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Determining Lyapunov Exponents from a Time Series
 Physica
, 1985
"... We present the first algorithms that allow the estimation of nonnegative Lyapunov exponents from an experimental time series. Lyapunov exponents, which provide a qualitative and quantitative characterization of dynamical behavior, are related to the exponentially fast divergence or convergence of n ..."
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We present the first algorithms that allow the estimation of nonnegative Lyapunov exponents from an experimental time series. Lyapunov exponents, which provide a qualitative and quantitative characterization of dynamical behavior, are related to the exponentially fast divergence or convergence of nearby orbits in phase space. A system with one or more positive Lyapunov exponents is defined to be chaotic. Our method is rooted conceptually in a previously developed technique that could only be applied to analytically defined model systems: we monitor the longterm growth rate of small volume elements in an attractor. The method is tested on model systems with known Lyapunov spectra, and applied to data for the BelousovZhabotinskii reaction and CouetteTaylor flow. Contents 1.
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 ..."
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We discuss procedures based on nonparametric regression for estimating the dominant Lyapunov exponent Al from timeseries data generated by a system x t
Modulated waves in TaylorCouette flow Part 2. Numerical simulation
 J. Fluid Mech
, 1992
"... quasiperiodic fluid flows, with application to the flow between concentric cylinders (TaylorCouette flow). In this paper we present numerical simulations of TaylorCouette flow, motivated by laboratory experiments which show that periodic rotating waves become unstable to quasiperiodic modulated ..."
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quasiperiodic fluid flows, with application to the flow between concentric cylinders (TaylorCouette flow). In this paper we present numerical simulations of TaylorCouette flow, motivated by laboratory experiments which show that periodic rotating waves become unstable to quasiperiodic modulated waves as the Reynolds number is increased. We find that several branches of quasiperiodic solutions exist, and not all of them occur as direct bifurcations from rotating waves. We compute solutions to the NavierStokes equations for both rotating waves and two branches of modulated waves; those discussed by Gorman & Swinney (1979, 1982), and those discovered by Zhang & Swinney (1985). A simple physical process is associated with both types of modulation. In the rotating frame the quasiperiodic disturbance field (defined as the total flow minus the timeaveraged, steadystate piece) appears as a set of compact, coherent vortices confined almost entirely to the outflow jet region. They are advected (with some distortion) in the azimuthal direction along the outflow between adjacent Taylor vortices, with the frequency of modulation directly related to the mean drift speed. The azimuthal wavelength of the disturbance field determines the second wavenumber. We argue that the quasiperiodic flow arises as an instability of the essentially axisymmetric vortex outflow jet. Experimentally, modulated waves appear to bifurcate directly to lowdimensional chaos. Neither the mathematical nor the physical mechanisms of this translation have been well understood. Our numerical work suggests that the observed transition to chaos results from the existence of physically distinct Floquet modes arising from instability of the outflow jet. 1.
Numerical analysis of dynamical systems
, 2000
"... This paper presents a brief overview of algorithms that aid in the analysis of dynamical systems and their bifurcations. The viewpoint is geometric and the goal is to describe ..."
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This paper presents a brief overview of algorithms that aid in the analysis of dynamical systems and their bifurcations. The viewpoint is geometric and the goal is to describe
Analysis of Numerical Methods Suitable for Computing Lyapunov Exponents
"... Two standard methods for numerically estimating Lyapunov exponents are reviewed and it is noted that a numerical integration scheme that preserves orthonormality is required. A procedure is introduced for modifying arbitrary rth order numerical schemes to preserve orthonormality. Convergence is show ..."
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Two standard methods for numerically estimating Lyapunov exponents are reviewed and it is noted that a numerical integration scheme that preserves orthonormality is required. A procedure is introduced for modifying arbitrary rth order numerical schemes to preserve orthonormality. Convergence is shown for the particular case when explicit Euler's method is taken as the arbitrary method. This motivates looking at more general systems of ordinary differential equations which conserve orthonormality. An arbitrary convergent numerical method is modified to preserve orthonormality and convergence discussed in this general case. In each case numerical results are presented for the estimation of Lyapunov exponents for the Lorenz equations and results compared with those of other authors. Short Title : Numerical methods for Lyapunov Exponents AMS(MOS) Subject Classification : 65L 1 Introduction Lyapunov exponents are widely used to characterize the asymptotic behaviour of dynamical systems. I...
Analysis of Relations in Stochastic Systems
"... this article. The exact definitions of the terms like Lyapunov exponents, fractal dimension, Kolmogorov  Sinai entropy are out of the range of this paper. For better understanding the problematic we can recommend the literature mentioned above. ..."
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this article. The exact definitions of the terms like Lyapunov exponents, fractal dimension, Kolmogorov  Sinai entropy are out of the range of this paper. For better understanding the problematic we can recommend the literature mentioned above.
BRAIN DYNAMICS AT MULTIPLE SCALES: CAN ONE RECONCILE THE APPARENT LOWDIMENSIONAL CHAOS OF MACROSCOPIC VARIABLES WITH THE SEEMINGLY STOCHASTIC BEHAVIOR OF SINGLE NEURONS?
