emergent dynamics of winfree oscillators on locally coupled networks abstract the winfree model is the first mathematical model for synchronization of weakly coupled oscillators compared to the well known kuramoto model the winfree model does not conserve the total phase this leads to rich dynamic features compared to those produced by other phase models in this paper we study the emergent dynamics of the winfree model on a locally coupled static network a randomly chosen phase configuration undergoes several dynamic phase transitions such as incoherence partial locking complete locking partial oscillator death and complete oscillator death as the coupling strength increases we provide several rigorous analytical results on the emergence of these dynamic features we also provide several numerical simulations and compare their results to the analytical results

identification of physical parameters in structures systems abstract linear time invariant systems with a limited number of unknowns corresponding to physical component values often exhibit certain structural conditions on the way in which these values appear in the transfer functions in this paper the problem of identifying these unknowns is considered the structure translates to an associated identification problem which is multilinear in the unknowns this is exploited in the design of two algorithms which use measurements of input and output and knowledge of the polynomial coefficients of the multilinear combinations of the system parameters conditions are established for the uniform asymptotic stability of these algorithms

hydrodynamic effects on a predator approaching a group of preys abstract a numerical approach to predict the hydrodynamics involving a predator approaching a group of 100 preys is presented a collective behavioural model is adopted to predict the two dimensional space time evolution of the predator preys system that is supposed to be immersed in a fluid the preys manifest mutual repulsion attraction and orientation while the predator is idealized as an individual to be strongly repulsed during the motion the predator experiences a resistance induced by the encompassing fluid such effect is accounted for by computing the hydrodynamic force and by modifying the predator s velocity given by the behavioural equations a numerical campaign is carried out by varying the predator s drag coefficient moreover analyses characterized by progressively wider predator s perception areas are performed thus highlighting the role of the hydrodynamics over the behavioural interactions in order to estimate the predator s performance an ad hoc parameter is proposed moreover findings in terms of trajectories and angular momentum of the group of preys are discussed present findings show that the sole collective behavioural equations are insufficient to predict the performance of a predator that is immersed in a fluid since its motion is drastically affected by the resistance of the surrounding fluid

ergodic theory for markov fibred systems and parabolic rational maps a paraboiic rational map of the riemann sphere admits a non atomic h conformal measure on its julia set where h the hausdorff dimension of the julia set and satisfies 1 2 h 2 with respect to this measure the rational map is conservative exact and there is an equivalent a finite invariant measure finiteness of the measure is characterised central limit theorems are proved in the case of a finite invariant measure and return sequences are identified in the case of an infinite one a theory of markov fibred systems is developed and parabolic rational maps are considered within this framework

area and hausdorff dimension of julia sets of entire functions we show the julia set of a sin z has positive area and the action of a sin z on its julia set is not ergodic the julia set of a exp z has hausdorff dimension two but in the presence of an attracting periodic cycle its area is zero

an algorithmic approach to collective behavior the emergence of collective structure from the decentralized interaction of autonomous agents remains with notable exceptions a mystery while powerful tools from dynamics and statistical mechanics have been brought to bear sometimes with great success an algorithmic perspective has been lacking viewing collective behavior through the lens of natural algorithms offers potential benefits this article examines the merits and challenges of an algorithmic approach to the emergence of collective order

non uniform hyperbolicity in complex dynamics we say that a rational function f satisfies the summability condition with exponent if for every critical point c which belongs to the julia set j there exists a positive integer nc so that sum n 1 infty f n prime f n c c alpha infty and f has no parabolic periodic cycles let max be the maximal multiplicity of the critical points

geometry and ergodic theory of non hyperbolic exponential maps we deal with all the maps from the exponential family e z such that the orbit of zero escapes to infinity sufficiently fast in particular all the parameters a 1 e are included we introduce as our main technical devices the projection f of the map f to the infinite cylinder q c 2 iz and an appropriate conformal measure m we prove that j r f essentially the set of points in q returning infinitely often to a compact region of q disjoint from the orbit of 0 q has the hausdorff dimension h 1 2 that the h dimensional hausdorff measure of j r f is positive and finite and that the h dimensional packing measure is locally infinite at each point of j r f we also prove the existence and uniqueness of a borel probability f invariant ergodic measure equivalent to the conformal measure m as a byproduct of the main course of our considerations we reprove the result obtained independently by lyubich and rees that the limit set under f of lebesgue almost every point in c coincides with the orbit of zero under the map f finally we show that the the function a h 1 e is continuous

attractors for graph critical rational maps we call a rational map f graph critical if any critical point either belongs to an invariant finite graph g or has minimal limit set or is non recurrent and has limit set disjoint from g we prove that for any conformal measure either for almost every point of the julia set j f its limit set coincides with j f or for almost every point of j f its limit set coincides with the limit set of a critical point of f

parabolic orbifolds and the dimension of the maximal measure for rational maps let f g g be a rational map of the riemann sphere d e g f 2 a natural invariant measure m the measure of maximal entropy was constructed by ljubich lju and independently by freire lopes and mafi6 flm the aim of this paper is to compare this measure with some hausdorff measures first recall the following definition for a probabili ty measure v on or more generally on a smooth manifold the hausdorff dimension of v is defined by a formula hd v inf hd y y v y l

