a dual to lyapunov s stability theorem lyapunov s second theorem is a standard tool for stability analysis of ordinary differential equations here we introduce a theorem which can be viewed as a dual to lyapunov s result from existence of a scalar function satisfying certain inequalities it follows that almost all trajectories of the system tend to zero the scalar function has a physical interpretation as the stationary density of a substance that is generated in all points of the state space and flows along the system trajectories if the stationary density is bounded everywhere except at a singularity in the origin then almost all trajectories tend towards the origin the weaker notion of stability allows for applications also in situations where lyapunov s theorem cannot be used moreover the new criterion has a striking convexity property related to control synthesis

analysis of a generalized dispersive model coupled to a dgtd method with application to nanophotonics in this paper we are concerned with the numerical modelling of the propagation of electromagnetic waves in dispersive materials for nanophotonics applications we focus on a generalized model that allows for the description of a wide range of dispersive media the underlying differential equations are recast into a generic form and we establish an existence and uniqueness result we then turn to the numerical treatment and propose an appropriate discontinuous galerkin time domain framework we obtain the semidiscrete convergence and prove the stability and to a larger extent convergence of a runge kutta 4 fully discrete scheme via a technique relying on energy principles finally we validate our approach through two significant nanophotonics test cases

stability and asymptotic properties of a linearized hydrodynamic medium model for dispersive media in nanophotonics abstract we analyze the stability of a linearized hydrodynamical model describing the response of nanometric dispersive metallic materials illuminated by optical light waves that is the situation occurring in nanoplasmonics this model corresponds to the coupling between the maxwell system and a pde describing the evolution of the polarization current of the electrons in the metal we show the well posedness of the system polynomial stability and optimal energy decay rate we also investigate the numerical stability for a discontinuous galerkin type approximation and several explicit time integration schemes

isostables isochrons and koopman spectrum for the action angle representation of stable fixed point dynamics abstract for asymptotically periodic systems a powerful phase reduction of the dynamics is obtained by computing the so called isochrons i e the sets of points that converge toward the same trajectory on the limit cycle motivated by the analysis of excitable systems a similar reduction has been attempted for non periodic systems admitting a stable fixed point in this case the isochrons can still be defined but they do not capture the asymptotic behavior of the trajectories instead the sets of interest that we call isostables are defined in the literature as the sets of points that converge toward the same trajectory on a stable slow manifold of the fixed point however it turns out that this definition of the isostables holds only for systems with slow fast dynamics also efficient methods for computing the isostables are missing the present paper provides a general framework for the definition and the computation of the isostables of stable fixed points which is based on the spectral properties of the so called koopman operator more precisely the isostables are defined as the level sets of a particular eigenfunction of the koopman operator through this approach the isostables are unique and well defined objects related to the asymptotic properties of the system also the framework reveals that the isostables and the isochrons are two different but complementary notions which define a set of action angle coordinates for the dynamics in addition an efficient algorithm for computing the isostables is obtained which relies on the evaluation of laplace averages along the trajectories the method is illustrated with the excitable fitzhugh nagumo model and with the lorenz model finally we discuss how these methods based on the koopman operator framework relate to the global linearization of the system and to the derivation of special lyapunov functions

nonlinear eigenvalue approach to differential riccati equations for contraction analysis in this paper we extend the eigenvalue method of the algebraic riccati equation to the differential riccati equation dre in contraction analysis one of the main results is showing that solutions to the dre can be expressed as functions of nonlinear eigenvectors of the differential hamiltonian matrix moreover under an assumption for the differential hamiltonian matrix real symmetry regularity and positive semidefiniteness of solutions are characterized by nonlinear eigenvalues and eigenvectors

a min max medial axis transformation blum s medial axis transformation mat of the set s of 1 s in a binary picture can be defined by an iterative shrinking and reexpanding process which detects corners on the contours of constant distance from s and thereby yields a skeleton of s for unsegmented gray level pictures one can use an analogous definition in which local min and max operations play the roles of shrinking and expanding to compute a mmmat value at each point of the picture the set of points having high values defines a good skeleton for the set of high gray level points in the given picture

spectral decomposition of a fokker planck equation at criticality the mean field for a complex network consisting of a large but finite number of random two state elements m has been shown to satisfy a nonlinear langevin equation the noise intensity is inversely proportional to sqrt m in the limiting case m infty the solution to the langevin equation exhibits a transition from exponential to inverse power law relaxation as criticality is approached from above or below the critical point when m infty the inverse power law is truncated by an exponential decay with rate vargamma the evaluation of which is the main purpose of this article an analytic numeric approach is used to obtain the lowest order eigenvalues in the spectral decomposition of the solution to the corresponding fokker planck equation and its equivalent schrodinger equation representation

