Chaos and Stochasticity in Deterministically Generated Multifractal Measures

Abstract

Successful usage of a large family of deterministically generated measures to model complex nonlinear phenomena (e.g. rainfall, turbulence and groundwater contaminant transport) has been reported recently. 1-7 As these measures, generated as derived distributions of multifractal measures via fractal interpolating functions (FIF), i.e. the fractal-multifractal (FM) approach, have been found to share the inherent character of natural sets, the present study further investigates their dynamical (chaotic or stochastic) properties. It is shown, through a variety of examples and via the use of power spectrum, mass exponents and false nearest neighbors in the state-space, that the FM approach indeed generates deterministic measures whose behavior (depending on their parameters) may be classified as low-dimensional and chaotic or as highdimensional and stochastic. These results suggest the general suitability of the FM approach for understanding and modeling nonlinear natural phenomena.

Estimating Stochasticity of Acoustic Signals

“…Addressing an important question, "Chaos or Stochasticity" [10], [11] is beyond the scope of the present paper. According to [11], [12], we assume here that: "Stochasticity means Randomness" and "Randomness implies a lack of Predictability".…”

Encoding hydrologic information via a fractal geometric approach and its extensions

“…This is also evident from a comparison of their statistical and multifractal characteristics. The four sets share in fact similar decays in their autocorrelation functions and reasonable correlation scales s e (the lag where 1/e is reached) ranging from 80 to 200 lags; all signals exhibit power-law scaling in their power spectrum (S(w) * w -b ) with spectral exponents b that include common values ranging from 1.15 to 1.37 (computed up to a logarithmic frequency scale of 2.2); all sets possess similar multifractal properties as reflected by their parabolic multifractal spectra (f vs. a curve), leading to similar entropy dimensions D 1 ranging from 0.85 to 0.96, as defined by the intersection of the parabola and the f = a line; and also it can be shown (not included) that all signals define lowdimensional chaotic dynamic systems of similar dimensions (Puente and Obregón 1996;Puente et al 2002). As the sets in Fig.…”

“…Sivakumar et al 2007), which may lead to an increased understanding of hydrologic and climatic regimes and the interrelation between rainfall and other relevant climatic variables. Finally, as the analysis of derived distributions leads to the identification of 'chaotic' and also 'random' dynamics for suitable regions in the FMFP parameter space (Puente et al 2002) and as one may perhaps study the dynamics of rainfall via the successive FMFP parameters corresponding to successive sets, the present results also suggest that the general framework explained herein may indeed serve as a sensible 'middle-ground' approach to hydrologic modeling, especially in tandem with a chaotic dynamic framework [see also Sivakumar (2004Sivakumar ( , 2009) for some details], one not requiring stochastic partial differential equations but rather geometric trends.…”

“…rainfall, turbulence, width function, chaotic signals) as measured in time at a given site (e.g. Obregón 1996, 1999;Puente et al 2002;Puente and Sivakumar 2003), and (ii) two-dimensional spatial patterns (radar images, contamination plumes) as gathered over a region (e.g. Puente et al 2001a, b).…”

Nonlinear determinism in river flow: prediction as a possible indicator

“…However, as river flow dynamics at some time-/space scales are not as irregular and as complex as those at other time-/space scales, the need and appropriateness of the stochastic process concept for 'every' river flow (and hydrologic and geomorphic) series may be questioned (see, for example, Puente and Obregon, 1996;Porporato and Ridolfi, 1997;Puente et al, 2002, andSivakumar, 2003, for details). The limitation of the stochastic models may be explained with reference to, for instance, the often deliberate removal of the simple and easily predictable components of river flow (e.g.…”