fractal basin boundaries abstract basin boundaries for dynamical systems can be either smooth or fractal this paper investigates fractal basin boundaries one practical consequence of such boundaries is that they can lead to great difficulty in predicting to which attractor a system eventually goes the structure of fractal basin boundaries can be classified as being either locally connected or locally disconnected examples and discussion of both types of structures are given and it appears that fractal basin boundaries should be common in typical dynamical systems lyapunov numbers and the dimension for the measure generated by inverse orbits are also discussed

complexity in hamiltonian driven dissipative chaotic dynamical systems the existence of symmetry in chaotic dynamical systems often leads to one or several low dimensional invariant subspaces in the phase space we demonstrate that complex behaviors can arise when the dynamics in the invariant subspace is hamiltonian but the full system is dissipative in particular an infinite number of distinct attractors can coexist these attractors can be quasiperiodic strange nonchaotic and chaotic with different positive lyapunov exponents finite perturbations in initial conditions or parameters can lead to a change from nonchaotic attractors to chaotic attractors however arbitrarily small perturbations can lead to dynamically distinct chaotic attractors this work demonstrates that the interplay between conservative and dissipative dynamics can give rise to rich complexity even in physical systems with a few degrees of freedom

multi dimensioned intertwined basin boundaries basin structure of the kicked double rotor abstract using numerical computations on a map which describes the time evolution of a particular mechanical system in a four dimensional phase space the kicked double rotor we have found that the boundaries separating basins of attraction can have different properties in different regions and that these different regions can be intertwined on arbitrarily fine scale in particular for the double rotor map if one chooses a restricted region of the phase space and examines the basin boundary in that region then either one observes that the boundary is a smooth three dimensional surface or one observes that the boundary is fractal with dimension d 3 9 and which of these two possibilities applies depends on the particular phase space region chosen for examination furthermore for any region no matter how small for which d 3 9 one can choose subregions within it for which d 3 hence d 3 9 region and d 3 region are intertwined on arbitrarily fine scale other examples will also be presented and analyzed to show how this situation can arise these include one dimensional map cases a map of the plane and the lorenz equations in one of our one dimensional map cases the boundary will be fractal everywhere but the dimension can take on either of two different values both of which lie between 0 and 1 these examples lead us to conjecture that basin boundaries typically can have at most a finite number of possible dimension values more specifically let these values be denoted d 1 d 2 d n choose a volume region of phase space whose interior contains some part of the basin boundary and evaluate the dimension of the boundary in that region then our conjecture is that for all typical volume choices the evaluated dimension within the chosen volume will be one of the values d 1 d 2 d n for example in our double rotor map it appears that n 2 and d 1 3 0 and d 2 3 9

on the number of common factors with high frequency data summaryin this paper we introduce a local principal component analysis approach to determining the number of common factors of a continuous time factor model with time varying factor loadings using high frequency data the model is approximated locally on shrinking blocks using discrete time factor models the number of common factors is estimated by minimizing the penalized aggregated mean squared residual error over all shrinking blocks while the local mean squared residual error on each block converges at rate min n 1 4 p where n is the sample size and p is the dimension the aggregated mean squared residual error converges at rate min n 1 2 p this achieves the convergence rate of the penalized criterion function of the global principal component analysis method assuming restrictive constant factor loading an estimator of the number of factors based on the local principal component analysis is consistent simulation results justify the performance of our estimator a real financial dataset is analysed

determining the number of factors in approximate factor models in this paper we develop some econometric theory for factor models of large dimensions the focus is the determination of the number of factors r which is an unresolved issue in the rapidly growing literature on multifactor models we first establish the convergence rate for the factor estimates that will allow for consistent estimation of r we then propose some panel criteria and show that the number of factors can be consistently estimated using the criteria the theory is developed under the framework of large cross sections n and large time dimensions t no restriction is imposed on the relation between n and t simulations show that the proposed criteria have good finite sample properties in many configurations of the panel data encountered in practice

on time varying factor models estimation and testing abstract conventional factor models assume that factor loadings are fixed over a long horizon of time which appears overly restrictive and unrealistic in applications in this paper we introduce a time varying factor model where factor loadings are allowed to change smoothly over time we propose a local version of the principal component method to estimate the latent factors and time varying factor loadings simultaneously we establish the limiting distributions and uniform convergence of the estimated factors and factor loadings in the standard large n and large t framework we also propose a bic type information criterion to determine the number of factors which can be used in models with either time varying or time invariant factor models based on the comparison between the estimates of the common components under the null hypothesis of no structural changes and those under the alternative we propose a consistent test for structural changes in factor loadings we establish the null distribution the asymptotic local power property and the consistency of our test simulations are conducted to evaluate both our nonparametric estimates and test statistic we also apply our test to investigate stock and watson s 2009 u s macroeconomic data set and find strong evidence of structural changes in the factor loadings

