Controversy in Financial Chaos Research and Nonlinear Dynamics: A Short Literature Review

Chaos Analysis Framework: How to Safely Identify and Quantify Time-Series Dynamics

“…Finally, the Hurst exponent or in the case of non-stationary data, the detrended fluctuation analysis (DFA) alpha value is calculated to obtain in-depth information about the evolutionary nature of the dynamical system [54,77,78]. In an ongoing debate, the interpretation of the Hurst exponent and its initial interpretation by Benoit Mandelbrot (see [79,80]) is challenged [16,25,44].…”

“…Recent trends within chaotic dynamical analysis have led to a proliferation of publications, stating structural nonlinear models to be capable of displaying instabilities and chaos to be able to mimic empirical time-series properties 4 [22]. Henceforth, a crucial pillar in nonlinear forecasting for over 40 years is the revelation of whether the considered time-series data sets are generated via deterministic or stochastic 5 dynamical systems since their respective mathematical operations differ noticeable (see the bibliometric review of Vogl [25]) [23,24,26]. Speaking in a mathematical sense, a chaotic dynamical system has a dense collection of points with periodic orbits, sensitivity to initial conditions and topological transitivity, which is discussed in Eckmann and Ruelle [27,Devaney [28], BenSaïda and Litimi [29].…”

“…Furthermore, since the determination of the (empirical) DGP by the proposed framework is by far trivial in its application, owing to a large variety of advanced algorithms to be implemented, this chapter is purposed to provide a clear and distinct practical guideline on how to successfully implement the denoted analysis framework. Particularly, the determination of 'scaling regions' via correlation sum and correlation dimensional schemes (refer to [45][46][47]) and the reconstruction algorithms of (strange, fractal) attractors of complex dynamical systems with chaotic traits (see [10,48]) represent one of the main emphases, among others, since many erroneous conductions are possible and are widely dispersed throughout the scientific literature [25,40,45]. The distinct goal and aim of this chapter are to provide the researcher and practitioner with an empirical-practical guide on how to implement the presented chaos analysis framework successfully, thus, determining the (empirical) DGP and reconstructing potentially existing system attractors out of a scalar timeseries given [40,44].…”