between sub and supercritical bifurcations in agreement with earlier asymptotic expansions of the bifurcating solutions our formulation makes it possible to use robust numerical boundary value algorithms based on orthogonal collocation rather than shooting techniques which greatly expands its applicability the actual implementation is described in detail we include three numerical examples in which codim 2 singularities are detected along branches of codim 1 bifurcations of limit cycles as zeros of the periodic normal form coefficients

hopf bifurcation at 1 1 resonance on etudie un probleme concernant la bifurcation de hopf pour une famille a 1 parametre d equations differentielles dans un cas de resonance

a perturbation method for computing the simplest normal forms of dynamical systems abstract a previously developed perturbation method is generalized for computing the simplest normal form at each level of computation the minimum number of terms are retained of general n dimensional differential equations this direct approach combining the normal form theory with center manifold theory in one unified procedure can be used to systematically compute the simplest or unique normal form two particular singularities of the jacobian of the system are considered in this paper the first one is associated with one pair of purely imaginary eigenvalues hopf type singularity and the other corresponds to a simple zero and a pair of purely imaginary eigenvalues hopf zero type singularity the approach can be easily formulated and implemented using a computer algebra system maple programs have been developed in this paper which can be automatically executed by a user without the knowledge of computer algebra a physical oscillator model is studied in detail to show the computational efficiency of the direct method and the advantage of using the simplest normal form which greatly simplifies the analysis on dynamical systems in particular for bifurcations and stability

on the structure of nilpotent normal form modules abstract the set of vector fields that are in normal form with respect to a given linear part has the structure of a module and is best described by giving the stanley decomposition of that module an algorithm is presented that produces a stanley decomposition for the module of equivariants of the flow of a nilpotent linear vector field given a stanley decomposition for the ring of invariants this reduces the study of nilpotent normal forms to classical invariant theory plus this one additional algorithm both the inner product normal form elphick and the sl 2 normal form cushman sanders are covered within a single theory and simplified non equivariant versions of each normal form are presented

comment on asymptotic properties of coupled langevin equations the asymptotic behavior of coupled langevin equations in the limit of weak noise is studied by general normal form techniques in the vicinity of a pitchfork bifurcation the non gaussian behavior of the critical variable is established the conditional probability of the noncritical variable around the center manifold is determined it is shown that in certain cases the distribution of this later variable may be non gaussian

existence of travelling wave front solutions of a two dimensional anisotropic model this paper considers a two dimensional anisotropic email protected email protected 0 email protected k 0 2 email protected k 0 email protected y email protected email protected 0 email protected x email protected y email protected j 3 introduced by pesch and kramer 1986 28 assume that j travels with a speed c in the propagation direction x and is periodic in the transverse direction y this model is formulated as a spatial dynamic system in which the variable x is a time like variable a center manifold reduction technique and a normal form analysis are applied to show that this dynamic system can be reduced to a system of ordinary differential equations a bifurcation analysis yields the persistence of the heteroclinic orbit for the reduced system when higher order terms are added and the speed c is small enough which establishes the existence of travelling wave front solutions in order to overcome the difficulty caused by the irreversibility some appropriate constants are adjusted

codimension one and two bifurcations in a symmetrical ring network with delay abstract in this paper we discussed all the possible codimension one and partial codimension two bifurcations existing in a symmetrical ring network using the symmetric bifurcation theory of delay differential equations coupled with the representation theory of lie groups and dynamical analysis methods we have not only figured out the pattern of each bifurcation but also predicted the directions and stability analysis of the bifurcated solutions according to structure of the system numerical simulations were then given to verify our theoretical analysis and investigate the complexity of codimension two mode intersections comments were also presented to highlight some future work

space periodically driven codimension two bifurcations an amplitude equation is derived for a one dimensional extended dynamical system undergoing a codimension two bifurcation subjected to a weak near resonant spatially periodic forcing it is proved that in some appropriate limit the dynamics is of the propagative phase type and it is governed by the time dependent sine gordon equation with weak dissipation

