Analysis and Data Based Reconstruction of Complex Nonlinear Dynamical Systems : using the Methods of Stochastic Processes 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 analyse 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 1 Introduction 2 Introduction to Stochastic Processes 3 Kramers Moyal Expansion and Fokker Planck Equation 4 Continuous Stochastic Process 5 The Langevin Equation and Wiener Process 6 Stochastic Integration It^o and Stratonovich Calculi 7 Equivalence of Langevin and Fokker Planck Equations 8 Examples of Stochastic Calculus 9 Langevin Dynamics in Higher Dimensions 10 Levy Noise Driven Langevin Equation and its Time Series Based Reconstruction 11 Stochastic Processes with Jumps and Non Vanishing Higher Order Kramers Moyal Coefficients 12 Jump Diffusion Processes 13 Two Dimensional (Bivariate) Jump Diffusion Processes 14 Numerical Solution of Stochastic Differential Equations: Diffusion and Jump Diffusion Processes 15 The Friedrich Peinke Approach to Reconstruction of Dynamical Equation for Time Series: Complexity in View of Stochastic Processes 16 How To Set Up Stochastic Equations For Real World Processes: Markov Einstein Time Scale 17 Reconstruction of Stochastic Dynamical Equations: Exemplary Stationary Diffusion and Jump Diffusion Processes 18 The Kramers Moyal Coefficients of Non Stationary Time series in The Presence of Microstructure (Measurement) Noise 19 Influence of Finite Time Step in Estimating of the Kramers Moyal Coefficients 20 Distinguishing Diffusive and Jumpy Behaviors in Real World Time Series 21 Reconstruction of Langevin and Jump Diffusion Dynamics From Empirical Uni and Bivariate Time Series 22 Applications and Outlook 23 Epileptic Brain Dynamics