New method for the analysis of combined chaotic - stochastic processes
Valeria Bondarenko, Ina Taralova
The understanding of the dynamic behavior in real physical or industrial system is of almost importance, for analysis, synthesis, prediction, etc.
It’s sensible to consider that the behavior of many physical systems like phytoplankton ,solar activity, oscillation of waves is a combination between chaotic or stochastic processes, which can be successfully used for prediction of health applications, meteorological phenomena etc.
Many physical/ chemical or sometimes financial phenomena are considered as being only chaotic ((ex. Belousov–Zhabotinsky_reaction 1) or purely stochastic (stock model price, integral Ito, Black-Scholes model), but in fact they are both deterministic and stochastic (1-2).
So it is of utmost interest to find new models taking into account both behaviors, stochastic and chaotic, to understand and predict better the real physical phenomena, but also to model data for biomedical applications like (ECG, IRM, …. To be completed) The original idea in this paper is to juxtapose methods from stochastic signal analysis (nonstationary Gaussian processes, statistics from limit theorems by Nordin, Hurst exponent), and nonlinear (chaotic) dynamical system analysis (phase portrait, phase delayed plot, Lyapunov exponents), to develop a common methodology to analyze complex time series. Assuming that these two behaviors are inherently correlated, we are analyzing if there exists a correlation exists between the stochastic quantifiers (Hurst exponent, Garch method,ARMA) and chaotic quantifiers (Lyapunov exponents). To do that, different kind of stochastic-chaotic mixed processes shall be modeled and analyzed from different points of view to be developed.