computation of normal forms abstract in this paper a method of computing a normal form for a system of ordinary differential equations is given a program using the symbolic manipulator macsyma is used on several examples

non linear stability analysis of a complex rotor stator contact system in this paper a non linear strategy based on the centre manifold the rational approximants and the alternating frequency time domain method has been developed in order to study the non linear dynamical behaviour of a system in the neighbourhood of a critical steady state equilibrium point the stability analysis and the non linear dynamics of a complex braking system with a non linear rotor stator contact are presented moreover one of the most important steps of this paper is the determination of the non linear behaviour and the limit cycle amplitudes of this complex system in order to conduct this study the dynamic response is evaluated by using applying the centre manifold the rational approximants and the alternating frequency time domain method that permit to obtain rapidly and efficiently the non linear behaviour of the system the dynamic response obtained by applying this method is compared with that evaluated through numerical integration

capillary gravity waves on the free surface of an inviscid fluid of infinite depth existence of solitary waves permanent capillary gravity waves on the free surface of a two dimensional inviscid fluid of infinite depth are investigated an application of the hodograph transform converts the free boundary value problem into a boundary value problem for the cauchy riemann equations in the lower halfplane with nonlinear differential boundary conditions this can be converted to an integro differential equation with symbol k2 4 k 4 1 where is a bifurcation parameter a normal form analysis is presented which shows that the boundary value problem can be reduced to an integrable system of ordinary differential equations plus a remainder term containing nonlocal terms of higher order for small this normal form system has been studied thoroughly by several authors iooss kirchgassner 8 iooss peroueme 10 dias iooss 5 it admits a pair of solitary wave solutions which are reversible in the sense ofkirchgassner 11 by applying a method introduced in 11 it is shown that this pair of reversible solitary waves persists for the boundary value problem and that the decay at infinity of these solitary waves is at least like 1 x

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

on a parameter estimation method for gibbs markov random fields this correspondence is about a gibbs markov random field gmrf parameter estimation technique proposed by derin and elliott 1987 we will refer to this technique as the histogramming h method first the relation of the h method to the conditional maximum likelihood method is considered second a bias reduction based modification of the h method is proposed to improve its performance especially in the case of small amounts of image data

explicit transversality conditions and local bifurcation diagrams for bogdanov takens bifurcation on center manifolds abstract this application oriented study is concerned with the derivation of parameter dependent normal forms for the codimension two bogdanov takens bifurcation of n dimensional m parameterized systems on the basis of the homological method in the case of an enduring equilibrium simple formulas are obtained for the transformation of parameters enabling the formulation of explicit transversality conditions and bifurcation diagrams to at most the second order moreover in z 2 symmetric systems the calculation can be further limited within certain subspaces in the general case existing results are re derived and a revision necessary for determining the bifurcation diagrams to the second order is then provided these results facilitate the derivation of normal forms check of transversality and depiction of bifurcation diagrams for the bogdanov takens bifurcation

existence and stability of limit cycles in a macroscopic neuronal population model abstract we present rigorous results concerning the existence and stability of limit cycles in a macroscopic model of neuronal activity the specific model we consider is developed from the ki set methodology popularized by walter freeman in particular we focus on a specific reduction of the kii sets denoted rkii sets we analyse the unfolding of supercritical hopf bifurcations via consideration of the normal forms and centre manifold reductions subsequently we analyse the global stability of limit cycles on a region of parameter space and this is achieved by applying a new methodology termed global analysis of piecewise linear systems the analysis presented may also be used to consider coupled systems of this type a number of macroscopic mean field approaches to modelling human eeg may be considered as coupled rkii networks hence developing a theoretical understanding of the onset of oscillations in models of this type has important implications in clinical neuroscience as limit cycle oscillations have been demonstrated to be critical in the onset of certain types of epilepsy

orbits homoclinic to exponentially small periodic orbits for a class of reversible systems application to water waves in this paper a class of reversible analytic vector fields is investigated near an equilibrium for these vector fields the part of the spectrum of the differential at the equilibrium which lies near the imaginary axis comes from the perturbation of a double eigenvalue 0 and two simple eigenvalues