the birth of strange nonchaotic attractors abstract a mechanism is described for the development of strange nonchaotic attractors from two frequency quasiperiodic attractors in quasiperiodically driven maps this mechanism is intimately tied to the phenomenon of torus doubling the transition to strange nonchaotic behavior occurs when a period doubled torus collides with its unstable parent torus as the collision is approached the period doubled torus becomes extremely wrinkled ultimately becoming fractal at the collision the lyapunov exponent remains negative through the collision these collisions are shown to be a new type of attractor merging crisis the new feature is the possibility of nonchaotic attractors taking part in the crisis this mechanism is illustrated via numerical and analytical studies of a quasiperiodically driven logistic map

testing against constant factor loading matrix with large panel high frequency data abstract in this paper we introduce a nonparametric test against the constancy of the factor loading matrix of a high dimensional continuous time factor model using high frequency data the central limit theorems on the test statistics with and without perturbation are established under the null hypothesis that the factor loading matrix is constant as time evolves the tests perform well in size and power interestingly the test statistic without perturbation converges at a rate that depends not only on the sample size but also on the dimension through the cross sectional dependence of the residual process which is a distinctive feature that contrasts with the low dimensional setting extensive numerical studies including monte carlo simulations and real data analysis validate the performance of our test

strange non chaotic attractor in a quasiperiodically forced circle map abstract we show that in the quasiperiodically forced circle map strange non chaotic attractors can appear for non linearities far from the border of chaos the destruction of a two frequency quasiperiodic torus connected with the appearance of a strange non chaotic attractor is described in detail this strange non chaotic attractor is characterized by logarithmically slow diffusion of the phase it is shown that in this regime the high order phase locking states disappear and the rotation number varies rather smoothly with the parameters

quasiperiodically forced dynamical systems with strange nonchaotic attractors abstract we discuss the existence and properties of strange nonchaotic attractors of quasiperiodically forced nonlinear dynamical systems we do this by examining a particular model differential equation cos e cos 2 t where is a two frequency quasiperiodic function of t when 0 the analysis of the equation is facilitated since then it can be related to the schrodinger equation with quasiperiodic potential we show that the equation does indeed exhibit strange nonchaotic attractors and we consider the important question of whether these attractors are typical in the sense that they exist on a set of positive lebesgue measure in parameter space the equation also exhibits two and three frequency quasiperiodic behavior we also show that the strange nonchaotic attractors have distinctive frequency spectrum this property might make them experimentally observable

a minimum variance of utility approach to ideal point estimation in certain biased groups abstract this paper discusses a problem of group decision making called ideal point estimation this problem is equivalent to all individuals of the group or a representative committee attempting to estimate the maximum point for an unknown group multiattribute value function the paper concentrates on a type of systematic error which is possible in such processes and proposes a method of estimation the minimum variance of utility method which is capable of debiasing such estimates in certain cases a model of group error called the dome error process is shown to provide a concrete instance of this type of systematic bias the method is illustrated numerically and some applications are discussed

designing realised kernels to measure the ex post variation of equity prices in the presence of noise this paper shows how to use realised kernels to carry out efficient feasible inference on the expost variation of underlying equity prices in the presence of simple models of market frictions the issue is subtle with only estimators which have symmetric weights delivering consistent estimators with mixed gaussian limit theorems the weights can be chosen to achieve the best possible rate of convergence and to have an asymptotic variance which is close to that of the maximum likelihood estimator in the parametric version of this problem realised kernels can also be selected to i be analysed using endogenously spaced data such as that in databases on transactions ii allow for market frictions which are endogenous iii allow for temporally dependent noise the finite sample performance of our estimators is studied using simulation while empirical work illustrates their use in practice

controlling chaotic dynamical systems abstract we describe a method that converts the motion on a chaotic attractor to a desired attracting time periodic motion by making only small time dependent perturbations of a control parameter the time periodic motion results from the stabilization of one of the infinite number of previously unstable periodic orbits embedded in the attractor the present paper extends that of ott grebogi and yorke phys rev lett 64 1990 1196 allowing for a more general choice of the feedback matrix and implementation to higher dimensional systems the method is illustrated by an application to the control of a periodically impulsively kicked dissipative mechanical system with two degrees of freedom resulting in a four dimensional map the double rotor map a key issue addressed is that of the dependence of the average time to achieve control on the size of the perturbations and on the choice of the feedback matrix