geometric decompositions of collective motion collective motion in nature is a captivating phenomenon revealing the underlying mechanisms which are of biological and theoretical interest will require empirical data modelling and analysis techniques here we contribute a geometric viewpoint yielding a novel method of analysing movement snapshots of collective motion are portrayed as tangent vectors on configuration space with length determined by the total kinetic energy using the geometry of fibre bundles and connections this portrait is split into orthogonal components each tangential to a lower dimensional manifold derived from configuration space the resulting decomposition when interleaved with classical shape space construction is categorized into a family of kinematic modes including rigid translations rigid rotations inertia tensor transformations expansions and compressions snapshots of empirical data from natural collectives can be allocated to these modes and weighted by fractions of total kinetic energy such quantitative measures can provide insight into the variation of the driving goals of a collective as illustrated by applying these methods to a publicly available dataset of pigeon flocking the geometric framework may also be profitably employed in the control of artificial systems of interacting agents such as robots

a macroscopic model for a system of swarming agents using curvature control in this paper we study the macroscopic limit of a new model of collective displacement the model called ptwa is a combination of the vicsek alignment model vicsek et al in phys rev lett 75 6 1226 1229 1995 and the persistent turning walker ptw model of motion by curvature control degond and motsch in j stat phys 131 6 989 1021 2008 gautrais et al in j math biol 58 3 429 445 2009 the ptw model was designed to fit measured trajectories of individual fish gautrais et al in j math biol 58 3 429 445 2009 the ptwa model persistent turning walker with alignment describes the displacements of agents which modify their curvature in order to align with their neighbors the derivation of its macroscopic limit uses the non classical notion of generalized collisional invariant introduced in degond and motsch in math models methods appl sci 18 1 1193 1215 2008 the macroscopic limit of the ptwa model involves two physical quantities the density and the mean velocity of individuals it is a system of hyperbolic type but is non conservative due to a geometric constraint on the velocity this system has the same form as the macroscopic limit of the vicsek model degond and motsch in math models methods appl sci 18 1 1193 1215 2008 the vicsek hydrodynamics but for the expression of the model coefficients the numerical computations show that the numerical values of the coefficients are very close the vicsek hydrodynamic model appears in this way as a more generic macroscopic model of swarming behavior as originally anticipated

swarming on random graphs ii we consider an individual based model where agents interact over a random network via first order dynamics that involve both attraction and repulsion in the case of all to all coupling of agents in mathbb r d this system has a lowest energy state in which an equal number of agents occupy the vertices of the d dimensional simplex the purpose of this paper is to sharpen and extend a line of work initiated in 56 which studies the behavior of this model when the interaction between the n agents occurs according to an erd s renyi random graph mathcal g n p instead of all to all coupling in particular we study the effect of randomness on the stability of these simplicial solutions and provide rigorous results to demonstrate that stability of these solutions persists for probabilities greater than np o log n in other words only a relatively small number of interactions are required to maintain stability of the state the results rely on basic probability arguments together with spectral properties of random graphs

self organized hydrodynamics with congestion and path formation in crowds a continuum model for self organized dynamics is numerically investigated the model describes systems of particles subject to alignment interaction and short range repulsion it consists of a non conservative hyperbolic system for the density and velocity orientation short range repulsion is included through a singular pressure which becomes infinite at the jamming density the singular limit of infinite pressure stiffness leads to phase transitions from compressible to incompressible dynamics the paper proposes an asymptotic preserving scheme which takes care of the singular pressure while preventing the breakdown of the courant friedrichs lewy cfl stability condition near congestion it relies on a relaxation approximation of the system and an elliptic formulation of the pressure equation numerical simulations of impinging clusters show the efficiency of the scheme to treat congestions a two fluid variant of the model provides a model of path formation in crowds

spectral method for a kinetic swarming model in this paper we present the first numerical method for a kinetic description of the vicsek swarming model the kinetic model poses a unique challenge as there is a distribution dependent collision invariant to satisfy when computing the interaction term we use a spectral representation linked with a discrete constrained optimization to compute these interactions to test the numerical scheme we investigate the kinetic model at different scales and compare the solution with the microscopic and macroscopic descriptions of the vicsek model we observe that the kinetic model captures key features such as vortex formation and traveling waves

julia sets and bifurcation diagrams for exponential maps on decrit certaines des bifurcations qui apparaissent dans la famille des applications entieres e z exp z pour 1 on sait que j e c on montre qu il y a beaucoup d autres valeurs pour lesquelles ceci arrive

obstacle and predator avoidance in a model for flocking abstract the modeling and investigation of the dynamics and configurations of animal groups is a subject of growing attention in this paper we present a deterministic continuum model of flocking and use it to investigate the linearized response of an infinite extent flock to an obstacle or an attacking predator we show that the steady state flock response is in the form of density disturbances that resemble mach cones whose configuration is determined by the anisotropic propagation of waves through the flock we analytically and numerically test relations that predict the mach wedge angles disturbance heights and wake widths we find that these expressions are insensitive to many of the parameters of the model