toroidal normal forms for bifurcations in retarded functional differential equations i multiple hopf and transcritical multiple hopf interaction abstract for finite dimensional bifurcation problems it is well known that it is possible to compute normal forms which possess nice symmetry properties oftentimes these symmetries may allow for a partial decoupling of the normal form into a so called radial part and an angular part analysis of the radial part usually gives an enormous amount of valuable information about the bifurcation and its unfoldings in this paper we are interested in the case where such bifurcations occur in retarded functional differential equations and we revisit the realizability and restrictions problem for the class of radial equations by nonlinear delay differential equations our analysis allows us to recover and considerably generalize recent results by faria and magalhaes t faria l t magalhaes restrictions on the possible flows of scalar retarded functional differential equations in neighborhoods of singularities j dynam differential equations 8 1996 35 70 and by buono and belair p l buono j belair restrictions and unfolding of double hopf bifurcation in functional differential equations j differential equations 189 2003 234 266

further reductions of normal forms for dynamical systems abstract we propose in this paper a method for obtaining a significant refinement of normal forms for dynamical systems or vector fields with concrete and interesting applications we use lower order nonlinear terms in the normal form for the simplifications of higher order terms our approach is applicable for both the non nilpotent and the nilpotent cases for dynamical systems of dimensions 2 and 3 we give an algorithm that leads to interesting finite order normal forms which are optimal or unique with respect to equivalence by formal near identity transformations we can compute at the same time a formal diffeormorphism that realizes the normalization comparisons with other methods are given for several examples

parametric stabilization of a gyroscopic system this paper studies the stabilization of a gyroscopic system using parametric stabilization near a combination resonance the gyroscopic system is near its primary instability i e the bifurcation parameter is such that the system possesses a double zero eigenvalue the stability of the system is studied for the linear hamiltonian system the damped linear system the forced linear hamiltonian system and finally the damped and forced linear system the addition of the periodic excitation near the critical combination resonance provides the system with an extended stability region when the excitation frequency is slightly above the combination resonance a non linear numerical example shows that these results may persist for the non linear problem the results of this work are then discussed in relation to an example gyroscopic problem a rotating shaft with periodically perturbed rotation rate

on the convergence of normalizing transformations in the presence of symmetries it is shown that under suitable conditions involving in particular the existence of analytic constants of motion the presence of lie point symmetries can ensure the convergence of the transformation of a vector field or dynamical system into normal form

normal forms and nonlinear symmetries we give some general theorems and extensions of previous results concerning the problem of transforming an algebra of vector fields into poincare normal form by means of a unifying algebraic language we show the possibility of obtaining either a parallel or joint normal form of the vector fields in a definite way which simplifies the construction of normal forms providing a precise restriction on their structure the application to finite dimensional dynamical systems and their lie point symmetries is also discussed

an investigation of methods for determining depth from focus the concept of depth from focus involves calculating distances to points in an observed scene by modeling the effect that the camera s focal parameters have on images acquired with a small depth of field this technique is passive and requires only a single camera the most difficult segment of calculating depth from focus is deconvolving the defocus operator from the scene and modeling it most current methods for determining the defocus operator employ inverse filtering the authors reveal some fundamental problems with inverse filtering inaccuracies in finding the frequency domain representation windowing effects and border effects a general matrix based method using regularization is presented which eliminates these problems the new method is confirmed experimentally with the results showing an rms error of 1 3

convergent normal forms of symmetric dynamical systems it is shown that the presence of lie point symmetries of non hamiltonian dynamical systems can ensure the convergence of the coordinate transformations which take the dynamical system or vector field into poincare dulac normal form

a unified approach to superresolution and multichannel blind deconvolution this paper presents a new approach to the blind deconvolution and superresolution problem of multiple degraded low resolution frames of the original scene we do not assume any prior information about the shape of degradation blurs the proposed approach consists of building a regularized energy function and minimizing it with respect to the original image and blurs where regularization is carried out in both the image and blur domains the image regularization based on variational principles maintains stable performance under severe noise corruption the blur regularization guarantees consistency of the solution by exploiting differences among the acquired low resolution images several experiments on synthetic and real data illustrate the robustness and utilization of the proposed technique in real applications