strange attractors that are not chaotic abstract it is shown that in certain types of dynamical systems it is possible to have attractors which are strange but not chaotic here we use the word strange to refer to the geometry or shape of the attracting set while the word chaotic refers to the dynamics of orbits on the attractor in particular the exponential divergence of nearby trajectories we first give examples for which it can be demonstrated that there is a strange nonchaotic attractor these examples apply to a class of maps which model nonlinear oscillators continuous time which are externally driven at two incommensurate frequencies it is then shown that such attractore are persistent under perturbations which preserve the original system type i e there are two incommensurate external driving frequencies this suggests that for systems of the typw which we have considered nonchaotic strange attractors may be expected to occur for a finite interval of parameter values on the other hand when small perturbations which do not preserve the system type are numerically introduced the strange nonchaotic attractor is observed to be converted to a periodic or chaotic orbit thus we conjecture that in general continuous time systems flows which are not externally driven at two incommensurate frequencies should not be expected to have strange nonchaotic attractors except possibly on a set of measure zero in the parameter space

a test for the rank of the volatility process the random perturbation approach in this paper we present a test for the maximal rank of the matrix valued volatility process in the continuous it o semimartingale framework our idea is based upon a random perturbation of the original high frequency observations of an it o semimartingale which opens the way for rank testing we develop the complete limit theory for the test statistic and apply it to various null and alternative hypotheses finally we demonstrate a homoscedasticity test for the rank process

the realized laplace transform of volatility we introduce and derive the asymptotic behavior of a new measure constructed from high frequency data which we call the realized laplace transform of volatility the statistic provides a nonparametric estimate for the empirical laplace transform function of the latent stochastic volatility process over a given interval of time and is robust to the presence of jumps in the price process with a long span of data that is under joint long span and infill asymptotics the statistic can be used to construct a nonparametric estimate of the volatility laplace transform as well as of the integrated joint laplace transform of volatility over different points of time we derive feasible functional limit theorems for our statistic both under fixed span and infill asymptotics as well as under joint long span and infill asymptotics which allow us to quantify the precision in estimation under both sampling schemes

the transition to chaotic attractors with riddled basins abstract recently it has been shown that there are chaotic attractors whose basins are such that any point in the basin has pieces of another attractor basin arbitrarily nearby the basin is riddled with holes here we consider the dynamics near the transition to this situation as a parameter is varied using a simple analyzable model we obtain the characteristic behaviors near this transition numerical tests on a more typical system are consistent with the conjecture that these results are universal for the class of systems considered

correlations and spectra of strange non chaotic attractors we consider correlations and spectra of strange nonchaotic attractors in quasiperiodically driven nonlinear systems it is demonstrated that a self similar autocorrelation function and a singular continuous spectrum may be observed in such systems the properties of correlations and spectra depend very subtly on the rotation number of the quasiperiodic force and on the parameters of the system for some parameter values the usual discrete spectrum is restored

controlling the false discovery rate a practical and powerful approach to multiple testing summary the common approach to the multiplicity problem calls for controlling the familywise error rate fwer this approach though has faults and we point out a few a different approach to problems of multiple significance testing is presented it calls for controlling the expected proportion of falsely rejected hypotheses the false discovery rate this error rate is equivalent to the fwer when all hypotheses are true but is smaller otherwise therefore in problems where the control of the false discovery rate rather than that of the fwer is desired there is potential for a gain in power a simple sequential bonferronitype procedure is proved to control the false discovery rate for independent test statistics and a simulation study shows that the gain in power is substantial the use of the new procedure and the appropriateness of the criterion are illustrated with examples

anova for diffusions and ito processes ito processes are the most common form of continuous semimartingales and include diffusion processes this paper is concerned with the nonparametric regression relationship between two such ito processes we are interested in the quadratic variation integrated volatility of the residual in this regression over a unit of time such as a day a main conceptual finding is that this quadratic variation can be estimated almost as if the residual process were observed the difference being that there is also a bias which is of the same asymptotic order as the mixed normal error term the proposed methodology anova for diffusions and ito processes can be used to measure the statistical quality of a parametric model and non parametrically the appropriateness of a one regressor model in general on the other hand it also helps quantify and characterize the trading hedging error in the case of financial applications

inferential theory for factor models of large dimensions this paper develops an inferential theory for factor models of large dimensions the principal components estimator is considered because it is easy to compute and is asymptotically equivalent to the maximum likelihood estimator if normality is assumed we derive the rate of convergence and the limiting distributions of the estimated factors factor loadings and common components the theory is developed within the framework of large cross sections n and a large time dimension t to which classical factor analysis does not apply we show that the estimated common components are asymptotically normal with a convergence rate equal to the minimum of the square roots of n and t the estimated factors and their loadings are generally normal although not always so the convergence rate of the estimated factors and factor loadings can be faster than that of the estimated common components these results are obtained under general conditions that allow for correlations and heteroskedasticities in both dimensions stronger results are obtained when the idiosyncratic errors are serially uncorrelated and homoskedastic a necessary and sufficient condition for consistency is derived for large n but fixed t