decentralised consensus based formation tracking of multiple differential drive robots abstractthis article investigates the control problem for formation tracking of multiple nonholonomic robots under distributed manner which means each robot only needs local information exchange a class of general state and input transform is introduced to convert the formation tracking issue of multi robot systems into the consensus like problem with time varying reference the distributed observer based protocol with nonlinear dynamics is developed for each robot to achieve the consensus tracking of the new system which namely means a group of nonholonomic mobile robots can form the desired formation configuration with its centroid moving along the predefined reference trajectory the finite time stability of observer and control law is analysed rigorously by using the lyapunov direct method algebraic graph theory and matrix analysis numerical examples are finally provided to illustrate the effectiveness of the theory results proposed in this paper

anomalous scaling of a passive scalar advected by the synthetic compressible flow abstract the field theoretic renormalization group and operator product expansion are applied to the problem of a passive scalar advected by a gaussian nonsolenoidal velocity field with finite correlation time in the presence of large scale anisotropy the energy spectrum of the velocity in the inertial range has the form e k k1 e and the correlation time at the wave number k scales as k 2 it is shown that depending on the values of the exponents e and the model exhibits various types of inertial range scaling regimes with nontrivial anomalous exponents explicit asymptotic expressions for the structure functions and other correlation functions are obtained they are represented by superpositions of power laws with nonuniversal amplitudes and universal independent of the anisotropy anomalous exponents calculated to the first order in e and in any space dimension these anomalous exponents are determined by the critical dimensions of tensor composite operators built of the scalar gradients and exhibit a kind of hierarchy related to the degree of anisotropy the less is the rank the less is the dimension and consequently the more important is the contribution to the inertial range behavior the leading terms of the even odd structure functions are given by the scalar vector operators the anomalous exponents depend explicitly on the degree of compressibility

zero beta function for a model of diffusion in potential random field a rigorous perturbative proof is given of both the renormalisability of the field theoretic model of diffusion in potential random field and the triviality of the renormalisation group beta function

information transfer between dynamical system components we present a rigorous formalism of information transfer for systems with dynamics fully known this follows from an accurate classification of the mechanisms for the entropy change of one component into a self evolution plus a transfer from the other component the formalism applies to both continuous flows and discrete maps the resulting transfer measure possesses a property of asymmetry and is qualitatively consistent with the classical measures it is further validated with the baker transformation and the henon map

logarithmic violation of scaling in strongly anisotropic turbulent transfer of a passive vector field inertial range asymptotic behavior of a vector e g magnetic field passively advected by a strongly anisotropic turbulent flow is studied by means of the field theoretic renormalization group and the operator product expansion the advecting velocity field is gaussian not correlated in time with the pair correlation function of the form t t k d 1 where k k and k is the component of the wave vector perpendicular to the distinguished direction direction of the flow the d dimensional generalization of the ensemble introduced by avellaneda and majda commun math phys 131 381 1990 the stochastic advection diffusion equation for the transverse divergence free vector field includes as special cases the kinematic dynamo model for magnetohydrodynamic turbulence and the linearized navier stokes equation in contrast to the well known isotropic kraichnan s model where various correlation functions exhibit anomalous scaling behavior with infinite sets of anomalous exponents here the dependence on the integral turbulence scale l has a logarithmic behavior instead of powerlike corrections to ordinary scaling determined by naive canonical dimensions the anomalies manifest themselves as polynomials of logarithms of l the key point is that the matrices of scaling dimensions of the relevant families of composite operators appear nilpotent and cannot be diagonalized the detailed proof of this fact is given for the correlation functions of arbitrary order

mean field approximation and a small parameter in turbulence theory numerical and physical experiments on two dimensional 2d turbulence show that the differences of transverse components of velocity field are well described by gaussian statistics and kolmogorov scaling exponents in this case the dissipation fluctuations are irrelevant in the limit of small viscosity in general one can assume the existence of a critical space dimensionality d d c at which the energy flux and all odd order moments of velocity difference change sign and the dissipation fluctuations become dynamically unimportant at dl d c the flow can be described by the mean field theory leading to the observed gaussian statistics and kolmogorov scaling of transverse velocity differences it is shown that in the vicinity of d d c the ratio of the relaxation and translation characteristic times decreases to zero thus giving rise to a small parameter of the theory the expressions for pressure and dissipation contributions to the exact equation for the generating function of transverse velocity differences are derived in the vicinity of d d c the resulting equation describes experimental data on two dimensional turbulence and demonstrates the onset of intermittency as d ensuremath d c g0 and r stackrel ensuremath rightarrow l 0 in three dimensional flows in close agreement with experimental data in addition some exact relations between correlation functions of velocity differences are derived it is also predicted that the single point probability density function of transverse velocity components in developing as well as in the large scale stabilized two dimensional turbulence is a gaussian

distributed formation control of multi agent systems using complex laplacian the paper concentrates on the fundamental coordination problem that requires a network of agents to achieve a specific but arbitrary formation shape a new technique based on complex laplacian is introduced to address the problems of which formation shapes specified by inter agent relative positions can be formed and how they can be achieved with distributed control ensuring global stability concerning the first question we show that all similar formations subject to only shape constraints are those that lie in the null space of a complex laplacian satisfying certain rank condition and that a formation shape can be realized almost surely if and only if the graph modeling the inter agent specification of the formation shape is 2 rooted concerning the second question a distributed and linear control law is developed based on the complex laplacian specifying the target formation shape and provable existence conditions of stabilizing gains to assign the eigenvalues of the closed loop system at desired locations are given moreover we show how the formation shape control law is extended to achieve a rigid formation if a subset of knowledgable agents knowing the desired formation size scales the formation while the rest agents do not need to re design and change their control laws

