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43
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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Cited by 495 (1 self)
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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.
Equations of motion from a data series
 Complex Systems
, 1987
"... Abstract. Temporal pattern learning, control and prediction, and chaotic data analysis share a common problem: deducing optimal equations of motion from observations of timedependent behavior. Each desires to obtain models of the physical world from limited information. We describe a method to reco ..."
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Cited by 58 (15 self)
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Abstract. Temporal pattern learning, control and prediction, and chaotic data analysis share a common problem: deducing optimal equations of motion from observations of timedependent behavior. Each desires to obtain models of the physical world from limited information. We describe a method to reconstruct the deterministic portion of the equations of motion directly from a data series. These equations of motion represent a vast reduction of a chaotic data set’s observed complexity to a compact, algorithmic specification. This approach employs an informational measure of model optimality to guide searching through the space of dynamical systems. As corollary results, we indicate how to estimate the minimum embedding dimension, extrinsic noise level, metric entropy, and Lyapunov spectrum. Numerical and experimental applications demonstrate the method’s feasibility and limitations. Extensions to estimating parametrized families of dynamical systems from bifurcation data and to spatial pattern evolution are presented. Applications to predicting chaotic data and the design of forecasting, learning, and control systems, are discussed. 1.
State space reconstruction parameters in the analysis of chaotic time series  the role of the time window length
 Physica D, 95:13
, 1996
"... dimension The most common state space reconstruction method in the analysis of chaotic time series is the Method of Delays (MOD). Many techniques have been suggested to estimate the parameters of MOD, i.e. the time delay τ and the embedding dimension m. We discuss the applicability of these techniqu ..."
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Cited by 25 (3 self)
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dimension The most common state space reconstruction method in the analysis of chaotic time series is the Method of Delays (MOD). Many techniques have been suggested to estimate the parameters of MOD, i.e. the time delay τ and the embedding dimension m. We discuss the applicability of these techniques with a critical view as to their validity, and point out the necessity of determining the overall time window length, τw, for successful embedding. Emphasis is put on the relation between τw and the dynamics of the underlying chaotic system, and we suggest to set τw ≥ τp, the mean orbital period; τp is approximated from the oscillations of the time series. The procedure is assessed using the correlation dimension for both synthetic and real data. For clean synthetic data, values of τw larger than τp always give good results given enough data and thus τp can be considered as a lower limit (τw ≥ τp). For noisy synthetic data and real data, an upper limit is reached for τw which approaches τp for increasing noise amplitude. 1
On Periodic Points for Systems of Weakly Coupled 1Dim Maps
, 1997
"... In paper we present the topological method of proving the existence of periodic in multidimensional nonuniformly hyperbolic dynamical systems. We apply our method to the maps of the form F (x1 ; : : : ; xn) := (f(x1); f(x2); : : : ; f(xn)) +G : R n ! R n where f : R! R is either logistic or tent ..."
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Cited by 10 (1 self)
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In paper we present the topological method of proving the existence of periodic in multidimensional nonuniformly hyperbolic dynamical systems. We apply our method to the maps of the form F (x1 ; : : : ; xn) := (f(x1); f(x2); : : : ; f(xn)) +G : R n ! R n where f : R! R is either logistic or tent map in the chaotic region and G is sufficiently small in C 0 norm. We prove that F have inifinite number periodic points with symbolic dynamics. We prove also the result for spatial distribution of periodic points for F in the invariant set. Another considered example is Rossler walking stick diffeomorphism, for which we prove the existence of inifinite number of periodic points. Keywords: fixed point index, chaos, nonuniformly hyperbolic systems AMS classification scheme numbers: 58F15, 58G10 1 Introduction The aim of this paper is to develop the method of TSmaps (topological shifts) introduced by the author in [?] for proving existence of infinite number of periodic points in multidi...
Synchronized chaos and other coherent states for two coupled neurons
 Physica D
, 1999
"... Synchronized chaos and other coherent states for two coupled neurons by ..."
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Cited by 10 (3 self)
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Synchronized chaos and other coherent states for two coupled neurons by
Synchronous and Asynchronous Chaos in Coupled Neuromodules
 International Journal of Bifurcation and Chaos
, 1999
"... The parametrized timediscrete dynamics of two recurrently coupled neuromodules is studied analytically and by computer simulations. Conditions for the existence of synchronized dynamics are derived and periodic as well as quasiperiodic and chaotic attractors constrained to a synchronization man ..."
