testing and modeling multivariate threshold models abstract threshold autoregressive models in which the process is piecewise linear in the threshold space have received much attention in recent years in this article i use predictive residuals to construct a test statistic for detecting threshold nonlinearity in a vector time series and propose a procedure for building a multivariate threshold model the thresholds and the model are selected jointly based on the akaike information criterion the finite sample performance of the proposed test is studied by simulation the modeling procedure is then used to study arbitrage in security markets and results in a threshold cointegration between logarithms of future contracts and spot prices of a security after adjusting for the cost of carrying the contracts in this particular application thresholds are determined in part by the transaction costs i also apply the proposed procedure to u s monthly interest rates and two river flow series of iceland

self weighted lad based inference for heavy tailed threshold autoregressive models abstract the least squares estimator of the threshold autoregressive tar model may not be consistent when its tail is less than or equal to 2 neither theory nor methodology can be applied to model fitting in this case this paper is to develop a systematic procedure of statistical inference for the heavy tailed tar model we first investigate the self weighted least absolute deviation estimation for the model it is shown that the estimated slope parameters are n consistent and asymptotically normal and the estimated thresholds are n consistent each of which converges weakly to the smallest minimizer of a compound poisson process based on this theory the wald test statistic is considered for testing the linear restriction of slope parameters and a procedure is given for inference of threshold parameters we finally construct a sign based portmanteau test for model checking simulations are carried out to assess the performance of our procedure and a real example is given

a connection between a system of random walks and rumor transmission abstract we establish a relationship between the phenomenon of rumor transmission on a population and a probabilistic model of interacting particles on the complete graph more precisely we consider variations of the maki thompson epidemic model and the frog model of random walks which were introduced in the scientific literature independently and in different contexts we analyze the markov chains which describe these models and show a coupling between them our connection shows how the propagation of a rumor in a closed homogeneously mixing population can be described by a system of random walks on the complete graph additionally we discuss further applications of the random walk model which are relevant to the modeling of different biological dynamics

the discrete frenkel kontorova model and its extensions i exact results for the ground states abstract we present a rigorous study of the classical ground states under boundary conditions of a class of one dimensional models generalizing the discrete frenkel kontorova model the extremalization equations of the energy of these models turn out to define area preserving twist maps which exhibits periodic quasi periodic and chaotic orbits for all boundary conditions we select among all the extremum solutions of the energy of the model those which correspond to the ground states of the infinite system we prove that these ground states are either periodic commensurate or quasi periodic incommensurate but are never chaotic we also prove the existence of elementary discommensurations which are minimum energy configuration of the model for certain special boundary conditions the topological structure of the whole set of ground states is described in details in addition to physical applications consequences for twist map homeomorphisms are mentioned part ii s aubry p y ledaeron and g andre will be mostly devoted to exact results on the transition by breaking of analyticity which occurs on the incommensurate ground states when the model parameters vary and on its connection with the stochasticity threshold in the corresponding twist map

asymptotic inference in multiple threshold double autoregressive models abstract this paper investigates a class of multiple threshold models called multiple threshold double ar mtdar models a sufficient condition is obtained for the existence and uniqueness of a strictly stationary and ergodic solution to the first order mtdar model we study the quasi maximum likelihood estimator qmle of the mtdar model the estimated thresholds are shown to be n consistent asymptotically independent and to converge weakly to the smallest minimizer of a two sided compound poisson process the remaining parameters are n consistent and asymptotically multivariate normal in particular these results apply to the multiple threshold arch model with or without ar part and to the multiple threshold ar models with arch errors a score based test is also presented to determine the number of thresholds in mtdar models the limiting distribution is shown to be distribution free and is easy to implement in practice simulation studies are conducted to assess the performance of the qmle and our score based test in finite samples the results are illustrated with an application to the quarterly us real gnp data over the period 1947 2013

critical behavior of propagation on small world networks we report numerical evidence that an epidemiclike model which can be interpreted as the propagation of a rumor exhibits critical behavior at a finite randomness of the underlying small world network the transition occurs between a regime where the rumor dies in a small neighborhood of its origin and a regime where it spreads over a finite fraction of the whole population critical exponents are evaluated through finite size scaling analysis and the dependence of the critical randomness with the network connectivity is studied the behavior of this system as a function of the small network randomness bears noticeable similarities with an epidemiological model reported recently m kuperman and g abramson phys rev lett 86 2909 2001 in spite of substantial differences in the respective dynamical rules