self similar decay in the kraichnan model of a passive scalar we study the two point correlation function of a freely decaying scalar in kraichnan s model of advection by a gaussian random velocity field that is stationary and white noise in time but fractional brownian in space with roughness exponent 0 2 appropriate to the inertial convective range of the scalar we find all self similar solutions by transforming the scaling equation to kummer s equation it is shown that only those scaling solutions with scalar energy decay exponent a d 1 are statistically realizable where d is space dimension and 2 an infinite sequence of invariants jp p 0 1 2 is pointed out where j0 is corrsin s integral invariant but the higher invariants appear to be new we show that at least one of the invariants j0 or j1 must be nonzero possibly infinite for realizable initial data initial datum with a finite nonzero invariant the first being jp converges at long times to a scaling solution p with a d p p 0 1 the latter belongs to an exceptional series of self similar solutions with stretched exponential decay in space however the domain of attraction includes many initial data with power law decay when the initial datum has all invariants zero or infinite and also it exhibits power law decay then the solution converges at long times to a nonexceptional scaling solution with the same power law decay these results support a picture of a two scale decay with breakdown of self similarity for a range of exponents d a d 2 analogous to what has recently been found in the decay of burgers turbulence

passive advection of a vector field anisotropy finite correlation time exact solution and logarithmic corrections to ordinary scaling in this work we study the generalization of the problem considered in phys rev e 91 013002 2015 to the case of finite correlation time of the environment velocity field the model describes a vector e g magnetic field passively advected by a strongly anisotropic turbulent flow inertial range asymptotic behavior is studied by means of the field theoretic renormalization group and the operator product expansion the advecting velocity field is gaussian with finite correlation time and preassigned pair correlation function due to the presence of distinguished direction n all the multiloop diagrams in this model vanish so that the results obtained are exact the inertial range behavior of the model is described by two regimes the limits of vanishing or infinite correlation time that correspond to the two nontrivial fixed points of the rg equations their stability depends on the relation between the exponents in the energy spectrum e k 1 and the dispersion law k 2 in contrast to the well known isotropic kraichnan s model where various correlation functions exhibit anomalous scaling behavior with infinite sets of anomalous exponents here the corrections to ordinary scaling are polynomials of logarithms of the integral turbulence scale l

scalings of scalar structure functions in a velocity field with coherent vortical structures in planar turbulence modeled as an isotropic and homogeneous collection of two dimensional noninteracting compact vortices the structure functions s p r of a statistically stationary passive scalar field have the following scaling behavior in the limit where the peclet number pe s n r const in r l pe 1 3 for l pe 1 3 r l s p r r l pe 1 3 6 1 d for l pe 1 2 r l pe 1 3 where l is a large scale and d is the fractal codimension of the spiral scalar structures generated by the vortices d 2 3 note that l pe 1 2 is the scalar taylor microscale that stems naturally from our analytical treatment of the advection diffusion equation the essential ingredients of our theory are the locality of interscale transfer and lundgren s time average assumption a phenomenological theory explicitly based only on these two ingredients reproduces our results and a generalization of this phenomenology to spatially smooth chaotic flows yields k ln k 1 generalized power spectra for the advected scalar fields

heat equilibrium distribution in a turbulent flow we consider a shear flow of a scale invariant gaussian random velocity field that does not depend on the coordinates in the direction of the flow we investigate a heat advection coming from a gaussian random homogeneous source we discuss a relaxation at large time of a temperature distribution determined by the forced advection diffusion equation we represent the temperature correlation functions by means of the feynman kac formula jensen inequalities are applied for lower and upper bounds on the correlation functions we show that at finite time there is no velocity dependence of long range temperature correlations low momentum asymptotics in the direction of the flow but the equilibrium heat distribution has large distance correlations low momentum behavior with an index depending on the scaling index of the random flow and of the index of the random forcing if the velocity has correlations growing with the distance a turbulent flow then the large distance correlations depend in a crucial wa

renormalization group operator product expansion and anomalous scaling in models of turbulent advection recent progress on the anomalous scaling in models of turbulent heat and mass transport is reviewed with the emphasis on the approach based on the field theoretic renormalization group rg and operator product expansion ope in that approach the anomalous scaling is established as a consequence of the existence in the corresponding field theoretic models of an infinite number of dangerous composite fields operators with negative critical dimensions which are identified with the anomalous exponents this allows one to calculate the exponents in a systematic perturbation expansion similar to the e expansion in the theory of critical phenomena the rg and ope approach is presented in a self contained way for the example of a passive scalar field temperature concentration of an impurity etc advected by a self similar gaussian velocity ensemble with vanishing correlation time the so called kraichnan s rapid change model where the anomalous exponents are known up to order o e3 effects of anisotropy compressibility and the correlation time of the velocity field are discussed passive advection by non gaussian velocity field governed by the stochastic navier stokes equation and passively advected vector e g magnetic fields are considered