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Cited by 7 (6 self)
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The parametrized timediscrete dynamics of two recurrently coupled neuromodules is studied analytically and by computer simulations. Conditions for the existence of synchronized dynamics are derived and periodic as well as quasiperiodic and chaotic attractors constrained to a synchronization manifold M are observed. Stability properties of the synchronized dynamics is discussed by using Lyapunov exponents parallel and transversal to the synchronization manifold. Simulation results are presented for selected sets of parameters. It is observed that locally stable synchronous dynamics often coexists with asynchronous periodic, quasiperiodic or even chaotic attractors. MPIMIS preprint 24/99, and International Journal of Bifurcation and Chaos, 9, 19571968 (1999). 1 1 Introduction Ever since the feasibility of synchronizing chaotic systems has been established also by Pecora & Carroll [1990], this phenomenon has been investigated in many articles. Part of the work on synch...
ACTIVE CONTROLLER DESIGN FOR GLOBAL CHAOS SYNCHRONIZATION OF HYPERCHAOTIC BAO AND HYPERCHAOTIC XU SYSTEMS
"... ABSTRACT This paper investigates the global chaos synchronization of identical hyperchaotic Bao systems (Bao and Liu, 2008) , identical hyperchaotic Xu systems ..."
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Cited by 6 (6 self)
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ABSTRACT This paper investigates the global chaos synchronization of identical hyperchaotic Bao systems (Bao and Liu, 2008) , identical hyperchaotic Xu systems
A new application of multistage homotopy perturbation method to the Chaotic Rossler system,”
 in Proceedings of the 2nd International Conference on Fundamental and Applied Sciences (ICFAS ’12), vol. 1482 ofAIPConference Proceedings,
, 2012
"... The multistage homotopyperturbation method (MHPM) is applied to the nonlinear chaotic and hyperchaotic Lü systems. MHPM is a technique adapted from the standard homotopyperturbation method (HPM) where the HPM is treated as an algorithm in a sequence of time intervals. To ensure the precision of t ..."
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Cited by 4 (0 self)
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The multistage homotopyperturbation method (MHPM) is applied to the nonlinear chaotic and hyperchaotic Lü systems. MHPM is a technique adapted from the standard homotopyperturbation method (HPM) where the HPM is treated as an algorithm in a sequence of time intervals. To ensure the precision of the technique applied in this work, the results are compared with a fourthorder RungeKutta method and the standard HPM. The results show that the MHPM is an efficient and powerful technique in solving both chaotic and hyperchaotic systems.
CONTROLLED PROJECTIVE SYNCHRONIZATION IN NONPARTIALLYLINEAR CHAOTIC SYSTEMS
, 2001
"... Projective synchronization (PS), in which the state vectors synchronize up to a scaling factor, is usually observable only in partially linear systems. We show that PS could, by means of control, be extended to general classes of chaotic systems with nonpartial linearity. Performance of PS may also ..."
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Cited by 4 (0 self)
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Projective synchronization (PS), in which the state vectors synchronize up to a scaling factor, is usually observable only in partially linear systems. We show that PS could, by means of control, be extended to general classes of chaotic systems with nonpartial linearity. Performance of PS may also be manipulated by controlling the scaling factor to any desired value. In numerical experiments, we illustrate the applications to a Rössler system and a Chua’s circuit. The feasibility of the control for high dimensional systems is demonstrated in a hyperchaotic system.
State Space Reconstruction: Method of Delays vs Singular Spectrum Approach
 N. Christophersen, http://citeseer.nj.nec.com/kugiumtzis97state.html
, 1997
"... Abstract The analysis of chaotic time series requires proper reconstruction of the state space from the available data in order to successfully estimate invariant properties of the embedded attractor. Using the correlation dimension, we discuss the applicability of the two most common methods of re ..."
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Abstract The analysis of chaotic time series requires proper reconstruction of the state space from the available data in order to successfully estimate invariant properties of the embedded attractor. Using the correlation dimension, we discuss the applicability of the two most common methods of reconstruction, the method of delays (MOD) and the Singular Spectrum Approach (SSA). Contrary to previous discussions, we found that the two methods perform equivalently in practice for noisefree data provided the parameters of the two methods are properly related. In fact, the quality of the reconstruction is in both cases determined by the choice of the time window length τw and is independent of the selected method. However, when the data are noisy, we find that SSA outperforms MOD.