flocking dynamics and mean field limit in the cucker smale type model with topological interactions abstract we introduce a cucker smale type model for flocking where the strength of interaction between two agents depends on their relative separation called topological distance in previous works which is the number of intermediate individuals separating them this makes the model scale free and is motivated by recent extensive observations of starling flocks suggesting that the interaction ruling animal collective behavior depends on topological rather than the metric distance we study the conditions leading to asymptotic flocking in the topological model defined as the convergence of the agents velocities to a common vector the shift from metric to topological interactions requires development of new analytical methods taking into account the graph theoretical nature of the problem moreover we provide a rigorous derivation of the mean field limit of large populations recovering kinetic and hydrodynamic descriptions in particular we introduce the novel concept of relative separation in continuum descriptions which is applicable to a broad variety of models of collective behavior as an example we shortly discuss a topological modification of the attraction repulsion model and illustrate with numerical simulations that the modified model produces interesting new pattern dynamics

stochastic stability of continuous time consensus protocols a unified approach to studying convergence and stochastic stability of continuous time consensus protocols cps is presented in this work our method applies to networks with directed information flow both cooperative and noncooperative interactions networks under weak stochastic forcing and those whose topology and strength of connections may vary in time the graph theoretic interpretation of the analytical results is emphasized we show how the spectral properties such as algebraic connectivity and total effective resistance as well as the geometric properties such as the dimension and the structure of the cycle subspace of the underlying graph shape stability of the corresponding cps in addition we explore certain implications of spectral graph theory to cp design in particular we point out that expanders sparse highly connected graphs generate cps whose performance remains uniformly high when the size of the network grows unboundedly similarly we highlight the benefits of using random

gravity versus radiation models on the importance of scale and heterogeneity in commuting flows we test the recently introduced radiation model against the gravity model for the system composed of england and wales both for commuting patterns and for public transportation flows the analysis is performed both at macroscopic scales i e at the national scale and at microscopic scales i e at the city level it is shown that the thermodynamic limit assumption for the original radiation model significantly underestimates the commuting flows for large cities we then generalize the radiation model introducing the correct normalization factor for finite systems we show that even if the gravity model has a better overall performance the parameter free radiation model gives competitive results especially for large scales

a new model for self organized dynamics and its flocking behavior we introduce a model for self organized dynamics which we argue addresses several drawbacks of the celebrated cucker smale c s model the proposed model does not only take into account the distance between agents but instead the influence between agents is scaled in term of their relative distance consequently our model does not involve any explicit dependence on the number of agents only their geometry in phase space is taken into account the use of relative distances destroys the symmetry property of the original c s model which was the key for the various recent studies of c s flocking behavior to this end we introduce here a new framework to analyze the phenomenon of flocking for a rather general class of dynamical systems which covers systems with non symmetric influence matrices in particular we analyze the flocking behavior of the proposed model as well as other strongly asymmetric models with leaders

the fibonacci unimodal map the fibonacci recurrence of the critical orbit appeared in the work of branner and hubbard on complex cubic polynomials bh 12 and in yoccoz s work yl y2 on quadratic ones as the worst pattern of recurrence on the other hand a real quadratic fibonacci map was suggested by hofbauer and keller hk as a possible candidate for a map having a wild attractor that is a set a which is the w limit set for lebesgue almost every orbit but is strictly smaller than the w limit set for a generic orbit the w limit set of the critical orbit in hk possesses all known topological properties of wild attractors compare bl2 in fact we will see below that the quadratic fibonacci map does not have a wild attractor however the corresponding question for a map with a degenerate critical point remains open actually the first indication of the fibonacci map appeared in the numerical work of tsuda t related to the belousova zhabotinskii reaction and also in numerical work of shibayama sh more precisely they studied the sequence of fibonacci bifurcations creating the fibonacci map this paper will study topological geometrical and measure theoretical properties of the real fibonacci map our goal was to figure out if this type of recurrence really gives any pathological examples and to compare it with the infinitely renormalizable patterns of recurrence studied by sullivan s it turns out that the situation can be understood completely and is of quite regular nature in particular any fibonacci map with negative schwarzian and nondegenerate critical point has an absolutely continuous invariant measure so we deal with a regular type of chaotic dynamics it turns out also that geometrical properties of the closure of the critical orbit are quite different from those of the feigenbaum map its hausdorff dimension is equal to zero and its geometry is not rigid but depends on one parameter branner and hubbard introduce the concept of a tableau in order to describe recurrence of critical orbits their fibonacci tableau is a basic example which corresponds to one particularly close and regular pattern of recurrence if a complex quadratic map z 1 z2 c realizes this fibonacci tableau then the

symmetry and reduction in collectives low dimensional cyclic pursuit we investigate low dimensional examples of cyclic pursuit in a collective wherein each agent employs a constant bearing cb steering law relative to exactly one other agent for the case of three agents in the plane we characterize relative equilibria and pure shape equilibria of associated closed loop dynamics re scaling time yields a reduction of phase space to two dimensions and effective tools for stability analysis study of bifurcation of a family of collinear equilibria dependent on a single cb control parameter reveals the presence of a rich collection of trajectories that are periodic in shape and undergo precession in physical space for collectives in three dimensions with an appropriate notion of cb pursuit strategy and corresponding steering law the two agent case proves to be explicitly integrable these results suggest control schemes for small teams of mobile robotic agents engaged in area coverage tasks such as search and rescue and raise interesting possibilities for behaviour in biological contexts