anomalous scaling in the n point functions of a passive scalar a recent analysis of the four point correlation function of the passive scalar advected by a time decorrelated random flow is extended to the n point case it is shown that all stationary state inertial range correlations are dominated by homogeneous zero modes of singular operators describing their evolution we compute analytically the zero modes governing the n point structure functions and the anomalous dimensions corresponding to them to the linear order in the scaling exponent of the two point function of the advecting velocity field the implications of these calculations for the dissipation correlations are discussed textcopyright 1996 the american physical society

formation stability analysis of unmanned multi vehicles under interconnection topologies in this paper the overall formation stability of an unmanned multi vehicle is mathematically presented under interconnection topologies a novel definition of formation error is first given and followed by the proposed formation stability hypothesis based on this hypothesis a unique extension decomposition aggregation scheme is then employed to support the stability analysis for the overall multi vehicle formation under a mesh topology it is proved that the overall formation control system consisting of n number of nonlinear vehicles is not only asymptotically stable but also exponentially stable in the sense of lyapunov within a neighbourhood of the desired formation this technique is shown to be applicable for a mesh topology but is equally applicable for other topologies a simulation study of the formation manoeuvre of multiple aerosonde uavs unmanned aerial vehicles in 3 d space is finally carried out verifying the achieved formation stability result

a rigorous formalism of information transfer between dynamical system components i discrete mapping we put the concept of information transfer on a rigorous footing and establish for it a formalism within the framework of discrete maps the resulting transfer measure possesses a property of directionality or transfer asymmetry as emphasized by schreiber t schreiber measuring information transfer phys rev lett 85 2 2000 461 it also verifies the transfer measure for two dimensional systems which was obtained by liang and kleeman x s liang r kleeman information transfer between dynamical system components phys rev lett 95 24 2005 244101 through a different avenue connections to classical formalisms are explored and applications presented we find that in the context of the baker transformation there is always information flowing from the stretching direction to the folding direction while no transfer occurs in the opposite direction we also find that within the henon map system the transfer from the quadratic component to the linear component is of a simple form as expected on physical grounds this latter result is unique to our formalism

structure of the three point correlation function of a passive scalar in the presence of a mean gradient the three point correlation function of a passive scalar advected by a random incompressible velocity field in the presence of a mean gradient is investigated by means of phenomenological hopf equations numerical solutions are provided in the case where the velocity is gaussian white in time and with a power law in space v stackrel ensuremath rightarrow r ensuremath v stackrel ensuremath rightarrow 0 2 ensuremath sim r 2 ensuremath ensuremath epsilon and in the model introduced by shraiman and siggia c r acad sci 321 279 1995 anomalous scaling exponents are found in both models the numerics agrees with all the available analytic perturbative results in addition the angular dependence of the correlation function is explicitly determined in the batchelor limit of random advection by a smooth velocity field the exponent is found to remain very close to 1 as found experimentally in this limit the three point correlation function is found to be very well represented away from collinearity by an explicit integral representation

anomalous scaling nonlocality and anisotropy in a model of the passively advected vector field a model of the passive vector quantity advected by the gaussian velocity field with the covariance ensuremath propto ensuremath delta t ensuremath t ensuremath mathbf x ensuremath mathbf x ensuremath ensuremath varepsilon is studied the effects of pressure and large scale anisotropy are discussed the inertial range behavior of the pair correlation function is described by an infinite family of scaling exponents which satisfy exact transcendental equations derived explicitly in d dimensions by means of the functional techniques the exponents are organized in a hierarchical order according to their degree of anisotropy with the spectrum unbounded from above and the leading minimal exponent coming from the isotropic sector this picture extends to higher order correlation functions like in the scalar model the second order structure function appears nonanomalous and is described by the simple dimensional exponent s 2 ensuremath propto r 2 ensuremath ensuremath varepsilon for the higher order structure functions s 2n ensuremath propto r n 2 ensuremath ensuremath varepsilon ensuremath delta n the anomalous scaling behavior is established as a consequence of the existence in the corresponding operator product expansions of dangerous composite operators whose negative critical dimensions determine the anomalous exponents ensuremath delta n l0 a close formal resemblance of the model with the stirred navier stokes equation reveals itself in the mixing of relevant operators and is the main motivation of the paper using the renormalization group the anomalous exponents are calculated in the o ensuremath varepsilon approximation in large d dimensions for the even structure functions up to the twelfth order

calculation of the anomalous exponents in the rapid change model of passive scalar advection to order epsilon 3 the field theoretic renormalization group and operator product expansion are applied to the model of a passive scalar advected by the gaussian velocity field with zero mean and correlation function approximately equal to delta t t k d epsilon inertial range anomalous exponents identified with the critical dimensions of various scalar and tensor composite operators constructed of the scalar gradients are calculated within the epsilon expansion to order epsilon 3 three loop approximation including the exponents in anisotropic sectors the main goal of the paper is to give the complete derivation of this third order result and to present and explain in detail the corresponding calculational techniques the character and convergence properties of the epsilon expansion are discussed the improved inverse epsilon expansion is proposed and the comparison with the existing nonperturbative results is given

exact results for conditional means of a passive scalar in certain statistically homogeneous flows for a passive scalar t r t randomly advected by a statistically homogeneous flow the probability density function pdf of its fluctuation can in general be expressed in terms of two conditional means 2 t t and t 2 t we find that in some special cases either one of the two conditional means can be obtained explicitly from the equation of motion in the cases when there is no external source and that the normalized fluctuation reaches a steady state or when a steady state results from a negative damping 2 t t t 2 t 2 t the linearity of the conditional mean in these cases follows directly from the fact that the advection equation of a passive scalar is linear on the other hand when the scalar is supported by a homogeneous white in time external source t 2 t t 2

