realization by inspection we investigate which first order representations can be obtained from high order representations of linear systems by inspection that is just by rearrangement of the data under quite weak conditions it is possible to obtain minimal realizations in the so called pencil form under stronger conditions one can obtain minimal realizations in standard state space form by inspection the development is based on a reformulation of the realization problem as a problem of finding a complete set of basis vectors for the nullspace of a given constant matrix since no numerical computation is needed the realization method in particular is suitable for situations in which some of the coefficients are symbolic rather than numerical

from time series to linear system part i finite dimensional linear time invariant systems abstract in the first part of this paper the definition of a dynamical system as simply consisting of a family of time series will be developed in this context the notions of linearity time invariance and finite dimensionality will be introduced it will be shown that a given family of time series may be represented by a system of ar equations riw t l rl 1w t l 1 r0w t 0 or equivalently by a finite dimensional linear time invariant system x t 1 ax t bu t y t cx t du t w u y if and only if this family is linear shift invariant and complete or as is equivalent closed in the topology of pointwise convergence this yields a very high level and elegant set of axioms which characterize these familiar objects it is emphasized however that no a priori choice is made as to which components of w are inputs and which are outputs such a separation always exists in any specific linear time invariant model starting from these definitions the structural indices of such systems are introduced and it is shown how an ar representation of a system having a given behaviour can be constructed these results will be used in a modelling context in part ii of the paper

practical global optimization for multiview geometry this paper presents a practical method for finding the provably globally optimal solution to numerous problems in projective geometry including multiview triangulation camera resectioning and homography estimation unlike traditional methods which may get trapped in local minima due to the non convex nature of these problems this approach provides a theoretical guarantee of global optimality the formulation relies on recent developments in fractional programming and the theory of convex underestimators and allows a unified framework for minimizing the standard l 2 norm of reprojection errors which is optimal under gaussian noise as well as the more robust l 1 norm which is less sensitive to outliers even though the worst case complexity of our algorithm is exponential the practical efficacy is empirically demonstrated by good performance on experiments for both synthetic and real data an open source matlab toolbox that implements the algorithm is also made available to facilitate further research

multiple view geometry under the l_ infty norm this paper presents a new framework for solving geometric structure and motion problems based on the linfin norm instead of using the common sum of squares cost function that is the l2 norm the model fitting errors are measured using the linfin norm unlike traditional methods based on l2 our framework allows for the efficient computation of global estimates we show that a variety of structure and motion problems for example triangulation camera resectioning and homography estimation can be recast as quasi convex optimization problems within this framework these problems can be efficiently solved using second order cone programming socp which is a standard technique in convex optimization the methods have been implemented in matlab and the resulting toolbox has been made publicly available the algorithms have been validated on real data in different settings on problems with small and large dimensions and with excellent performance

branch and bound methods for euclidean registration problems in this paper we propose a practical and efficient method for finding the globally optimal solution to the problem of determining the pose of an object we present a framework that allows us to use point to point point to line and point to plane correspondences for solving various types of pose and registration problems involving euclidean or similarity transformations traditional methods such as the iterative closest point algorithm or bundle adjustment methods for camera pose may get trapped in local minima due to the nonconvexity of the corresponding optimization problem our approach of solving the mathematical optimization problems guarantees global optimality the optimization scheme is based on ideas from global optimization theory in particular convex underestimators in combination with branch and bound methods we provide a provably optimal algorithm and demonstrate good performance on both synthetic and real data we also give examples of where traditional methods fail due to the local minima problem

global optimization through rotation space search this paper introduces a new algorithmic technique for solving certain problems in geometric computer vision the main novelty of the method is a branch and bound search over rotation space which is used in this paper to determine camera orientation by searching over all possible rotations problems can be reduced to known fixed rotation problems for which optimal solutions have been previously given in particular a method is developed for the estimation of the essential matrix giving the first guaranteed optimal algorithm for estimating the relative pose using a cost function based on reprojection errors recently convex optimization techniques have been shown to provide optimal solutions to many of the common problems in structure from motion however they do not apply to problems involving rotations the search method described in this paper allows such problems to be solved optimally apart from the essential matrix the algorithm is applied to the camera pose problem providing an optimal algorithm the approach has been implemented and tested on a number of both synthetically generated and real data sets with good performance

paradigms and puzzles in the theory of dynamical systems a self contained exposition is given of an approach to mathematical models in particular to the theory of dynamical systems the basic ingredients form a triptych with the behavior of a system in the center and behavioral equations with latent variables as side panels the author discusses a variety of representation and parametrization problems in particular questions related to input output and state models the proposed concept of a dynamical system leads to a new view of the notions of controllability and observability and of the interconnection of systems in particular to what constitutes a feedback control law the final issue addressed is that of system identification it is argued that exact system identification leads to the question of computing the most powerful unfalsified model