hypernormal form theory foundations and algorithms abstract hypernormal forms unique normal forms simplest normal forms are investigated both from the standpoint of foundational theory and algorithms suitable for use with computer algebra the baider theory of the campbell hausdorff group is refined by a study of its subgroups to determine the smallest substages into which the hypernormalization process can be divided this leads to a linear algebra algorithm to compute the generators needed for each substage with the least amount of work a concrete interpretation of jan sanders spectral sequence for hypernormal forms is presented examples are given and a proof is given for a little known theorem of belitskii expressing the hypernormal form space in the inner product style as the kernel of a higher order differential operator

box products in nilpotent normal form theory the factoring method abstract let n be a nilpotent matrix and consider vector fields x n x v x in normal form then v is equivariant under the flow e n t for the inner product normal form or e m t for the sl 2 normal form these vector equivariants can be found by finding the scalar invariants for the jordan blocks in n or m taking the box product of these to obtain the invariants for n or m itself and then boosting the invariants to equivariants by another box product these methods developed by murdock and sanders in 2007 are here given a self contained exposition with new foundations and new algorithms yielding improved simpler stanley decompositions for the invariants and equivariants ideas used include transvectants from classical invariant theory stanley decompositions from commutative algebra and integer cones from integer programming this approach can be extended to covariants of sl 2 k for k 1 known as slocc in quantum computing

initial conditions for models of dynamical systems abstract the long time behaviour of many dynamical systems may be effectively predicted by a low dimensional model that describes the evolution of a reduced set of variables we consider the question of how to equip such a low dimensional model with appropriate initial conditions so that it faithfully reproduces the long term behaviour of the orginal high dimensional dynamical system our method involves putting the dynamical system into normal form which not only generates the low dimensional model but also provides the correct initital conditions for the model we illustrate the method with several examples

resonant response functions for nonlinear oscillators with polynomial type nonlinearities in this paper we consider the steady state response of forced damped weakly nonlinear oscillators with polynomial type nonlinearities in particular we define general expressions that can be used to compute resonant response functions which define the steady state constant amplitude oscillatory response at the primary resonance and the associated harmonics the resonant response functions are derived using a normal form transformation which is carried out directly on the second order nonlinear oscillator the example of a forced van der pol oscillator with an additional cubic stiffness nonlinearity is used to demonstrate how the general analysis can be applied

variational approach to closure of nonlinear dynamical systems autonomous case a general approach for the derivation of nonlinear parameterizations of neglected scales is presented for nonlinear systems subject to an autonomous forcing in that respect dynamically based formulas are derived subject to a free scalar parameter to be determined per mode to parameterize for each high mode this free parameter is obtained by minimizing a cost functional a parameterization defect depending on solutions from direct numerical simulation dns but over short training periods of length comparable to a characteristic recurrence or decorrelation time of the dynamics an important class of dynamically based formulas for our parameterizations to optimize are obtained as parametric variations of manifolds approximating the invariant ones to better appreciate the origins of the modified manifolds thus obtained the standard approximation theory of invariant manifolds is revisited in part i of this article a special emphasis is put on backward forward bf systems naturally associated with the original system whose asymptotic integration provides the leading order approximation of invariant manifolds part ii presents then i the modifications of these approximating manifolds based also on integration of the same bf systems but this time over a finite time tau and ii the variational approach aimed at making an efficient selection of tau per mode to parameterize the parametric class of leading interaction approximation lia of the high modes obtained this way is completed by another parametric class built from the quasi stationary approximation qsa close to the first criticality the qsa is an approximation to the lia but it differs as one moves away from criticality rigorous results are derived that show that given a cutoff dimension the best manifolds that can be obtained through our variational approach are manifolds which are in general no longer invariant the minimizers are objects called the optimal parameterizing manifolds pms that are intimately tied to the conditional expectation of the original system i e the best vector field of the reduced state space resulting from averaging of the unresolved variables with respect to a probability measure conditioned on the resolved variables applications to the closure of low order models of atmospheric primitive equations and rayleigh benard convection are then discussed the approach is finally illustrated in the context of the kuramoto sivashinsky turbulence as providing efficient closures without slaving for a cutoff scale k mathfrak c placed within the inertial range and the reduced state space is just spanned by the unstable modes without inclusion of any stable modes whatsoever the underlying optimal pms obtained by our variational approach are far from slaving and allow for remedying the excessive backscatter transfer of energy to the low modes encountered by the lia or the qsa parameterizations in their standard forms when they are used at this cutoff wavelength

