Dynamics, Noise, Delays and the Gibbs and Conditional Entropy

In this paper we review the concepts of the Gibbs and conditional entropies and examine their dynamic behaviour when the underlying dynamics are described by ordinary differential equations or stochastic differential equations. We then go on to introduce techniques for the analogous examination when the dynamics involve delays and noise. It is found that the effects of stochastic perturbations and/or delayed dynamics may be such that the approach of the entropies to equilibrium are not necessarily monotone and are dependent on system parameters.