desynchronization transitions in nonlinearly coupled phase oscillators abstract we consider the nonlinear extension of the kuramoto model of globally coupled phase oscillators where the phase shift in the coupling function depends on the order parameter a bifurcation analysis of the transition from fully synchronous state to partial synchrony is performed we demonstrate that for small ensembles it is typically mediated by stable cluster states that disappear with creation of heteroclinic cycles while for a larger number of oscillators a direct transition from full synchrony to a periodic or a quasiperiodic regime occurs

encoding via conjugate symmetries of slow oscillations for globally coupled oscillators we study properties of the dynamics underlying slow cluster oscillations in two systems of five globally coupled oscillators these slow oscillations are due to the appearance of structurally stable heteroclinic connections between cluster states in the noise free dynamics in the presence of low levels of noise they give rise to long periods of residence near cluster states interspersed with sudden transitions between them moreover these transitions may occur between cluster states of the same symmetry or between cluster states with conjugate symmetries given by some rearrangement of the oscillators we consider the system of coupled phase oscillators studied by hansel et al phys rev e 48 3470 1993 in which one can observe slow noise driven oscillations that occur between two families of two cluster periodic states in the noise free case there is a robust attracting heteroclinic cycle connecting these families the two families consist of symmetric images of two inequivalent periodic orbits that have the same symmetry for n 5 oscillators one of the periodic orbits has one unstable direction and the other has two unstable directions examining the behavior on the unstable manifold for the two unstable directions we observe that the dimensionality of the manifold can give rise to switching between conjugate symmetry orbits by applying small perturbations to the system we can easily steer it between a number of different marginally stable attractors finally we show that similar behavior occurs in a system of phase energy oscillators that are a natural extension of the phase model to two dimensional oscillators we suggest that switching between conjugate symmetries is a very efficient method of encoding information into a globally coupled system of oscillators and may therefore be a good and simple model for the neural encoding of information

globally optimal estimates for geometric reconstruction problems we introduce a framework for computing statistically optimal estimates of geometric reconstruction problems while traditional algorithms often suffer from either local minima or non optimality or a combination of both we pursue the goal of achieving global solutions of the statistically optimal cost function our approach is based on a hierarchy of convex relaxations to solve non convex optimization problems with polynomials these convex relaxations generate a monotone sequence of lower bounds and we show how one can detect whether the global optimum is attained at a given relaxation the technique is applied to a number of classical vision problems triangulation camera pose homography estimation and last but not least epipolar geometry estimation experimental validation on both synthetic and real data is provided in practice only a few relaxations are needed for attaining the global optimum

bifurcations in phase oscillator networks with a central element abstract a system of phase oscillators with identical natural frequencies and the star like architecture of connections is considered interaction functions are described by two terms of fourier expansion bifurcation analysis of small systems containing 3 or 4 oscillators has been performed the results are summarized in bifurcation diagrams that provide a full description of the boundaries between regions with different dynamics and the types of bifurcations that lead to the changes in the topology of phase space the bifurcations include changes of fixed point stability and formation destruction of limit and heteroclinic cycles for the system with 4 oscillators chaotic behaviour has been investigated the results can be useful to control system dynamics through an appropriate choice and variation of parameter values the generalization of the results to the systems with an arbitrary number of oscillators and application of the results in computational neuroscience are discussed

synchronization and multi frequency oscillations in the low dimensional chain of the self oscillators abstract the problem of growing complexity of the dynamics of the coupled phase oscillators as the number of oscillators in the chain increases is considered the organization of the parameter space parameter of the frequency detuning between the second and the first oscillators versus parameter of dissipative coupling is discussed the regions of complete synchronization quasi periodic regimes of different dimensions and chaos are identified we discuss transformation of the domains of different dynamics as the number of oscillators grows we use the method of charts of lyapunov exponents and modification of the method of the chart of dynamical regimes to visualize two frequency regimes of different type limits of applicability of the quasi harmonic approximation and the features of the dynamics of the original system which are not described by the approximate phase equations are discussed for the case of three coupled oscillators

slow switching in globally coupled oscillators robustness and occurrence through delayed coupling the phenomenon of slow switching in populations of globally coupled oscillators is discussed this characteristic collective dynamics which was first discovered in a particular class of the phase oscillator model is a result of the formation of a heteroclinic loop connecting a pair of clustered states of the population we argue that the same behavior can arise in a wider class of oscillator models with the amplitude degree of freedom we also argue how such heteroclinic loops arise inevitably and persist robustly in a homogeneous population of globally coupled oscillators although a heteroclinic loop might seem to arise only exceptionally we find that it appears rather easily by introducing time delay into a population which would otherwise exhibit perfect phase synchrony we argue that the appearance of the heteroclinic loop induced by the delayed coupling is then characterized by transcritical and saddle node bifurcations slow switching arises when a system with a heteroclinic loop is weakly perturbed this will be demonstrated with a vector model by applying weak noises other types of weak symmetry breaking perturbations can also cause slow switching

