Basin entropy: a new tool to analyze uncertainty in dynamical systems

In nonlinear dynamics, basins of attraction link a given set of initial conditions to its corresponding final states. This notion appears in a broad range of applications where several outcomes are possible, which is a common situation in neuroscience, economy, astronomy, ecology and many other disciplines. Depending on the nature of the basins, prediction can be difficult even in systems that evolve under deterministic rules. From this respect, a proper classification of this unpredictability is clearly required. To address this issue, we introduce the basin entropy, a measure to quantify this uncertainty. Its application is illustrated with several paradigmatic examples that allow us to identify the ingredients that hinder the prediction of the final state. The basin entropy provides an efficient method to probe the behavior of a system when different parameters are varied. Additionally, we provide a sufficient condition for the existence of fractal basin boundaries: when the basin entropy of the boundaries is larger than log2, the basin is fractal.

The log 2 criterion is a sufficient condition to prove the fractality of the basin boundaries.
It is based on the concept of boundary basin entropy, defined as where N b is the number of boxes containing more than one color, that is, the number of boxes in the boundaries, and Now we assume that the boundaries separating the basins are smooth. In this case, the number of boxes lying in the boundary separating two basins grows as where D is the dimension of the phase space. For D = 2, the boundary would be a line, for D = 3, it would be a surface and so forth. However, there might be some boxes N k lying in the boundaries of k > 2 different basins. These boxes are in the intersection of at least two subspaces of dimension D − 1, that is, they are in the intersection of two smooth boundaries. For instance, when D = 2, it simply means that two or more smooth curves intersect in a point or collection of points, and when D = 3, two or more smooth surfaces intersect forming smooth curves. Thus, the dimension of the subspace separating more than two basins must be D − 2, and the boxes N k belonging to this subspace must grow as Taking into account that the total number of boxes grows as N =ñε −D , we can express N 2 in terms of N as and for the boundary boxes separating more than two basins N k , we have At this point, we recall that the maximum possible value of S in a box with m different colors is S = log m, which is the Boltzmann expression for the entropy of m equiprobable microstates. Then, we can find that all the boxes in the boundary of two basins have S log 2, while for boxes in the boundary of k basins, k > 2, we have that S log k.
Notice that the equality of the previous equations would be possible only in a pathological case where all the boxes in the boundaries have equal proportions of the different colors.
Then, the basin entropy S bb for this hypothetical system with smooth boundaries is By substituting N 2 and N k by Eqs. S.5-S.6, we obtain the following expression which can be simplified as S bb ≤ n 2 N log 2 + n kñ log k n 2 N + n kñ , (S.9) whereñ, n 2 , n k are constants. Finally, we can take the limit of the previous inequality for a large number of boxes, that is when N → ∞, leading to lim N →∞ S bb ≤ log 2. (S.10) Therefore, we have proven that if the boundaries are smooth, then S bb ≤ log 2, which is the same as to say that if S bb > log 2, then the boundaries are not smooth, i.e., they are fractal.
This is what we call the log 2 criterion.
This criterion is especially useful for experimental situations where the resolution cannot be arbitrarily chosen. In these cases we have a fixed value ε > 0. Nevertheless, if we take a sufficient large number of boxes N, then the log 2 criterion holds. Moreover, the equality of Eq. S.10 never takes place, so that there is some room for the possible deviations caused by the impossibility of making an infinite number of simulations or experiments.