spectral decomposition of nonlinear systems with memory we present an alternative approach to the analysis of nonlinear systems with long term memory that is based on the koopman operator and a levy transformation in time memory effects are considered to be the result of interactions between a system and its surrounding environment the analysis leads to the decomposition of a nonlinear system with memory into modes whose temporal behavior is anomalous and lacks a characteristic scale on average the time evolution of a mode follows a mittag leffler function and the system can be described using the fractional calculus the general theory is demonstrated on the fractional linear harmonic oscillator and the fractional nonlinear logistic equation when analyzing data from an ill defined black box system the spectral decomposition in terms of mittag leffler functions that we propose may uncover inherent memory effects through identification of a small set of dynamically relevant structures that would otherwise be obscured by conventional spectral methods consequently the theoretical concepts we present may be useful for developing more general methods for numerical modeling that are able to determine whether observables of a dynamical system are better represented by memoryless operators or operators with long term memory in time when model details are unknown

closing the hierarchy of moment equations in nonlinear dynamical systems the moment equations associated with the evolution of the probability density are known to form an infinite hierarchy of coupled equations in nonlinear dynamical systems in the present paper a systematic approach for closing this hierarchy is proposed based on the ansatz that in the long time limit there exist groups of moments varying on the same time scale the method is applied to a one dimensional vector field in the presence of noise and to two prototypes of chaotic behavior excellent agreement with numerical results is obtained special emphasis is placed on the role of symmetries and on the origin of the composite oscillations found for certain types of moments in the chaotic systems

stability in the almost everywhere sense a linear transfer operator approach abstract the problem of almost everywhere stability of a nonlinear autonomous ordinary differential equation is studied using a linear transfer operator framework the infinitesimal generator of a linear transfer operator perron frobenius is used to provide stability conditions of an autonomous ordinary differential equation it is shown that almost everywhere uniform stability of a nonlinear differential equation is equivalent to the existence of a non negative solution for a steady state advection type linear partial differential equation we refer to this non negative solution verifying almost everywhere global stability as lyapunov density a numerical method using finite element techniques is used for the computation of lyapunov density

pulse based control using koopman operator under parametric uncertainty in applications such as biomedicine and systems synthetic biology technical limitations in actuation complicate implementation of time varying control signals in order to alleviate some of these limitations it may be desirable to derive simple control policies such as step functions with fixed magnitude and length or temporal pulses in this technical note we further develop a recently proposed pulse based solution to the convergence problem i e minimizing the convergence time to the target exponentially stable equilibrium for monotone systems in particular we extend this solution to monotone systems with parametric uncertainty our solutions also provide worst case estimates on convergence times furthermore we indicate how our tools can be used for a class of nonmonotone systems and more importantly how these tools can be extended to other control problems we illustrate our approach on switching under parametric uncertainty and regulation around a saddle point problems in a genetic toggle switch system

linearization in the large of nonlinear systems and koopman operator spectrum abstract according to the hartman grobman theorem a nonlinear system can be linearized in a neighborhood of a hyperbolic stationary point here we extend this linearization around stable unstable equilibria or periodic orbits to the whole basin of attraction for both discrete diffeomorphisms and flows we discuss the connection of the linearizing transformation to the spectrum of koopman operator

a ridge following algorithm for finding the skeleton of a fuzzy image abstract a typical method for finding the skeleton of a grayscale image is to threshold the image then thin the result with one of several binary thinning algorithms this process loses information in the intensity dimension by reducing all intensities to either one of two values the loss of intensity information causes the skeleton of non uniform grayscale images to be centrally located within the thresholded image but may not lie along the maximum intensity ridge line in this paper we propose a method that avoids thresholding instead we transform the grayscale image into a fuzzy set and use the degree of membership in the underlying object to find the skeleton along the maximum intensity ridge line topographical labeling methods find a ridge line by topographically labeling pixels the ride line is the union of peak ridge and saddle point pixels these methods require preprocessing or postprocessing to find connected single pixel skeletons we show the results of our algorithm on images varying in uniformity from a gaussian smoothed binary rectangle to a scanned image of a grayscale object we compare results with the threshold and thin and topographical labeling methods