asymptotic theory of multidimensional chaos a delay differential equation u t u t f u t 1 0 t and its generalization are investigated in the limit 0 when the attractor s dimension increases infinitely it is shown that a number of statistical characteristics are asymptotically independent of as for the attractor it can be regarded as a direct product ofo 1 equivalent subattractors their statistical characteristics being asymptotically independent of the results enable one to predict some characteristics of the attractor with fractal dimensiond 1 for the case 1 when they are inaccessible numerically the approach developed seems to be applicable for a wide class of spatiotemporal systems

anomalous scaling of passive scalar fields advected by the navier stokes velocity ensemble effects of strong compressibility and large scale anisotropy the field theoretic renormalization group and the operator product expansion are applied to two models of passive scalar quantities the density and the tracer fields advected by a random turbulent velocity field the latter is governed by the navier stokes equation for compressible fluid subject to external random force with the covariance t t k 4 d y where d is the dimension of space and y is an arbitrary exponent the original stochastic problems are reformulated as multiplicatively renormalizable field theoretic models the corresponding renormalization group equations possess infrared attractive fixed points it is shown that various correlation functions of the scalar field its powers and gradients demonstrate anomalous scaling behavior in the inertial convective range already for small values of y the corresponding anomalous exponents identified with scaling critical dimensions of certain composite fields operators in the quantum field terminology can be systematically calculated as series in y the practical calculation is performed in the leading one loop approximation including exponents in anisotropic contributions it should be emphasized that in contrast to gaussian ensembles with finite correlation time the model and the perturbation theory presented here are manifestly galilean covariant the validity of the one loop approximation and comparison with gaussian models are briefly discussed

two different rapid decorrelation in time limits for turbulent diffusion a turbulent diffusion model in which the velocity field is gaussian and rapidly decorrelating in time grdt has been widely used recently in an endeavor to understand the emergence of anomalous scaling behavior of physical fields in fluid mechanics from the underlying stochastic partial differential equations the utility of the grdt model is the fact that correlation functions of the passive scalar field solve closed partial differential equations the usual moment closure obstacle is averted we study here the sense in which the grdt model describes turbulent diffusion by a general non gaussian velocity field with nontrivial temporal structure in the limit in which the correlation time of the velocity field is taken to zero when the velocity field is rescaled in a particular manner in this rapid decorrelation limit then a limit theorem of khas minskii indeed shows that the passive scalar statistics are described asymptotically by the grdt model for a broad class of velocity field models we provide however an explicit example of a poisson blob model velocity field which has two different well defined rapid decorrelation in time limits in one the passive scalar correlation functions converge to those of the grdt model and in the other they converge to a distinct nontrivial limit in which the correlation functions do not solve closed pde s we provide both mathematical and heuristic explanations for the differences between these two limits the conclusion is that the grdt model provides a universal description of the rapid decorrelation in time limit of general non gaussian velocity field models only when the velocity field is rescaled in a particular manner during the limit process

anomalous scaling of a passive scalar in the presence of strong anisotropy field theoretic renormalization group and the operator product expansion are applied to a model of a passive scalar quantity straight theta t x advected by the gaussian strongly anisotropic velocity field with the covariance infinity delta t t x x epsilon inertial range anomalous scaling behavior is established and explicit asymptotic expressions for the structure functions s n r identical with are obtained they are represented by superpositions of power laws the corresponding anomalous exponents which depend explicitly on the anisotropy parameters are calculated to the first order in epsilon in any space dimension d in the limit of vanishing anisotropy the exponents are associated with tensor composite operators built of the scalar gradients and exhibit a kind of hierarchy related to the degree of anisotropy the less is the rank the less is the dimension and consequently the more important is the contribution to the inertial range behavior the leading terms of the even odd structure functions are given by the scalar vector operators for the finite anisotropy the exponents cannot be associated with individual operators which are essentially mixed in renormalization but the aforementioned hierarchy survives for all the cases studied the second order structure function s2 is studied in more detail using the renormalization group and zero mode techniques the corresponding exponents and amplitudes are calculated within the perturbation theories in epsilon 1 d and in the anisotropy parameters if the anisotropy of the velocity is strong enough the skewness factor s 3 s 3 2 2 increases going down towards the depth of the inertial range the higher order odd ratios increase even if the anisotropy is weak

statistics of fourier modes in a turbulent flow fourier series are often used to discuss the properties of a homogeneous turbulent field we investigate the statistics of fourier modes of the turbulent velocity field and of a passive scalar the statistics of individual fourier modes is known to be gaussian when the size of the system l is much larger that the integral correlation size l 0 the case where the integral size is of the order of the system size l approximately l 0 is studied by direct numerical simulations in the range 20 or approximately rlambda or approximately 80 at a given rlambda we find that the probabilities of large fluctuations become larger when the wave number increases in qualitative agreement with the notion of intermittency as the reynolds number increases however the probability density functions become closer to gaussian in sharp contrast with the behavior of velocity increments we also show that in a simple model of cascade the fourier series decomposition is not appropriate to capture intermittency effects last we discuss other issues related to our results