lines and points in three views and the trifocal tensor this paper discusses the basic role of the trifocal tensor in scene reconstruction from three views this 3 times 3 times 3 tensor plays a role in the analysis of scenes from three views analogous to the role played by the fundamental matrix in the two view case in particular the trifocal tensor may be computed by a linear algorithm from a set of 13 line correspondences in three views it is further shown in this paper that the trifocal tensor is essentially identical to a set of coefficients introduced by shashua to effect point transfer in the three view case this observation means that the 13 line algorithm may be extended to allow for the computation of the trifocal tensor given any mixture of sufficiently many line and point correspondences from the trifocal tensor the camera matrices of the images may be computed and the scene may be reconstructed for unrelated uncalibrated cameras this reconstruction will be unique up to projectivity thus projective reconstruction of a set of lines and points may be carried out linearly from three views

infinitely renormalizable quadratic polynomials we prove that the julia set of a quadratic polynomial which admits an infinite sequence of unbranched simple renormalizations with complex bounds is locally connected the method in this study is three dimensional puzzles

a new interaction potential for swarming models abstract we consider a self propelled particle system which has been used to describe certain types of collective motion of animals such as fish schools and bird flocks interactions between particles are specified by means of a pairwise potential repulsive at short ranges and attractive at longer ranges the exponentially decaying morse potential is a typical choice and is known to reproduce certain types of collective motion observed in nature particularly aligned flocks and rotating mills we introduce a class of interaction potentials that we call quasi morse for which flock and rotating mills states are also observed numerically however in that case the corresponding macroscopic equations allow for explicit solutions in terms of special functions with coefficients that can be obtained numerically without solving the particle evolution we compare the obtained solutions with long time dynamics of the particle systems and find a close agreement for several types of flock and mill solutions

bifurcations in the space of exponential maps this article investigates the parameter space of the exponential family textit z mapsto exp textit z kappa we prove that the boundary in of every hyperbolic component is a jordan arc as conjectured by eremenko and lyubich as well as baker and rippon in fact we prove the stronger statement that the exponential bifurcation locus is connected in an analog of douady and hubbard s celebrated theorem that the mandelbrot set is connected we show furthermore that is not accessible through any nonhyperbolic queer stable component the main part of the argument consists of demonstrating a general squeezing lemma which controls the structure of parameter space near infinity we also prove a second conjecture of eremenko and lyubich concerning bifurcation trees of hyperbolic components

shrinkwrapping and the taming of hyperbolic 3 manifolds we introduce a new technique for finding cat 1 surfaces in hyperbolic 3 manifolds we use this to show that a complete hyperbolic 3 manifold with finitely generated fundamental group is geometrically and topologically tame

new tools for characterizing swarming systems a comparison of minimal models abstract we compare three simple models that reproduce qualitatively the emergent swarming behavior of bird flocks fish schools and other groups of self propelled agents by using a new set of diagnosis tools related to the agents spatial distribution two of these correspond in fact to different implementations of the same model which had been previously confused in the literature all models appear to undergo a very similar order to disorder phase transition as the noise level is increased if we only compare the standard order parameter which measures the degree of agent alignment when considering our novel quantities however their properties are clearly distinguished unveiling previously unreported qualitative characteristics that help determine which model best captures the main features of realistic swarms additionally we analyze the agent clustering in space finding that the distribution of cluster sizes is typically exponential at high noise and approaches a power law as the noise level is reduced this trend is sometimes reversed at noise levels close to the phase transition suggesting a non trivial critical behavior that could be verified experimentally finally we study a bi stable regime that develops under certain conditions in large systems by computing the probability distributions of our new quantities we distinguish the properties of each of the coexisting metastable states our study suggests new experimental analyses that could be carried out to characterize real biological swarms

time delay effect on the flocking in an ensemble of thermomechanical cucker smale particles abstract we study the communication time delay effect on the emergent dynamics of thermodynamic cucker smale tcs ensemble with temperature as an internal state in the presence of time delay conservation laws for momentum and energy do not hold so that the analytic tool based on the energy method cannot be applied for the tcs model as it is we show that temperature stays positive in the presence of small time delay as long as temperatures stay positive the tcs model is well posed by direct application of cauchy lipschitz theory when the size of time delay and initial data are sufficiently small we can derive differential inequalities for velocity and temperature diameters using these differential inequalities we show that temperature and velocity diameters tend to zero exponentially fast whereas the position diameter is uniformly bounded we also present several numerical simulations and compare them with analytic results

stable adaptive observers for nonlinear time varying systems an adaptive observer identifier for single input single output observable nonlinear systems that can be transformed to a certain observable canonical form is described sufficient conditions for stability of this observer are provided these conditions are in terms of the structure of the system and canonical form the boundedness of the parameter variations and the sufficient richness of some signals the scope of the canonical form and the use of the observer identifier is motivated by the presentation of applications to time invariant bilinear systems nonlinear systems in phase variable form a biotechnological process and a robot manipulator in each case the specific stability conditions are presented

self similarity of siegel disks and hausdorff dimension of julia sets let f z ez z where is an irrational number of bounded type according to siegel f is linearizable on a disk containing the origin in this paper we show the hausdorff dimension of the julia set j f is strictly less than two and if is a quadratic irrational such as the golden mean then the siegel disk for f is self similar about the critical point in the latter case we also show the rescaled first return maps converge exponentially fast to a system of commuting branched coverings of the complex plane