resonances in a chaotic attractor crisis of the lorenz flow local bifurcations of stationary points and limit cycles have successfully been characterized in terms of the critical exponents of these solutions lyapunov exponents and their associated covariant lyapunov vectors have been proposed as tools for supporting the understanding of critical transitions in chaotic dynamical systems however it is in general not clear how the statistical properties of dynamical systems change across a boundary crisis during which a chaotic attractor collides with a saddle this behavior is investigated here for a boundary crisis in the lorenz flow for which neither the lyapunov exponents nor the covariant lyapunov vectors provide a criterion for the crisis instead the convergence of the time evolution of probability densities to the invariant measure governed by the semigroup of transfer operators is expected to slow down at the approach of the crisis such convergence is described by the eigenvalues of the generator of this semigroup which can be divided into two families referred to as the stable and unstable ruelle pollicott resonances respectively the former describes the convergence of densities to the attractor or escape from a repeller and is estimated from many short time series sampling the state space the latter is responsible for the decay of correlations or mixing and can be estimated from a long times series invoking ergodicity it is found numerically for the lorenz flow that the stable resonances do approach the imaginary axis during the crisis as is indicative of the loss of global stability of the attractor on the other hand the unstable resonances and a fortiori the decay of correlations do not flag the proximity of the crisis thus questioning the usual design of early warning indicators of boundary crises of chaotic attractors and the applicability of response theory close to such crises

liouvillian dynamics of the hopf bifurcation two dimensional vector fields undergoing a hopf bifurcation are studied in a liouville equation approach the liouville equation rules the time evolution of statistical ensembles of trajectories issued from random initial conditions but evolving under the deterministic dynamics the time evolution of the probability densities of such statistical ensembles can be decomposed in terms of the spectrum of the resonances i e the relaxation rates of the liouvillian operator or the related frobenius perron operator the spectral decomposition of the liouvillian operator is explicitly constructed before at and after the hopf bifurcation because of the emergence of time oscillations near the hopf bifurcation the resonance spectrum turns out to be complex and defined by both relaxation rates and oscillation frequencies the resonance spectrum is discrete far from the bifurcation and becomes continuous at the bifurcation this continuous spectrum is caused by the critical slowing down of the oscillations occurring at the hopf bifurcation and it leads to power law relaxation as 1 square root of t of the probability densities and statistical averages at long times t infinity moreover degeneracy in the resonance spectrum is shown to yield a jordan block structure in the spectral decomposition

classical nonlinear response of a chaotic system ii langevin dynamics and spectral decomposition the spectrum of a strongly chaotic system consists of discrete complex ruelle pollicott rp resonances we interpret the rp resonances as eigenstates and eigenvalues of the fokker planck operator obtained by adding an infinitesimal diffusion term to the first order liouville operator we demonstrate how the deterministic expression for the linear response is reproduced in the limit of vanishing noise for the second order response function we establish an equivalence of the spectral decomposition in the limit of vanishing noise and the long time asymptotic expansion in the deterministic case

some applications of hausdorff dimension inequalities for ordinary differential equations upper bounds are obtained for the hausdorff dimension of compact invariant sets of ordinary differential equations which are periodic in the independent variable from these are derived sufficient conditions for dissipative analytic n dimensional periodic differential equations to have only a finite number of periodic solutions for autonomous equations the same conditions ensure that each bounded semi orbit converges to a critical point these results yield some information about the lorenz equation and the forced duffing equation

spectral signature of the pitchfork bifurcation liouville equation approach the time evolution of probability densities of one dimensional nonlinear vector fields is studied using a liouville equation approach it is shown that the liouville operator admits a discrete spectrum of eigenvalues of decaying type if the vector field is far from bifurcation the associated right and left eigenvectors are explicitly constructed for simple models and shown to be distributions rather than regular functions on the other hand the spectrum of the liouville operator may become continuous at the bifurcation point a phenomenon illustrated explicitly in the paper in the case of the pitchfork bifurcation the relationship between the spectral decompositions of the liouville and of the fokker planck equations is discussed in particular the spectral decompositions constructed here for the liouville equation are obtained as the noiseless limit of the well known spectral decompositions of the fokker planck equation of the associated stochastic process

a spectral characterization of nonlinear normal modes belgian network dysco dynamical systems control and optimization funded by the interuniversity attraction poles programme initiated by the belgian science policy office

lyapunov measure for almost everywhere stability this paper is concerned with the analysis and computational methods for verifying global stability of an attractor set of a nonlinear dynamical system based upon a stochastic representation of deterministic dynamics a lyapunov measure is proposed for these purposes this measure is shown to be a stochastic counterpart of stability transience just as an invariant measure is a counterpart of the attractor recurrence it is a dual of the lyapunov function and is useful for the study of more general weaker and set wise notions of stability in addition to the theoretical framework constructive methods for computing approximations to the lyapunov measures are presented these methods are based upon set oriented numerical approaches several equivalent descriptions including a series formula and a system of linear inequalities are provided for computational purposes these descriptions allow one to carry over the intuition from the linear case with stable equilibrium to nonlinear systems with globally stable attractor sets finally in certain cases the exact relationship between lyapunov functions and lyapunov measures is also given