pattern dynamics in spatiotemporal chaos pattern selection diffusion of defect and pattern competition intermettency abstract a class of coupled map lattices is investigated as a model for the spatiotemporal chaos as the parameters are changed phase transitions among the patterns are observed pattern selection through the chaotic motion of domain is observed chaos is suppressed by the selection selection of the zigzag pattern is studied where localized defects which separate the two anti phase domains are found they change chaotically in time and move randomly in space the diffusion of defect is related with its chaotic motion collapse of zigzag pattern and the defect turbulence are studied in connection with the crisis in high dimensional dynamical systems pattern selection with longer wavelengths is found in the stronger coupling regime transition from pattern selection phase to fully developed turbulence occurs via the intermittency selective flicker like noise and the pareto distribution of lifetime of pattern are observed at the intermittency regime for the quantitative characterization of each phase pattern distributions pattern transition matrix static and dynamical entropies lifetime distribution of pattern spatiotemporal power spectra and lyapunov spectra are calculated phase transition from the pattern selection to fully developed turbulence is investigated with the use of these quantifiers where some order parameters and critical indeces are introduced

anomalous scaling in two and three dimensions for a passive vector field advected by a turbulent flow a model of the passive vector field advected by the uncorrelated in time gaussian velocity with power like covariance is studied by means of the renormalization group and the operator product expansion the structure functions of the admixture demonstrate essential power like dependence on the external scale in the inertial range the case of an anomalous scaling the method of finding independent tensor invariants in the cases of two and three dimensions is proposed to eliminate linear dependencies between the operators entering into the operator product expansions of the structure functions the constructed operator bases which include the powers of the dissipation operator and the enstrophy operator provide the possibility of calculating the exponents of the anomalous scaling

sailing the deep blue sea of decaying burgers turbulence we study lagrangian trajectories and scalar transport statistics in decaying burgers turbulence we choose velocity fields solutions of the inviscid burgers equation whose probability distributions are specified by kida s statistics they are time correlated and neither time reversal invariant nor gaussian we discuss in some detail the effect of shocks on trajectories and transport equations we derive the inviscid limit of these equations using a formalism of operators localized on shocks we compute the probability distribution functions of the trajectories although they do not define markov processes as physically expected these trajectories are statistically well defined but collapse with probability one at infinite time we point out that the advected scalars enjoy inverse energy cascades we also make a few comments on the connection between our computations and persistence problems

on local isotropy of passive scalars in turbulent shear flows an assessment of local isotropy and universality in high reynolds number turbulent flows is presented the emphasis is on the behaviour of passive scalar fields advected by turbulence but a brief review of the relevant facts is given for the turbulent motion itself experiments suggest that local isotropy is not a natural concept for scalars in shear flows except perhaps at such extreme reynolds numbers that are of no practical relevance on earth yet some type of scaling exists even at moderate reynolds numbers the relation between these two observations is a theme of this paper

exact calculation of the anomalous dimension of the diffusion coefficient for a model of a random walk in a random potential renormalisation of a model of classical diffusion in a random potential is analysed it is shown that at two dimensions the one loop expression for the anomalous dimension of the diffusion coefficient is perturbatively exact leading to the coupling dependent value 1 nu 2 g 4 pi of the exponent nu

turbulence without pressure we develop exact field theoretic methods to treat turbulence when the effect of pressure is negligible we find explicit forms of certain probability distributions demonstrate that the breakdown of galilean invariance is responsible for intermittency and establish the operator product expansion we also indicate how the effects of pressure can be turned on perturbatively

anomalous scaling regimes of a passive scalar advected by the synthetic velocity field the field theoretic renormalization group rg is applied to the problem of a passive scalar advected by the gaussian self similar velocity field with finite correlation time and in the presence of an imposed linear mean gradient the energy spectrum in the inertial range has the form e k ensuremath propto k 1 ensuremath ensuremath varepsilon and the correlation time at the wave number k scales as k ensuremath 2 ensuremath eta it is shown that depending on the values of the exponents ensuremath varepsilon and ensuremath eta the model in the inertial convective range exhibits various types of scaling regimes associated with the infrared stable fixed points of the rg equations diffusive type regimes for which the advection can be treated within ordinary perturbation theory and three nontrivial convection type regimes for which the correlation functions exhibit anomalous scaling behavior the explicit asymptotic expressions for the structure functions and other correlation functions are obtained the anomalous exponents determined by the scaling dimensions of the scalar gradients are calculated to the first order in ensuremath varepsilon and ensuremath eta in any space dimension for the first nontrivial regime the anomalous exponents are the same as in the rapid change version of the model for the second they are the same as in the model with time independent frozen velocity field in these regimes the anomalous exponents are universal in the sense that they depend only on the exponents entering into the velocity correlator for the last regime the exponents are nonuniversal they can depend also on the amplitudes however the nonuniversality can reveal itself only in the second order of the rg expansion a brief discussion of the passive advection in the non gaussian velocity field governed by the nonlinear stochastic navier stokes equation is also given

anisotropic nonperturbative zero modes for passively advected magnetic fields an analytic assessment of the role of anisotropic corrections to the isotropic anomalous scaling exponents is given for the d dimensional kinematic magnetohydrodynamics problem in the presence of a mean magnetic field the velocity advecting the magnetic field changes very rapidly in time and scales with a positive exponent xi inertial range anisotropic contributions to the scaling exponents zeta j of second order magnetic correlations are associated with zero modes and have been calculated nonperturbatively for d 3 the limit xi 0 yields zeta j j 2 xi 2j 3 j 2 5j 4 2 4j 2 1 where j j or 2 is the order in the legendre polynomial decomposition applied to correlation functions conjectures on the fact that anisotropic components cannot change the isotropic threshold to the dynamo effect are also made

