An introduction to chaos

Rössler, Otto E.

doi:10.1002/int.4550100103

Cited by 4 publications

(4 citation statements)

References 12 publications

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“…For further testing, another Rössler system with its most commonly employed parameter set as [ α , β , δ ] = [0.1, 0.1, 14] (shown in Figure 7a) [ Rössler , 1995] was realized with a smaller variability of the random component ɛ ∼ Φ(0, 1/4 σ x 2 ) than the previous Rössler system ( σ x 2 ). The extension of the truncated last 300 values (Figure 7b) indicates that the future evolution of the nonlinear chaotic system is well reproduced by covering the variability of the system as shown in Figure 7c.…”

Section: Resultsmentioning

confidence: 99%

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Prediction of climate nonstationary oscillation processes with empirical mode decomposition

Lee¹,

Ouarda²

2011

J. Geophys. Res.

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[1] Long-term nonstationary oscillations (NSOs) are commonly observed in climatological data series such as global surface temperature anomalies (GSTA) and low-frequency climate oscillation indices. In this work, we present a stochastic model that captures NSOs within a given variable. The model employs a data-adaptive decomposition method named empirical mode decomposition (EMD). Irregular oscillatory processes in a given variable can be extracted into a finite number of intrinsic mode functions with the EMD approach. A unique data-adaptive algorithm is proposed in the present paper in order to study the future evolution of the NSO components extracted from EMD. To evaluate the model performance, the model is tested with the synthetic data set from Rössler attractor and with GSTA data. The results of the attractor show that the proposed approach provides a good characterization of the NSOs. For GSTA data, the last 30 observations are truncated and compared to the generated data. Then the model is used to predict the evolution of GSTA data over the next 50 years. The results of the case study confirm the power of the EMD approach and the proposed NSO resampling (NSOR) method as well as their potential for the study of climate variables.Citation: Lee, T., and T. B. M. J. Ouarda (2011), Prediction of climate nonstationary oscillation processes with empirical mode decomposition,

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Section: Resultsmentioning

confidence: 99%

“…To validate the model performance, the model was tested with the synthetic data from a nonlinear chaotic system, the Rössler attractor [ Rössler , 1976, 1995]. As a case study, the proposed model was applied to GSTA data.…”

Section: Introductionmentioning

confidence: 99%

Prediction of climate nonstationary oscillation processes with empirical mode decomposition

Lee¹,

Ouarda²

2011

J. Geophys. Res.

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“…One of the most famous and chaotic nonlinear dynamic systems, the Rössler attractor [ Rössler , 1976, 1995], was selected in order to test the performance of the suggested stochastic simulation model. This attractor was intended to behave similarly to the Lorenz attractor [ Lorenz , 1963] but with a better qualitative understanding of the chaotic flow.…”

Section: Model Validationmentioning

confidence: 99%

Stochastic simulation of nonstationary oscillation hydroclimatic processes using empirical mode decomposition

Lee¹,

Ouarda²

2012

Water Resources Research

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[1] Nonstationary oscillation (NSO) processes are observed in a number of hydroclimatic data series. Stochastic simulation models are useful to study the impacts of the climatic variations induced by NSO processes into hydroclimatic regimes. Reproducing NSO processes in a stochastic time series model is, however, a difficult task because of the complexity of the nonstationary behaviors. In the current study, a novel stochastic simulation technique that reproduces the NSO processes embedded in hydroclimatic data series is presented. The proposed model reproduces NSO processes by utilizing empirical mode decomposition (EMD) and nonparametric simulation techniques (i.e., k-nearestneighbor resampling and block bootstrapping). The model was first tested with synthetic data sets from trigonometric functions and the Rössler system. The North Atlantic Oscillation (NAO) index was then examined as a real case study. This NAO index was then employed as an exogenous variable for the stochastic simulation of streamflows at the Romaine River in the province of Quebec, Canada. The results of the application to the synthetic data sets and the real-world case studies indicate that the proposed model preserves well the NSO processes along with the key statistical characteristics of the observations. It was concluded that the proposed model possesses a reasonable simulation capacity and a high potential as a stochastic model, especially for hydroclimatic data sets that embed NSO processes.Citation: Lee, T., and T. B. M. J. Ouarda (2012), Stochastic simulation of nonstationary oscillation hydroclimatic processes using empirical mode decomposition, Water Resour. Res., 48, W02514,

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“…In order to validate the proposed M-NSOR model, one of the most famous and chaotic nonlinear dynamic systems, the Rössler attractor, is used (Rössler, 1976(Rössler, , 1995. The attractor is designed to behave similarly to the Lorenz attractor (Lorenz, 1963) but with a better understanding of the chaotic flow.…”

Section: Application Methodologymentioning

confidence: 99%

Multivariate Nonstationary Oscillation Simulation of Climate Indices With Empirical Mode Decomposition

Lee

Ouarda²

2019

Water Resources Research

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The objective of the current study is to build a stochastic model to simulate climate indices that are teleconnected with the hydrologic regimes of large-scale water resources systems such as the Great Lakes system. Climate indices generally contain nonstationary oscillations (NSOs). We adopted a stochastic simulation model based on Empirical Mode Decomposition (EMD). The procedure for the model is to decompose the observed series and then to simulate the decomposed components with the NSO resampling (NSOR) technique. Because the model has only been previously applied to single variables, a multivariate version of NSOR (M-NSOR) is developed to consider the links between the climate indices and to reproduce the NSO process. The proposed M-NSOR model is tested in a simulation study on the Rössler system. The simulation results indicate that the M-NSOR model reproduces the significant oscillatory behaviors of the system and the marginal statistical characteristics. Subsequently, the M-NSOR model is applied to three climate indices (i.e., Arctic Oscillation, El Niño-Southern Oscillation, and Pacific Decadal Oscillation) for the annual and winter data sets. The results of the proposed model are compared to those of the Contemporaneous Shifting Mean and Contemporaneous Autoregressive Moving Average model. The results indicate that the proposed M-NSOR model is superior to the Contemporaneous Shifting Mean and Contemporaneous Autoregressive Moving Average model for reproducing the NSO process, while the other basic statistics are comparatively well preserved in both cases. The current study concludes that the proposed M-NSOR model can be a good alternative to simulate NSO processes and their teleconnections with climate indices.

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An introduction to chaos

Cited by 4 publications

References 12 publications

Prediction of climate nonstationary oscillation processes with empirical mode decomposition

Prediction of climate nonstationary oscillation processes with empirical mode decomposition

Stochastic simulation of nonstationary oscillation hydroclimatic processes using empirical mode decomposition

Multivariate Nonstationary Oscillation Simulation of Climate Indices With Empirical Mode Decomposition

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