Topological time-series analysis of a string experiment and its synchronized model

Tufillaro, Nicholas B.; Wyckoff, P.; Brown, Robert G.; Schreiber, Thomas; Molteno, T. C. A.

doi:10.1103/physreve.51.164

Cited by 49 publications

(30 citation statements)

References 41 publications

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“…Topological analysis is a powerful technique to characterize the structure underlying a phase portrait. Although limited to 3D spaces, there are many examples of such an analysis applied to experimental data (see Mindlin and Gilmore, 1992;Tufillaro et al, 1995;Letellier et al, 1998 among others). Unfortunately, less than 20 cycles is far too little to be able to safely perform topological analysis and, consequently, the topological structure underlying the solar dynamics cannot be extracted directly from the sunspot numbers.…”

Section: Model Characterizationmentioning

confidence: 98%

Forecasting the Time Series of Sunspot Numbers

2008

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Forecasting the solar cycle is of great importance for weather prediction and environmental monitoring, and also constitutes a difficult scientific benchmark in nonlinear dynamical modeling. This paper describes the identification of a model and its use in the forecasting the time series comprised of Wolf's sunspot numbers. A key feature of this procedure is that the original time series is first transformed into a symmetrical space where the dynamics of the solar dynamo are unfolded in a better way, thus improving the model. The nonlinear model obtained is parsimonious and has both deterministic and stochastic parts. Monte Carlo simulation of the whole model produces very consistent results with the deterministic part of the model but allows for the determination of confidence bands. The obtained model was used to predict cycles 24 and 25, although the forecast of the latter is seen as a crude approximation, given the long prediction horizon required. As for the 24th cycle, two estimates were obtained with peaks of 65 ± 16 and of 87 ± 13 units of sunspot numbers. The simulated results suggest that the 24th cycle will be shorter and less active than the preceding one.

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Section: Model Characterizationmentioning

confidence: 98%

Forecasting the Time Series of Sunspot Numbers

2008

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“…It has been demonstrated that a set of ordinary differential equations can be automatically reconstructed from a deterministic scalar time series (Gouesbet and Maquet, 1992;Brown et al 1994;Gouesbet and Letellier, 1994;Tufillaro et al, 1995;Letellier et al 1995, and references therein), a process which is called global vector field reconstruction leading to a model. As for a phase space reconstruction, the model may be equivalently reconstructed by using delay coordinates (Brown et al, 1994;Serre et al, 1996), derivative coordinates (Gouesbet and Letellier, 1994) since it has been shown that all these choices of coordinates are equivalent (Gibson et al, 1992).…”

Section: Global Vector Field Reconstruction Theoretical Backgroundmentioning

confidence: 99%

“…The phase portrait is first investigated by using classical techniques discovered by Poincart (1952) such as Poincart sections and first-return maps. Besides, we also use more recent tools such as symbolic dynamics (Collet and Eckman, 1980;Bai Lin, 1989;Hall, 1994) or global vector field reconstructions (Brown et al, 1994;Gouesbet and Letellier, 1994;Tufillaro et al, 1995;Letellier et al, 1995) to characterize the dynamics of the velocity field. In particular, the success of a global vector field reconstruction may be seen as a confirmation of the existence of a deterministic behavior.…”

Section: The Velocity Field In a Standard Mixing Reactor With A Rushtmentioning

confidence: 99%

Recovering deterministic behavior from experimental time series in mixing reactor

et al. 1997

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“…Due to the limited amount of data and the influence of external noise, the determination of the kneading sequence is rather inaccurate for short time series [Tufillaro et al, 1995]. An empirical procedure may however allow one to determine the kneading sequence [Tufillaro et al, 1995;Fang, 1994] as described below.…”

Section: Symbolic Dynamics and Topological Characterizationmentioning

confidence: 99%

“…For finite data sets, the symbolic plane coordinates are approximated by taking D = 16 [Tufillaro et al, 1995]. We can then use a finite symbol string from a chaotic trajectory to generate a sequence of points in the symbolic plane.…”

Section: Symbolic Dynamics and Topological Characterizationmentioning

confidence: 99%

Analysis of a Nonsynchronized Sinusoidally Driven Dynamical System

Ménard

Letellier

Maquet

et al. 2000

Int. J. Bifurcation Chaos

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A nonautonomous system, i.e. a system driven by an external force, is usually considered as being phase synchronized with this force. In such a case, the dynamical behavior is conveniently studied in an extended phase space which is the product of the phase space R m of the undriven system by an extra dimension associated with the external force. The analysis is then performed by taking advantage of the known period of the external force to define a Poincaré section relying on a stroboscopic sampling. Nevertheless, it may so happen that the phase synchronization does not occur. It is then more convenient to consider the nonautonomous system as an autonomous system incorporating the subsystem generating the driving force. In the case of a sinusoidal driving force, the phase space is R m+2 instead of the usual extended phase space R m × S 1 . It is also demonstrated that a global model may then be obtained by using m + 2 dynamical variables with two variables associated with the driving force. The obtained model characterizes an autonomous system in contrast with a classical input/output model obtained when the driving force is considered as an input.

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Topological time-series analysis of a string experiment and its synchronized model

Cited by 49 publications

References 41 publications

Forecasting the Time Series of Sunspot Numbers

Forecasting the Time Series of Sunspot Numbers

Recovering deterministic behavior from experimental time series in mixing reactor

Analysis of a Nonsynchronized Sinusoidally Driven Dynamical System

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