anomalous scaling of a passive vector advected by the navier stokes velocity field using the field theoretic renormalization group and the operator product expansion the model of a passive vector field a weak magnetic field in the framework of the kinematic mhd advected by the velocity field which is governed by the stochastic navier stokes equation with the gaussian random stirring force correlated in time and with the correlator proportional to k4 d 2e is investigated to the first order in e one loop approximation it is shown that the single time correlation functions of the advected vector field have anomalous scaling behavior and the corresponding exponents are calculated in the isotropic case as well as in the case with the presence of large scale anisotropy the hierarchy of the anisotropic critical dimensions is briefly discussed and the persistence of the anisotropy inside the inertial range is demonstrated on the behavior of the skewness and hyperskewness dimensionless ratios of correlation functions as functions of the reynolds number re it is shown that even though the present model of a passive vector field advected by the realistic velocity field is mathematically more complicated than on one hand the corresponding models of a passive vector field advected by synthetic gaussian velocity fields and on the other hand than the corresponding model of a passive scalar quantity advected by the velocity field driven by the stochastic navier stokes equation the final one loop approximate asymptotic scaling behavior of the single time correlation or structure functions of the advected fields of all models are defined by the same anomalous dimensions up to normalization

whitening as a tool for estimating mutual information in spatiotemporal data sets we address the issue of inferring the connectivity structure of spatially extended dynamical systems by estimation of mutual information between pairs of sites the well known problems resulting from correlations within and between the time series are addressed by explicit temporal and spatial modelling steps which aim at approximately removing all spatial and temporal correlations i e at whitening the data such that it is replaced by spatiotemporal innovations this approach provides a link to the maximum likelihood method and for appropriately chosen models removes the problem of estimating probability distributions of unknown possibly complicated shape a parsimonious multivariate autoregressive model based on nearest neighbour interactions is employed mutual information can be reinterpreted in the framework of dynamical model comparison i e likelihood ratio testing since it is shown to be equivalent to the difference of the log likelihoods of coupled and uncoupled models for a pair of sites and a parametric estimator of mutual information can be derived we also discuss within the framework of model comparison the relationship between the coefficient of linear correlation and mutual information the practical application of this methodology is demonstrated for simulated multivariate time series generated by a stochastic coupled map lattice the parsimonious modelling approach is compared to general multivariate autoregressive modelling and to independent component analysis ica

geometric analysis of the formation problem for autonomous robots in the formation control problem for autonomous robots a distributed control law steers the robots to the desired target formation a local stability result of the target formation can be derived by methods of linearization and center manifold theory or via a lyapunov based approach besides the target formation the closed loop dynamics of the robots feature various other undesired invariant sets such as nonrigid formations this note addresses a global stability analysis of the closed loop formation control dynamics we pursue a differential geometric approach and derive purely algebraic conditions for local stability of invariant embedded submanifolds these theoretical results are then applied to the well known example of a cyclic triangular formation and result in instability of all invariant sets other than the target formation

turbulent compressible fluid renormalization group analysis scaling regimes and anomalous scaling of advected scalar fields we study a model of fully developed turbulence of a compressible fluid based on the stochastic navier stokes equation by means of the field theoretic renormalization group in this approach scaling properties are related to the fixed points of the renormalization group equations previous analysis of this model near the real world space dimension 3 identified a scaling regime n v antonov et al theor math phys 110 305 1997 tmphah0040 577910 1007 bf02630456 the aim of the present paper is to explore the existence of additional regimes which could not be found using the direct perturbative approach of the previous work and to analyze the crossover between different regimes it seems possible to determine them near the special value of space dimension 4 in the framework of double y and expansion where y is the exponent associated with the random force and 4 d is the deviation from the space dimension 4 our calculations show that there exists an additional fixed point that governs scaling behavior turbulent advection of a passive scalar density field by this velocity ensemble is considered as well we demonstrate that various correlation functions of the scalar field exhibit anomalous scaling behavior in the inertial convective range the corresponding anomalous exponents identified as scaling dimensions of certain composite fields can be systematically calculated as a series in y and all calculations are performed in the leading one loop approximation

normal and anomalous scaling of the fourth order correlation function of a randomly advected passive scalar for a delta correlated velocity field simultaneous correlation functions of a passive scalar satisfy closed equations we analyze the equation for the four point function to describe a solution completely one has to solve the matching problems at the scale of the source and at the diffusion scale we solve both the matching problems and thus find the dependence of the four point correlation function on the diffusion and pumping scale for large space dimensionality d it is shown that anomalous scaling appears in the first order of 1 d perturbation theory anomalous dimensions are found analytically both for the scalar field and for it s derivatives in particular for the dissipation field

spatio temporal structure in coupled map lattices two point correlations versus mutual information a one dimensional lattice of logistic maps is investigated in the case of strong nonlinearity and strong coupling although the dynamics may be classified as fully developed turbulence spatio temporal structure can be detected by computing time delayed mutual information and two point correlations the correlation is found to be superior in detecting weak structure an improved algorithm for mutual information is described