tracking invariant manifolds with differential forms in singularly perturbed systems abstract a technique is developed for finding homoclinic orbits in singularly perturbed systems the desired orbit lies near a singular solution that has both fast and slow time scales the orbit is to be constructed as the transverse intersection of a center stable and center unstable manifold which requires tracking the invariant manifolds near the singular solution the main technical advance is the exchange lemma which allows one to convert information about transversality of manifolds associated with the singular limit equations to information about behavior of the invariant manifold as it leaves a neighborhood of a slow segment of the singular solution the computations for this lemma are done with the use of differential forms and the lemma is applied to prove the existence of a homoclinic orbit in a class of singularly perturbed equations

the existence of localized vegetation patterns in a systematically reduced model for dryland vegetation abstract in this paper we consider the 2 component reaction diffusion model that was recently obtained by a systematic reduction of the 3 component gilad et al model for dryland ecosystem dynamics gilad et al 2004 the nonlinear structure of this model is more involved than other more conceptual models such as the extended klausmeier model and the analysis a priori is more complicated however the present model has a strong advantage over these more conceptual models in that it can be more directly linked to ecological mechanisms and observations moreover we find that the model exhibits a richness of analytically tractable patterns that exceeds that of klausmeier type models our study focuses on the 4 dimensional dynamical system associated with the reaction diffusion model by considering traveling waves in 1 spatial dimension we use the methods of geometric singular perturbation theory to establish the existence of a multitude of heteroclinic homoclinic periodic orbits that jump between normally hyperbolic slow manifolds representing various kinds of localized vegetation patterns the basic 1 front invasion patterns and 2 front spot gap patterns that form the starting point of our analysis have a direct ecological interpretation and appear naturally in simulations of the model by exploiting the rich nonlinear structure of the model we construct many multi front patterns that are novel both from the ecological and the mathematical point of view in fact we argue that these orbits patterns are not specific for the model considered here but will also occur in a much more general singularly perturbed reaction diffusion setting we conclude with a discussion of the ecological and mathematical implications of our findings

limiting properties of the least squares estimator of a continuous threshold autoregressive model the continuous threshold autoregressive model is a sub class of the threshold autoregressive model subject to the requirement that the piece wise linear autoregressive function be continuous everywhere in contrast with the discontinuous case it is shown that under suitable regularity conditions the conditional least squares estimator of the parameters including the threshold parameter is root n consistent and asymptotically normally distributed the theory is illustrated by a simulation study and is applied to the quarterly u s unemployment rates

the maki thompson rumor model on infinite cayley trees in this paper we study the maki thompson rumor model on infinite cayley trees the basic version of the model is defined by assuming that a population represented by a graph is subdivided into three classes of individuals ignorants spreaders and stiflers a spreader tells the rumor to any of its nearest ignorant neighbors at rate one at the same rate a spreader becomes a stifler after a contact with other nearest neighbor spreaders or stiflers in this work we study this model on infinite cayley trees which is formulated as a continuous times markov chain and we extend our analysis to the generalization in which each spreader ceases to propagate the rumor right after being involved in a given number of stifling experiences we study sufficient conditions under which the rumor either becomes extinct or survives with positive probability

bifurcations to travelling planar spots in a three component fitzhugh nagumo system in this article we analyse bifurcations from stationary stable spots to travelling spots in a planar three component fitzhugh nagumo system that was proposed previously as a phenomenological model of gas discharge systems by combining formal analyses center manifold reductions and detailed numerical continuation studies we show that in the parameter regime under consideration the stationary spot destabilizes either through its zeroth fourier mode in a hopf bifurcation or through its first fourier mode in a pitchfork or drift bifurcation whilst the remaining fourier modes appear to create only secondary bifurcations pitchfork bifurcations result in travelling spots and we derive criteria for the criticality of these bifurcations our main finding is that supercritical drift bifurcations leading to stable travelling spots arise in this model which does not seem possible for its two component version

