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.

We discuss procedures based on nonparametric regression for estimating the dominant Lyapunov exponent Al from time-series data generated by a system x t

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In the previous three chapters we have developed a theory which can be applied to calculate the nonlinear response of an arbitrary phase variable to an applied external field. We have described several different representations for the-particle, nonequilibrium distribution function, : the Kubo representation

In the previous three chapters we have developed a theory which can be applied to calculate the nonlinear response of an arbitrary phase variable to an applied external field. We have described several different representations for the N-particle, nonequilibrium distribution function, f (Γ, t): the Kubo representation (§7.1) which is

We consider the period-doubling and intermittency transitions in iterated nonlinear one-dimensional maps to corroborate unambiguously the validity of Tsallis ’ non-extensive statistics at these critical points. We study the map xn+1 = xn + u |xn | z, z> 1, as it describes generically the neighborhood of all of these transitions. The exact renormalization group (RG) fixed-point map and perturbation static expressions match the corresponding expressions for the dynamics of iterates. The time evolution is universal in the RG sense and the nonextensive entropy SQ associated to the fixed-point map is maximum with respect to that of the other maps in its basin of attraction. The degree of non-extensivity- the index Q in SQ- and the degree of nonlinearity z are equivalent and the generalized Lyapunov exponent λq, q = 2 − Q −1, is the leading map expansion coefficient u. The corresponding deterministic diffusion problem is similarly interpreted. We discuss our results. 1