information transfer in continuous processes we discuss a recently proposed quantity called transfer entropy which uses time series data to measure the amount of information transferred from one process to another in order to understand its foundation merits and limitations we review some aspects of information theoretic functionals while for symbol sequences these measures have an intuitive interpretation their application to continuous state processes and in particular their estimation from finite data sets is problematic for mutual information finite length scale estimates converge from below and can thus be used to reject the assumption that the observed processes are independent however mutual information does not provide any directional information conversely transfer entropy does resolve the directionality of information exchange but no similar monotonic convergence seems to hold thus only in the case of zero transfer entropy in one direction we can reliably infer an asymmetry of the information exchange

anomalous scaling of a passive scalar advected by the navier stokes velocity field two loop approximation the field theoretic renormalization group and operator product expansion are applied to the model of a passive scalar quantity advected by a non gaussian velocity field with finite correlation time the velocity is governed by the navier stokes equation subject to an external random stirring force with the correlation function proportional to ensuremath delta t ensuremath t ensuremath k 4 ensuremath d ensuremath 2 ensuremath epsilon it is shown that the scalar field is intermittent already for small ensuremath epsilon its structure functions display anomalous scaling behavior and the corresponding exponents can be systematically calculated as series in ensuremath epsilon the practical calculation is accomplished to order ensuremath epsilon 2 two loop approximation including anisotropic sectors as for the well known kraichnan rapid change model the anomalous scaling results from the existence in the model of composite fields operators with negative scaling dimensions identified with the anomalous exponents thus the mechanism of the origin of anomalous scaling appears similar for the gaussian model with zero correlation time and the non gaussian model with finite correlation time it should be emphasized that in contrast to gaussian velocity ensembles with finite correlation time the model and the perturbation theory discussed here are manifestly galilean covariant the relevance of these results for real passive advection and comparison with the gaussian models and experiments are briefly discussed

stabilisation of infinitesimally rigid formations of multi robot networks this article considers the design of a formation control for multivehicle systems that uses only local information the control is derived from a potential function based on an undirected infinitesimally rigid graph that specifies the target formation a potential function is obtained from the graph from which a gradient control is derived under this controller the target formation becomes a manifold of equilibria for the multivehicle system it is shown that infinitesimal rigidity is a sufficient condition for local asymptotical stability of the equilibrium manifold a complete study of the stability of the regular polygon formation is presented and results for directed graphs are presented as well finally the controller is validated experimentally

bounded controllers for formation stabilization of mobile agents with limited sensing ranges a constructive method is presented to design bounded cooperative controllers that force a group of n mobile agents with limited sensing ranges to stabilize at a desired location and guarantee no collisions between the agents the control development is based on new general potential functions which attain the minimum value when the desired formation is achieved and are equal to infinity when a collision occurs a p times differential bump function is introduced and embedded into the potential functions to deal with the agent limited sensing ranges an alternative to barbalat s lemma is developed to analyze stability of the closed loop system extension to formation tracking is also addressed

lyapunov based approach to multiagent systems with switching jointly connected interconnection this note addresses a coordination problem of a multiagent system with jointly connected interconnection topologies neighbor based rules are adopted to realize local control strategies for these continuous time autonomous agents described by double integrators although the interagent connection structures vary over time and related graphs may not be connected a sufficient condition to make all the agents converge to a common value is given for the problem by a proposed lyapunov based approach and related space decomposition technique

renormalization group operator product expansion and anomalous scaling in a model of advected passive scalar field theoretical renormalization group methods are applied to the obukhov kraichnan model of a passive scalar advected by the gaussian velocity field with the covariance mathbf v t mathbf x mathbf v t ensuremath mathbf x ensuremath mathbf v t mathbf x mathbf v t ensuremath mathbf x ensuremath ensuremath propto ensuremath delta t ensuremath t ensuremath mathbf x ensuremath mathbf x ensuremath ensuremath varepsilon inertial range anomalous scaling for the structure functions and various pair correlators is established as a consequence of the existence in the corresponding operator product expansions of certain essential or dangerous composite operators powers of the local dissipation rate whose negative critical dimensions determine anomalous exponents the main technical result is the calculation of the anomalous exponents in the order ensuremath varepsilon 2 of the ensuremath varepsilon expansion generalization of the results obtained to the case of a slow velocity field is also presented

lyapunov analysis and information flow in coupled map lattices abstract lyapunov analysis for the coupled map lattices is presented the co moving lyapunov exponent is calculated which is related with the propagation of the disturbance in space the propagation speed agrees with the zero crossing point of the co moving lyapunov exponent the co moving mutual information flow is introduced which shows the selective transmission of the information at some speed lyapunov spectra and vectors are calculated spatial structures of the vectors are investigated possible analogy with the anderson localization problem is discussed

intermittency and anomalous scaling of passive scalars in any space dimension we establish exact inequalities for the structure function scaling exponents of a passively advected scalar in both the inertial convective and viscous convective ranges these inequalities involve the scaling exponents of the velocity structure functions and in a refined form an intermittency exponent of the convective range scalar flux they are valid for three dimensional navier stokes turbulence and satisfied within errors by present experimental data the inequalities also hold for any synthetic turbulent velocity statistics with a finite correlation in time we show that for time correlation exponents of the velocity smaller than the local turnover exponent the scalar spectral exponent is strictly less than that in kraichnan s phys rev lett 72 1016 1994 phys fluids 11 945 1968 j fluid mech 64 737 1974 62 305 1974 soluble rapid change model with velocity ensuremath delta correlated in time our results include as a special case an exponent inequality derived previously by constantin and procaccia nonlinearity 7 1045 1994 but with a more direct proof the inequalities in their simplest form follow from a kolmogorov type relation for the turbulent passive scalar valid in each space dimension d our improved inequalities are based upon a rigorous version of the refined similarity hypothesis for passive scalars these are compared with the relations implied by fusion rules hypothesized for scalar gradients textcopyright 1996 the american physical society