threshold autoregressive models for interval valued time series data abstract modeling and forecasting symbolic data especially interval valued time series its data has received considerable attention in statistics and related fields the core of available methods on its analysis is based on various applications of conventional linear modeling however few works have considered possible nonlinearities in its data in this paper we propose a new class of threshold autoregressive interval tari models for its data by matching the interval model with interval observations we develop a minimum distance estimation method for tari models and establish the asymptotic theory for the proposed estimators we show that the threshold parameter estimator is t consistent and follows an asymptotic compound poisson process as the sample size t and the estimators for other tari model parameters are root t consistent and asymptotically normal simulation studies show that the proposed tari model provides more accurate out of sample forecasts than the existing center radius self exciting threshold cr setar model for its data in the literature empirical applications to the s p 500 price index document significant asymmetric reactions of the stock markets in japan u k and france to shocks from the u s stock market and that incorporating this asymmetric effect yield better out of sample forecasts than a variety of popular models available in the literature

orbits homoclinic to resonances with an application to chaos in a model of the forced and damped sine gordon equation abstract in this paper we develop new global perturbation techniques for detecting homoclinic and heteroclinic orbits in a class of four dimensional ordinary differential equations that are perturbations of completely integrable two degree of freedom hamiltonian systems our methods are fundamentally different than other global perturbation methods e g standard melnikov theory in that we are seeking orbits homoclinic and heteroclinic to fixed points that are created in a resonance resulting from the perturbation our methods combine the higher dimensional melnikov theory with geometrical singular perturbation theory and the theory of foliations of invariant manifolds we apply our methods to a modified model of the forced and damped sine gordon equation developed by bishop et al we give explicit conditions in terms of the system parameters for the model to possess a symmetric pair of homoclinic orbits to a fixed point of saddle focus type chaotic dynamics follow from a theorem of silnikov this provides a mechanism for chaotic dynamics geometrically similar to that observed by bishop et al namely a random jumping between two spatially dependent states with an intermediate passage through a spatially independent state however in order for this type of silnikov dynamics to exist we require a different and unphysical type of damping compared to that used by bishop et al

on the least squares estimation of multiple regime threshold autoregressive models abstract this paper studies the least squares estimator lse of the multiple regime threshold autoregressive tar model and establishes its asymptotic theory it is shown that the lse is strongly consistent when the autoregressive function is discontinuous over each threshold the estimated thresholds are n consistent and asymptotically independent each of which converges weakly to the smallest minimizer of a one dimensional two sided compound poisson process the remaining parameters are n consistent and asymptotically normal the theory of chan 1993 is revisited and a numerical approach is proposed to simulate the limiting distribution of the estimated threshold via simulating a related compound poisson process based on the numerical result one can construct a confidence interval for the unknown threshold this issue is not straightforward and has remained as an open problem since the publication of chan 1993 this paper provides not only a solution to this long standing open problem but also provides methodological contributions to threshold models simulation studies are conducted to assess the performance of the lse in finite samples the results are illustrated with an application to the quarterly u s real gnp data over the period 1947 2009

orbits homoclinic to resonances the hamiltonian case abstract in this paper we develop methods to show the existence of orbits homoclinic or heteroclinic to periodic orbits hyperbolic fixed points or combinations of hyperbolic fixed points and or periodic orbits in a class of two degree of freedom integrable hamiltonian systems subject to arbitrary hamiltonian perturbations our methods differ from previous methods in that the invariant sets periodic orbits fixed points are created and become hyperbolic as a result of the interaction of the perturbation with a resonance in the unperturbed system this results in a very degenerate situation that requires a combination of geometric singular perturbation theory higher dimensional melnikov type methods and transversality theory we establish a simple energy phase criterion which gives a fairly complete picture of the complex dynamics associated with orbits homoclinic to the resonance we apply our methods to a two mode truncation of the driven nonlinear schrodinger equation first studied by bishop et al in this example we show that as the energy is increased at resonance orbits homoclinic to hyperbolic periodic orbits are created in pairs in a global bifurcation that is best described as a saddle node bifurcation of homoclinic orbits