the convergence of bird flocking we bound the time it takes for a group of birds to stabilize in a standard flocking model each bird averages its velocity with its neighbors lying within a fixed radius we resolve the worst case complexity of this natural algorithm by providing asymptotically tight bounds on the time to equilibrium we reduce the problem to two distinct questions in computational geometry and circuit complexity

a hierarchy of heuristic based models of crowd dynamics we derive a hierarchy of kinetic and macroscopic models from a noisy variant of the heuristic behavioral individual based model of ngai et al disaster med public health prep 3 191 195 2009 where pedestrians are supposed to have constant speeds this ibm supposes that pedestrians seek the best compromise between navigation towards their target and collisions avoidance we first propose a kinetic model for the probability distribution function of pedestrians then we derive fluid models and propose three different closure relations the first two closures assume that the velocity distribution function is either a dirac delta or a von mises fisher distribution respectively the third closure results from a hydrodynamic limit associated to a local thermodynamical equilibrium we develop an analogy between this equilibrium and nash equilibria in a game theoretic framework in each case we discuss the features of the models and their suitability for practical use

high gain observer based estimation of parameter variations with delay alignment this technical note analyzes the estimation delay in a high gain observer where the state estimates may lag behind the actual states due to the observer s non zero phase response the technical note proves that for a slowly time varying system subject to bounded noises the estimation delay depends on the observer gain but is independent of the variations of system parameters rather than estimating the delay a novel method is proposed to calculate the delay from the observer s phase response in terms of system identification the delay is compensated by aligning other measurements with the lagged estimate so that they have the same lag the simulation results of an aero engine model show significant improvements in estimation on one hand the proposed approach improves the estimation accuracy and on the other hand it removes the assumption of zero delay and gives a new insight into the high gain observer design

kinetic models for topological nearest neighbor interactions we consider systems of agents interacting through topological interactions these have been shown to play an important part in animal and human behavior precisely the system consists of a finite number of particles characterized by their positions and velocities at random times a randomly chosen particle the follower adopts the velocity of its closest neighbor the leader we study the limit of a system size going to infinity and under the assumption of propagation of chaos show that the limit kinetic equation is a non standard spatial diffusion equation for the particle distribution function we also study the case wherein the particles interact with their k closest neighbors and show that the corresponding kinetic equation is the same finally we prove that these models can be seen as a singular limit of the smooth rank based model previously studied in 10 the proofs are based on a combinatorial interpretation of the rank as well as some concentration of measure arguments

consensus over numerosity constrained random networks we analyze the discrete time consensus problem for a group of agents that communicate through a stochastic directed network with fixed out degree this network construction models the perceptual phenomenon of numerosity observed in animal groups exhibiting collective behavior we find necessary and sufficient conditions for mean square consentability of the averaging protocol and we derive a closed form expression for the asymptotic convergence factor analytical results are illustrated through simulations

on the consensus protocol of conspecific agents abstract we present a class of consensus protocols over groups of agents with stochastically switching directed and weighted communication topologies in this protocol an agent s traits that is the cardinality of its neighbor set and the weight assigned to its neighbors in the updating process are given by two jointly distributed random variables and the neighbors of an agent are selected with equal probability we provide closed form results for the asymptotic convergence rate and for the steady state mean square deviation in the presence of additive noise these results are specialized to consensus protocols based on erd s renyi and numerosity constrained networks

indirect techniques for adaptive input output linearization of non linear systems abstract a technique of indirect adaptive control based on certainty equivalence for input output linearization of non linear systems is proven convergent it does not surfer from the overparameterization drawbacks of the direct adaptive control techniques on the same plant this paper also contains a semi indirect adaptive controller which has several attractive features of both the direct and indirect schemes

promoting collective motion of self propelled agents by discarding short range interactions abstract we study the collective motion of self propelled agents with the restricted view the field of view of each agent is an annulus bounded by the outer radius r and inner radius r where is a tunable parameter we find that there exists an optimal value of leading to the highest degree of direction consensus this phenomenon indicates that there exists superfluous communication in the collective motion of self propelled agents and short range interactions hinder the direction consensus of the system the value of optimal decreases as the absolute velocity increases while it increases as the outer radius r and the system size increase for a fixed value of direction consensus is enhanced when the absolute velocity is small the outer radius or the system size is large

collective behavior models with vision geometrical constraints truncated noises and propagation of chaos abstract we consider large systems of stochastic interacting particles through discontinuous kernels which has vision geometrical constrains we rigorously derive a vlasov fokker planck type of kinetic mean field equation from the corresponding stochastic integral inclusion system more specifically we construct a global in time weak solution to the stochastic integral inclusion system and derive the kinetic equation with the discontinuous kernels and the inhomogeneous noise strength by employing the 1 wasserstein distance

feigenbaum coullet tresser universality and milnor s hairiness conjecture we prove the feigenbaum coullet tresser conjecture on the hyperbolicity of the renormalization transformation of bounded type this gives the first computer free proof of the original feigenbaum observation of the universal parameter scaling laws we use the hyperbolicity theorem to prove milnor s conjectures on self similarity and hairiness of the mandelbrot set near the corresponding parameter values we also conclude that the set of real infinitely renormalizable quadratics of type bounded by some n 1 has hausdorff dimension strictly between 0 and 1 in the course of getting these results we supply the space of quadratic like germs with a complex analytic structure and demonstrate that the hybrid classes form a complex codimension one foliation of the connectedness locus