stirring up trouble multi scale mixing measures for steady scalar sources abstract the mixing efficiency of a flow advecting a passive scalar sustained by steady sources and sinks is naturally defined in terms of the suppression of bulk scalar variance in the presence of stirring relative to the variance in the absence of stirring these variances can be weighted at various spatial scales leading to a family of multi scale mixing measures and efficiencies we derive a priori estimates on these efficiencies from the advection diffusion partial differential equation focusing on a broad class of statistically homogeneous and isotropic incompressible flows the analysis produces bounds on the mixing efficiencies in terms of the peclet number a measure of the strength of the stirring relative to molecular diffusion we show by example that the estimates are sharp for particular source sink and flow combinations in general the high peclet number behavior of the bounds scaling exponents as well as prefactors depends on the structure and smoothness properties of and length scales in the scalar source and sink distribution the fundamental model of the stirring of a monochromatic source sink combination by the random sine flow is investigated in detail via direct numerical simulation and analysis the large scale mixing efficiency follows the upper bound scaling within a logarithm at high peclet number but the intermediate and small scale efficiencies are qualitatively less than optimal the peclet number scaling exponents of the efficiencies observed in the simulations are deduced theoretically from the asymptotic solution of an internal layer problem arising in a quasi static model

anomalous scaling for passively advected magnetic fields the kraichnan model 1 for passive scalar advection is by now considered a paradigm for intermittency problems a first systematic understanding of the scalar field scaling properties has been attained the crucial property of the model is that equal time correlation functions obey closed equations of motion this stems from the small correlation time of the velocity field advecting the scalar it has been recognized that a general mechanism for intermittency and anomalous scaling is the presence of nontrivial zero modes of the closed equations satisfied by the correlation functions 2 3 4 such mechanism can be illustrated in its simplest form by considering the following linear differential equation frac d 2 y r d r 2 frac a r frac dy r dr b frac y r r 2 f frac r l 1 where r varies on the positive axis y r is the unknown function a and b are two constants and f r l varies over distances of order l and is almost constant for r l in the physical situations l is the integral scale and the mechanisms maintaining the turbulence are supposed to act at this scale as for intermittency in navier stokes turbulence we are interested in the scaling behavior in the inertial range i e for r l note that the l h s of 1 is invariant under scale transformations reflecting the scale invariance of the general solution of 1 is given by a non homogeneous and a linear combination of two homogeneous solutions the combination is fixed by the conditions that y r being a correlation function should be regular and vanish at infinity

towards a geometrical classification of statistical conservation laws in turbulent advection abstract the paper revisits the compressible kraichnan model of turbulent advection in order to derive explicit quantitative relations between scaling exponents and lagrangian particle configuration geometry

scaling properties of passive scalars in one dimension this paper is a set of notes about work in progress since the work is directed toward scaling it seems quite appropriate to report this in a context devoted to ben widom s scientific contributions model equations for the motion of a passive scalar in one dimension are set up and some scaling properties of their solutions are derived all these models follow the approach set up by kraichnan in which the driving velocities have gaussian correlations with scaling properties in space but zero range correlations in time the simplest and most natural versions of the model fail to satisfy an incompressibility condition in another version the model does have an incompressible flow to permit the incompressibility there are two pipes each of which is described by temperature and velocity fields which are functions of a single coordinate x and time flow from one pipe to the other transfers both mass and heat this model has two conserved quantities in the limit of zero dissipation in still another version fluid elements are interchanged in position so that there are an infinite number of conservation laws in the no dissipation limit scaling properties of the resulting two point functions are derived

leader following formation control of multi agent systems under fixed and switching topologies in this article two kinds of leader following formation control problems for second order nonlinear multi agent systems are investigated that is the cases with fixed topology and with switching topology for the former by constructing an appropriate lyapunov functional and utilising linear matrix inequality lmi method we propose a formation control algorithm which makes the nonlinear multi agent systems converge to a desired formation in addition a formation control algorithm is also developed for coupled double integrators with a constant reference velocity then we extend these results to the case when the interaction topology is switching numerical simulations are presented finally to demonstrate the effectiveness of the proposed results

renormalization group and anomalous scaling in a simple model of passive scalar advection in compressible flow field theoretical renormalization group methods are applied to a simple model of a passive scalar quantity advected by the gaussian non solenoidal compressible velocity field with the covariance propto delta t t x x epsilon convective range anomalous scaling for the structure functions and various pair correlators is established and the corresponding anomalous exponents are calculated to the order epsilon 2 of the epsilon expansion these exponents are non universal as a result of the degeneracy of the rg fixed point in contrast to the case of a purely solenoidal velocity field obukhov kraichnan model the correlation functions in the case at hand exhibit nontrivial dependence on both the ir and uv characteristic scales and the anomalous scaling appears already at the level of the pair correlator the powers of the scalar field without derivatives whose critical dimensions determine the anomalous exponents exhibit multifractal behaviour the exact solution for the pair correlator is obtained it is in agreement with the result obtained within the epsilon expansion the anomalous exponents for passively advected magnetic fields are also presented in the first order of the epsilon expansion