versal normal form of the hamiltonian function of the restricted problem of three bodies near l 4 abstract when one studies matrices depending on parameters the transformation into jordan canonical form can become singular when multiple eigenvalues occur the versal normal form for a matrix was introduced by arnol d 1971 in order to overcome this difficulty unfortunately no method was given on how to construct this transformation for a special class of matrices we describe a simple procedure to accomplish this task we use the one to one resonance at the lagrangian triangular equilibrium point in the restricted problem of three bodies as an example to demonstrate our method we then extend the idea of versal normal form to higher order terms in the hamiltonian function after performing this normalization we are able to analyze the behavior of the periodic orbits near the equilibrium point

quasi homogeneous normal forms in this paper we extend the concepts of the normal form theory for vector fields that are expanded in quasi homogeneous components of a fixed type these expansions have been used by some authors in the analysis of the determinacy of a given singularity also the use of reparametrizations in the time are considered namely beyond the use of c conjugation to determine normal forms we present a method useful to determine how much a vector field can be simplified by using c equivalence the results obtained are applied in the case of the bogdanov takens singularity firstly using c conjugation and later showing the improvements provided by the c equivalence

a variational approach to recovering depth from defocused images in this paper we propose a regularized solution to the depth from defocus dfd problem using the space frequency representation sfr framework a smoothness constraint is imposed on the estimates of the blur parameter and a variational approach to the dfd problem is developed among the numerous sfrs we study the applicability of the complex spectrogram and the wigner distribution in particular for depth recovery the performance of the proposed variational method is tested on both synthetic and real images the method yields good results and the quality of the estimates is significantly better than that obtained without the smoothness constraint on the blur parameter

painleve analysis and normal forms theory nonlinear vector fields have two important types of singularities the fixed points in phase space and the time singularities in the complex plane the first singularities are locally analyzed via normal form theory whereas the second ones are studied by the painleve analysis in this paper normal form theory is used to describe the solutions around their complex time singularities to do so a transformation mapping the local series around the singularities to the local series around a fixed point of a new system is introduced regular normal form theory is then used in this new system it is shown that a vector field has the painleve property only if the associated system is locally linearizable around its fixed points a problem analogous to the classical problem of the center moreover the connection between partial and complete integrability and the structure of local series around both types of singularities are established a new proof of the convergence of the local psi series is given and an explicit method to prove the existence of finite time blow up manifold in phase space is presented

homoclinic orbits in reversible systems and their applications in mechanics fluids and optics abstract this survey article reviews the theory and application of homoclinic orbits to equilibria in reversible continuous time dynamical systems where the homoclinic orbit and the equilibrium are both reversible the focus is on even order reversible systems in four or more dimensions local theory generic argument and global existence theories are examined for each qualitatively distinct linearisation several recent results such as coalescence caused by non transversality and the reversible orbit flip bifurcation are covered a number of open problems are highlighted applications are reviewed to systems arising from a variety of disciplines with the aid of numerical methods three examples are presented in detail one of which is infinite dimensional