structural instability of exponential functions we first prove some equivalent statements on stability of families of critically finite entire functions then with these in hand a conjecture con cerning stability of the family of exponential functions is affirmatively answered in some cases

short range interactions versus long range correlations in bird flocks bird flocks are a paradigmatic example of collective motion one of the prominent traits of flocking is the presence of long range velocity correlations between individuals which allow them to influence each other over the large scales keeping a high level of group coordination a crucial question is to understand what is the mutual interaction between birds generating such nontrivial correlations here we use the maximum entropy me approach to infer from experimental data of natural flocks the effective interactions between individuals compared to previous studies we make a significant step forward as we retrieve the full functional dependence of the interaction on distance and find that it decays exponentially over a range of a few individuals the fact that me gives a short range interaction even though its experimental input is the long range correlation function shows that the method is able to discriminate the relevant information encoded in such correlations and single out a minimal number of effective parameters finally we show how the method can be used to capture the degree of anisotropy of mutual interactions

the hausdorff dimension of the boundary of the mandelbrot set and julia sets it is shown that the boundary of the mandelbrot set m has hausdorff dimension two and that for a generic c e am the julia set of z i z2 c also has hausdorff dimension two the proof is based on the study of the bifurcation of parabolic periodic points

hausdorff dimension and conformal measures of feigenbaum julia sets 1 1 statement of the results one of the first questions usually asked about a fractal subset of r is whether it has the maximal possible hausdorff dimension n it certainly happens if the set has positive lebesgue measure on the other hand it is easy to construct fractal sets of zero measure but of dimension n moreover this phenomenon is often observable for fractal sets produced by conformal dynamical systems iterated rational functions or kleinian groups in particular the analogy with kleinian groups suggested that the julia sets of feigenbaum maps should have hausdorff dimension two in this paper we will show that this is not always the case a quadratic polynomial or more generally a quadratic like map is called feigenbaum if it is infinitely renormalizable of bounded combinatorial type with a priori bounds see 2 for a precise definition feigenbaum polynomials are remarkable dynamical systems whose geometry has been a focus of research for the past 25 years by analogy with kleinian groups it was anticipated that their julia sets have dimension 2 and this problem has been around for a while however in this paper we will show that this is not always the case

cucker smale flocking under rooted leadership with free will agents abstract motivated by the recent works of cucker smale and shen we study the emergent behavior of discrete time cucker smale flocking under rooted leadership with free will agents we prove that if the free will accelerations of the agents relative to the leader are summable then the individual velocities will converge to the velocity of leader under suitable initial conditions in particular we prove that the unconditional convergence is true if the decay exponent of the communication weight is below some critical value which depends on the interaction topology we also present some numerical examples to illustrate the main result

multi agent flocking under topological interactions in this paper we consider a multi agent system consisting of mobile agents with second order dynamics the communication network is determined by the so called topological interaction rule agents interact with a fixed number of their closest neighbors this rule comes from observations of real flock of starlings the non symmetry of the interactions adds to the difficulty of the analysis the goal of this paper is to determine practical conditions on the initial positions and velocities of agents ensuring that the agents asymptotically agree on a common velocity i e a flocking behavior is achieved for this purpose we define a notion of hierarchical structure in the interaction graph which allows to establish such conditions building upon previous work on multi agent systems with switching communication networks though conservative our approach gives conditions that can be verified a priori our result is illustrated through simulations

puzzle geometry and rigidity the fibonacci cycle is hyperbolic we describe a new and robust method to prove rigidity results in complex dynamics the new ingredient is the geometry of the critical puzzle pieces under control of geometry and complex bounds two generalized polynomial like maps which admit a topological conjugacy quasiconformal outside the filled in julia set are indeed quasiconformally conjugate the proof uses a new abstract removability type result for quasiconformal maps following ideas of heinonen and koskela and of kallunki and koskela optimized for applications in complex dynamics we prove as the first application of this new method that for even criticalities distinct from two the period two cycle of the fibonacci renormalization operator is hyperbolic with dimensional unstable manifold

robust adaptive regulation with minimal prior knowledge a robust adaptive regulator is designed under minimal prior knowledge namely the order of a nominal plant model in a well defined sense the involved robustness means that the closed loop stability is guaranteed in the presence of some class of time varying parameters and unmodeled dynamics the main design feature of the proposed adaptive regulator consists in its self excitation capability together with an appropriate identification stabilization time splitting unlike in the available literature concerning robust adaptive control the involved self excitation is established independently of the parameter estimator properties the use of the robustness oriented modifications as deadzone and parameter projection or contraction which are aimed at achieving the parameter estimates boundedness is no longer required beforehand thereby reducing the underlying prior knowledge