shape ambiguities in structure from motion this paper examines the fundamental ambiguities and uncertainties inherent in recovering structure from motion by examining the eigenvectors associated with null or small eigenvalues of the hessian matrix we can quantify the exact nature of these ambiguities and predict how they affect the accuracy of the reconstructed shape our results for orthographic cameras show that the bas relief ambiguity is significant even with many images unless a large amount of rotation is present similar results for perspective cameras suggest that three or more frames and a large amount of rotation are required for metrically accurate reconstruction

bifurcation to heteroclinic cycles and sensitivity in three and four coupled phase oscillators abstract we study the bifurcation and dynamical behaviour of the system of n globally coupled identical phase oscillators introduced by hansel mato and meunier in the cases n 3 and n 4 this model has been found to exhibit robust slow switching oscillations that are caused by the presence of robust heteroclinic attractors this paper presents a bifurcation analysis of the system in an attempt to better understand the creation of such attractors we consider bifurcations that occur in a system of identical oscillators on varying the parameters in the coupling function these bifurcations preserve the permutation symmetry of the system we then investigate the implications of these bifurcations for the sensitivity to detuning i e the size of the smallest perturbations that give rise to loss of frequency locking for n 3 we find three types of heteroclinic bifurcation that are codimension one with symmetry on varying two parameters in the coupling function we find three curves giving a an s 3 transcritical homoclinic bifurcation b a saddle node heteroclinic bifurcation and c a z 3 heteroclinic bifurcation we also identify several global bifurcations with symmetry that organize the bifurcation diagram these are codimension two with symmetry for n 4 oscillators we determine many but not all codimension one bifurcations with symmetry including those that lead to a robust heteroclinic cycle a robust heteroclinic cycle is stable in an open region of parameter space and unstable in another open region furthermore we verify that there is a subregion where the heteroclinic cycle is the only attractor of the system while for other parts of the phase plane it can coexist with stable limit cycles we finish with a discussion of bifurcations that appear for this coupling function and general n as well as for more general coupling functions

intermittent switching for three repulsively coupled oscillators we study intermittent switching behaviors in a system with three identical oscillators coupled diffusively and repulsively to clarify a bifurcation scenario which generates such intermittent switching behaviors we use the stuart landau oscillator which is a general form of hopf bifurcation and can describe both cases limit cycle and inactive i e non self oscillatory cases from a numerical study of the bifurcation structure two different routes to chaos which has s 3 symmetry were found one is the sudden appearance of chaos as pomeau manneville intermittency which is found for the inactive case in this case a trajectory shows switching among three mutually symmetric tori when a parameter exceeds critical value the other route which appears for the limit cycle case consists of two parts first chaos with lower symmetry appears through period doubling and after the two successive attractor merging crises chaos which has s 3 symmetry appears at each crisis the attractor changes its symmetry

quasiconvex optimization for robust geometric reconstruction geometric reconstruction problems in computer vision are often solved by minimizing a cost function that combines the reprojection errors in the 2d images in this paper we show that for various geometric reconstruction problems their reprojection error functions share a common and quasiconvex formulation based on the quasiconvexity we present a novel quasiconvex optimization framework in which the geometric reconstruction problems are formulated as a small number of small scale convex programs that are readily solvable our final reconstruction algorithm is simple and has intuitive geometric interpretation in contrast to existing local minimization approaches our algorithm is deterministic and guarantees a predefined accuracy of the minimization result the quasiconvexity also provides an intuitive method to handle directional uncertainties and outliers in measurements for a large scale problem or in a situation where computational resources are constrained we provide an efficient approximation that gives a good upper bound but not global minimum on the reconstruction error we demonstrate the effectiveness of our algorithm by experiments on both synthetic and real data

slow switching in a population of delayed pulse coupled oscillators we show that peculiar collective dynamics called slow switching arises in a population of leaky integrate and fire oscillators with delayed all to all pulse couplings by considering the stability of cluster states and symmetry possessed by our model we argue that saddle connections between a pair of the two cluster states are formed under general conditions slow switching appears as a result of the system s approach to the saddle connections it is also argued that such saddle connections are easy to arise near the bifurcation point where the state of perfect synchrony loses stability we develop an asymptotic theory to reduce the model into a simpler form with which an analytical study of the cluster states becomes possible

constants of motion for superconducting josephson arrays abstract we show that series arrays of n identical overdamped josephson junctions have extremely degenerate dynamics in particular we prove that such arrays have n 3 constants of motion for all n 3 the analysis is based on a coordinate transformation that reduces the governing equations to an n 3 parameter family of low dimensional systems in the weak coupling limit the reduced equations can be analyzed completely either all solutions approach the synchronous state or they converge to a continuous family of incoherent oscillations depending on a certain parameter value at the transitional value of this parameter the system becomes completely integrable then the phase space is foliated by invariant two dimensional tori for any n 3 the infinite n limit of the system is an integro partial differential equation with rigorously low dimensional dynamics it supports solitons in the integrable case and chaotic waves in the non integrable case