rational filter design for depth from defocus the paper describes a new simple procedure to determine the rational filters that are used in the depth from defocus dfd procedure previously researched by watanabe and nayar 1998 4 their dfd uses two differently defocused images and the filters accurately model the relative defocus in the images and provide a fast calculation of distance this paper presents a simple method to determine the filter coefficients by separating the m p ratio into a linear and a cubic error correction model the method avoids the previous iterative minimisation technique and computes efficiently the model has been verified by comparison with the theoretical m p ratio the proposed filters have been compared with the previous for frequency response closeness of fit to m p rotational symmetry and measurement accuracy experiments were performed for several defocus conditions it was observed that the new filters were largely insensitive to object texture and modelled the blur more precisely than the previous experiments with real planar images demonstrated a maximum rms depth error of 1 18 for the proposed compared to 1 54 for the previous filters complicated objects were also accurately measured

modulating pulse solutions for a class of nonlinear wave equations we consider modulating pulse solutions for a nonlinear wave equation on the infinite line such a solution consists of a permanent pulse like envelope steadily advancing in the laboratory frame and modulating an underlying wave train the problem is formulated as an infinite dimensional dynamical system with one stable one unstable and infinitely many neutral directions using a partial normal form and invariant manifold theory we establish the existence of modulating pulse solutions which decay to small amplitude disturbances at large distances

a bayesian framework for image segmentation with spatially varying mixtures a new bayesian model is proposed for image segmentation based upon gaussian mixture models gmm with spatial smoothness constraints this model exploits the dirichlet compound multinomial dcm probability density to model the mixing proportions i e the probabilities of class labels and a gauss markov random field mrf on the dirichlet parameters to impose smoothness the main advantages of this model are two first it explicitly models the mixing proportions as probability vectors and simultaneously imposes spatial smoothness second it results in closed form parameter updates using a maximum a posteriori map expectation maximization em algorithm previous efforts on this problem used models that did not model the mixing proportions explicitly as probability vectors or could not be solved exactly requiring either time consuming markov chain monte carlo mcmc or inexact variational approximation methods numerical experiments are presented that demonstrate the superiority of the proposed model for image segmentation compared to other gmm based approaches the model is also successfully compared to state of the art image segmentation methods in clustering both natural images and images degraded by noise

symbolic numerical methods for the computation of normal forms of pdes the center manifold and the normal forms are effective tools for the study of local bifurcations occurring in evolution equations the computation of the center manifold and the normal form amounts after more or less complex algebraic transformations to solve in a recursive way a hierarchy of linear equations we present a method and computer programs for the computation of normal forms of some nonlinear parabolic pdes these computations are performed using the symbolic algebra system maple matlab and exploiting the compatibility of these two systems here the linear equations to be solved are infinite dimensional and we use the finite element method for this purpose the use of the finite element method allows to consider problems with complex shape domains in our programs maple takes care of the algebraic manipulations delivering the set of linear equations to be solved and writes some parts of the matlab code for their resolution we give three applications a pitchfork bifurcation in a semilinear parabolic equation a hopf bifurcation and a bifurcation to rotating and standing waves in a reaction diffusion system

nonlinear approach for enhancement of image focus volume in shape from focus mostly shape from focus algorithms use local averaging using a fixed rectangle window to enhance the initial focus volume in this linear filtering the window size affects the accuracy of the depth map a small window is unable to suppress the noise properly whereas a large window oversmoothes the object shape moreover the use of any window size smoothes focus values uniformly consequently an erroneous depth map is obtained in this paper we suggest the use of iterative 3 d anisotropic nonlinear diffusion filtering andf to enhance the image focus volume in contrast to linear filtering andf utilizes the local structure of the focus values to suppress the noise while preserving edges the proposed scheme is tested using image sequences of synthetic and real objects and results have demonstrated its effectiveness

bayesian depth from defocus with shading constraints we present a method that enhances the performance of depth from defocus dfd through the use of shading information dfd suffers from important limitations namely coarse shape reconstruction and poor accuracy on textureless surfaces that can be overcome with the help of shading we integrate both forms of data within a bayesian framework that capitalizes on their relative strengths shading data however is challenging to accurately recover from surfaces that contain texture to address this issue we propose an iterative technique that utilizes depth information to improve shading estimation which in turn is used to elevate depth estimation in the presence of textures the shading estimation can be performed in general scenes with unknown illumination using an approximate estimate of scene lighting with this approach we demonstrate improvements over existing dfd techniques as well as effective shape reconstruction of textureless surfaces

