Shannon entropy diffusion estimates: sensitivity on the parameters of the method

Abstract

In the present effort, we revisit the Shannon entropy approach for the study of both the extent and the rate of diffusion in a given dynamical system. In particular, we provide a theoretical and numerical study of the dependence of the formulation on the parameters of the method. We succeed in deriving not only a diffusion coefficient, D S , but also an estimate of the macroscopical instability time for the system under study. Dealing with a toy model, namely a 4D symplectic application that represents the dynamics around a junction of resonances of different order, and an a particular case of the planar three-body problem, the HD20003 planetary system, we obtain numerical evidence that D S is a robust measure of the diffusion rate, no significant dependence on the free parameter of the entropy formulation (the size of the elements of the partition) being observed. Moreover, successful results concerning the estimation of macroscopical instability times obtained from D S are presented in both cases.

The Shannon entropy: An efficient indicator of dynamical stability

“…Thus, from (7) 88 On the other hand, in case of a trajectory γ c confined to a small 94 domain of S, as it was discussed in [30,31], the distribution of the 95 n k approaches a delta, δ(n k − λ), and thus estimating |ξ k | ≈ 1/2 96 (see [30]), it follows from (4) that 97…”

“…In this section we summarize the formulation given in [25,27,28,31] regarding the Shannon entropy as a dynamical indicator as well as a measure of the diffusion rate in action space of high-dimensional Hamiltonian systems or symplectic maps. For a general background on the Shannon entropy we refer to [32,33] as well as [34].…”

“…Following [31], a local diffusion coefficient for γ in the interval 100 (t, t+δt) can be estimated from the time derivative of S whenever 101 Σ being the area of S where the partition is defined, so that Σ/q provides the size of the cells in action dimensions (see below for an alternative definition of Σ). The estimate (11) rests on the assumption that locally the variation of S is due to changes in the number of occupied cells, i.e.…”

“…The estimate (11) rests on the assumption that locally the variation of S is due to changes in the number of occupied cells, i.e. due to variations in the actions in the interval (t, t + δt), in such a way that (see [27,31])…”

Uncertainty Measure of Basic Probability Assignment Based on Renyi Entropy and Its Application in Decision-Making

“…Shannon introduced entropy theory into the field of information, which broadened the research of uncertainty measurement [27]. Since Shannon entropy was proposed, entropy theory has been widely developed and improved [45], [46], [47], [48], [49]. However, in Dempster-Shafer evidence theory, how to calculate the uncertainty of BPA is still a developmental problem.…”