the periodic points of renormalization it will be shown that the renormalization operator acting on the space of smooth unimodal maps with critical exponent a 1 has periodic points of any combinatorial type a central question in the theory of dynamical systems is whether small scale geometrical properties of dynamical systems are determined by the combinatorial properties of the system indeed such universality of small scale geometry was discovered by coullet tresser and feigenbaum they studied infinitely renormalizable unimodal maps of the period doubling type and observed that the geometry of the invariant cantor set of such maps converges when looking at smaller and smaller scales furthermore they observed that the limiting geometry was universal in the sense that the small scale geometry of these cantor sets depends only on the local behavior of the map around the critical point this local behavior is specified by the critical exponent to explain the universality of geometry they defined the period doubling renormalization operator on a suitable space of unimodal maps this operator acts like a microscope the image under the renormalization operator is a unimodal map describing the geometry and dynamics on a smaller scale the universality of geometry was understood by conjecturing that the renormalization operator has a unique hyperbolic fixed point in particular the infinitely renormalizable unimodal maps form the stable manifold of the fixed point of the renormalization operator the first step in proving these conjectures is showing the existence of a renormalization fixed point

perturbing misiurewicz parameters in the exponential family in one dimensional real and complex dynamics a map whose post singular or post critical set is bounded and uniformly repelling is often called a misiurewicz map in results hitherto perturbing a misiurewicz map is likely to give a non hyperbolic map as per jakobson s theorem for unimodal interval maps this is despite genericity of hyperbolic parameters at least in the interval setting we show the contrary holds in the complex exponential family z mapsto lambda exp z misiurewicz maps are lebesgue density points for hyperbolic parameters as a by product we also show that lyapunov exponents almost never exist for exponential misiurewicz maps the lower lyapunov exponent is almost everywhere the upper lyapunov exponent is non negative and depends on the choice of metric

diffusive influence systems influence systems seek to model how influence broadly defined spreads across a dynamic network we build a general analytical framework which we then use to prove that while turing complete influence dynamics of the diffusive type is almost surely asymptotically periodic in addition to resolving the dynamics of a widely used family of multiagent systems we introduce a general renormalization method for the bifurcation analysis of multiagent systems

observer based parameter identifiers for nonlinear systems with parameter dependencies a class of nonlinear dynamical systems whose dynamics depend linearly on the unknown parameter vector is considered after reviewing a standard observer based identifier for estimating the unknown parameters we propose a family of new identifiers which exploit the a priori known parameter dependencies we then establish under mild assumptions on the dynamical system that 1 the proposed identifiers are stable and 2 the weighted norm of state parameter errors using the proposed identifiers are less than the corresponding errors using the standard identifier for a length of time after t 0 the main contribution of this paper is that it introduces a family of observer based identifiers which exhibit better transient performance than the standard identifier

interaction ruling animal collective behavior depends on topological rather than metric distance evidence from a field study numerical models indicate that collective animal behavior may emerge from simple local rules of interaction among the individuals however very little is known about the nature of such interaction so that models and theories mostly rely on aprioristic assumptions by reconstructing the three dimensional positions of individual birds in airborne flocks of a few thousand members we show that the interaction does not depend on the metric distance as most current models and theories assume but rather on the topological distance in fact we discovered that each bird interacts on average with a fixed number of neighbors six to seven rather than with all neighbors within a fixed metric distance we argue that a topological interaction is indispensable to maintain a flock s cohesion against the large density changes caused by external perturbations typically predation we support this hypothesis by numerical simulations showing that a topological interaction grants significantly higher cohesion of the aggregation compared with a standard metric one

decay of geometry for unimodal maps an elementary proof we prove that a nonrenormalizable smooth unimodal interval map with critical order between 1 and 2 displays decay of geometry by an elementary and purely real argument this completes a real approach to milnor s attractor problem for smooth unimodal maps with critical order not greater than 2

conservative exact rational maps of the sphere abstract in this paper we investigate conditions for conservativity and exactness ofd to 1 endomorphisms we prove that all rational maps of degree greater than 1 are either conservative and exact with respect to normalized lebesgue measure on the sphere or have an attracting property inside the julia set we provide examples of rational maps which can be shown to be conservative and exact by using the previous result which was not formerly known to have this property

symmetry and reduction in collectives cyclic pursuit strategies we specify and analyse models that capture the geometry of purposeful motion of a collective of mobile agents with a focus on planar motion dyadic strategies and attention graphs which are static directed and cyclic strategies are formulated as constraints on joint shape space and are implemented through feedback laws for the actions of individual agents here modelled as self steering particles by reduction to a labelled shape space using a redundant parametrization to account for cycle closure constraints and a further reduction through time rescaling we characterize various special solutions relative equilibria and pure shape equilibria for cyclic pursuit with a constant bearing cb strategy this is accomplished by first proving convergence of the nonlinear dynamics to an invariant manifold the cb pursuit manifold and then analysing the closed loop dynamics restricted to the invariant manifold for illustration we sketch some low dimensional examples this formulation involving strategies attention graphs and sensor driven steering laws and the resulting templates of collective motion are part of a broader programme to interpret the mechanisms underlying biological collective motion

