Tabar M.R.R. Analysis and Data-Based Reconstruction of Complex Nonlinear Dynamical Systems: Using the Methods of Stochastic Processes

Springer, 2019. — 289 p. — (Understanding Complex Systems). — ISBN: 978-3-030-18471-1.This book focuses on a central question in the field of complex systems: Given a fluctuating (in time or space), uni- or multi-variant sequentially measured set of experimental data (even noisy data), how should one analyze non-parametrically the data, assess underlying trends, uncover characteristics of the fluctuations (including diffusion and jump contributions), and construct a stochastic evolution equation?
Here, the term "non-parametrically" exemplifies that all the functions and parameters of the constructed stochastic evolution equation can be determined directly from the measured data.
The book provides an overview of methods that have been developed for the analysis of fluctuating time series and of spatially disordered structures. Thanks to its feasibility and simplicity, it has been successfully applied to fluctuating time series and spatially disordered structures of complex systems studied in scientific fields such as physics, astrophysics, meteorology, earth science, engineering, finance, medicine and the neurosciences, and has led to a number of important results.
The book also includes the numerical and analytical approaches to the analyses of complex time series that are most common in the physical and natural sciences. Further, it is self-contained and readily accessible to students, scientists, and researchers who are familiar with traditional methods of mathematics, such as ordinary, and partial differential equations.
The codes for analysing continuous time series are available in an R package developed by the research group Turbulence, Wind energy and Stochastic (TWiSt) at the Carl von Ossietzky University of Oldenburg under the supervision of Prof. Dr. Joachim Peinke. This package makes it possible to extract the (stochastic) evolution equation underlying a set of data or measurements.Introduction to Stochastic Processes
Kramers–Moyal Expansion and Fokker–Planck Equation
Continuous Stochastic Processes
The Langevin Equation and Wiener Process
Stochastic Integration, Itô and Stratonovich Calculi
Equivalence of Langevin and Fokker–Planck Equations
Example of Stochastic Calculus
Langevin Dynamics in Higher Dimensions
Lévy Noise-Driven Langevin Equation and Its Time Series–Based Reconstruction
Stochastic Processes with Jumps and Non-vanishing Higher-Order Kramers–Moyal Coefficients
Jump-Diffusion Processes
Two-Dimensional (Bivariate) Jump-Diffusion Processes
Numerical Solution of Stochastic Differential Equations: Diffusion and Jump-Diffusion Processes
The Friedrich–Peinke Approach to Reconstruction of Dynamical Equation for Time Series: Complexity in View of Stochastic Processes
How to Set Up Stochastic Equations for Real World Processes: Markov–Einstein Time Scale
The Kramers–Moyal Coefficients of Non-stationary Time Series and in the Presence of Microstructure (Measurement) Noise
Influence of Finite Time Step in Estimating of the Kramers–Moyal Coefficients
Distinguishing Diffusive and Jumpy Behaviors in Real-World Time Series
Reconstruction Procedure for Writing Down the Langevin and Jump-Diffusion Dynamics from Empirical Uni- and Bivariate Time Series
Reconstruction of Stochastic Dynamical Equations: Exemplary Diffusion, Jump-Diffusion Processes and Lévy Noise-Driven Langevin Dynamics
Applications and Outlook
Epileptic Brain Dynamics