Analysis and Data-Based Reconstruction of Complex Nonlinear Dynamical Systems

Chapter 1 Introduction

Chapter 2 Introduction to Stochastic Processes

Chapter 3 Kramers–Moyal Expansion and Fokker–Planck Equation

Chapter 4 Continuous Stochastic Processes

Chapter 5 The Langevin Equation and Wiener Process

Chapter 6 Stochastic Integration, Itô and Stratonovich Calculi

Chapter 7 Equivalence of Langevin and Fokker–Planck Equations

Chapter 8 Example of Stochastic Calculus

Chapter 9 Langevin Dynamics in Higher Dimensions

Chapter 10 Lévy Noise-Driven Langevin Equation and Its Time Series–Based Reconstruction

Chapter 11 Stochastic Processes with Jumps and Non-vanishing Higher-Order Kramers–Moyal Coefficients

Chapter 12 Jump-Diffusion Processes

Chapter 13 Two-Dimensional (Bivariate) Jump-Diffusion Processes

Chapter 14 Numerical Solution of Stochastic Differential Equations: Diffusion and Jump-Diffusion Processes

Chapter 15 The Friedrich–Peinke Approach to Reconstruction of Dynamical Equation for Time Series: Complexity in View of Stochastic Processes

Chapter 16 How to Set Up Stochastic Equations for Real World Processes: Markov–Einstein Time Scale

Chapter 17 The Kramers–Moyal Coefficients of Non-stationary Time Series and in the Presence of Microstructure (Measurement) Noise

Chapter 18 Influence of Finite Time Step in Estimating of the Kramers–Moyal Coefficients

Chapter 19 Distinguishing Diffusive and Jumpy Behaviors in Real-World Time Series

Chapter 20 Reconstruction Procedure for Writing Down the Langevin and Jump-Diffusion Dynamics from Empirical Uni- and Bivariate Time Series

Chapter 21 Reconstruction of Stochastic Dynamical Equations: Exemplary Diffusion, Jump-Diffusion Processes and Lévy Noise-Driven Langevin Dynamics

Chapter 22 Applications and Outlook

Chapter 23 Epileptic Brain Dynamics