a fast two dimensional entropic thresholding algorithm abstract two dimensional 2d entropic thresholding is one of the important thresholding techniques for image segmentation the selection of the global threshold vector is usually through a maximin optimization procedure a fast two phase 2d entropic thresholding algorithm is proposed in order to reduce the computation time first 9l 2 3 candidate threshold vectors are estimated from a quantized image of the original the global threshold vector is then obtained by checking candidates only the optimal computation complexity is o l 2 3 by quantizing the gray level into l 2 3 levels experimental results show that the processing time of each image is reduced from more than 2 h to about 2 min the required memory space is also greatly reduced

reaction diffusion theory in the presence of an attractive harmonic potential problems involving the capture of a moving entity by a trap occur in a variety of physical situations the moving entity being an electron an excitation an atom a molecule a biological object such as a receptor cluster a cell or even an animal such as a mouse carrying an epidemic theoretical considerations have almost always assumed that the particle motion is translationally invariant we study here the case when that assumption is relaxed in that the particle is additionally subjected to a harmonic potential this tethering to a center modifies the reaction diffusion phenomenon using a smoluchowski equation to describe the system we carry out a study which is explicit in one dimension but can be easily extended for arbitrary dimensions interesting features emerge depending on the relative location of the trap the attractive center and the initial placement of the diffusing particle

a differential lyapunov framework for contraction analysis lyapunov s second theorem is an essential tool for stability analysis of differential equations the paper provides an analog theorem for incremental stability analysis by lifting the lyapunov function to the tangent bundle the lyapunov function endows the state space with a finsler structure incremental stability is inferred from infinitesimal contraction of the finsler metrics through integration along solutions curves

a reduced order dg formulation based on pod method for the time domain maxwell s equations in dispersive media abstract in this work a proper orthogonal decomposition pod method is applied to time domain maxwell s equations coupled to a drude dispersion model which are discretized in space by a discontinuous galerkin dg method an auxiliary differential equation ade method is used to represent the constitutive relation for the dispersive medium a pod dgtd formulation with lower dimension and sufficiently high accuracy is established together with the description of the pod reduced order basis its construction from a snapshot set and its application to the solution of the time domain maxwell s equations the overall goal is to reduce the computational time while maintaining an acceptable level of accuracy in order to obtain an efficient time domain solver to be used as a starting point for an optimization strategy we provide results from numerical experiments for two dimensional problems that illustrate the capabilities of the proposed pod dgtd formulation and assess its efficiency

a skeletonization algorithm by maxima tracking on euclidean distance transform abstract a simple and efficient algorithm using the maxima tracking approach on euclidean distance transform to detect skeleton points is presented the advantages of the skeleton obtained are 1 connectivity preservation 2 single pixel in width and 3 its locations as close as to the most symmetrical axes besides the condition of the least slope change of skeleton is used to ensure the fairness of the digital medial axes with the least effort the algorithm can be modified to eliminate non significant short skeletal branches originating from the object contour while the critical shape informative medial axes are preserved

global stability analysis using the eigenfunctions of the koopman operator we propose a novel operator theoretic framework to study global stability of nonlinear systems based on the spectral properties of the so called koopman operator our approach can be regarded as a natural extension of classic linear stability analysis to nonlinear systems the main results establish the necessary and sufficient relationship between the existence of specific eigenfunctions of the koopman operator and the global stability property of hyperbolic fixed points and limit cycles these results are complemented with numerical methods which are used to estimate the region of attraction of the fixed point or to prove in a systematic way global stability of the attractor within a given region of the state space

dynamics of coarse grained variables in spatially extended systems abstract some new ways to probe the dynamics of spatially extended systems giving rise to spatio temporal chaos are explored coarse grained observables suitable for describing large scale behavior are introduced by locally averaging the original variables over a spatial region and their stability spatial correlation function and statistical properties are compared to those of the original variables in general these observables do not obey closed form equations truncation schemes in which high order fourier modes or variables other than the coarse grained ones are discarded are developed and their sensitivity to the parameters is assessed

trace formula for noisy flows a trace formula is derived for the fokker planck equation associated with ito stochastic differential equations describing noisy time continuous nonlinear dynamical systems in the weak noise limit the trace formula provides estimations of the eigenvalues of the fokker planck operator on the basis of the pollicott ruelle resonances of the noiseless deterministic system which is assumed to be non bifurcating at first order in the noise amplitude the effect of noise on a periodic orbit is given in terms of the period and the derivative of the period with respect to the pseudo energy of the onsager machlup freidlin wentzell scheme