homoclinic orbits near heteroclinic cycles with one equilibrium and one periodic orbit we analyze homoclinic orbits near codimension 1 and 2 heteroclinic cycles between an equilibrium and a periodic orbit for ordinary differential equations in three or higher dimensions the main motivation for this study is a self organized periodic replication process of travelling pulses which has been observed in reaction diffusion equations we establish conditions for existence and uniqueness of countably infinite families of curve segments of 1 homoclinic orbits which accumulate at codimension 1 or 2 heteroclinic cycles the main result shows the bifurcation of a number of curves of 1 homoclinic orbits from such codimension 2 heteroclinic cycles which depends on a winding number of the transverse set of heteroclinic points in addition a leading order expansion of the associated curves in parameter space is derived its coefficients are periodic with one frequency from the imaginary part of the leading stable floquet exponents of the periodic orbit and one from the winding number

adaptive estimation of the threshold point in threshold regression this paper studies semiparametric efficient estimation of the threshold point in threshold regression the classical literature of semiparametric efficient estimation rests on the fact that the maximum likelihood estimator is efficient in any parametric submodel for a large class of loss functions however in threshold regression the maximum likelihood estimator is not efficient while the bayes estimators are efficient and different loss functions induce different efficient estimators for an additively separable loss function that separates the efficiency problem of the threshold point from that of other parameters we show that the semiparametric and parametric efficiency risk bounds coincide then we design a semiparametric empirical bayes estimator to achieve this bound in consequence the threshold point can be adaptively estimated even under conditional moment restrictions we also provide a valid confidence interval called the nonparametric posterior interval for the threshold point simulation studies show that the semiparametric empirical bayes approach is substantially better than existing methods to illustrate our procedure in practice we apply it to an economic growth model for detecting different growth patterns

phase transition for the maki thompson rumour model on a small world network we consider the maki thompson model for the stochastic propagation of a rumour within a population in this model the population is made up of spreaders ignorants and stiflers any spreader attempts to pass the rumour to the other individuals via pair wise interactions and in case the other individual is an ignorant it becomes a spreader while in the other two cases the initiating spreader turns into a stifler in a finite population the process will eventually reach an equilibrium situation where individuals are either stiflers or ignorants we extend the original hypothesis of homogenously mixed population by allowing for a small world network embedding the model in such a way that interactions occur only between nearest neighbours this structure is realized starting from a k regular ring and by inserting in the average c additional links in such a way that k and c are tuneable parameters for the population architecture we prove that this system exhibits a transition between regimes of localization where the final number of stiflers is at most logarithmic in the population size and propagation where the final number of stiflers grows algebraically with the population size at a finite value of the network parameter c a quantitative estimate for the critical value of c is obtained via extensive numerical simulations

consistency and limiting distribution of the least squares estimator of a threshold autoregressive model it is shown that under some regularity conditions the least squares estimator of a stationary ergodic threshold autoregressive model is strongly consistent the limiting distribution of the least squares estimator is derived it is shown that the estimator of the threshold parameter is n consistent and its limiting distribution is related to a compound poisson process

multi bump orbits homoclinic to resonance bands we establish the existence of several classes of multi bump orbits homoclinic to resonance bands for completely integrable hamiltonian systems subject to small amplitude hamiltonian or dissipative perturbations each bump is a fast excursion away from the resonance band and the bumps are interspersed with slow segments near the resonance band the homoclinic orbits which include multi bump silnikov orbits connect equilibria and periodic orbits in the resonance band the main tools we use in the existence proofs are the exchange lemma with exponentially small error and the existence theory of orbits homoclinic to resonance bands which make only one fast excursion away from the resonance bands

limit theorems for a general stochastic rumour model we study a general stochastic rumour model in which an ignorant individual has a certain probability of becoming a stifler immediately upon hearing the rumour we refer to this special kind of stifler as an uninterested individual our model also includes distinct rates for meetings between two spreaders in which both become stiflers or only one does so that particular cases are the classical daley kendall and maki thompson models we prove a law of large numbers and a central limit theorem for the proportions of those who ultimately remain ignorant and those who have heard the rumour but become uninterested in it

