Information entropy of the inhomogeneous dynamical systems

Abstract

In the present paper describe a new method for calculation of entropy with regard of degree of homogeneity of the two-dimensional set. The urgency of the work: application of these methods for analysis of dynamic systems. We need to consider the relationship between entropy and order parameter of a dynamical system.

For this purpose, we used the expression for evolution of order parameter of a dynamical system, whichis a determining variable and for the description of the bifurcation regimes. This parameter was introduced earlier (Zhanabaev Z. Zh., 2007) in the form of the generalized metric characteristic.

Non-additive information entropy S of two-dimensional set which is the phase portrait of timerealization was defined. Non-additivity of entropy exists due to homogeneity of q. This parameter is defined as a statistical measure of deviation ofTsallisfrom the Gibbs statistics. For q = 1, entropy is an additive value.Tsallis entropy coincides with Renyi entropy.

Dependence of the non-additive entropy on evolution of the order parameter for dynamical systems of different types was shown. Evolution of fractal mapping implements asymmetric bifurcations of the type «gluing» and chaos whichsatisfies the entropy criteria of self-organization 0,567 <S <0.806, where S is a value of Kolmogorov -Sinai entropy. These self-affine and self-similar criteria have been established by (ZhanabaevZ. Zh., 1996) previously.