scholarly journals Probabilistic Description of Extreme Events in Intermittently Unstable Dynamical Systems Excited by Correlated Stochastic Processes

2015 ◽  
Vol 3 (1) ◽  
pp. 709-736 ◽  
Author(s):  
Mustafa A. Mohamad ◽  
Themistoklis P. Sapsis
Keyword(s):  
Dynamical Systems ◽  
Stochastic Processes ◽  
Extreme Events ◽  
Probabilistic Description

scholarly journals Potential Well in Poincaré Recurrence

Entropy ◽  
2021 ◽  
Vol 23 (3) ◽  
pp. 379
Author(s):  
Miguel Abadi ◽  
Vitor Amorim ◽  
Sandro Gallo
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Stochastic Processes ◽  
Recurrence Time ◽  
Potential Well ◽  
Exponential Approximation ◽  
Mixing Processes ◽  
System Perspective ◽  
Return Times ◽  
Error Terms

From a physical/dynamical system perspective, the potential well represents the proportional mass of points that escape the neighbourhood of a given point. In the last 20 years, several works have shown the importance of this quantity to obtain precise approximations for several recurrence time distributions in mixing stochastic processes and dynamical systems. Besides providing a review of the different scaling factors used in the literature in recurrence times, the present work contributes two new results: (1) For ϕ-mixing and ψ-mixing processes, we give a new exponential approximation for hitting and return times using the potential well as the scaling parameter. The error terms are explicit and sharp. (2) We analyse the uniform positivity of the potential well. Our results apply to processes on countable alphabets and do not assume a complete grammar.

Introduction to Stochastic Processes

2019 ◽  
pp. 1-24
Author(s):  
Florence Merlevède ◽  
Magda Peligrad ◽  
Sergey Utev
Keyword(s):  
Dynamical Systems ◽  
Stochastic Processes ◽  
Gaussian Approximation ◽  
Partial Sums ◽  
Stationary Sequences ◽  
Linear Processes ◽  
Limiting Behavior ◽  
Additive Functionals ◽  
Moderate Deviations Principle ◽  
Recursive Sequences

We start by stating the need for a Gaussian approximation for dependent structures in the form of the central limit theorem (CLT) or of the functional CLT. To justify the need to quantify the dependence, we introduce illustrative examples: linear processes, functions of stationary sequences, recursive sequences, dynamical systems, additive functionals of Markov chains, and self-interactions. The limiting behavior of the associated partial sums can be handled with tools developed throughout the book. We also present basic notions for stationary sequences of random variables: various definitions and constructions, and definitions of ergodicity, projective decomposition, and spectral density. Special attention is given to dynamical systems, as many of our results also apply in this context. The chapter also surveys the basic theory of the convergence of stochastic processes in distribution, and introduces the reader to tightness, finite-dimensional convergence, and the need for maximal inequalities. It ends with the concepts of the moderate deviations principle and its functional form.

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

DYNAMICAL ASPECTS OF INTERACTION NETWORKS

2005 ◽  
Vol 15 (11) ◽  
pp. 3467-3480 ◽  
Author(s):  
G. NICOLIS ◽  
A. GARCÍA CANTÚ ◽  
C. NICOLIS
Keyword(s):  
Phase Space ◽  
Dynamical Systems ◽  
Stochastic Processes ◽  
Lyapunov Exponents ◽  
Network Theory ◽  
Finite Size ◽  
Interaction Networks ◽  
Deterministic Systems ◽  
Connectivity Patterns ◽  
Probabilistic Process

A connection between dynamical systems and network theory is outlined based on a mapping of the dynamics into a discrete probabilistic process, whereby the phase space is partitioned into finite size cells. It is found that the connectivity patterns of networks generated by deterministic systems can be related to the indicators of the dynamics such as local Lyapunov exponents. The procedure is extended to networks generated by stochastic processes.

scholarly journals Buonomano against Bell: Nonergodicity or nonlocality?

2017 ◽  
Vol 15 (08) ◽  
pp. 1740010 ◽  
Author(s):  
Andrei Khrennikov
Keyword(s):  
Quantum Mechanics ◽  
Dynamical Systems ◽  
Stochastic Processes ◽  
Bell Inequality ◽  
Quantum Measurements ◽  
Neutron Interferometry ◽  
Bell’S Inequality ◽  
Weak Mixing ◽  
Bell's Inequality

The aim of this note is to attract attention of the quantum foundational community to the fact that in Bell’s arguments, one cannot distinguish two hypotheses: (a) quantum mechanics is nonlocal, (b) quantum mechanics is nonergodic. Therefore, experimental violations of Bell’s inequality can be as well interpreted as supporting the hypothesis that stochastic processes induced by quantum measurements are nonergodic. The latter hypothesis was discussed actively by Buonomano since 1980. However, in contrast to Bell’s hypothesis on nonlocality, it did not attract so much attention. The only experiment testing the hypothesis on nonergodicity was performed in neutron interferometry (by Summhammer, in 1989). This experiment can be considered as rejecting this hypothesis. However, it cannot be considered as a decisive experiment. New experiments are badly needed. We point out that a nonergodic model can be realistic, i.e. the distribution of hidden (local!) variables is well-defined. We also discuss coupling of violation of the Bell inequality with violation of the condition of weak mixing for ergodic dynamical systems.

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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