This paper investigates the dynamics in a simple present discounted value asset pricing model with heterogeneous beliefs. Agents choose from a finite set of predictors of future prices of a risky asset and revise their `beliefs' in each period in a boundedly rational way, according to a `fitness measure' such as past realized profits. Price fluctuations are thus driven by an evolutionary dynamics between different expectation schemes (`rational animal spirits'). Using a mixture of local bifurcation theory and numerical methods, we investigate possible bifurcation routes to complicated asset price dynamics. In particular, we present numerical evidence of strange, chaotic attractors when the intensity of choice to switch prediction strategies is high.

The analysis of uni- or multivariate time series provides crucial information to describe, understand, and predict climatic variability. The discovery and implementation of a number of novel methods for extracting useful information from time series has recently revitalized this classical eld of study. Considerable progress has also been made in interpreting the information so obtained in terms of dynamical systems theory.

. Matrix refinement equations are functional equations of the form f(x) = P N k=0 c k f(2x \Gamma k), where the coefficients c k are matrices and f is a vector-valued function. Refinement equations play key roles in wavelet theory and approximation theory. Existence and uniqueness properties of scalar refinement equations (where the coefficients c k are scalars) are known. This paper considers analogous questions for matrix refinement equations. Conditions for existence and uniqueness of compactly supported distributional solutions are given in terms of the convergence properties of an infinite product of the matrix \Delta = 1 2 P c k with itself. Fundamental differences between solutions of matrix equations and scalar refinement equations are examined. In particular, it is shown that "constrained" solutions of the matrix refinement equation can exist even when the infinite product diverges. The existence of constrained solutions is related to the eigenvalue structure of \Delta; so...

In this paper, we consider discrete and continuous QR algorithms for computing all of the Lyapunov exponents of a regular dynamical system. We begin by reviewing theoretical results for regular systems and present general perturbation results for Lyapunov exponents. We then present the algorithms, give an error analysis of them, and describe their implementation. Finally, we give several numerical examples and some conclusions.

In the past twenty years there has been much interest in the physical and biological sciences in nonlinear dynamical systems that appear to have random, unpredictable behavior. One important parameter of a dynamic system is the dominant Lyapunov exponent (LE). When the behavior of the system is compared for two similar initial conditions, this exponent is related to the rate at which the subsequent trajectories diverge. A bounded system with a positive LE is one operational definition of chaotic behavior. Most methods for determining the LE have assumed thousands of observations generated from carefully controlled physical experiments. Less attention has been given to estimating the LE for biological and economic systems that are subjected to random perturbations and observed over a limited amount of time. Using nonparametric regression techniques (Neural Networks and Thin Plate Splines) it is possible to consistently estimate the LE. The properties of these methods have been studied using simulated data and are applied to a biological time series: marten fur returns for the Hudson Bay Company (1820-1900). Based on a nonparametric analysis there is little evidence for lowdimensional chaos in these data. Although these methods appear to work well for systems perturbed by small amounts of noise, finding chaos in a system with a significant stochastic component may be difficult.

Detecting the presence of chaos in a dynamical system is an important problem that is solved by measuring the largest Lyapunov exponent. Lyapunov exponents quantify the exponential divergence of initially close state-space trajectories and estimate the amount of chaos in a system. We present a new method for calculating the largest Lyapunov exponent from an experimental time series. The method follows directly from the definition of the largest Lyapunov exponent and is accurate because it takes advantage of all the available data. We show that the algorithm is fast, easy to implement, and robust to changes in the following quantities: embedding dimension, size of data set, reconstruction delay, and noise level. Furthermore, one may use the algorithm to calculate simultaneously the correlation dimension. Thus, one sequence of computations will yield an estimate of both the level of chaos and the system complexity.

: We compare two theoretically distinct approaches to generating artificial (or "surrogate") data for testing hypotheses about a given data set. The first and more straightforward approach is to fit a single "best" model to the original data, and then to generate surrogate data sets that are "typical realizations" of that model. The second approach concentrates not on the model but directly on the original data; it attempts to constrain the surrogate data sets so that they exactly agree with the original data for a specified set of sample statistics. Examples of these two approaches are provided for two simple cases: a test for deviations from a gaussian distribution, and a test for serial dependence in a time series. Additionally, we consider tests for nonlinearity in time series based on a Fourier transform (FT) method and on more conventional autoregressive moving-average (ARMA) fits to the data. The comparative performance of hypothesis testing schemes based on these two approaches...

Abstract: Interest has been growing in testing for nonlinearity or chaos in economic data, but much controversy has arisen about the available results. This paper explores the reasons for these empirical difficulties. We designed and ran a single-blind controlled competition among five highly regarded tests for nonlinearity or chaos with ten simulated data series. The data generating mechanisms include linear processes, chaotic recursions, and nonchaotic stochastic processes; and both large and small samples were included in the experiment. The data series were produced in a single blind manner by the competition manager and sent by e-mail, without identifying information, to the experiment participants. Each such participant is an acknowledged expert in one of the tests and has a possible vested interest in producing the best possible results with that one test. The results of this competition provide much surprising information about the power functions of some of the best regarded tests for nonlinearity or noisy chaos.

In this paper, the problem of estimating the intrinsic dimension of a data set is investigated. A fractal-based approach using the Grassberger-Procaccia algorithm is proposed. Since the Grassberger-Procaccia algorithm performs badly on sets of high dimensionality, an empirical procedure, that improves the original algorithm, has been developed. The procedure has been tested on data sets of known dimensionality and on time series of Santa Fe competition.

Let Xn = fx 1 ; : : : ; xn g, be an i.i.d. sample having multivariate distribution P . We derive a.s. limits for the power weighted edge weight function of greedy approximations to a class of minimal graphs spanning k of the n samples. The class includes minimal k-point graphs constructed by the partitioning method of Ravi, Sundaram, Marathe, Rosenkrantz and Ravi [43] where the edge weight function satises the quasi-additive property of Redmond and Yukich [45]. In particular this includes greedy approximations to the k-point minimal spanning tree (k-MST), Steiner tree (k-ST), and the traveling salesman problem (k-TSP). An expression for the inuence function of the minimal weight function is given which characterizes the asymptotic sensitivity of the graph weight to perturbations in the underlying distribution. The inuence function takes a form which indicates that the k-point minimal graph in d > 1 dimensions has robustness properties in IR d which are analogous to those of rank order statistics in one dimension. A direct result of our theory is that the log-weight of the k-point minimal graph is a consistent nonparametric estimate of the Renyi entropy of the distribution P . Possible applications of this work include: analysis of random communication network topologies, estimation of the mixing coecient in -contaminated mixture models, outlier discrimination and rejection, clustering and pattern recognition, robust non-parametric regression, two sample matching and image registration.