focused image recovery from two defocused images recorded with different camera settings two new methods are presented for recovering the focused image of an object from only two blurred images recorded with different camera parameter settings the camera parameters include lens position focal length and aperture diameter first a blur parameter 0 is estimated using one of our two recently proposed depth from defocus methods then one of the two blurred images is deconvolved to recover the focused image the first method is based on a recently proposed spatial domain convolutioddeconvolution transform this method requires only the knowledge of 7 of the camera s point spread function psf it does not require information about the actual form of the camera s psf the second method in contrast to the first requires full knowledge of the form of the psf as part of the second method we present a calibration procedure for estimating the camera s psf for different values of the blur parameter 0 in the second method the focused image is obtained through deconvolution in the fourier domain using the wiener filter for both methods results of experiments on actual defocused images recorded by a ccd camera are given the first method requires much less computation than the second method the first method gives satisfactory results for up to medium levels of blur and the second method gives good results for up to relatively high levels of blur

new local and parallel finite element algorithm based on the partition of unity abstract in this study based on a combination of the two grid method and the partition of unity based domain decomposition method we propose a new local and parallel finite element algorithm for the elliptic boundary value problem the proposed method has three key features 1 it inherits the flexibility and controllability of domain decomposition based on the partition of unity 2 global fine grid correction is replaced by solving a series of locally defined approximate residual problems with homogeneous dirichlet boundary conditions on some finer grids 3 a global continuous finite element solution is constructed by solving a coarse grid correction problem and by assembling all the local solutions together using the partition of unity subordinate under appropriate assumptions the optimal error estimates in l 2 and the energy norms are proved by new analytical results in addition several numerical simulations are presented to demonstrate the high efficiency and flexibility of the new algorithm

a recursive algorithm for maximum likelihood based identification of blur from multiple observations a maximum likelihood based method is proposed for blur identification from multiple observations of a scene when the relations among the blurring functions are known the estimate of blur obtained using the proposed method is very good since direct computation of the likelihood function becomes difficult as the number of images increases we propose an algorithm to compute the likelihood function recursively

a simple global characterization for normal forms of singular vector fields we derive a new global characterization of the normal forms of amplitude equations describing the dynamics of competing order parameters in degenerate bifurcation problems using an appropriate scalar product in the space of homogeneous vector polynomials we show that the resonant terms commute with the group generated by the adjoint of the original critical linear operator this leads to a very efficient constructive method to compute both the nonlinear coefficients and the unfolding of the normal form explicit examples and results obtained when there are additional symmetries are also presented

normal form transforms separate slow and fast modes in stochastic dynamical systems modelling stochastic systems has many important applications normal form coordinate transforms are a powerful way to untangle interesting long term macroscale dynamics from insignificant detailed microscale dynamics we explore such coordinate transforms of stochastic differential systems when the dynamics has both slow modes and quickly decaying modes the thrust is to derive normal forms useful for macroscopic modelling of complex stochastic microscopic systems thus we not only must reduce the dimensionality of the dynamics but also endeavour to separate emph all slow processes from emph all fast time processes both deterministic and stochastic quadratic stochastic effects in the fast modes contribute to the drift of the important slow modes some examples demonstrate that the coordinate transform may be only locally valid or may be globally valid depending upon the dynamical system the results will help us accurately model interpret and simulate multiscale stochastic systems

a normal form of thin fluid film equations resolves the transient paradox abstract imagine two constant thickness thin films of fluid colliding together the transient flow forms a hump where they collide thereafter they slowly relax but apparently reliable lubrication models expressed only in the thickness of the fluid forecast that precisely nothing happens how can we resolve this paradox dynamical systems theory constructs a normal form of the navier stokes equations for the flow of a thin layer of fluid upon a solid substrate these normal form equations illuminate the fluid dynamics by decoupling the interesting long term lubrication flow from the rapid viscous decay of transient shear modes the normal form clearly shows the slow manifold of the lubrication model and demonstrates that the initial condition for the fluid thickness of the lubrication model is not the initial physical fluid thickness but instead is modified by any initial lateral shear flow with these initial conditions the lubrication model makes better forecasts this dynamical systems approach could similarly illuminate other models of complicated dynamics

