Results 1  10
of
17
Finding Chaos in Noisy Systems
, 1991
"... 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 comp ..."
Abstract

Cited by 50 (1 self)
 Add to MetaCart
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 (18201900). 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.
Estimating Lyapunov Exponents with Nonparametric Regression
, 1990
"... We discuss procedures based on nonparametric regression for estimating the dominant Lyapunov exponent Al from timeseries data generated by a system x t ..."
Abstract

Cited by 11 (1 self)
 Add to MetaCart
We discuss procedures based on nonparametric regression for estimating the dominant Lyapunov exponent Al from timeseries data generated by a system x t
Critical fluctuations, intermittent dynamics and Tsallis statistics
, 2004
"... It is pointed out that the dynamics of the order parameter at a thermal critical point obeys the precepts of the nonextensive Tsallis statistics. We arrive at this conclusion by putting together two welldefined statisticalmechanical developments. The first is that critical fluctuations are correctl ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
It is pointed out that the dynamics of the order parameter at a thermal critical point obeys the precepts of the nonextensive Tsallis statistics. We arrive at this conclusion by putting together two welldefined statisticalmechanical developments. The first is that critical fluctuations are correctly described by the dynamics of an intermittent nonlinear map. The second is that intermittency in the neighborhood of a tangent bifurcation in such map rigorously obeys nonextensive statistics. We comment on the implications of this result. Key words: critical fluctuations, intermittency, nonextensive statistics, anomalous stationary states PACS: 64.60.Ht, 75.10.Hk, 05.45.a, 05.10.Cc 1
Criticality in nonlinear onedimensional maps: RG universal map and nonextensive entropy
, 2003
"... We consider the perioddoubling and intermittency transitions in iterated nonlinear onedimensional maps to corroborate unambiguously the validity of Tsallis ’ nonextensive statistics at these critical points. We study the map xn+1 = xn + u xn  z, z> 1, as it describes generically the neighborhoo ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
We consider the perioddoubling and intermittency transitions in iterated nonlinear onedimensional maps to corroborate unambiguously the validity of Tsallis ’ nonextensive 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) fixedpoint 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 fixedpoint map is maximum with respect to that of the other maps in its basin of attraction. The degree of nonextensivity 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
10. Towards a Thermodynamics of Steady States
"... 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 theparticle, nonequilibrium distribution function, : the Kubo repre ..."
Abstract
 Add to MetaCart
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 theparticle, nonequilibrium distribution function, : the Kubo representation
10 TOWARDS A THERMODYNAMICS OF STEADY STATES 251 10. Towards a Thermodynamics of Steady States
"... 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 Nparticle, nonequilibrium distribution function, f (Γ, t): the ..."
Abstract
 Add to MetaCart
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 Nparticle, nonequilibrium distribution function, f (Γ, t): the Kubo representation (§7.1) which is
National Library of Australia
"... mechanics of nonequilibrium liquids statistical mechanics of nonequilibrium liquids ..."
Abstract
 Add to MetaCart
mechanics of nonequilibrium liquids statistical mechanics of nonequilibrium liquids
Intermittency at critical transitions and aging dynamics at edge of chaos
, 2005
"... We recall that, at both the intermittency transitions and at the Feigenbaum attractor in unimodal maps of nonlinearity of order ζ> 1, the dynamics rigorously obeys the Tsallis statistics. We account for the qindices and the generalized Lyapunov coefficients λq that characterize the universality cl ..."
Abstract
 Add to MetaCart
We recall that, at both the intermittency transitions and at the Feigenbaum attractor in unimodal maps of nonlinearity of order ζ> 1, the dynamics rigorously obeys the Tsallis statistics. We account for the qindices and the generalized Lyapunov coefficients λq that characterize the universality classes of the pitchfork and tangent bifurcations. We identify the Mori singularities in the Lyapunov spectrum at the edge of chaos with the appearance of a special value for the entropic index q. The physical area of the Tsallis statistics is further probed by considering the dynamics near criticality and glass formation in thermal systems. In both cases a close connection is made with states in unimodal maps with vanishing Lyapunov coefficients.