Shannon entropy, Fisher information and uncertainty relations for log-periodic oscillators

We calculate the time-dependent Shannon (S-x and S-p) entropy and Fisher (F-x and F-p) information of three log-periodic oscillators. We obtain a general expression for S-x,S-p and F-x,F-p in the state n = 0 in terms of rho, a c-number quantity satisfying a nonlinear differential equation. For two out of three oscillators S-x,S-p and F-x,F-p depend on time, but S-x + S-p and FxFp do not. The other oscillator behaves as the time-independent harmonic oscillator where S-x,S-p and F-x,F-p are all constants. Relations among the Fisher information and the Stam and Cramer-Rao inequalities are also discussed. (C) 2014 Elsevier B.V. All rights reserved.