Common and different features between the behavior of the chaotic dynamical systems and the 1/f^alpha(f) noise

Abstract

The major goal of the present paper is to find out the manifestation of the boundedness of fluctuations. Two different subjects are considered: (i) an ergodic Markovian process associated with a new type of large scaled fluctuations at spatially homogeneous reaction systems; (ii) simulated dynamical systems that posses strange attractors. Their common property is that the fluctuations are bounded. It is found out that the mathematical description of the stochasity at both types of systems is identical. Then, it is to be expected that it exhibits certain common features whose onset is the stochasticity, namely: (i) The power spectrum of a time series of length $T$ comprises a strictly decreasing band that uniformly fits the shape $1/f^α(f)$ where $α(1/T)=1$ and $α(f)$ strictly increases to the value $α(\inf)=p$ ($p>2$) as $f$ approaches infinity. Practically, at low frequencies this shape is $1/f$-like with high accuracy because the deviations of the non-constant exponent $α(f)$ from 1 are very small and become even smaller as the frequency tends to 1/T. The greatest advantage of the shape $1/α(f)$ is that it ensures a finite variance of the fluctuations. (ii) It is found out that the structure of a physical and strange attractor is identical and they are non-homogeneous. (iii) The Kolmogorov entropy is finite.