a spatial stochastic model for rumor transmission we consider an interacting particle system representing the spread of a rumor by agents on the d dimensional integer lattice each agent may be in any of the three states belonging to the set 0 1 2 here 0 stands for ignorants 1 for spreaders and 2 for stiflers a spreader tells the rumor to any of its nearest ignorant neighbors at rate at rate a spreader becomes a stifler due to the action of other nearest neighbor spreaders finally spreaders and stiflers forget the rumor at rate one we study sufficient conditions under which the rumor either becomes extinct or survives with positive probability

testing and modeling threshold autoregressive processes abstract the threshold autoregressive model is one of the nonlinear time series models available in the literature it was first proposed by tong 1978 and discussed in detail by tong and lim 1980 and tong 1983 the major features of this class of models are limit cycles amplitude dependent frequencies and jump phenomena much of the original motivation of the model is concerned with limit cycles of a cyclical time series and indeed the model is capable of producing asymmetric limit cycles the threshold autoregressive model however has not received much attention in application this is due to a the lack of a suitable modeling procedure and b the inability to identify the threshold variable and estimate the threshold values the primary goal of this article therefore is to suggest a simple yet widely applicable model building procedure for threshold autoregressive models based on some predictive residuals a simple statistic is proposed to test for threshold nonlinearity and specify the

assessing the robustness of spatial pattern sequences in a dryland vegetation model a particular sequence of patterns gaps labyrinth spots occurs with decreasing precipitation in previously reported numerical simulations of partial differential equation dryland vegetation models these observations have led to the suggestion that this sequence of patterns can serve as an early indicator of desertification in some ecosystems because parameter values in the vegetation models can take on a range of plausible values it is important to investigate whether the pattern sequence prediction is robust to variation for a particular model we find that a quantity calculated via bifurcation theoretic analysis appears to serve as a proxy for the pattern sequences that occur in numerical simulations across a range of parameter values we find in further analysis that the quantity takes on values consistent with the standard sequence in an ecologically relevant limit of the model parameter values this suggests that the standard sequence is a robust prediction of the model and we conclude by proposing a methodology for assessing the robustness of the standard sequence in other models and formulations

estimation and model selection based inference in single and multiple threshold models this paper evaluates the properties of a joint and sequential estimation procedure for estimating the parameters of single and multiple threshold models we initially proceed under the assumption that the number of regimes is known a priori but subsequently relax this assumption via the introduction of a model selection based procedure that allows the estimation of both the unknown parameters and their number to be performed jointly theoretical properties of the resulting estimators are derived and their finite sample properties investigated

a smoothed least squares estimator for threshold regression models abstract we propose a smoothed least squares estimator of the parameters of a threshold regression model our model generalizes that considered in hansen 2000 sample splitting and threshold estimation econometrica 68 575 603 to allow the thresholding to depend on a linear index of observed regressors thus allowing discrete variables to enter we also do not assume that the threshold effect is vanishingly small our estimator is shown to be consistent and asymptotically normal thus facilitating standard inference techniques based on estimated standard errors or standard bootstrap for the slope and threshold parameters

shadowing lemma and singularly perturbed boundary value problems a complete procedure is given to determine the outer and inner expansions of a singularly perturbed boundary value problem in mathbb r n the validity of such expansions is deduced from a gene

estimation of multiple regime threshold autoregressive models with structural breaks the threshold autoregressive tar model is a class of nonlinear time series models that have been widely used in many areas due to its nonlinear nature one major difficulty in fitting a tar model is the estimation of the thresholds as a first contribution this article develops an automatic procedure to estimate the number and values of the thresholds as well as the corresponding ar order and parameter values in each regime these parameter estimates are defined as the minimizers of an objective function derived from the minimum description length mdl principle a genetic algorithm ga is constructed to efficiently solve the associated minimization problem the second contribution of this article is the extension of this framework to piecewise tar modeling that is the time series is partitioned into different segments for which each segment can be adequately modeled by a tar model while models from adjacent segments are different for such piecewise tar modeling a procedure is developed to est