, 2008
"... Nonlinear time series analyses have suggested that the human electroencephalogram (EEG) may share statistical and dynamical properties with chaotic systems. During slowwave sleep or pathological states like epilepsy, correlation dimension measurements display low values, while in awake and attentiv ..."
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Nonlinear time series analyses have suggested that the human electroencephalogram (EEG) may share statistical and dynamical properties with chaotic systems. During slowwave sleep or pathological states like epilepsy, correlation dimension measurements display low values, while in awake and attentive subjects, there is no such low dimensionality, and the EEG is more similar to a stochastic variable. We briefly review these results and contrast them with recordings in cat cerebral cortex, as well as with theoretical models. In awake or sleeping cats, recordings with microelectrodes inserted in cortex show that global variables such as local field potentials (local EEG) are similar to the human EEG. However, neuronal discharges are highly irregular and exponentially distributed, similar to Poisson stochastic processes. To reconcile these results, we investigate models of randomlyconnected networks of integrateandfire neurons, and also contrast global (averaged) variables, with neuronal activity. The network displays different states, such as “synchronous regular ” (SR) or “asynchronous irregular ” (AI) states. In SR states, the global variables display coherent behavior with low dimensionality, while in AI states, the global activity is highdimensionally chaotic with exponentially distributed neuronal discharges, similar to awake cats. Scaledependent Lyapunov exponents and entropies show that the seemingly stochastic nature at small scales (neurons) can coexist with more coherent behavior at larger scales (averages). Thus, we suggest that brain activity obeys a similar scheme, with seemingly stochastic dynamics at small scales (neurons), while large scales (EEG) display more coherent behavior or highdimensional chaos.
NorthHolland, Amsterdam CHARACTERIZATION OF HYDRODYNAMIC STRANGE ATrRACTORS
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"... 78 J. D. Rodriguez and L. Sirovich / Lowdimensional dynamics for the complex GL equation present an analogous treatment of the Dirichlet case. This second case is one in which spatial as well as temporal chaos plays a role. The method of approach is based on the KarhunenLoeve (KL) [7, 81 procedure ..."
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78 J. D. Rodriguez and L. Sirovich / Lowdimensional dynamics for the complex GL equation present an analogous treatment of the Dirichlet case. This second case is one in which spatial as well as temporal chaos plays a role. The method of approach is based on the KarhunenLoeve (KL) [7, 81 procedure, which also plays a role in fluid turbulence [9, lo], and other chaotic problems [ll]. Briefly stated, the KL procedure generates an optimal system of basis functions, based on secondorder statistics. These best fitting functions are then used in a Gale&in approximation to the GL equation. Since the KL basis functions are derived for a specific set of parameter values, an important question is the range of accuracy of the derived dynamical system. This, as we show, is substantial in both instances. We begin with a brief review and summary of the KL procedure with particular attention to the role it plays in the types of problems under investigation. 2. Derivation of the near ideal basis The Jlow A(x, t) is assumed to be chaotic and the system sufficiently aged so that the system point moves on the chaotic attractor. We denote by V.(x)) = {4x&J) (2.1) an ensemble of snapshots of the flow at uncorrelated times { tn}. The ensemble of states (2.1) is supposed to be large enough so that the attractor is sufficiently sampled so that we can perform the necessary statistics. Since solutions to (1.1) are invariant under multiplication by complex numbers lying on the unit circle, G(A) =O*G(e”A) =0 (2.2) for c real, it follows from averaging over this group of transformations that The angular brackets in (2.3) and in what follows denote an ensemble average. A geometry is introduced into the space through the complex inner product (u,v) =/ii(x) u(x)dx, (2.4 where the integration is over the interval which in our case is (0, T). We seek the maximum of the most likely state on the chaotic attractor, C#I, defined by
and
, 1992
"... Abstract. The possible chaotic nature of the turbulence of the atmospheric boundary layer in and above a decidious forest is investigated. In particular, this work considers high resolution temperature and threedimensional wind speed measurements, gathered at six alternative levations at Camp Borde ..."
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Abstract. The possible chaotic nature of the turbulence of the atmospheric boundary layer in and above a decidious forest is investigated. In particular, this work considers high resolution temperature and threedimensional wind speed measurements, gathered at six alternative levations at Camp Borden, Ontario, Canada (Shaw et al., 1988). The goal is to determine whether these time series may be described (individually) by sets of deterministic nonlinear differential equations, such that: (i) the data's intrinsic (and seemingly random) irregularities are captured by suitable lowdimensional fractal sets (strange attractors), and (ii) the equation's lack of knowledge of initial conditions translates into unpredictable behavior (chaos). Analysis indicates that indeed all series exhibit chaotic behavior, with strange attractors whose (correlation) dimensions range from 4 to 7. These results upport the existence of a lowdimensional chaotic attractor in the lower atmosphere. 1.