“…Those authors proposed, for instance, to use the dense phase dynamic component to estimate the bed expansion ratio and, therefore, the subsequent bed height and void fraction. Furthermore, the development of nonlinear science has provided tools such as the state space analysis, which has demonstrated how useful it is for extracting the dynamic information hidden within the time series (Daw and Hallow, 1993;Hay et al, 1995;Huilin et al, 1995;Briongos and Guardiola, 2004). This paper proposes the use of the dimension of interaction dynamics, d int 2 (Wojcik et al, 2001), to study the nonlinear interactions between the different dynamic scales drawn from the HHTM.…”