fitting an error distribution in some heteroscedastic time series models this paper addresses the problem of fltting a known distribution to the innovation distribution in a class of stationary and ergodic time series models the asymptotic null distribution of the usual kolmogorov smirnov test based on the residuals generally depends on the underlying model parameters and the error distribution to overcome the dependence on the underlying model parameters we propose that tests be based on a vector of certain weighted residual empirical processes under the null hypothesis and under minimal moment conditions this vector of processes is shown to converge weakly to a vector of independent copies of a gaussian process whose covariance function depends only on the fltted distribution and not on the model under certain local alternatives the proposed test is shown to have nontrivial asymptotic power the monte carlo critical values of this test are tabulated when fltting standard normal and double exponential distributions the results obtained are shown to be applicable to garch and arma garch models the often used models in econometrics and flnance a simulation study shows that the test has satisfactory size and power for flnite samples at these models the paper also contains an asymptotic uniform expansion result for a general weighted residual empirical process useful in heteroscedastic models under minimal moment conditions a result of independent interest

local and global behavior near homoclinic orbits we study the local behavior of systems near homoclinic orbits to stationary points of saddle focus type we explicitly describe how a periodic orbit approaches homoclinicity and with the help of numerical examples discuss how these results relate to global patterns of bifurcations

sample splitting and threshold estimation threshold models have a wide variety of applications in economics direct applications include models of separating and multiple equilibria other applications include empirical sample splitting when the sample split is based on a continuously distributed variable such as firm size in addition threshold models may be used as a parsimonious strategy for nonparametric function estimation for example the threshold autoregressive model tar is popular in the nonlinear time series literature threshold models also emerge as special cases of more complex statistical frameworks such as mixture models switching models markov switching models and smooth transition threshold models it may be important to understand the statistical properties of threshold models as a preliminary step in the development of statistical tools to handle these more complicated structures despite the large number of potential applications the statistical theory of threshold estimation is undeveloped it is known that threshold estimates are super consistent but a distribution theory useful for testing and inference has yet to be provided this paper develops a statistical theory for threshold estimation in the regression context we allow for either cross section or time series observations least squares estimation of the regression parameters is considered an asymptotic distribution theory for the regression estimates the threshold and the regression slopes is developed it is found that the distribution of the threshold estimate is nonstandard a method to construct asymptotic confidence intervals is developed by inverting the likelihood ratio statistic it is shown that this yields asymptotically conservative confidence regions monte carlo simulations are presented to assess the accuracy of the asymptotic approximations the empirical relevance of the theory is illustrated through an application to the multiple equilibria growth model of durlauf and johnson 1995

panel threshold regressions with latent group structures abstract in this paper we consider the least squares estimation of a panel structure threshold regression pstr model where both the slope coefficients and threshold parameters may exhibit latent group structures we study the asymptotic properties of the estimators of the latent group structure and the slope and threshold coefficients we show that we can estimate the latent group structure correctly with probability approaching 1 and the estimators of the slope and threshold coefficients are asymptotically equivalent to the infeasible estimators that are obtained as if the true group structures were known we study likelihood ratio based inferences on the group specific threshold parameters under the shrinking threshold effect framework we also propose two specification tests one tests whether the threshold parameters are homogeneous across groups and the other tests whether the threshold effects are present when the number of latent groups is unknown we propose a bic type information criterion to determine the number of groups in the data simulations demonstrate that our estimators and tests perform reasonably well in finite samples we apply our model to revisit the relationship between capital market imperfection and the investment behavior of firms and to examine the impact of bank deregulation on income inequality we document a large degree of heterogeneous effects in both applications that cannot be captured by conventional panel threshold regressions

transversal heteroclinic and homoclinic orbits in singular perturbation problems abstract in this paper we give a geometric construction of heteroclinic and homoclinic orbits for singularly perturbed differential equations by using methods from invariant manifold theory we show that transversal intersection of stable and unstable manifolds of the reduced problem implies the existence of transversal heteroclinic or homoclinic orbits of the singularly perturbed problem we derive analytical conditions for transversality we show how these results can be used to prove the existence of heteroclinic and homoclinic orbits in singularly perturbed problems which depend on additional parameters we describe a configuration which implies transversal intersection of the stable and unstable manifolds of periodic orbits and the associated chaotic dynamics

