Extreme Events in Deterministic Dynamical Systems

2006 ◽  
Vol 97 (21) ◽  
Author(s):  
C. Nicolis ◽  
V. Balakrishnan ◽  
G. Nicolis
Keyword(s):  
Dynamical Systems ◽  
Extreme Events ◽  
Deterministic Dynamical Systems

scholarly journals Using machine learning to predict extreme events in complex systems

2019 ◽  
Vol 117 (1) ◽  
pp. 52-59 ◽  
Author(s):  
Di Qi ◽  
Andrew J. Majda
Keyword(s):  
Dynamical Systems ◽  
Learning Strategies ◽  
Extreme Events ◽  
Water Waves ◽  
Extreme Values ◽  
Training Data ◽  
Partition Functions ◽  
Grand Challenge ◽  
Natural Systems ◽  
Using Data

Extreme events and the related anomalous statistics are ubiquitously observed in many natural systems, and the development of efficient methods to understand and accurately predict such representative features remains a grand challenge. Here, we investigate the skill of deep learning strategies in the prediction of extreme events in complex turbulent dynamical systems. Deep neural networks have been successfully applied to many imaging processing problems involving big data, and have recently shown potential for the study of dynamical systems. We propose to use a densely connected mixed-scale network model to capture the extreme events appearing in a truncated Korteweg–de Vries (tKdV) statistical framework, which creates anomalous skewed distributions consistent with recent laboratory experiments for shallow water waves across an abrupt depth change, where a remarkable statistical phase transition is generated by varying the inverse temperature parameter in the corresponding Gibbs invariant measures. The neural network is trained using data without knowing the explicit model dynamics, and the training data are only drawn from the near-Gaussian regime of the tKdV model solutions without the occurrence of large extreme values. A relative entropy loss function, together with empirical partition functions, is proposed for measuring the accuracy of the network output where the dominant structures in the turbulent field are emphasized. The optimized network is shown to gain uniformly high skill in accurately predicting the solutions in a wide variety of statistical regimes, including highly skewed extreme events. The technique is promising to be further applied to other complicated high-dimensional systems.

scholarly journals Routes to extreme events in dynamical systems: Dynamical and statistical characteristics

2020 ◽  
Vol 30 (6) ◽  
pp. 063114 ◽  
Author(s):  
Arindam Mishra ◽  
S. Leo Kingston ◽  
Chittaranjan Hens ◽  
Tomasz Kapitaniak ◽  
Ulrike Feudel ◽  
...  
Keyword(s):  
Dynamical Systems ◽  
Extreme Events ◽  
Statistical Characteristics

scholarly journals Data-assisted reduced-order modeling of extreme events in complex dynamical systems

PLoS ONE ◽  
2018 ◽  
Vol 13 (5) ◽  
pp. e0197704 ◽  
Author(s):  
Zhong Yi Wan ◽  
Pantelis Vlachas ◽  
Petros Koumoutsakos ◽  
Themistoklis Sapsis
Keyword(s):  
Dynamical Systems ◽  
Extreme Events ◽  
Reduced Order Modeling ◽  
Complex Dynamical Systems ◽  
Reduced Order

Stochastically perturbed limit cycles

1978 ◽  
Vol 15 (02) ◽  
pp. 311-320
Author(s):  
Charles J. Holland
Keyword(s):  
Dynamical Systems ◽  
Invariant Measure ◽  
Probability Measure ◽  
Limit Cycles ◽  
Noise Intensity ◽  
Stable Limit Cycle ◽  
Stable Limit ◽  
Additive White Noise ◽  
Small Intensity ◽  
Deterministic Dynamical Systems

In this paper we examine the effects of perturbing certain deterministic dynamical systems possessing a stable limit cycle by an additive white noise term with small intensity. We place assumptions on the system guaranteeing that when noise is present the corresponding random process generates an ergodic probability measure. We then determine the behavior of the invariant measure when the noise intensity is small.

scholarly journals Model Error, Information Barriers, State Estimation and Prediction in Complex Multiscale Systems

Entropy ◽  
2018 ◽  
Vol 20 (9) ◽  
pp. 644 ◽  
Author(s):  
Andrew Majda ◽  
Nan Chen
Keyword(s):  
Information Theory ◽  
Dynamical Systems ◽  
State Estimation ◽  
Extreme Events ◽  
Nonlinear Modeling ◽  
Complex Dynamical Systems ◽  
Information Theoretic ◽  
Reduced Order ◽  
Information Barriers ◽  
Multiscale Systems

Complex multiscale systems are ubiquitous in many areas. This research expository article discusses the development and applications of a recent information-theoretic framework as well as novel reduced-order nonlinear modeling strategies for understanding and predicting complex multiscale systems. The topics include the basic mathematical properties and qualitative features of complex multiscale systems, statistical prediction and uncertainty quantification, state estimation or data assimilation, and coping with the inevitable model errors in approximating such complex systems. Here, the information-theoretic framework is applied to rigorously quantify the model fidelity, model sensitivity and information barriers arising from different approximation strategies. It also succeeds in assessing the skill of filtering and predicting complex dynamical systems and overcomes the shortcomings in traditional path-wise measurements such as the failure in measuring extreme events. In addition, information theory is incorporated into a systematic data-driven nonlinear stochastic modeling framework that allows effective predictions of nonlinear intermittent time series. Finally, new efficient reduced-order nonlinear modeling strategies combined with information theory for model calibration provide skillful predictions of intermittent extreme events in spatially-extended complex dynamical systems. The contents here include the general mathematical theories, effective numerical procedures, instructive qualitative models, and concrete models from climate, atmosphere and ocean science.

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