numerical normalization techniques for all codim 2 bifurcations of equilibria in ode s explicit computational formulas for the coefficients of the normal forms for all codim 2 equilibrium bifurcations of equilibria in autonomous odes are derived they include second order coefficients for the bogdanov takens bifurcation third order coefficients for the cusp and fold hopf bifurcations and coefficients of the fifth order terms for the generalized hopf bautin and double hopf bifurcations the formulas are independent on the dimension of the phase space and involve only critical eigenvectors of the jacobian matrix of the right hand sides and its transpose as well as multilinear functions from the taylor expansion of the right hand sides at the critical equilibrium the normal form coefficients for the fold hopf bifurcation in the new lorenz model are computed using the derived formulas proving the existence of a nontrivial invariant set in the system

nonequilibrium potentials near instabilities the existence of polynomial approximations for nonequilibrium potentials determined by a master equation near an instability of arbitrary codimension with diagonalizable linear part is studied it is shown that the approximations exist provided some relations are satisfied between the coefficients of the master equation

fold pitchfork bifurcation for maps with z 2 symmetry in pipe flow this study aims to provide a better understanding of recently identified transition scenarios exhibited by traveling wave solutions in pipe flow this particular family of solutions are invariant under certain reflectional symmetry transformations and they emerge from saddle node bifurcations within a two dimensional parameter space characterized by the length of the pipe and the reynolds number the present work precisely provides a detailed analysis of a codimension two saddle node bifurcation arising in discrete dynamical systems maps with z2 symmetry normal form standard techniques are applied in order to obtain the reduced map up to cubic order all possible bifurcation scenarios exhibited by this normal form are analyzed in detail finally a qualitative comparison of these scenarios with the ones observed in the aforementioned hydrodynamic problem is provided

normal forms for stochastic differential equations we address the following problem from the intersection of dynamical systems and stochastic analysis two sde dx t j 0 m f j x t dw t j and dx t j 0 m g j x t dw t j in d with smooth coefficients satisfying f j 0 g j 0 0 are said to be smoothly equivalent if there is a smooth random diffeomorphism coordinate transformation h with h 0 0 and dh 0 id which conjugates the corresponding local flows

amplitude equations for systems with competing instabilities we describe how to derive ordinary differential equations to predict the time dependence of a system governed by partial differential equations when that system is near to the polycritical condition for the onset of several instabilities the method is illustrated for the general case of two competing instabilities and then a specific physical example of this case is worked out in detail the basic idea is to extend the krylov bogolyubov mitropolsky method of nonlinear mechanics to bifurcation problems

space variant restoration of images degraded by camera motion blur we examine the problem of restoration from multiple images degraded by camera motion blur we consider scenes with significant depth variations resulting in space variant blur the proposed algorithm can be applied if the camera moves along an arbitrary curve parallel to the image plane without any rotations the knowledge of camera trajectory and camera parameters is not necessary at the input the user selects a region where depth variations are negligible the algorithm belongs to the group of variational methods that estimate simultaneously a sharp image and a depth map based on the minimization of a cost functional to initialize the minimization it uses an auxiliary window based depth estimation algorithm feasibility of the algorithm is demonstrated by three experiments with real images

embedding and splitting ordinary differential equations in normal form abstract we introduce an embedding of real or complex n dimensional space k n as an algebraic variety v which is determined by the action of a linear one parameter group every analytic vector field on k n corresponds to some embedded vector field on v for a symmetric vector field this embedded vector field splits into a reduced system and a direct sum of non autonomous linear systems examples and applications are mostly concerned with poincare dulac normal forms embeddings provide a natural setting for perturbations of symmetric systems in particular of systems in normal form up to some degree