An issue in the context of self-organization is the existence of bifurcation processes that are often observed in dissipative dynamical systems. The first bifurcation occurs when a stable fixed point becomes unstable as the parameter set of the system slightly changes. Then the system eventually paves their way to new stable branches. When the most ordered spatial patterns emerge, the system implies a high level of self-organization in the phase space. Exchange of energy/matter with its surroundings is allowed for in such "open" systems. Open systems cannot be analyzed in terms of usual H-theorem of Boltzmann since it is only valid for isolated systems. Thereby, the Boltzmann-Gibbs (BG) entropy (or Shannon) is not able to explain such systems which increase their order as a signature of self-organization. However, Klimontovich showed that the BG entropy can still be used in such systems after a renormalization procedure and he also introduced the renormalized entropy as a measure of the relative degree of order by using his S-theorem. In this paper, we analyze the Henan map and the Rossler oscillator as high dimensional dissipative systems. We show that the renormalized entropy detects transition points of the systems and explains changes in relative order through successive bifurcations or chaotic band-mergings. The correlation analysis also shows that the renormalized entropy is a better measure than the Shannon or Kullback-Leibler (KL) entropies. (C) 2018 Elsevier B.V. All rights reserved.