the hausdorff dimension of the limit set of a geometrically finite kleinian group we consider in this paper groups of mobius transformations of l r tj zc such a group is called kleinian if it acts discontinuously somewhere in 1 the action of g extends to the n d dimensional hyperbolic space h i r zc and g is geometrically finite if there is a hyperbolic fundamental polyhedron with a finite number of faces for the action of g in h t for a more precise definition see 15 1 b we prove in this paper that the hausdor f dimension dim l g of the limit set l g of a geometrically finite kleinian group g of 1 is less than n theorem d our proof of this theorem is based on the following obser ation assume that g is of compact t3 pe i e if izi i h iui then i l g g is compact then there is an integer q such that if we divide any n cube q of r into q equal subcubes then at least one of these subcubes does not touch l g let l4 q be the family of these subcubes which touch l g then the n measures of q e le q do not add up to the n measure of q and we get

reinforcement learning and wavelet adapted vortex methods for simulations of self propelled swimmers we present a numerical method for the simulation of collective hydrodynamics in self propelled swimmers swimmers in a viscous incompressible flow are simulated with a remeshed vortex method coupled with brinkman penalization and projection approach the remeshed vortex methods are enhanced via wavelet based adaptivity in space and time the method is validated on benchmark swimming problems furthermore the flow solver is integrated with a reinforcement learning algorithm such that swimmers can learn to adapt their motion so as to optimally achieve a specified goal such as fish schooling the computational efficiency of the wavelet adapted remeshed vortex method is a key aspect for the effective coupling with learning algorithms the suitability of this approach for the identification of swimming behaviors is assessed on a set of learning tasks

dynamic rays of bounded type entire functions we construct an entire function in the eremenko lyubich class b whose julia set has only bounded path components this answers a question of eremenko from 1989 in the negative on the other hand we show that for many functions in b in particular those of nite order every escaping point can be connected to 1 by a curve of escaping points this gives a partial positive answer to the aforementioned question of eremenko and answers a question of fatou from 1926

hausdorff dimension of julia sets of feigenbaum polynomials with high criticality we consider unimodal polynomials with feigenbaum topological type and critical points whose orders tend to infinity it is shown that the hyperbolic dimensions of their julia set go to 2 furthermore that the hausdorff dimensions of the basins of attraction of their feigenbaum attractors also tend to 2 the proof is based on constructing a limiting dynamics with a flat critical point

continuum modeling of the equilibrium and stability of animal flocks abstract groups of animals often tend to arrange themselves in flocks that have characteristic spatial attributes and temporal dynamics using a dynamic continuum model for a flock of individuals we find equilibria of finite spatial extent where the density goes continuously to zero at a well defined flock edge and we discuss conditions on the model that allow for such solutions we also demonstrate conditions under which as the flock size increases the interior density in our equilibria tends to an approximately uniform value motivated by observations of starling flocks that are relatively thin in a direction transverse to the direction of flight we investigate the stability of infinite planar sheet flock equilibria we find that long wavelength perturbations along the sheet are unstable for the class of models that we investigate this has the conjectured consequence that sheet like flocks of arbitrarily large transverse extent relative to their thickness do not occur however we also show that our model admits approximately sheet like pancake shaped three dimensional ellipsoidal equilibria with definite aspect ratios transverse length scale to flock thickness determined by anisotropic perceptual response characteristics of the flocking individuals and we argue that these pancake like equilibria are stable to the previously mentioned sheet instability

statistical mechanics for natural flocks of birds flocking is a typical example of emergent collective behavior where interactions between individuals produce collective patterns on the large scale here we show how a quantitative microscopic theory for directional ordering in a flock can be derived directly from field data we construct the minimally structured maximum entropy model consistent with experimental correlations in large flocks of starlings the maximum entropy model shows that local pairwise interactions between birds are sufficient to correctly predict the propagation of order throughout entire flocks of starlings with no free parameters we also find that the number of interacting neighbors is independent of flock density confirming that interactions are ruled by topological rather than metric distance finally by comparing flocks of different sizes the model correctly accounts for the observed scale invariance of long range correlations among the fluctuations in flight direction

k nearest neighbor interaction induced self organized pedestrian counter flow abstract a recent field study confirmed that animal crowd behavior is dominated by the interaction from the k nearest neighbors rather than all the neighbors in a given metric distance for the reason that systems with local interaction perform similar self organized phenomena we in this paper build two models i e a metric distance based model and a k nearest neighbor k nn counterflow model based on a simple discrete cellular automaton model entitled the basic model to investigate the fundamental interaction ruling pedestrian counter flow pedestrians move in a long channel and as a result are divided into left moving pedestrians and right moving pedestrians these pedestrians interact with each other in different forms in different models in the metric distance based model ones direction of chosen behavior is influenced by all those who are in a small metric distance and come from the opposite direction while in the k nn counterflow model ones direction of chosen behavior is influenced by the distribution of a fixed number of the k nearest neighbors coming from the opposite direction the self organized lane formation is captured and factors affecting the number of lanes formed in the channel are investigated results imply that with varying density the lane formation pattern is almost the same in the k nn counterflow model while it is not in the case of metric distance based model this means that the k nn interaction plays a more fundamental role in the emergence of collective pedestrian phenomena then the k nn counterflow model is further validated by comparing the lane formation pattern and the fundamental diagram with real pedestrian counter flow reasons for the lane formation and improvement of flow rate are discussed the relations among mean velocity occupancy and total entrance density of the model are also studied the results indicate that the k nn interaction provides a more efficient traffic condition and is able to quantify features such as segregation and phase transition at high density of pedestrian traffic