invariant manifolds and singularly perturbed boundary value problems boundary value problems for singularly perturbed equations often have singular solutions with internal and boundary layers this paper addresses the question of when singular solutions to such prob

theory of rumour spreading in complex social networks we introduce a general stochastic model for the spread of rumours and derive mean field equations that describe the dynamics of the model on complex social networks in particular those mediated by the internet we use analytical and numerical solutions of these equations to examine the threshold behaviour and dynamics of the model on several models of such networks random graphs uncorrelated scale free networks and scale free networks with assortative degree correlations we show that in both homogeneous networks and random graphs the model exhibits a critical threshold in the rumour spreading rate below which a rumour cannot propagate in the system in the case of scale free networks on the other hand this threshold becomes vanishingly small in the limit of infinite system size we find that the initial rate at which a rumour spreads is much higher in scale free networks than in random graphs and that the rate at which the spreading proceeds on scale free networks is further increased when assortative degree correlations are introduced the impact of degree correlations on the final fraction of nodes that ever hears a rumour however depends on the interplay between network topology and the rumour spreading rate our results show that scale free social networks are prone to the spreading of rumours just as they are to the spreading of infections they are relevant to the spreading dynamics of chain emails viral advertising and large scale information dissemination algorithms on the internet

a large deviations principle for the maki thompson rumour model abstract we consider the stochastic model for the propagation of a rumour within a population which was formulated by maki and thompson 20 sudbury 22 established that as the population size tends to infinity the proportion of the population never hearing the rumour converges in probability to 0 2032 watson 23 later derived the asymptotic normality of a suitably scaled version of this proportion we prove a corresponding large deviations principle with an explicit formula for the rate function

subsampling inference in threshold autoregressive models this paper discusses inference in self exciting threshold autoregressive setar models of main interest is inference for the threshold parameter it is well known that the asymptotics of the corresponding estimator depend upon whether the setar model is continuous or not in the continuous case the limiting distribution is normal and standard inference is possible in the discontinuous case the limiting distribution is non normal and it is not known how to estimate it consistently we show that valid inference can be drawn by the use of the subsampling method moreover the method can even be extended to situations where the dis continuity of the model is unknown in this case the inference for the regression parameters of the model also becomes difficult and subsampling can be used again in addition we consider an hypothesis test for the continuity of a setar model a simulation study examines small sample performance and an application illustrates how the proposed methodology works in practice

lyapunov schmidt reduction for unfolding heteroclinic networks of equilibria and periodic orbits with tangencies abstract this article concerns arbitrary finite heteroclinic networks in any phase space dimension whose vertices can be a random mixture of equilibria and periodic orbits in addition tangencies in the intersection of un stable manifolds are allowed the main result is a reduction to algebraic equations of the problem to find all solutions that are close to the heteroclinic network for all time and their parameter values a leading order expansion is given in terms of the time spent near vertices and if applicable the location on the non trivial tangent directions the only difference between a periodic orbit and an equilibrium is that the time parameter is discrete for a periodic orbit the essential assumptions are hyperbolicity of the vertices and transversality of parameters using the result conjugacy to shift dynamics for a generic homoclinic orbit to a periodic orbit is proven finally equilibrium to periodic orbit heteroclinic cycles of various types are considered

melnikov s method and arnold diffusion for perturbations of integrable hamiltonian systems we start with an unperturbed system containing a homoclinic orbit and at least two families of periodic orbits associated with action angle coordinates we use kolmogorov arnold moser kam theory to show that some of the resulting tori persist under small perturbations and use a vector of melnikov integrals to show that under suitable hypotheses their stable and unstable manifolds intersect transversely this transverse intersection is ultimately responsible for arnold diffusion on each energy surface the method is applied to a pendulum oscillator system

hysteretic autoregressive time series models this paper extends the classical two regime threshold autoregressive model by introducing hysteresis to its regime switching structure which leads to a new model the hysteretic autoregressive model the proposed model enjoys the piecewise linear structure of a threshold model but has a more flexible regime switching mechanism a sufficient condition is given for geometric ergodicity conditional least squares estimation is discussed and the asymptotic distributions of its estimators and information criteria for model selection are derived simulation results and an example support the model